Relativistic Quantum Physics

Relativistic quantum physics

Glenn Eric Johnson Corolla, NC E-mail: [email protected]

September 22, 2020

Abstract: A consistent unification of relativity with quantum mechanics follows from acceptance of quantum mechanical state descriptions as physical reality. Schrödinger’s1926 analysis of the linear harmonic oscillator, Newton and Wigner’s 1949 identification of the Hermitian relativistic location operator, Everett’s 1957 relative state formulation, and example realizations of relativistic quantum physics suggest a revision to the canonical formalism’s anticipated quantum-classical correspondences. Schrödinger’sanalysis uses classical dynamical variables as representatives for the support of appropriate states, and Newton and Wigner’s relativistic location operator is Hermitian but is not an exact elevation of classical location. Everett describes measurement in quantum mechanics without ad hoc accommodations that preserve classical descriptions. Application of these concepts to constructions of generalized functions realize the vacuum expectation values (VEV) of a quantum field theory. If the fields that label VEV are not Hermitian Hilbert space operators, then realizations are admitted. Elevation of the arguments of the VEV functionals to rigged Hilbert space operators that are also unitarily similar to free fields is problematic for interacting fields and is abandoned in this development. The alternative is to use scalar products constructed of explicit, Poincarécovariant, connected and clustering VEV functionals with support limited to positive energy mass shells to describe relativistic quantum physics. This development is realizable in four dimensional spacetime. Similarly to that the elevation of the spacetime arguments of functions that describe states are not the Hermitian relativistic location operators, the elevations of the arguments of the VEV functionals are not the Hermitian field operators. Nevertheless, quantum fields conditionally approximate classical fields. Example constructions include elementary particles characterized by any composition of free fields, and scattering amplitudes include asymptotes to Feynman series at weak coupling. New example realizations satisfy a strong cluster decomposition condition that implies independence of the local observables of non-entangled, spatially distant bodies. The revised quantum-classical correspondence, representatives for isolated, localized concentrations in the spatial support of states behave like bodies described in classical field theory, necessarily applies only when the quantum state is accurately represented by classical bodies and fields. The classical correspondences of the constructions are richer than anticipated

1 CONTENTS 2 by the canonical formalism. The reproduction of the low orders of Feynman series establishes a correspondence of the constructions with conventional high energy physics methods and the nonrelativistic dynamics of localized, isolated state descriptions correspond with Newtonian mechanics for −g/r pair potentials. This second classical correspondence is distinct from the correspondence of the scattering amplitudes. The behavior of states with significantly overlapping support is distinct from the dynamics of state descriptions with isolated and localized support. Example realizations illustrate how the persistent influence of classical concepts, in particular, the extrapolation of classical dynamics to describe quantum dynamics, has thwarted full exploitation of quantum mechanical possibilities for descriptions of high energy physics. Keywords: Foundations of quantum mechanics, relativistic quantum physics, constructive quantum field theory, high energy physics.

Contents

1 Background and summary 3 1.1 Realization and canonical quantization ...... 3 1.2 Descriptions of nature ...... 10 1.3 Extrapolation from classical concepts ...... 14 1.4 Less constrained alternatives ...... 16 1.5 Location and correspondences in relativistic quantum physics ...... 20

2 Quantum mechanical descriptions of nature 22 2.1 Quantum mechanical description of state ...... 23 2.2 Axioms for the VEV of relativistic fields ...... 35 2.3 The basis function space P and equivalent development for S ...... 43 2.4 Relativistic free fields ...... 47 2.5 Nontrivial realizations of relativistic fields ...... 49 2.6 Positive definiteness ...... 60 2.7 Cluster decomposition ...... 62 2.8 Regularity ...... 65 2.9 Example M(p), S(A) and D ...... 69 2.10 Hermitian Hilbert space operators, states and classical observables ...... 71 2.11 Heisenberg uncertainty ...... 77 2.12 Observation ...... 81 2.13 Localized observables, separation and independence ...... 90 2.14 Why does the flawed classical interpretation suffice in everyday life? ...... 96 2.15 Scattering amplitudes, the geometry of spacetime, and units ...... 105 1 BACKGROUND AND SUMMARY 3

3 Classical correspondences 110 3.1 A nonrelativistic classical correspondence ...... 112 3.2 The most classical-like state descriptions and −g/r potentials ...... 118 3.3 Two body classical trajectories ...... 126 3.4 VEV and nonrelativistic approximation of likelihoods ...... 128 3.5 Discussion of the classical correspondences ...... 140

4 Inconsistency of the classical description with nature 142

5 Technical concerns with canonical quantization 145

6 Conclusion 150

7 Appendices 151 7.1 The Dirac-von Neumann axioms for quantum mechanics ...... 151 7.2 Translations, position operators and relativistically invariant localized states . . . 154 7.3 Classical descriptions of nature ...... 163 7.4 Polar decomposition and type III operators ...... 170 7.5 Hilbert space ﬁeld operators ...... 181 7.6 Tensor products of linear vector spaces ...... 184 7.7 The degenerate scalar product for free ﬁeld VEV ...... 185 7.8 Evaluation of the scalar product hϕ2(0)|U(λ)ϕ2(λ)i ...... 187

1 Background and summary

1.1 Realization and canonical quantization Does a “quantization” procedure to elevate classical dynamical models to relativistically covariant scattering amplitudes suffice, or does quantum mechanics provide a comprehensive description of nature? To realize the latter, it is beneficial to reconsider: how is nature represented in quantum mechanics? The view expressed here is that a persistent classical predisposition has impeded realization of relativistic quantum physics. Indeed, despite evident quantum-classical correspondences and impressive phenomenological successes, the Feynman series development has not fully realized quantum mechanics [5] nor is gravity straightforwardly included. Lim- itations originate in the inconsistency of realization of relativistic quantum mechanics with canonical formalism-conjectured quantum-classical correspondences. Interpretation of quantum mechanics as physical reality is a persistently controversial topic [3, 5, 14, 45, 54, 67]. The classical description of nature has been observed not to be consistent with nature and understanding of the description that is consistent with observation, quan- 1 BACKGROUND AND SUMMARY 4 tum mechanics, is controversial even while the practice is well accepted. An example of the discomfort with quantum mechanics is: We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I’m an old enough man that I haven’t got to the point that this stuff is obvious to me. Okay, I still get nervous with it... You know how it always is, every new idea, it takes a generation or two until it becomes obvious that there’s no real problem. I cannot define the real problem, therefore I suspect there’s no real problem, but I’m not sure there’s no real problem. – Richard Feynman, 1982, p. 471 in [15]. Our intuition for nature often remains classical and this predisposition has hindered application of quantum mechanics. Classical concepts persist due to our familiarity with that description, its relative simplicity, the apparent reinforcement of that view in our everyday experience, and the many successful applications of the canonical formalism for quantum mechanics [11, 63]. However, the adoption of quantum mechanics is not a transition from classical to quantum dynamics for classically described bodies, as some attempt to interpret quantum mechanics. The alternative illustrated in these notes is that the transition is from a classical to a quantum mechanical description of nature. Although classical mechanics provides approximations in common instances, quantum mechanics supersedes classical descriptions. Quantum dynamical models need not “quantize” classical models: quantum dynamics need not elevate classical descriptions. Example realizations of relativistic quantum physics are developed below to illustrate alternatives to canonical quantization. No distinction is made here between operator-centric (Dyson) or path-integral (Feynman) methods in relativistic quantum physics, both elevate classical models and are designated as canonical formalism developments here. For my purposes, the canonical formalism refers broadly to reliance on extrapolations of classical dynamics to describe quantum dynamics. Much of the canonical formalism is indisputable, and the refinement pursued here concerns quantum-classical correspondences and the implied Hermiticity of particular operators in relativistic physics. In nonrelativistic physics, canonical quantization, described by the Dirac-von Neumann axioms discussed in appendix 7.1, applies. These notes are a study of quantum mechanical realizations that exhibit the physical characteristics of relativistic physics. There are realizations of quantum mechanics that exhibit the physical characteristics of relativistic physics but do not comply with a canonical quantization. Example realizations are presented in section 2 and [32, 34, 38]. The realizations describe vacuum expectation values (VEV) of quantum fields

hΦ(xk)κk ... Φ(x1)κ1 Ω|Φ(xk+1)κk+1 ... Φ(xn)κn Ωi (1) and the associated observable quantities. These VEV exhibit interaction and are defined as generalized functions (functionals) dual to a selected basis space of functions. These promising constructions exhibit necessary properties to realize relativistic quantum physics: the descriptions of states are elements of rigged Hilbert spaces; temporal evolution of these state 1 BACKGROUND AND SUMMARY 5 descriptions is likelihood preserving (unitary) and causal; the scalar products of elements are Poincaréinvariant; energies are nonnegative; evolution of features of the quantum description of states is approximated by classical mechanics in appropriate instances; and for the new examples in section 2, the local observables of non-entangled, distantly space-like separated bodies are independent. These conditions, more precisely stated in section 2.2, describe quantum mechanical realizations of relativistic physics [6, 59]. The constructed Hilbert space realizations are prospective descriptions of nature. Except, in contradiction to the canonical formalism-conjectured quantum-classical correspondences, the quantum field Φ(x)κ in (1) is not necessarily realized as Hermitian Hilbert space operators. These otherwise promising realizations of nontrivial relativistic quantum physics do not satisfy the conjecture that the Φ(x)κ in (1) elevate a classical field and are consequently realized as Hermitian operators. However, and illustrated by the lack of a Hermitian operator that exactly elevates location, an exact elevation of classical dynamical variable to Hermitian Hilbert space operator is not necessary for an appropriate quantum-classical correspondence. QFT is a phenomenologically successful development that lacks a Hermitian operator exactly corresponding with location [44]. This is the “localization problem” in QFT: relativity precludes an exact correspondence of classical location with a Hermitian operator [69]. Analogously, it is not necessary for the quantum field Φ(x)κ in (1) to be Hermitian Hilbert space operators to correspond appropriately with a classical field. This decisive distinction is developed below. Revisiting the quantum-classical correspondences for relativistic physics escapes troubling limitations in QFT [5]. The quantum field Φ(x)κ in (1) is well defined as an operation within the basis space of functions and describes states as elements of a rigged Hilbert space. The VEV describe the scalar products of states and by Born’s rule, describe the likelihoods of observations. States are within a completion of elements

n X X Z Y |fi = d(x)n fn((x)n)(κ)n | Φ(xk)κk Ωi (2) n (κ)n k=1

4n with each fn((x)n)(κ)n ∈ P(R ), the basis space of functions. In the examples, P is a subspace of the Schwartz tempered functions S.a The notation is from section 2.1. Satisfaction of established physical principles extends from the basis space of functions P to S to realize Hermitian Hilbert space ﬁeld operators for physically trivial constructions but the quantum ﬁeld in (1) is otherwise demonstrably non-Hermitian. Without extension, there are too few

aThere is an equivalent development displayed in (29) in section 2.3 with modified VEV and the Schwartz tempered functions S as the basis function space. The VEV are modified from the VEV displayed in section 2.5 and the modification results in non-Hermitian field operators. The modified VEV in (29) do not include free field VEV but, of course, VEV based on S include the free field and related physically trivial constructions that result in Hermitian field operators. The development displayed in these notes is based on P for commonality with the developments in [32, 34, 38] and physical insight. The two developments, basis space P with the VEV displayed in section 2.5 or basis space S with modified, non-Hermitian VEV, are equivalent. Without embellishment, both developments produce non-Hermitian fields in (1). 1 BACKGROUND AND SUMMARY 6

functions, no real functions, in the basis function space P for the Φ(x)κ in (1) to be Hermitian operators: extension is available for physically trivial constructions but introduces negative energies to the physically nontrivial constructions. In these notes, an elevation refers to the Hilbert space operator generated from a classical dynamical variable. An elevation has eigenstates that exactly correspond to the elevated classical dynamical variable, for example, the elevation of location is the Hilbert space operator with eigenstates that are Dirac delta functions. Dirac delta functions exactly correspond to the classically idealized locations of classical bodies. However, there can be a distinction between the Hermitian Hilbert space operator that corresponds to a classical dynamical variable, and elevations. Elevations are not necessarily Hermitian. Discussed in [11, 62] and section 2.10, to correspond to a classical dynamical variable, an elevation must be Hermitian but the Hilbert space scalar product is not necessarily compatible with the selection of eigenstates: a quantum- classical correspondence that elevates classical dynamical variables is not necessarily available. Eigenstates for a Hermitian Hilbert space operator are orthogonal in the Hilbert space scalar product [21]. The correspondence of Hermitian operator with classical dynamical variable may be approximate and conditional, and section 1.5 includes a procedure to construct a Hermi- tian operator that corresponds to a classical dynamical variable. The archetypal example of a non-Hermitian elevation is relativistic location: a scalar product with a Källén-Lehmannform two-point function achieves the physical properties of Lorentz invariance and positive energies but Dirac delta functions are not orthogonal in this scalar product.b Dirac delta functions are not the eigenfunctions of a Hermitian operator in relativistic physics [44]. Exact elevation excessively extrapolates a classical description into regimes where classical insight has not been observed to apply. A refinement to the canonical formalism’s quantum-classical correspondence by elevation is to view classical dynamical variables as representatives for the support of the functions that describe appropriate states. In these notes, rather than exact elevations, the quantum-classical correspondences are relaxed to include appropriate associations of observable features in the quantum description of state with classical dynamical variables. Features are the support of function sequences that describe the states. This correspondence is technically relaxed, more approximate and conditional, than the canonical formalism-anticipated exact elevation of classical dynamic variables to densely defined Hermitian Hilbert space operators. Function sequences are supported over space and field components, and the approximations and conditions describe the instances when representation of this support by classical dynamical variables is appropriate. States are labeled by equivalence classes of function sequences and the physically significant property of these functions is the Lebesgue summation over mea- 3 3 surable subsets of R and not the more classically conceived, point-wise relation R 7→ C.A correspondence of Lebesgue measurable subsets with idealized classical positions is necessarily approximate, and the association of the supports of a Gaussian delta sequence with a point

bR + + + dxdy ∆ (x − y)δ(x − xo)δ(y − yo) = ∆ (xo − yo) 6= 0 for xo 6= yo with ∆ (x) the positive frequency Pauli-Jordan function [6]. 1 BACKGROUND AND SUMMARY 7 is similarly inexact. This refinement to the quantum-classical correspondence, motivated and developed throughout these notes, provides the physical understanding of the example realizations of relativistic quantum physics. It is demonstrated here by construction that this well-founded revision to the quantum-classical correspondences results in decisive revision to the mathematical development of relativistic quantum physics. Insistence that the operators that correspond to classical dynamical variables are exact elevations excludes even relativistic free scalar fields from quantum mechanics. Physically justifiable relaxation of quantum-classical correspondences results in realizations. Freed of constraints imposed to satisfy a more classical understanding of nature, realizations of relativistic quantum physics become available. The approach to quantum mechanics considered in these notes is identified as unconstrained. The refined quantum-classical correspondence is adopted to relax constraints imposed by the canonical formalism but otherwise the Hilbert space realizations of states remain in compliance with established physical principles. In this context, unconstrained expresses that the development is conventional quantum mechanics as described in the Dirac-von Neumann axioms in appendix 7.1 except that: - canonical quantization’s “elevation of c-number to q-number” is relaxed to the refined, approximate and conditional, correspondence of classical with quantum state descriptions and

- no interaction picture, unitary similarity of free fields and fields that exhibit interaction is imposed. While realizable with the scalar product for nonrelativistic physics, location demonstrates that a canonical quantization is not generally available in relativistic physics. One significant distinction between relativistic and nonrelativistic physics is that the scalar product is invariant to velocity shifts in a nonrelativistic development but a finite speed of light precludes velocity shift invariance in relativistic developments. The elevation of x is Hermitian in nonrelativistic but not relativistic physics as a consequence.c In section 1.5, this archetypal example, relativistic location, illustrates the distinction between the unconstrained development and a canonical quantization: the distinction between the elevation of a classical dynamical variable and the Hermitian operator that corresponds with the classical dynamical variable. In section 1.5, three 3 quantities are distinguished: x, Xν, and Xbν. The vector x ∈ R specifies a classical location. Xν for ν = 1, 2, 3 are operators that elevate the three components of x, operators with generalized eigenstates labeled by Dirac delta functions. And, the Xbν are Hermitian Hilbert space operators with generalized eigenstates labeled by the relativistically invariant localized functions of Newton and Wigner [44]. There are simultaneous eigenstates of the three Xbν centered on every location x. These generalized eigenstates describe the optimally localized states in a relativistic free field development. These generalized eigenstates are limits of elements from the

c iko·x iko·x As a consequence of he f(x)|e g(x)i = hf(x)|g(x)i for the L2 scalar product, the generator of velocity shifts, the elevation of x, is Hermitian. 1 BACKGROUND AND SUMMARY 8

Hilbert space just as Dirac delta functions are approximated by L2 functions in nonrelativistic developments. Anticipation from classical intuition, expressed in the “elevation of c-number to q-number” conjecture, is that the Xν should be Hermitian operators. The contradiction to the Hermiticity of Xν in relativistic physics, included in section 5, is the “localization problem” of QFT. Relativistically invariant localized functions for distinct locations are orthogonal for the Hilbert space scalar product but only approximate Dirac delta functions. The Hermitian d operator associated with location Xbν becomes an approximation to the elevation of location Xν in nonrelativistic instances (mc large with respect to the magnitudes kpk of the momenta).

Xν ≈ Xbν. Perception determines the meaning of “localized” in relativistic quantum mechanics. The support of the relativistically invariant localized functions is not concentrated at a point but the support is dominantly within a volume of diameter proportional to the Compton wavelength. The support of functions is discussed in section 2.1. The perception of functions dominantly supported near x as supported solely on x is the refined correspondence of quantum descriptions with classical ideals. That a small neighborhood of a location x includes all perceptions of location to arbitrarily great likelihood suffices physically as the description of localized. The characterization of the support of a sufficiently localized state by one representative location is a physically appropriate correspondence. This example of location in relativistic physics is also discussed in [44], sections 1.5 and 2.1, and appendix 7.2. The example nontrivial realizations of relativistic quantum physics apply a similar distinction to fields. Three fields: A(x)κ, Φ(x)κ, and Φ(b x)κ are distinguished in the unconstrained development. Functions A(x)κ are classical fields. Operators Φ(x)κ are the quantum fields in (1) and (2), and describe elementary particles. In section 2.1, the quantum field Φ(x)κ is displayed as a multiplication (14) in the basis space of function sequences [7]. This operation provides VEV for the quantum field (1) that determine the kernel functionals in the scalar product of the Hilbert space realization. The classical fields A(x)κ appropriately correspond to Hermitian Hilbert space field operators Φ(b x)κ. The classical fields provide approximations to the evolution of states when classical correspondences apply. In the unconstrained development with the refined quantum-classical correspondences, the behavior of bodies establishes correspondences of classical fields A(x)κ with the example realizations of relativistic quantum physics. For appropriate states with body-like features, the behavior of the quantum description of states is compared with corresponding locations and momenta of classically described bodies from a classical field theory. These correspondences are discussed further in sections 1.5 e and 3. The Hermitian field operators Φ(b x)κ do not elevate the classical field. Anticipation

d 1/2 −1/2 From appendix 7.2, the differential operator ω d/dpν ω ≈ d/dpν for functions of sufficient decline dominantly supported near kpk = 0. ω is from (26), the positive root of ω2 = m2c2 + p2. e An eigenstate for the elevation of a classical field would be a simultaneous generalized eigenstate of all Φ(x)κ R such that |Φ(x)κ si = A(x)κ|si in the sense that |Φ(f)κsi = dx f(x)A(x)κ|si for all f(x) in the basis function space. It is demonstrable that there are no such eigenstates, section 5. 1 BACKGROUND AND SUMMARY 9

from the canonical formalism is that Φ(x)κ = Φ(b x)κ despite that: no fully realized, canonical formalism-based development of nontrivial relativistic quantum physics has been discovered; the quantum field Φ(x)κ has no eigenstates; and relativistic location is a counterexample to Hermitian elevations. The Lamb shift from the Feynman series in quantum electrodynamics (QED) suggests that the quantum field Φ(x)κ may approximately elevate classical fields, that analogous to Xν ≈ Xbν, Φ(x)κ ≈ Φ(b x)κ.

In free field examples, Φ(b x)κ = Φ(x)κ at least to the extent that Φ(x)κ is Hermitian after extension of the basis function space to S. There are no eigenstates of the quantum field Φ(x)κ. Conjecture for the properties of Hilbert space operators is avoided in the unconstrained development. Or more precisely, since there is a one-to-one correspondence of subspaces of states and projection operators [40], the emphasis is on the orthogonal projection operators and characterization of when classical dynamical variables represent perception of the states in particular subspaces. The properties and understanding of operators are results, for example: the spectral theory for rigged Hilbert space operators [19, 40]; and Stone’s theorem [24] for unitarily realized groups of Hilbert space transformations. Spectral theory displays Hermitian operators as limits of linear combinations of projections weighted by representative values of the observable associated with subspaces of states. The “localization problem” in relativistic quantum physics can be considered a correspondence problem, a problem that is overcome by adopting more appropriate quantum-classical correspondences. The conjectured equality of the Φ(x)κ that describes states (2) with a Her- mitian Φ(b x)κ that corresponds to a classical field A(x)κ has decisive impact on realization of relativistic quantum physics. There is no necessity for the quantum fields in (1) to be Hermitian to establish an appropriate quantum-classical correspondence. With this technical change to quantum-classical correspondences, relativistic quantum physics becomes realizable. In section 2 below, new examples of nontrivial realizations are constructed. A stronger cluster decomposition property than the uniqueness of the vacuum condition used in [32, 34, 38] applies in these new examples and provides that the truncated functionals [6] are connected functionals. The developments in [32, 34, 38] preserved satisfaction of the established formal Hermiticity condition, W.a in section 2.2. Formal Hermiticity is convenient to demonstrate that the field Φ(x)κ in (1) is a (non-Hermitian) Hilbert space operator, but it is not necessary and formal Hermiticity is not applied to the new constructions in section 2.5. Developed in sections 2.12 and 2.13, this stronger cluster decomposition condition implies: the essential independence of the local observables of non-entangled, spatially distant bodies; that states with sufficiently isolated and spatially separated support are described by free particles; and a unique vacuum. In these notes, topics are introduced in only enough detail to motivate and illustrate concepts, and references must be consulted for more complete developments and original citations. The emphasis is fully quantum mechanical example realizations and the elementary physics described by the union of the principles of relativity and quantum mechanics. Throughout the 1 BACKGROUND AND SUMMARY 10 text, the terminology and concepts for observation of the quantum mechanical description of state are clarified and illustrated. The intent is to discuss the foundations of quantum mechanics in sufficient detail to illustrate but not to digress to fully develop results. The endeavor aims to explain relativistic quantum physics and satisfy physicists’ inquiries yet provide an introduction for a more general audience.

1.2 Descriptions of nature Both classical and quantum mechanics describe nature. Contrasts of classical and quantum dynamics begin with consideration of these distinct descriptions: classical physics describes 3 distinguishable, geometric bodies located in an R configuration space; and quantum physics 3 describes the states of nature as complex-valued functions over R . These functions label elements of an infinite dimensional, rigged Hilbert space. In both cases, evolution of the state is parametrized by time. The quantum mechanical description is more comprehensive: quantum descriptions may be approximately represented by or deviate substantially from classical descriptions. In classical mechanics, bodies are described by dynamical variables, for example: position, orientation, mass and charges describe a rigid body. At each instant, a classical body is thought to have a single, determined value for each dynamical variable. This classical description is conceived as transparent to observation: the state can in principle be perceived without disruption. The laws of classical physics determine the evolution of the state; a determined future flows continuously from a known present. The temporal evolution of each body is followed. The quantum mechanical descriptions of states are functions. Rather than position of a body as a single point, many possible locations and velocities weighted by likelihood derive from the functions that describe a state, and the derived likelihoods apply to many indistinguishable bodies. Our perceptions of these functions are determined by the Hilbert space description of nature: - observers are systems also described by quantum mechanics and the relevant description of the system that results from an observation is entangled with a corresponding description of the observer as discussed in [9] and section 2.12;

- chance, we typically perceive one from among many possibilities;

- and our preconceptions, we often describe our perceptions in classical terms. Unlike the classical description of state as determined and immutable except as it evolves due to classical forces, in quantum mechanics, generally observation randomly disturbs the relevant description of state. To understand quantum mechanics as reality, this is perhaps the greatest distinction between classical and quantum concepts. In classical physics, the observer is essentially omnipotent while in quantum mechanics, the observer is another body participating in the dynamics. In quantum physics, one perceives a state that derives from the initial state but this observed state is also determined by an entanglement of the observer’s particular final state 1 BACKGROUND AND SUMMARY 11 with the corresponding state of the observed body, and chance. Distinct states of an observer are entangled with each of the possibilities for perception, and one of these states is distinguished as describing the observer for subsequent observations. Consistency and universality demand that observers are also described by quantum mechanics. Localized states, states that appear to satisfy classical description, result from localizing interactions whether the initial state is localized or not. States do not appear localized because they are classical bodies but rather, the process of observation often results in a perception interpreted as a classical body. Localized states appear to satisfy classical description because of our typical inability to perceive the spread of a sufficiently localized function. Spatial resolution is inescapably limited by the wave nature of state descriptions and particle creation. We do not perceive the initial state; we observe one of the possible results of our interaction with the initial state. The likelihoods of subsequent observations are determined by the results of previous observation. Observation is discussed throughout the text but predominantly in sections 2.10-2.14 and appendix 7.2. This description of nature by functions is the result of an extraordinary effort to rectify observation with prediction begun in the early twentieth century. The initial effort is notably due to Max Planck, Albert Einstein, Max Born, Werner Heisenberg, Erwin Schrödinger,Paul Dirac, Wolfgang Pauli, Pascual Jordan, Neils Bohr, Hermann Weyl, John von Neumann and Eugene Wigner, and efforts to understand the resulting formalism of quantum mechanics are notably due to Hugh Everett III and Bryce DeWitt. One attempt at a statistical interpretation of the quantum formalism views the function that describes a state as a superposition over classical states, but this consideration is profoundly misleading. No classical descriptions are among the quantum descriptions of states. The classical and quantum descriptions of states differ fundamentally. Quantum mechanical descriptions of bodies do not allow simultaneous, arbitrarily precise determinations of location and velocity, nor the identification of trajectories except approximately in special, albeit common, instances. Classical dynamical variables often approximate selected features of quantum states although classical descriptions of states are not included among the quantum descriptions. The classical view is not supported by our broadened observations of nature. The classical description is a perception facilitated by often excellent approximation. The support of functions was identified as observable features of state descriptions in early studies of quantum mechanics. In appropriate instances, classical dynamical variables are representative of the support of the functions that describe states. Published in 1926, Erwin Schrödingercontrasted classical and quantum mechanical descriptions of the states of a linear harmonic oscillator in a nonrelativistic development [52]. This study, illustrated in section 3 and [35], compared quantum and classical descriptions of states. Schrödinger’sdevelopment compared features of the quantum and classical descriptions of state for particular, minimum packet functions. This is an alternative to association of the eigenfunctions of Hermitian Hilbert space operators with the classical dynamical variables, position and momentum. The minimum packet states are the most classical-like in the sense that simultaneous knowledge of position and momentum is maximal consistent with a quantum mechanical description of state. The 1 BACKGROUND AND SUMMARY 12 eigenfunctions are among the least classical-like states: simultaneous knowledge of velocity and position is minimal with divergent uncertainties of either position or momentum. The classical description of temporal evolution is Newton’s equation, and the quantum description is Schrödinger’sequation that describes the evolution of a wave function. In Schrödinger’sdevelopment, favorable comparisons are made between the support of the wave function that describes the distinguished, minimum packet states and the corresponding classical dynamic variables. An analogous comparison of observable features from the classical and quantum descriptions of states is adopted in the unconstrained development. The Einstein-Podolsky-Rosen (EPR) [14] paradox and confirmation of Bell’s inequalities [3] illustrate that a classical description is inconsistent with nature. The quantum mechanical description of nature has been described as “bizarre” but, of course, this description reflects that quantum mechanics demands rejection of well established classical concepts. The EPR paradox [14] illustrates the quantum mechanical description of nature and contradicts classical concepts. Einstein, Podolsky and Rosen develop the argument that quantum mechanics must be incomplete because the quantum description conflicts with a classical concept of state. Spin angular momentum is quantized with the same discrete values on any axis of observation. The paradox arises if a spin zero particle decays into two spin one-half particles that subsequently fly apart. Sufficiently separated, we can determine the spin state of an arbitrarily distant particle by observing the paired particle: the distant spin is the one that paired with the near spin conserves angular momentum. The paradox is that we determine the spin of the non-causally related distant particle differently as determined by our selection of measurement axis. If the distant particle is classically described, then it has a determined spin unaffected by our observation of the nearby particle. The resolution to the conflict is entanglement, a concept in quantum mechanics that is not supported in a classical description. That is, the resolution of the EPR paradox is not a contradiction to quantum mechanics but a contradiction to classical concepts, to the classical description of state. Entanglement is natural in the Hilbert space description of multiple particle states by functions [53]. The paradox is resolved by the realization that a classical description is not consistent with nature. Although Einstein motivated quantum mechanics with studies of the photoelectric effect and the heat capacity of solids [58], he is perhaps most celebrated for his understanding of time and motion described in special relativity and geometrodynamical gravity. Einstein’s derivations of both special relativity and gravity employ classical descriptions of nature. Expressed in the EPR paradox, Einstein did not em- brace implications of quantum mechanics, in particular, the entanglement of states. I presume that Einstein was persuaded by his immensely successful insights into time and motion that employ classical concepts in conflict with quantum mechanics. Also, more consistent perspectives on quantum mechanics [9] developed later. The principle of equivalence, that acceleration is equivalent to a gravitational force, relies on the classical description of a body characterized by a trajectory. In a quantum mechanical description, the characterization of the velocity of a body deteriorates with enhancements in location accuracy: both location and velocity are never known with a precision that exceeds the Heisenberg uncertainty bound. We cannot arbitrarily 1 BACKGROUND AND SUMMARY 13 precisely associate the acceleration of a body with its locations.f Einstein pursued classical, geometrodynamical unification of the forces, an effort that continues to this day. It is a great irony that just as problematic aspects of Newtonian mechanics were resolved with dynamical, relativistic time and the equivalence of acceleration with gravitation, observation necessitated the new mechanics that supersedes a classical development. Flaws in classical physics such as the non-causal radiation reaction force in classical electrodynamics [30] can be tolerated since classical models only approximate the more fundamental, and causal, quantum descriptions. Indicated in Feynman’s quote above, it could be anticipated that the transition from the well established and successful practice of classical physics to quantum mechanics would be slow. Even though it was observation that necessitated the development of the new mechanics, adoption of quantum mechanics is inhibited by our established reliance on classical concepts. Technically, classical mechanics provides approximations that are often of great utility; conceptually, the descriptions of nature in classical and quantum mechanics are violently incompatible. Even in education today, the concepts of classical physics are established in students before quantum mechanics is introduced despite that classical physics is contradicted by nature. Quantum mechanics is the more comprehensive model and supported by our observations. Although the classical perspective is an accurate approximation of the quantum description in our common experience, the classical perspective is contradicted by nature. Of course, one man’s description of nature is another’s mystery. Descriptions are in the contexts and preconceptions of the commentator.

Otherwise the “paradox” is only a conﬂict between reality and your feeling of what reality “ought to be.” – Richard Feynman, 1965, Sect. 18, p. 9 in [16].

Questions of interpretation have always stirred controversy. Is acceleration due to force more mysterious than that a curvature of space and time results from an energy-momentum density, or that exchanges of virtual particles determine an exhibition of interaction? “Turtles all the way down” is an anecdote within the perspective that the Earth is flat and the natural state of bodies is to fall. In this description, the Earth does not fall because it rests on the back of a World Turtle [68]. This turtle is itself standing on a turtle and so forth for a column of turtles. The issue is what is the final turtle standing upon and the anecdote quips that it is “turtles all the way down.” The World Turtle supports a flat Earth and implements the observation that earthbound bodies are supported from falling by other bodies. A World Turtle is replaced in a more comprehensive perspective. Albeit, explanations tend to ultimately settle on “that is just the way it is.” Here, a description of nature is considered satisfactory if it is logically consistent, relatively simple, and universal within the known context. Interpretation has always

f Although for typical nonrelativistic Hamiltonians, acceleration and location commute, [Xν , [H,Pν ]] = 0, an estimate for acceleration from a sequence of location measurements fails due to the lack of a trajectory associated with the body. Each observation of position contributes velocity uncertainty. 1 BACKGROUND AND SUMMARY 14 been subject to discussion. Nevertheless, consistency, simplicity, and universality eliminate the classical description of nature. These notes are my understanding of the quantum mechanical description of nature.

1.3 Extrapolation from classical concepts Early in the development of quantum mechanics, wider acceptance of the new science was expedited by demonstrations of connections to established classical dynamics as well as by consistency with observation. The early development of quantum mechanics included results that described a quantization of previously developed classical physics models with applications on microscopic scales. In the nonrelativistic development, or if low order perturbation analyses of S-matrices suffice, the hypothesized quantization is spectacularly successful (e.g., the Lamb shift of the energy levels for the hydrogen atom) and provides many of the successes of modern physics. The quantum-classical correspondences of the canonical formalism, also denoted here as canonical quantization, hypothesizes an elevation of classical to quantum dynamics through the substitution of quantum mechanical, Hermitian Hilbert space operators for observable classical dynamical variables. This has been characterized as “elevation of c-number to q-number.” Hypothesized properties of these Hilbert space operators have been central to the established development of quantum dynamics [11, 60, 62]. This foundation is based upon elevation of classical dynamics despite that the approximations that classical mechanics provide for quantum mechanics are limited. The approximations to quantum dynamics provided by classical dynamics necessarily apply only for isolated, localized bodies with nonrelativistic relative velocities. It is not evident that extrapolation from these limited instances should determine the dynamics of all states [4]. This remarkable, mysterious extrapolation from classical models to the more general quantum theory is a wondrous conjecture. However, this canonical quantization development appears to be inconsistent with relativistic physics. The established quantum-classical correspondences of the canonical formalism are demonstrably inconsistent with example realizations of relativistic quantum physics introduced in [32, 34, 38] and section 2, and with relativistic location [44]. And, in Feynman series results, application of established quantum-classical correspondences is limited to scattering over infinite intervals [5]. Although contradictions can be developed in the nonrelativistic development, an appropriately focused development results in the successful model of nonrelativistic physics. Concerns with the quantum-classical correspondences of the canonical formalism are discussed in sections 2 and 5. However, canonical quantization encountered significant difficulty with electrodynamics, a relativistic classical field theory [5, 6, 59, 66, 69]. Developments of Ernst Stueckelberg, Richard Feynman, Freeman Dyson, Julian Schwinger, Sin-Itiro Tomonaga and others [4, 55, 63] resulted in rules to predict results for electrodynamics and these rules make some of the most precisely correct predictions known in physics. Canonical quantization’s correspondence of quantum with classical field includes consideration of a field operator assigned to each spacetime point. This correspondence is despite that Poincarécovariance is incompatible with VEV 1 BACKGROUND AND SUMMARY 15

that are functions [6]. The quantum field Φ(x)κ has been designated an operator-valued distribution [65], that is, the VEV of the field are generalized functions and field operators Φ(f)κ are labeled by functions f(x) from a selected basis space of functions. The action of the quantum field operators Φ(f)κ is defined in a rigged Hilbert space with elements labeled by functions from the basis function space [6, 7]. In a canonical formulation development, a field Φ(ct, x)κ is formally expressed as interaction picture unitary transformations of free fields, −1 Φ(ct, x)κ = U(t, t0)Φo(ct0, x)κU(t, t0) (3) and this assignment leads to the Feyman rules for scattering amplitudes in a Dyson series development [23, 55, 63]. In (3), U(t, t0) is expanded in powers of an interaction Hamiltonian expressed in ordered products of creation and annihilation components of a free field Φo(x)κ. In a plane at coinciding times, t = t0 in (3), the fields Φ(x)κ equal free fields. This expression (3) is not evidently compatible with both relativity and interaction. One constant time plane has been distinguished. The VEV implied by (3) result in scalar products that are not relativistically invariant unless time differences are infinite or vanish, or U(t, t0) is trivial. This interaction picture development succeeds in nonrelativistic physics but in relativistic physics, consideration of multiple moving observers suggests that the equality with free fields extends to non-coinciding times. Creation and annihilation of particles are events that observers agree upon. The Haag (Haag-Hall-Wightman-Greenberg) theorem [6, 59, 66] is a precise development with a similar conclusion: fields that are unitary similar to free fields do not exhibit interaction. For unitary equivalence, realizations must also satisfy assumptions concerning operator domains [24]. Restriction of quantum field theory to a rules-based (ansatz) evaluation of Lorentz covariant S-matrix elements evades the conflict with relativity but dismisses evaluation of the interpolating states [5, 63]. The rules do not describe the scalar products of states except at equal times or infinitely differing times. We do not observe transitions over infinite intervals but the differences of infinite and sufficient finite intervals are considered negligible. Interpolating states describe the state evolution over finite intervals and the descriptions in (3) are trivial or not relativistically covariant. And, products of generalized functions are not generally defined, e.g., δ(x)2. Indeed, terms in the expansion of (3) diverge. These divergences are regularized by counterterms and renormalizations that may introduce anomalies: each term in the regularized Feynman series expansion of scattering amplitudes is finite in renormalizable cases. The rules achieve a “quantization” of classical developments but: countenances the lack of a description for the interpolating states [5]; the elevation of gravity is not renormalizable; and indications are that the resulting expansions diverge [6, 59, 66]. The Feynman series rules indulge classical intuition: interactions are analyzed by considering determined numbers of particles that transition into distinct numbers and species of outgoing particles at vertices. Nevertheless, the series are intriguing and well established since many phenonemologically successful outcomes result. The Feynman rules have not been rectified with quantum mechanics although the rules employ concepts and methods from quantum mechanics. James Bjorken and Sidney Drell in their text on relativistic quantum physics (sec. 15.5, [4]) remark that, “It is never hard to find 1 BACKGROUND AND SUMMARY 16 trouble in field theory...” Their remark is focused but expresses a sentiment widely applicable to the quantization of classical field theories. An originator of quantum field theory stated:

These rules of renormalisation give surprisingly, excessively good agreement with experiments. Most physicists say that these working rules are, therefore, correct. I feel that is not an adequate reason. Just because the results happen to be in agreement with observation does not prove that one’s theory is correct. – Paul Dirac, 1984, in “The inadequacies of quantum ﬁeld theory,” Proceedings of Loyola University Symposium, New Orleans.

Many years of concerted effort [1, 2, 6, 31, 39, 41, 59, 61] have not yet resulted in a relativistic realization of physical interest from canonical quantization. While the Feynman rules produce many phenomenologically successful results, significant deficits remain between the rules and a statement of principles for quantum mechanics.

1.4 Less constrained alternatives Throughout much of the development of quantum mechanics, the question has been how to develop quantum mechanics from classical mechanics. An alternative perspective is to understand classical mechanics as an approximation to quantum mechanics that applies in appropriate instances. Quantum mechanics is the more general model and supersedes classical concepts. Study of mathematical structures that might realize relativistic quantum physics, including studies by John von Neumann, Rudolf Haag, Res Jost, Arthur Wightman, Léonvan Hove, Nikolay Bogolubov, Hans-JürgenBorchers, Huzihiro Araki and Raphael Høegh-Krohn, has not yet identified examples of quantum mechanics that realize nontrivial relativistic physics. Meth- ods include the proposal of axioms to describe the general properties of relativistic quantum physics. Axioms are used to define concepts, derive general results, and assess the consistency of the axioms with additional assumptions. Significantly, the only realizations discovered for established, prospective relativistic quantum physics axioms are physically trivial. Established axioms include the Wightman-Borchers (Wightman) functional analytic axioms [6, 7, 59, 64], the G˚arding-Wightman axioms for field operators [6, 59, 65], and the Haag-Kastler (Araki- Haag-Kastler) algebraic axioms for bounded, local Hermitian Hilbert space operators [6, 70]. These axioms all consider Hilbert space realizations in addition to Fock space but the realizations that have been constructed do not exhibit interaction. The example realizations of relativistic quantum physics in section 2.5 and [32, 34, 38] satisfy a technical revision to the Wightman-Borchers axioms that preserves the physical properties of these established axioms. The example realizations also satisfy the Haag-Kastler axioms with a trivial realization for isotonyg [70] and the indicated refinement to the quantum-classical correspondences. The revised Wightman-Borchers axioms are presented in section 2.2.

gDiscussed in section 2.3, there are no states with support strictly limited to bounded spatial volumes. 1 BACKGROUND AND SUMMARY 17

The significant enhancement of the unconstrained development over canonical quantization is elimination of conjecture for Hilbert space operators by refinement to the quantum-classical correspondence. Elimination of conjecture from a development loses nothing. A more general development includes canonical quantization, should it be realizable. Revision of the quantum- classical correspondence is motivated to consistently describe observables such as location and fields in example nontrivial realizations of relativistic quantum physics. The field in (1) is not Hermitian for the example realizations of nontrivial relativistic quantum physics in section 2.5 and since a physically equivalent development of the free field lacks Hermitian field operators [32], it is questionable whether Hermitian fields are essential. The refined quantum-classical correspondence discussed in section 1.1 is an approximation that is essentially physically equivalent yet decisively distinct mathematically: the field in (1) is not required to be Hermitian. The Wightman-Borchers functional analytic axioms for VEV of quantum fields include formal Hermiticity and totality conditionsh that imply that the real, observable components of the quantum field in (1) are densely defined Hermitian Hilbert space operators. The formal Hermiticity and totality conditions are discussed in section 2.2 and the real, observable components of a field are identified below (72) in section 2.10. Formal Hermitic- ity and totality implement the quantum-classical correspondence of the canonical formalism but lack a compelling phenomenological basis, and these two conditions apparently preclude realization of nontrivial relativistic quantum fields. Satisfaction of these two conditions distinguish known physically trivial examples, free fields, monomials and generalized free fields, from the nontrivial constructions. Without these two conditions, the remaining Wightman-Borchers axioms are realizable. Demonstrated in section 2.5 and [32, 34, 38], there are nontrivial, local, causal, Poincarécovariant generalized functions that define the scalar product of states supported solely on positive energies. These conditions: nontriviality, locality, causality, Poincaré covariance and positive energies, are more firmly motivated characteristics of relativistic physics than Hermiticity of the field in (1). The totality constraint, that there must be a state in nature for every sequence of tempered test functions, precludes the nontrivial example realizations, and the formal Hermiticity condition precludes nonzero spins in the nontrivial example realizations. Without the totality and formal Hermiticity conditions, there are classes of nontrivial relativistic quantum physics realizations for every free field or composite of free fields. The canonical formalism’s exact elevation as the quantum-classical correspondence is the impediment to the example realizations of relativistic quantum physics. The nontrivial example realizations introduce connected contributions to the VEV (1) such as n ˜ ˜ Y 2 2 2 h... Φ(p1)Ω| ... Φ(pn)Ωi = ... + cnδ(p1 +p2 + ... +pn) δ(pj −m c ) (4) j=1 in the case of a neutral scalar field Φ(x)(Nc = 1), for n ≥ 4 and with Φ(˜ p) the Fourier transform

hNot called out as axioms but formal Hermiticity and totality are assumed within a statement of the axioms [6, 7, 59]. 1 BACKGROUND AND SUMMARY 18 of Φ(x) [32, 34]. The selected basis function space P includes only states with positive energy: P includes only functions with Fourier transforms that vanish on the negative energy mass shells. Fourier transforms of elements of P vanish if E < 0 within the support of the VEV. The physical properties of relativistic quantum physics are exhibited in the Hilbert space with elements labeled by functions based on P. At least 3+1 dimensional spacetime is necessary to include massless particles and define (4) as a generalized function, and 3+1 dimensional spacetime suffices for the nontrivial examples. The interaction captured in these unconstrained constructions exhibits distinct classical correspondences in different regimes: plane wave scattering exhibits a short range Yukawa-like equivalent potential in first Born approximation [32, 34]; and isolated, particle-like concentrations in the support of states follow the trajectories of long range g/r potentials in nonrelativistic approximations [33]. This suggests that the evolution of states is approximated on relatively large scales by the motions of classical particles in long range interactions such as gravity and electromagnetism while at short ranges, interactions such as the strong and weak nuclear forces dominate. The unconstrained dynamics is described by effective, non-free field Hamiltonians over limited duration intervals in nonrelativistic limits in section 3 and [33], and by first Born approximation in plane wave scattering for weak coupling in [34, 36, 38]. Significantly, except for a phase difference between the forward and scattered contributions, plane wave scattering amplitudes include asymptotes at weak coupling to Feynman series results. In the example of a neutral scalar field, scattering cross sections coincide with first order expansions for a :P(Φ)4 : interaction. Example realizations provide explicit scattering amplitudes that agree with Feynman series results for sufficiently weak interactions [32, 34] and in some examples, with corrections at small distances (large exchange momenta) [36]. The unconstrained constructions exhibit promising connections to established QFT developments. Do realizations that include VEV such as (4) exhibit interaction “at a point” or are they string theories? Of course, such characterizations are not intrinsic to quantum mechanics but are imposed by classically-inspired, canonical formalism conjecture: by a predisposition for classical descriptions. The generator of temporal translations, the Hamiltonian, is a self-adjoint operator and in the example realizations of relativistic quantum physics, even when interaction is manifest, the Hamiltonians coincide with what is identified in the canonical formalism as free field Hamiltonians [34]. The canonical formalism-identified interaction Hamiltonian is zero. The VEV including (4) are solutions to the Klein-Gordon equation yet exhibit interaction. The exhibition of interaction is captured in the scalar product determined by VEV that include terms like (4). Mass shell singularities of these VEV together with the lack of a constraint that momenta must be equal in pairs result in nontrivial scattering. A theorem derived within the canonical formalism, a result of the assertion that the quantum field in (1) elevates a classical field and is consequently Hermitian, implies that solutions of the Klein-Gordon equation provide trivial physics. The canonical formalism-derived theorem provides that if the fields in the VEV (1) are densely defined Hermitian Hilbert space operators and exhibit the physical properties of relativistic quantum physics, A.1-5 in section 2.2, then VEV that satisfy the Klein- 1 BACKGROUND AND SUMMARY 19

Gordon equation are free field VEV. This result is the Jost-Schroer theoremi [6, 59]. However, this theorem does not apply in the unconstrained development. Without the conjectured exact quantum-classical correspondences of the canonical formalism, without assertion that the quantum field in (1) is realized as Hermitian Hilbert space operators, solutions to the Klein-Gordon equation are of interest as demonstrated by (4) and coincidence of the scattering predictions with first order Feynman series results. This suggests that the canonical formalism conjecture for quantum-classical correspondences precludes nontrivial realization of relativistic quantum physics. The refined, more approximate and conditional quantum-classical correspondence that does not require Hermitian operators in (1) admits nontrivial realizations like (4). Alternative descriptions of quantum dynamics, alternatives to extrapolation of classical descriptions, are introduced by the refined quantum-classical correspondence. That an interaction Hamiltonian rather than the scalar product implements interaction is a classical predisposition. The unconstrained constructions in section 2 and [32, 34, 36, 38] exhibit both interaction and the physical properties of relativistic quantum physics. The physical properties of relativistic quantum physics are described in section 2.2. The example constructions realize the additional possibilities sought in an unconstrained development of relativistic quantum physics. In QFT, the endeavor has been to identify and regularize Lagrangians to produce a series expansion for S-matrix elements [5]. Indications are that the series expansions do not converge. The Lagrangians are selected to elevate classical developments in the case of the long range QED and gravity, except that gravity does not regularize acceptably. This established Lagrangian development should lead to a set of Wightman-Borchers axiom-compliant VEV except that one of the failings of canonical quantization applied to QFT is that the scalar products of intermediate states constructed from (3) are not Lorentz covariant. In contrast to operator-based Lagrangian developments, the evolution of general states are explicitly constructed in the unconstrained development. Interaction is captured in nonzero likelihoods of observing distinct momenta and particle species. The finite likelihoods of observing changes to momenta and particle creations or annihilations are expressed in the nontrivial, plane wave scattering amplitudes [32, 34, 36, 38]. The nontrivial scattering amplitudes indicate that the evolution of intermediate states is nontrivial. State descriptions, whether the support of functions are overlapping or distantly separated, whether interaction is manifest or attenuated by cluster decomposition, determine likelihoods. Cluster decomposition implies that the interaction of localized, isolated concentrations in the support of states weakens with increased spatial separation. In particular, sufficiently isolated, localized particles satisfy free field-like descriptions and in these instances it is taken as axiomatic (A.7 in section 2.2) that the particles propagate nearly freely. This implementation of interaction is natural to quantum mechanics but is less analogous than the canonical formalism to classical concepts. The classical concept is that a Hamiltonian generator of time translation implements interaction. Interpretation of state descriptions is revised to

iThe Jost-Schroer theorem refers to results including that if the two-point functional of a local Hermitian scalar field (2) with a cyclic vacuum is the Pauli-Jordan function, then all the VEV are free field VEV. 1 BACKGROUND AND SUMMARY 20 be conditional in the unconstrained development: the interpretation of a state as particles is indeterminant unless the support of each constituent particle is isolated and localized.j With the interaction described by the scalar product, the unconstrained development takes fuller advantage of quantum mechanical possibilities. The endeavor in the unconstrained development is to understand the physics implied by constructed VEV and in particular, to describe the quantum-classical correspondences including bound states. Selection of VEV includes an identification of the elementary particles. Scattering amplitudes are explicit. Example plane wave scattering amplitudes are presented in section 2.15 and [34, 36, 38].

1.5 Location and correspondences in relativistic quantum physics Although the elevation of location to Hermitian operator succeeds in nonrelativistic physics, this exact elevation is precluded by relativity even for free fields as discussed in section 5. But, clearly, location remains an observable. The classical dynamical variables x do not elevate to Hermitian operators but the spatial support of functions that describe states is still interpreted as corresponding to locations of elementary particles, at least in free field developments and, as a consequence of cluster decomposition, in instances with interaction if the supports of the functions that describe states represent sufficiently isolated and localized particles. The Hermitian Hilbert space operator associated with an observable classical dynamic variable that describes a body results from the correspondence:

1. particular values of the observable are representative of the supports of functions that label elements of subspaces of states, e.g., there is a subspace of states essentially localized in a neighborhood of x,

2. an association of orthogonal projection operators with these subspaces is bijective,

3. linear combinations of the projections with coeﬃcients that are representative values of the observable for the subspaces approximate the desired Hermitian operators,

4. a limit as the subspaces become optimally associated with values of the classical dynamic variable is the corresponding Hermitian operator.

The Hermitian operators that correspond to location derive from descriptions of single elementary particles that are localized consistently with relativity. These states are described by the relativistically invariant localized functions of Theodore Newton and Eugene Wigner [44]. The relativistically invariant localized functions describe a body located near x at time t and the general forms of Hermitian rigged Hilbert space operators result in the Hermitian

jThe state transforms under Lorentz transformation as an n-body state but the observable properties of the state, its support, is not necessarily interpretable as n bodies. From the discussion in section 2.1, there can be a signiﬁcant likelihood of the state being perceived as ` 6= n bodies. From Born’s rule, likelihoods are the squared norms of scalar products and the example nontrivial scalar products are demonstrably Lorentz invariant. 1 BACKGROUND AND SUMMARY 21 operator associated with location. Within the subspace of one particle states, the resulting Hermitian location operator is Z Xbν := dx xνEx. (5) R3 The Ex = |ψxihψx| map test functions to the relativistically invariant localized functions (17), the generalized eigenfunctions of Xbν,

Xbνψx(x1) = xνψx(x1), and Z Qχ := dx Ex χ are projection operators that provide a resolution of unity. ν = 1, 2, 3,

x = (x0, x1, x2, x3) = (ct, x),

3 3 dx is Lebesgue measure on R , and χ is a measurable neighborhood within R . Properties of these functions ψx(x1) are discussed below in section 2.1. The Qχ are projections (idempotent, self-adjoint operators)k in the one particle subspace of a Hilbert space with a relativistically invariant scalar product and, as a consequence, the eigenfunctions of Xbν do not include Dirac delta functions. Dirac delta functions are the generalized eigenfunctions of the non-Hermitian multiplication by x. Self-adjoint, rigged Hilbert space operators are realized as real linear combinations of projection operators such as (5) (theorem 1, appendix to section 4 [19], lemma 5.6.7 [40]). The designations of Hilbert space operators as self-adjoint or Hermitian are discussed in section 2.1. Orthogonal projection operators are Birkhoff and von Neumann’s propositions [62] and in the unconstrained constructions, the ranges of the projections Qχ are subspaces of positive energy, Poincarécovariant, essentially localized states. The functions ψx(x1) are essentially localized but not strictly localized and indeed, there are no states strictly limited to bounded spatial volumes described by functions in the basis function space P. P, a subspace of the Schwartz tempered functions S, is discussed in section 2.3. There are many linear differential operators that canonically commute with the Hermitian momentum operators Pν. Examples are constructed in appendix 7.2 and include: the established Hermitian elevation of x that applies in the L2 Hilbert spaces of nonrelativistic physics; the Hermitian operator that has Newton and Wigner’s relativistically invariant localized functions [44] as eigenfunctions and applies in the established Fock space development of the relativistic free field; and the Hermitian operator that applies in the Hilbert space based on P in

k 0 0 0 0 ExEx0 = 0 if x 6= x and t = t , and then QχQχ0 = 0 if χ ∩ χ = ∅ for spatial regions χ, χ within a constant time plane. The functions ψx(x1) and ψx0 (x1) are orthogonal at coincident times. However, evaluated at distinct 0 times t 6= t , ExEx0 6= 0 and the regions with QχQχ0 = 0 are insufficient to conclude from Bogolubov’s edge 0 4 of the wedge theorem that the Qχ = 0. If Q∆Q∆0 = 0 for projections onto spacetime volumes ∆, ∆ ⊂ R 0 with ∆ ∩ ∆ = ∅ and space-like separated, then the Q∆ = 0 [69]. There are no strictly localized states in these examples of nontrivial relativistic quantum physics. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 22 the unconstrained development of the relativistic free field [32]. The operators corresponding to location are determined in the Hilbert space context as well as by classical correspondence. For a free field, the Hermitian operator Xbν in (5) that applies in the one-particle subspace extends to multiple particle states by second quantization. With interaction, the Hilbert space is not Fock space and if the interaction of states is significant, then the multiple argument components of states are not necessarily interpretable as a determined number of particles. If the correspondence of state description with particles is obscure, then the extensions of these single particle, Hermitian location operators Xbν to multiple argument subspaces do not necessarily correspond with a particular species of particle’s location. The number and species of particles and even whether the state is perceived as particles is not evident unless the contributions of the interaction are negligible for the state of interest. That is, a free field location operator applies approximately for states that are well described by classical particles. Free field states are readily interpreted as consisting of particular numbers and species of particle. Otherwise, the dominant supports of functions correspond to likely locations but the numbers and species of particles at those locations is indeterminant. Discussed in section 2.1, a classical particle interpretation is a perception that applies to particular states included within the more general quantum description of states. Quantum-classical correspondences for fields may similarly rely on associations of states. With consideration limited to the observation of features associated with bodies, if we calculate that classical bodies move according to a force due to a field with the sources on the bodies (e.g., the Lorentz force and Maxwell’s equations in electrodynamics, or geometrodynamical gravity) and this classical dynamics produces an accurate approximation to the quantum dynamics of the support of the functions that describe bodies, then we would say that the quantum model corresponds with the classical field theory. This correspondence would be valid regardless of whether the elevations of the observable field components are Hermitian operators or the description of the quantum dynamics results from a “quantization” of the associated classical dynamics. The correspondence of classical field theory with quantum dynamics is established by approximation of the motions of observed bodies. The correspondence of the Hermitian Hilbert space operator Φ(b x)κ from section 1.4 with a classical field is then established by the approximations for the dynamics of features of the states that are interpretable as classical bodies. The approximations locally associate states with the classical field that produced the dynamics. States necessarily describe bodies when the dominant support of the functions are isolated and localized [33].

2 Quantum mechanical descriptions of nature

Example realizations of relativistic quantum physics that exhibit interaction are constructed in this section. The description of the states of nature by functions and Hilbert space scalar products of these functions are developed. The constructed VEV (1) satisfy a revision to the 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 23

Wightman-Borchers axioms for functions from a basis space P. In section 2.1, the Hilbert space realizations are described, and the revised axioms are presented in section 2.2. The selection of basis space P is described in section 2.3. In section 2.4 and appendix 7.7, the well established VEV for the relativistic free ﬁeld are discussed in this unconstrained context. The example VEV that exhibit interaction are constructed in section 2.5, and satisfaction of the axioms and physical correspondences are discussed in sections 2.6-2.15.

2.1 Quantum mechanical description of state In current understanding, the natural world is described by quantum mechanics. In quantum mechanics, states are described by complex-valued functions over three dimensional space. These functions are parametrized by time. The resulting functions over a 3+1 spacetime label elements of a rigged (equipped) Hilbert space. This description can be very different than, or quite similar to, the “a ball is a ball” description of classical physics. A richer range of state descriptions is included in quantum mechanics and the perception of states by observers is a key consideration. The unconstrained development emphasizes realization of state descriptions as elements of Hilbert spaces over conjecture for the properties of particular operators. The equipped or rigged Hilbert spaces of interest are completions of linear vector spaces of equivalence classes of function sequences [6, 10, 59]. The completion and equivalence classes are in the Hilbert space norm described below in (6). The properties of states, likelihoods of observations, quantum-classical correspondences and properties of the field all derive from the choice of vacuum expectation values (VEV) for products of the field and the choice of basis function space [6, 7, 32, 34, 38, 59]. For realizations of relativistic quantum physics: states are elements of this rigged Hilbert space; temporal evolution of these state descriptions is likelihood preserving (unitary) and causal; the temporal evolution of features of the quantum descriptions of state are approximated by classical mechanics in appropriate instances; the likelihoods of observation are given by Born’s rule; and the local observables of non-entangled, distantly space-like separated bodies are essentially independent. A realization of relativistic quantum physics is the specification of a basis function space P and VEV, and a physical understanding of the state descriptions. The VEV specify the scalar product hf|gi of Hilbert space elements described by sequences of functions f, g ∈ P or their limits, and as a consequence of Born’s rule, specify likelihoods. In the nonrelativistic development, time is a universal parameter and many Hamiltonians are compatible with the nonrelativistic scalar product [55] while in relativistic physics, the VEV determine the time translations of states, and hence, the Hamiltonian. This Hilbert space description fulfills essential properties of nature [11, 62]. In a modern approach [9, 45], quantum mechanics is taken as the complete description of nature: there is no need for observers external to the development nor for the description of quantum dynamics to originate in classical dynamics. There is no “macroscopic” or “classical” domain governed by distinct physical principles. The state, an element of an appropriate Hilbert space, is the entire state description and the simple, comprehensive, and consistent description of nature. Perceptions 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 24 of these states by observers and the temporal evolution of states are the concerns of mechanics. The observable features of the quantum mechanical descriptions of the states of nature are extracted from the support of these functions and their Fourier transforms. Complex-valued functions supported solely in isolated, localized neighborhoods describe what is typically interpreted as classical bodies. Functions also describe waves that may be widely distributed over space. Multiple bodies are described by multiple argument functions although with interaction, the number of bodies described coincides with the number of arguments only when states are accurately represented by a classical description. The association of bodies with function arguments is a classical interpretation of the quantum description of state. Multiple component fields introduce distinct particle species and spin polarizations. In quantum mechanics: the possible values and likelihoods of perceptions of the positions and momenta of bodies all derive from the functions that describe the state and as a consequence, the position and velocity of a body can never both be arbitrarily precisely known; and similarly described bodies are indistinguishable. Bodies with the same mass, charges and polarizations are indistinguishable if they are not isolated. That is, when the spatial support of the functions that describe two or more similar bodies overlap, observation at those locations cannot distinguish which body was actually seen. States are perceived as point-like in interactions that localize the description and as wave-like in, for example, propagation through a Michelson interferometer or Young’s double slit. The interpretation of a body as wave-like occurs before we detect and perceive a localized, point-like function. This quantum description of state as a Lebesgue summable function is a primary source of discomfort with quantum mechanics. We typically perceive bodies; our preconceptions dissuade us from a description of observations as functions. Classical bodies are perceptions rather than descriptions of reality. Central to these perceptions is consideration of the likelihoods of observation, the dominant support of functions, and the typical results of observation. Here, the support of a function refers to contiguous subsets of the domain that include an arbitrarily large proportion of the Hilbert space norm of the function: supports are volumes of space or momenta such that summation of the function over the support nearly 3 equals the summation over the entire domain R . The quantities of physical significance are Lebesgue summations of the functions that describe states: states are labeled by any function from within Hilbert space norm-equivalent classes. In mathematics, the support of a function typically refers to the subset of the domain that is not mapped to zero. Here, lesser locally Hilbert space norm-averaged magnitudes of the functions are considered zero in determination of the support, and negligible values are determined by the choice of likelihood and volumes. Functions with sufficiently localized and isolated spatial support are perceived as classical bodies, and in cases such as Young’s double slit or Michelson’s interferometer when the state is described by a function more widely distributed over space, we do not perceive bodies until detection, after the wave effects have been expressed. If we try to observe whether a particle is within one slit or one interferometer arm, the relevant state description becomes localized by the observation and wave effects are suppressed. Observation determines the relevant state description in quantum mechanics; we do not necessarily perceive the initial state descriptions 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 25 nor even a state close to the initial state. Rather, the relevant descriptions that emerge from observation and determine the possibilities for future observations result from the interaction of state descriptions, observer and observed, that constitutes observation, and happenstance. Our everyday experience is in a world awash in photons that constantly localize our perceptions of relatively massive bodies. The process of observation is fundamental and necessary in quantum mechanics. Evident in our daily experience, our perceptions of states and their temporal evolution are often well described by the classical model. Localized, isolated support of functions are features of a quantum description of state that approximately correspond with the idealized classical concept of a body. Much of these notes is devoted to this approximation of quantum mechanics provided by classical mechanics. The approximation does not apply to all state descriptions but it applies in common cases. Exceptions from classical approximations occur particularly for light mass and relativistic bodies but are also evident in entanglement of state descriptions and in the indistinguishability of similarly described bodies. Although much of our daily experience appears to be described by classical mechanics, more probing and comprehensive observations contradict the classical description. Indeed, it was not until the early twentieth century that classical concepts were contradicted. The necessity for a quantum mechanical description of nature did not emerge until physics turned its attention to the spectra of electromagnetic emissions, the heat capacity of solids, and the photoelectric effect. Hilbert spaces: A Hilbert space H is characterized by the number of linearly independent elements. The descriptions of states are elements of the Hilbert space described by sequences of functions that can depict the vacuum, a single body, multiple bodies or unparticle-like [26] states. The locations and velocities of bodies are perceptions extracted from these functions in appropriate cases. For every two elements labeled f, g ∈ P in a Hilbert space of interest in quantum mechanics, there is a complex number hf|gi,

f, g ∈ H 7→ hf|gi ∈ C designated the scalar product of the elements. Properties of this scalar product include that hg|fi is the complex conjugate of hf|gi and the scalar product is linear in the second argument, hf|αg + βhi = αhf|gi + βhf|hi for α, β ∈ C, and as a consequence, complex conjugate linear in the ﬁrst argument. Scalar products are nonnegative, hf|fi ≥ 0, and this provides satisfaction of the Cauchy-Schwarz-Bunyakovsky inequality,

|hg|fi|2 ≤ hg|gihf|fi.

In particular, if hf|fi = 0, then hg|fi = 0 for every element |gi of the Hilbert space. This zero element is unique in a Hilbert space. A degenerate scalar product is deﬁned for pairs of function sequences from the basis space P considered as a linear vector space. A degenerate scalar product has all the properties of a scalar product except for uniqueness of the zero element. The Hilbert spaces of interest are the completions of linear vector spaces with elements that are equivalence classes of vectors labeled by function sequences [10]. The elements of the Hilbert space may be 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 26 labeled by any function sequence in an equivalence class. An isometry extends the degenerate scalar product to a scalar product (the elements of the Hilbert space are equivalence classes of functions equal in the norm (6) and hf|fi = 0 states that the function sequence f is an element of the equivalence class of zero). The norm is provided by the scalar product, q kfk := hf|fi. (6)

The separation kg − fk is zero for two equivalent function sequences, and two is the maximum separation of normalized state descriptions (kfk = kgk = 1). The maximum separation oc- √ curs when normalized state descriptions are antipodal (hg|fi = −1). The separation is 2 for orthogonal, normalized states. Two states f, g are orthogonal if hg|fi = 0. In quantum mechanics, every element on a ray |afi with nonzero a ∈ C describes the same physical state. States |afi are physically equivalent: overall phase and finite amplitude are physically irrelevant but relative phase and amplitude are observable. Linearity and completeness are characteristic of Hilbert spaces. A Hilbert space H is complete. The limit of every Cauchy sequence of elements |g i ∈ H is included in H. That is, if ν kg − g k → 0 ν n for ν, n > N → ∞, there is an element |gi ∈ H such that |gi = lim |g i. In a separable ν→∞ ν Hilbert space, every element is arbitrarily well approximated by a denumerably indexed sum of N linearly independent elements. X |fi = c`|eì ` with ` ∈ N, the natural numbers, and c` ∈ C. A (closed) subspace of a Hilbert space is the linear span of a subset of elements. If there are only N linearly independent elements, the Hilbert space or subspace is finite dimensional of dimension N, and if the number of linearly independent elements is unbounded but includes a denumerable, dense set of elements, the Hilbert space or subspace is denoted infinite dimensional and separable. The Hilbert spaces of interest here are separable [7]. The elements of the rigged Hilbert spaces of interest are described either by a terminating function sequence from P or the limit of a Cauchy sequence of elements from P. The element |fi is labeled by any equivalent sequence of functions g (kf − gk = 0), or any element on the ray |afi with a nonzero a ∈ C. There is an uncountably infinite number of functions in each of the countable number of equivalence classes. Likelihoods: Born’s rule provides the likelihood of observing a normalized state described by |gi from an initial normalized state |ψi. This likelihood is the squared magnitude of the scalar product, Likelihood := |hg|ψi|2 ≤ 1 if kgk = kψk = 1 with kψk2 = hψ|ψi. Born’s rule is inherent to a Hilbert space realization: no additional assumptions for the forms or properties of operators are required to evaluate 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 27 likelihoods. Discussed in section 5, the association of observations with elements of the rigged Hilbert space is more robust than conjecture for elevations of the classical dynamical variables to Hermitian Hilbert space operators. Observers interpret particular states |gi in classical terms and the descriptions of the states that interact and constitute measurement determine the possibilities for the observed states |gi. For example, a measurement may entangle an observer with states of bodies dominantly located within neighborhoods of particular locations or traveling with momenta within neighborhoods of particular momenta. Examples of observations that convey simultaneous knowledge of both position and momentum are sequences of localizing observations that exhibit the orbit of a planet or the track of a high energy particle. When observation conveys knowledge of both location and momentum, the more massive a body the better the approximation of state by a classical idealization. In the case of a particle detector, the observations entangle the observer with a time sequence of states localized nearby the observed track and with momenta nearly conveyed by the track. This detection results in neither collapse to eigenstates of location nor to eigenstates of momentum although imprecise knowledge of both location and momenta is conveyed. If the observations caused collapse to position eigenstates, knowledge of the momentum would be lost. If the observations caused collapse to momentum eigenstates, knowledge of the location would be lost. And, neither eigenstates of location nor eigenstates of momentum are elements of the rigged Hilbert spaces of interest although these eigenstates are approximated by elements. Observed states |gi are neither eigenstates of location nor momentum. Entanglement: Quantum mechanics includes the concept of entangled states. The descriptions of states are elements of Hilbert spaces and elements are expressed as linear combinations of other elements. The linear expansion of states is a property of quantum physics distinct from classical physics. In classical physics, the state of each body is determined and independently specified. A simple example of entanglement is four states: |up, upi, |up, downi, |down, upi, and |down, downi that span the spin states of two spin one-half bodies. The states describe the four possibilities for the measurement of spin polarizations along a particular axis. Any linear combination of these four states describes a state. For example, 1 1 √ |up, downi + √ |down, upi 2 2 is one possible state. For this example state, the polarizations are entangled: up for the first body occurs only with down for the second body, and down for the first body occurs only with up for the second body. The state of the first body is not independent of the state for the second and total spin adds to zero in any observation. Here, the two bodies are assumed to be identifiable by other characteristics such as mass, charge or isolated locations. Also, the states are not determined, the first body can be detected as either spin up or spin down, with equal likelihoods in this example. But, knowledge of one of the spins implies knowledge of the other due to the entanglement of states. This entanglement is established when the particles are causally related and the entanglement persists as the particles become acausally separated. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 28

The entanglement of bodies need not be perfect. For example,

a |up, up, downi + b |up, down, upi + c |down, up, upi is an example with a particular angular momentum but no perfectly entangled pair. In section 2.12, the states that result from an observation are described by linear combinations of states P of an observer entangled with the possible results of observation, |ψi = k ck |ok, ski. In the example of Schr¨odinger’scat thought experiment [54], o1 would be “observed a live cat,” o2 would be “observed a dead cat,” s1 would be “a live cat,” s2 would be “a dead cat” and these descriptions are entangled c1|o1, s1i+c2|o2, s2i. Anticipation on physical grounds is that |o1, s2i and |o2, s1i do not persist in the evolution of states. States: The representations for states in relativistic quantum physics are sequences of functions that generally describe multiple bodies. The elements |fi of the Hilbert space HP are described by sequences of multiple argument functions [6, 7].

f := (f0, f1(x1)1, f1(x1)2, . . . f1(x1)Nc , . . . fn((x)n)(κ)n ,...) (7)

4n with fn((x)n)(κ)n : R 7→ C, f0 ∈ C the component of the state in the vacuum state labeled 4 Ω := (1, 0, 0 ...), the Nc f1(x)κ ∈ P(R ) describe single particle states with masses mκ, and n each of the (Nc) 4n fn((x)n)(κ)n := fn(x1, x2, . . . xn)κ1κ2...κn ∈ P(R ) with n ≥ 2 are associated with n-particles if states are well approximated by classical descriptions, for example, in the absence of interaction. Each integer κj ∈ {1,Nc} and Nc is the number of ﬁeld components. Multiple argument functions include descriptions of multiple particles although in the unconstrained development, state descriptions generally correspond to an undetermined number of particles when there is interaction. Functions with n arguments are generally not orthogonal to k 6= n argument functions for the nontrivial scalar products. The basis function space P, discussed in section 2.3 below, is a subspace of the space of terminating sequences of Schwartz tempered functions S. P ⊂ S provides that S0 ⊂ P0 for the duals [18] but for these example constructions, the constructed VEV lie within S0. In the sequence (7), n there are (Nc) functions in the subsequence with n arguments,

fn((x)n)1...11, fn((x)n)1...12, . . . fn((x)n)NcNc...Nc .

The labels on arguments do not identify particular bodies. Gibb’s paradox is resolved in quantum mechanics by the indistinguishability of similarly described bodies. If the spatial support of the functions fn((x)n)(κ)n that describe two or more bodies with the same mass, charge and polarization overlap, observation of a body at those locations cannot distinguish which body was actually seen. Unless the description of the support of a state is isolated and localized to great likelihood, a trajectory can not be reliably associated with a particle. This indistinguishability is implemented by symmetry of the scalar product with interchange of 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 29

argument labels. If xj − xk is space-like in the support of fn((x)n)(κ)n , then f(. . . xj, xk,...)(κ)n 4n is equivalent to either ±f(. . . xk, xj,...)(κ)n . For free field VEV and functions fn ∈ P(R ), an n-argument function describes an n-particle state [32]. More generally in relativistic physics when interaction is manifest, particle number is not conserved and states with different numbers of particles are generally not orthogonal. With interaction and if n ≥ 2, an n-argument function is not generally associated with any fixed number of bodies k ≥ 2. But even with interaction, there are states well approximated by n-particle descriptions. States described by n-argument functions with localized and isolated support in each argument are locally well approximated by the dynamics of n freely propagating bodies [33]. In these instances, the greatest likelihood is that an n-particle state is described by an n-argument function. But, more generally, since scalar products of functions with different numbers of arguments do not vanish, there are nonzero likelihoods of observing k 6= n particles and distinct particle species for a state described by an n-argument function. The Scalar Product: To achieve Poincaréinvariance of likelihoods and limit the support of states to nonnegative energies, kernel functionals Wk,n−k((x)n)(κ)n implement the degenerate scalar product for the sequences of functions that describe states (7). The degenerate scalar product is X X W(f ∗, g) := W (f ∗ , g ) n,m n,(κ)n m,(κ)n+1,n+m (κ)n+m n,m (κ) n+mZ X X (8) = d(x)n+m (D·)nWn,m((x)n+m)(κ)n+m n,m (κ)n+m

×fn(xn, . . . x1)κn...κ1 gm(xn+1, . . . xn+m)(κ)n+1,n+m 4n with f, g ∈ P and z the complex conjugate of z ∈ C. f, g ∈ P implies that each gn, fn ∈ P(R ) 4 and g0, f0 ∈ C. Each spacetime Lorentz vector xk is summed over R and each κj ∈ N is summed from 1 to Nc. n and m are summed over the nonnegative integers, {0, ∞}. The 4n Wk,n−k((x)n)(κ)n are generalized functions dual to Schwartz tempered functions S(R ) and generalize the Wightman functionals [7]. Multiple arguments are denoted

(x)j,k := xj, xj+1, . . . xk in the ascending case, (x)j,k := xj, xj−1, . . . xk otherwise and (x)n := (x)1,n. The ∗-morphism of function sequences uses complex conjugation, argument transpositions, and a nonsingular Nc × Nc linear transformation D. ˜∗ T ˜ fn((p)n)(κ)n := (D ·)nfn(−pn,... − p1)κn...κ1 (9) T in matrix notation with (D )ij := Dji, each Dji ∈ C and N N N Xc Xc Xc (D·)nV(κ)n+m := ... Dκ1`1 Dκ2`2 ...Dκn`n V`1...`nκn+1...κn+m . (10) `1=1 `2=1 `n=1 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 30

The space of function sequences becomes an algebra with a product

n X f × g := (f0g0,..., f`(x1, . . . x`)(κ)1,` gn−`(x`+1, . . . xn)(κ)`+1,n ,... ). (11) `=0 D, designated here as Dirac conjugation, satisﬁes

DD = I (12)

∗∗ with I the Nc × Nc identity and as a consequence, the ∗-morphism satisﬁes f = f and (g × f)∗ = f ∗ × g∗. This ∗-morphism is an involution of the algebra of function sequences if it is an automorphism. But, the ∗-morphism is not an automorphism for the P selected in the unconstrained constructions: P∗ 6= P due to constraints on the energy support of the elements 4n 4n of P(R ). Introduced in section 2.3, the elements of P(R ) have zeros on the negative energy mass shells and the ∗-morphism maps these zeros to the positive energy mass shells.l The displayed form for (8) follows from the convenient identity

X ∗ X † ∗ Wn,m(((V ·)nfn) , gm) = (DV D·)nWn,m (fn, gm). (κ)n+m (κ)n+m

† † V is the Hermitian transpose of V ,(V )jk = V kj. The scalar product results from the isometry

hf|gi = W(f ∗, g) that associates elements of the Hilbert space with equivalence classes of sequences of functions. Then the norm (6) is kfk = (W(f ∗, f))1/2. A sequence of kernel functionals is denoted

W := (1, W1,0,, W0,1,..., Wn,0, Wn−1,1 ..., W0,n,...). (13)

n The arguments of Wk,n−k are (x)n, (κ)n and similarly to (7), there are (Nc) functionals in the subsequence Wk,n−k distinguished by the values of the κj ∈ {1,Nc}, j ∈ {1, n}.

The Quantum Field and VEV: The kernel functionals Wk,n−k((x)n)(κ)n provide the vacuum expectation values (VEV) of the quantum field (1). The quantum field Φ(f) is defined as the multiplication (11) in the algebra of function sequences [7].

Φ(f) g := f × g (14)

lR dx e−ipxψ(x) = R dx eipxψ(x) = ψ˜(−p). 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 31 with f := (0,..., 0, f(x1)κ, 0,...), g ∈ P. Only one, one argument component κ of this f is nonzero and formally Z Φ(f) = dx Φ(x)κf(x)κ

(without a summation convention). The designation Z Φ(f)κ := dx Φ(x)κf(x) is also used with an f ∈ P selected for the mass mκ of field component κ ∈ {1,Nc}. The scalar product (8) and definition of the quantum field (14) result in

(D·)kWk,n−k((x)n)(κ)n := hΦ(xk)κk ... Φ(x1)κ1 Ω|Φ(xk+1)κk+1 ... Φ(xn)κn Ωi (15) using the notation (10). Use of the VEV in the scalar product equates a general state description with a linear combination of products of the quantum field applied to the vacuum (2). Then for the example constructions, cluster decomposition and elemental stability (A.6 and A.7 in section 2.2) provides that when the support of a state is localized and isolated, the state is well approximated by a free field state consisting of determined numbers of the elementary particle species. This associates the quantum field with elementary particles. Discussed in section 3, the association of quantum field with classical field is determined by the predicted motions of bodies, whether the trajectories of classical bodies corresponding to appropriate states follow trajectories predicted by a classical field theory. Fields (14) may be Hilbert space operators but are generally not Hermitian in the example nontrivial developments. States of a single particle: The one argument component f1(x)κ of a state description f is identified as the description of an elementary particle state. This associates the components of the field with particle species and polarizations. From axiom A.7 of section 2.2 below, one argument functions are orthogonal to all but other one argument functions when vacuum polarization vanishes. Then, as a consequence of cluster decomposition A.6 of section 2.2, this description as one particle states extends to sufficiently isolated and localized support of components of multiple argument states. This extension is an approximation in likelihood. A function ψ(x) := f1(x)κ of a single argument describes a single body of mass mκ and determines 3 a complex value for every point x ∈ R at each time t ∈ R. The arguments of the function ψ(x) are distances x = ct, x for a time t, spatial location x and the speed of light c. From Born’s rule, the likelihood of observing a one particle state of mass mκ described by ψ = (0,..., 0, ψ(x), 0, 0 ...) from a one particle state of the same charge, polarization and mass mκ described by g with ϕ(x) = g1(x)κ is the squared magnitude of Z hψ|gi = dxdy W1,1(x, y)κκψ(x)ϕ(y) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 32 from the scalar product (8) for this example. For the free scalar field [23, 55, 63] or unconstrained scalar field examples [32, 34, 38], Z dp hψ|gi = ψ˜(ω, p)ϕ ˜(ω, p) (16) 2ω

p 2 2 2 ˜ with ω determined by the mass mκ as deﬁned below in (26), ω = mκc + p . ψ(p) designates the Fourier transform (24) of the function ψ(x). The notation p is used for either wave-number with units of inverse length or momentum, the reduced Planck constant times wave-number. The notation is resolved by context. With the emphasis on descriptions of states rather than operators, the focus now is to identify those states that are perceived as localized elementary particles. The development continues with the case of a neutral scalar ﬁeld, Nc = 1. The relativistically invariant localized functions of Newton and Wigner [44],

1 ˜ 2 −ipxo ψxo (p) = (2ω) e , (17)

4 are mutually orthogonal functions labeled by spacetime points xo ∈ R . Evaluated for equal times, xo,0 = yo,0, the ψxo (x) are orthogonal for the scalar product (16),

3 hψxo |ψyo i = (2π) δ(xo − yo).

Since dp/ω and pxo are Lorentz invariants, the relativistically invariant localized functions are orthogonal on constant time planes in any inertial reference frame if the parameters transform 0 −1 xo = Λ (xo − a) under Poincarétransformation. In the conventions of (22) below for this scalar field case, Poincarétransformation is

(a, Λ)ψ˜(p) = e−ipa ψ˜(Λ−1p).

These functions ψxo (x) are localized. Discussed in [44] and appendix 7.2, the spatial extent of the relativistically invariant localized functions (17) is in a neighborhood of x = xo characterized by 5 (1) 4 (λc/r) H 5 (ir/λc) 4 (1) 2 2 − 5 for the Hankel function Hν and r = (x − xo) . The r 2 divergence for r ≈ 0 and the decline at large r/λc demonstrates that the dominant support of ψxo (x) is in a neighborhood of xo of size proportional to the reduced Compton wavelength,

λ = ~ . (18) c mc

The nonrelativistic approximation to (16) is an L2 scalar product and then the ψxo (x) approximate the classical ideal, Dirac delta functions. In the nonrelativistic limit, m2c2 p2 and 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 33

ω ≈ mc. In relativistic cases, the Dirac delta functions are not mutually orthogonal and are not the generalized eigenfunctions of a Hermitian Hilbert space operator in HP . The functions

ψxo (x) label eigenfunctions of Hermitian Hilbert space operators developed in [44] and appendix 7.2, and are the optimally localized, orthogonal functions for (16). As discussed in section 1.5, the functions ψxo (x) associate a Hermitian Hilbert space operator with location. Interpretation of the functions ψ(x) as providing likelihoods of locations of elementary particles derives from Born’s rule, axiom A.7 below, and the dominant support of these relativistically invariant localized functions (16). In the notation of section 1.5 and for a normalized, broadly supported one particle state described by ϕ(x), Born’s rule assigns a likelihood of observing the associated body in a neighborhood χ of a location xo at a time t as Z Z 2 hϕ|Qχϕi = dxo hϕ|ψxo ihψxo |ϕi ≈ dxo |ϕ(xo)| , χ χ the squared magnitudes of complex numbers averaged over the neighborhood in space using

Lebesgue measure. The ψxo (x) are appropriately normalized for a projection Qχ and the states are evaluated at equal times. For this approximation, ϕ(x) is nearly constant over a Compton wavelength. The likelihood that a classical body will be perceived in the neighborhood χ of a point results from the Lebesgue measure weighted squared magnitudes of the complex-valued function ϕ(x) that describes the one particle state. These states ψxo (x) and associated projections Qχ apply to observation of isolated one particle states and more generally, whether an n-argument state can be accurately interpreted as consisting of n particles must be considered. Identification of the quantum descriptions of state as particles is a classical interpretation. Ob- servation of location in relativistic quantum physics is also addressed in sections 1.5, 2.11 and 2.12, and appendix 7.2. Rigged Hilbert spaces: Rigged (equipped) Hilbert spaces are appropriate settings for quantum mechanics. VEV that provide a Poincaréinvariant scalar product (8) must be functionals: the VEV can not be functions [6]. In the example of Wightman’s development of the free field [59], the elements are labeled by function sequences in the completion of the Schwartz tempered functions S in the Hilbert space norm. The tempered functions S are smooth (infinitely differentiable) and exhibit rapid decline for large values of their arguments [18]. The basis function space P used in the examples of the unconstrained development includes only those elements from S with Fourier transforms that vanish on the negative energy mass shells. Three classes of functions, denoted a Gelfand triple after Israel Gelfand, describe a rigged Hilbert space. The elements of a countably normed function space are denoted test functions [6, 17, 18]. A particularly useful space of test functions are the Schwartz tempered functions S: the space of Fourier transforms of S coincides with S. The associated class of generalized functions S0 are the linear functionals dual to S. Linear functionals T (x) map functions to complex numbers. 0 T (x) ∈ S : ψ(x) ∈ S 7→ T (ψ) ∈ C. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 34

S0 is usefully conceived as limits of functions T (x) such that Z T (ψ) = dx T (x)ψ(x) is ﬁnite when ψ(x) ∈ S with acknowledgment that this concept includes generalized functions T (x) that in the limit are not summable using Lebesgue measure. Indeed, the generalized functions dual to the functions of bounded support can be represented

X Z dnψ(x) T (ψ) = dµ (x) k dxn n with the summation over a finite number of terms using measures µk on the real numbers [18]. Dirac (generalized) functions δ(x) have point support. Z Z y dψ(x) dx δ(x − y)ψ(x) := dx = ψ(y) dx with the understanding that although δ(x) is not a summable function, δ(x) is well defined as a functional and a limit of a sequence of functions from S [18]. HS consists of S plus the Cauchy 0 sequences convergent in the norm (6). As a consequence, the Gelfand triple (S, HS , S ) satisfies

0 S ⊂ HS ⊂ S .

The eigenfunctions of location and momentum are not elements of HS but are elements of 0 S . The elements of P ⊂ S are test functions and their limits within HS include generalized functions. The generalized eigenfunctions of a linear operator A defined in S are the generalized 0 functions Tλ(x) ∈ S such that Tλ(Af) = λTλ(f) for any f ∈ S. S suffices for many models of relativistic quantum physics that lack interaction but it has not been possible to achieve both interaction and satisfaction of the physical conditions in S. The basis function spaces P described in section 2.3 suffice to achieve both interaction and satisfy the physical conditions for the examples with interaction introduced below in section 2.5. The function spaces P ⊂ S are a union of nuclear, countably normed spaces [19]. Hilbert space operators: Hilbert space operators are linear maps of elements of a Hilbert space H to other elements within the Hilbert space [21, 40]. A is a Hilbert space operator if

A : |fi ∈ H 7→ |Afi ∈ H for a subset of elements |fi ∈ DA ⊆ H denoted the domain of A. For a bounded operator,

kAψk ≤ kAk kψk with kAk := sup(kAψk : kψk ≤ 1) < ∞. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 35

A bounded operator is continuous, its domain is a (closed) subspace, and the domain can be extended to the entire Hilbert space. Unitary operators are bounded and preserve the scalar product, that is, for unitary U, hUg|Uψi = hg|ψi. Illustrated in section 5, an unbounded operator is not continuous, its domain is not a (closed) subspace, and the domain necessarily does not include the entire Hilbert space although the domain may be dense in the Hilbert space. The adjoint operator A∗ of a Hilbert space operator A is defined by the property that ∗ hA h|ψi := hh|Aψi for |ψi ∈ DA and |hi ∈ DA∗ . The set of h ∈ H such that there is a g ∈ H ∗ with hh|Aψi = hg|ψi is the domain DA∗ of A . An operator A with domain DA is Hermitian if hu|Avi = hAu|vi for every u, v ∈ DA [21], a Hermitian operator is symmetric if DA is dense in ∗ the Hilbert space, and a symmetric operator is self-adjoint if DA = DA∗ and Au = A u for every u ∈ DA. These designations correspond to Hermitian, maximal Hermitian, and hypermaximal respectively in von Neumann’s designation [62]. If the multiplication (14) preserves equivalence classes of function sequences then the field is realized as Hilbert space operators. That is, if khk = 0 implies that kΦ(f) hk = 0 then the field maps equivalence classes to equivalence classes (Φ(f)(g + h) = Φ(f)g if g + h = g, that is, Φ(f) maps |gi to |Φ(f)gi). If the basis function space is the ∗-algebra of tempered functions S, then g∗ × h ∈ S for g, h ∈ S and if the scalar product satisfies formal Hermiticity (W.a below), then the Cauchy-Schwarz-Bunyakovsky inequality demonstrates that the field (14) preserves equivalence classes. Neither a ∗-algebra nor formal Hermiticity is necessary to realize relativistic quantum physics and the algebra P used in the unconstrained development is not involutive for the ∗-morphism (9) [34, 38]. Demonstrated in section 2.10, if formal Hermiticity is satisfied then the properties of the product (11) and ∗-morphism (9) provide that the adjoint of the field operator is Φ(f)∗ = Φ(f ∗) with ∗ designating the Hilbert space operator adjoint on the left hand side and the ∗-morphism on the right-hand side. There are too few functions in the basis function space P for Hermiticity of the field operator. Except for the vacuum state aΩ = (a, 0, 0 ...) with a ∈ R, there are no real sequences in P [34]. Hermitian field operators for the real, observable components of a field appear to be peculiar to the free field and related constructs that lack interaction. The free field and related physically trivial constructs satisfy the axioms A.1-7 and W.a-b in section 2.2 for the basis function space S. Nevertheless, for many example nontrivial constructions, the field is realized as non-Hermitian Hilbert space operators. These examples include when formal Hermiticity W.a is satisfied. Appendix 7.5 includes the demonstration that the field (14) is a Hilbert space operator for example constructions that satisfy formal Hermiticity and identifies sufficient conditions for the field (14) to be a Hilbert space operator.

2.2 Axioms for the VEV of relativistic ﬁelds Realizations of relativistic quantum physics exhibit properties characteristic of nature [6, 7, 59, 70]. From Born’s rule, the scalar products (8) of function sequences from the selected basis 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 36 function space P determine likelihoods and these likelihoods satisfy physical conditions. The physical conditions that realize relativistic quantum physics include seven prospective axioms A.1-7 adapted from the Wightman-Borchers axioms [6, 7, 59]. A.1) Regularity: the vacuum expectation values (VEV) of quantum ﬁelds are generalized functions dual to a subspace P of the Schwartz tempered functions S,

0 4n (D·)kWk,n−k((x)n)(κ)n := hΦ(xk)κk ... Φ(x1)κ1 Ω|Φ(xk+1)κk+1 ... Φ(xn)κn Ωi ∈ P (R ) with Dirac conjugation D from (9), in the notation (10), and with P ⊆ S.

A.2) Positive deﬁniteness: states are described by sequences of functions f ∈ P that label elements of a Hilbert space. The VEV are kernel functionals that deﬁne a degenerate scalar product W(g∗, f) (8) of sequences of functions g, f ∈ P. P ⊆ S, the space of terminating sequences of Schwartz tempered functions.

A.3) Relativistic invariance: transition amplitudes are the same for all observers. The degenerate scalar product is invariant to proper orthochronous Poincar´etransformations of the function sequences.

A.4) Spectral support: the generators of translations are components of an energy-momentum Lorentz vector. The spectrum of the generators of translations lie in the closed forward (nonnegative energy) cone.

A.5) Local commutativity: ﬁelds are causally related and similarly described particles are indistinguishable. Up to a sign, scalar products are invariant with interchanges of the arguments of functions f when supports are space-like separated. The notation is developed in section 2.1 and [7]. Axioms A.2-5 with A.6-7 below are designated here as the physical conditions [32]. A.1 is the mathematical context and A.2 provides the Hilbert space realization characteristic of quantum mechanics. Regularity and positive deﬁniteness imply that states are realized as elements of a rigged Hilbert space. A condition that provides an interpretation of states as particles in appropriate instances and results in satisfaction of regularity is included as A.7. Causality includes the condition: A.6) Cluster decomposition: transition amplitudes for non-entangled, distantly space-like separated state descriptions are essentially independent. The degenerate scalar product of functions supported on distantly space-like separated volumes factors,

∗ ∗ ∗ ∗ W(g1 × g2 , f1 × f2) → W(g1 , f1)W(g2 , f2), (19)

as the supports of g1, f1 become arbitrarily distantly space-like separated from the supports of g2, f2. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 37

This condition manifests causality: great spatial separation implies independence of the local observables of subsystems that are not entangled. The constructions in [32, 34, 38] have unique vacua and satisfy formal Hermiticity W.a below. However, in section 2.5 it is illustrated that formal Hermiticity W.a conflicts with cluster decomposition A.6 in constructions with nonzero spin and in some scalar field examples. A.6 is stronger than the condition used in [32] that was also designated as cluster decomposition. Either condition, A.6 or the cluster decomposition condition used in [32, 34, 38], imply that the vacuum is the sole translation-invariant state. In common with Wightman’s development, formal Hermiticity was developed in [38] and resulted in truncated functions that are not connected. In this earlier development that satisfied formal Hermiticity, the cluster decomposition condition A.6 is not satisfied. Formal Hermiticity provides a simple demonstration that fields (14) are Hilbert space operators, appendix 7.5. The development in section 2.5 abandons formal Hermiticity to establish the causal cluster decomposition property A.6 and include nonzero spins. Cluster decomposition is a nonlinear condition and relates the description of interaction across orders of the VEV. There are no functions of strictly localized support in the example unconstrained developments in P and A.6 applies as an approximation that asymptotically approaches equality of likelihoods. In the example constructions, regularity A.1 is implied if isolated elementary particles are stable. This condition is:

A.7) Elemental stability: hg|Φ(f)Ωi = 0 and hg|Ωi = 0 if g ∈ P and g0 = g1(x)κ = 0. Ω := (1, 0, 0 ...) and Φ(f) is from (14). For the example nontrivial realizations, condition A.7 both enables satisfaction of regularity A.1 and associates the descriptions of states with elementary particles in appropriate instances. This condition together with A.6 and the selected form of A.1-5 compliant two-point functions provides that isolated, localized regions of support of the functions that describe states are perceived as classical particles. Isolated, localized concentrations of support propagate as nearly free particles until supports approach. The masses of the elementary particles are the mκ associated with each field component Φ(x)κ by the construction in section 2.5. A.7 provides that the vacuum and one-argument states are orthogonal to multiple argument states and this orthogonality results in satisfaction of positive definiteness A.2 together with regularity A.1 for the example constructions. Satisfaction of A.7 is not necessary for regularity for all example constructions. For example constructions that do not satisfy formal Hermiticity W.a, the kernel functionals can be selected to satisfy A.7 and positive definiteness A.2 provides that both Wn−k,k(xn, . . . x1)κn . . . κ1 = 0 and Wk,n−k(x1, . . . xn)κ1...κn = 0 if either equals zero. Satisfaction of formal Hermiticity W.a precludes setting W0,n = W1,n = 0 for n ≥ 2 but kinematic constraints can result in ∗ ∗ 4 4n W0,n(f0 , fn) = W1,n(f1 , fn) = 0 if f1 ∈ P(R ) and fn ∈ P(R ) for nonzero W0,n and W1,n. Kinematic constraints include conservation of energy-momentum and spin. For the example ∗ ∗ realizations with only a single mass, W0,n(f0 , fn) = W1,n(f1 , fn) = 0 if n ≥ 2 due to conservation of energy-momentum: neither the vacuum nor a single particle can create a particle pair with each particle of the same mass as the decaying particle due to the constrained energy 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 38

4n support of functions in P(R ) and the limited support of the kernel functionals Wk,n−k. In the constructions of section 2.5, the support of the kernel functionals Wk,n−k is limited to mass shells. In the example constructions, the mean values of boson field components hΩ|Φ(x)κΩi are asserted to vanish and axiom A.7 is stated for mean zero fields. More generally, mean values are independently specified without affecting satisfaction of A.1-7 with the appropriate revision to the statement of A.7. Nonzero mean values are discussed in section 2.5. Elemental stability A.7 expresses that if a state is prepared with only one elementary particle, or together with A.6 that if one elementary particle is greatly isolated from the support of every other body, then the particle is stable until it encounters another body. It does not provide whether an isolated state of the elementary particle can be prepared. Whether or which elementary particles are confined is not determined by the axiom and a determination of con- finement awaits better understanding of bound states within the unconstrained development. The condition A.7 is inconsistent with Hermitian field operators. If the field were Hermitian, then ∗ hΦ(f2)κ2 Φ(f1)κ1 Ω|gi = hΦ(f1)κ1 Ω|Φ(f2)κ2 gi

= hΦ(f1)κ Ω|Φ(f2)κ gi 1 2 (20)

= hΦ(f1)κ1 Ω|f2 × gi = 0 from the definition of adjoint operator, the assumed Hermiticity, and A.7. Thus, if the field is Hermitian for the Hilbert spaces constructed from the basis space P, kernel functionals Wk,n−k for k > 1 must also vanish. Formal Hermiticity W.a, described below, is necessary to Hermiticity of field operators. 4 For fj ∈ P(R ), the basis function space P introduced below in section 2.3 that enables nontrivial realization of A.1-5, and for free field VEV: the Φ(fj)κj in (14) equals the non- ∗ Hermitian creation operator a (fj)κj and the adjoint is the annihilation operator a(fj)κj = 0 4 for fj ∈ P(R ) [32]. There are no real test functions in the P selected to achieve nontrivial satisfaction of A.1-5 [34]. For the nontrivial unconstrained examples, selection of a basis function space such as the P in section 2.3 is necessary to satisfy A.4. The emphasis on realization of states and adoption of an association of features as the quantum-classical correspondence enables the example realizations of relativistic quantum physics. The anticipation that fields (14) are Hermitian Hilbert space operators is an example of an unnecessary reliance on classical concepts. For the example realizations in section 2.5 and [32, 34, 38], if the real, observable components of the field are realized by Hermitian Hilbert space operators, elements of a von Neumann or C∗-algebra, then the field is necessarily either physically trivial or violates spectral support A.4. In the free field case, the basis function space can be enlarged from P to S and then ∗ the real, observable components of the field are Hermitian for real functions, fj = fj = fj ∈ S from (9) and identification of the adjoint in (72). With the enlargement of the basis function 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 39

∗ ∗ space to S for the free field VEV, Φ(fj)κj = a (fj)κj + a(fj)κj and Φ (fj)κj = Φ(fj)κj for real fj. 4 ∗ The interpretation of Φ(fj)κj for fj ∈ P(R ) as the non-Hermitian creation operator a (fj)κj applies for free field VEV but not generally. For the example VEV that exhibit interaction, products of k fields acting on the vacuum state Ω produce k-argument states, but as discussed in section 2.1, a k-argument state describes a k-particle state only approximately in instances of isolated, localized state descriptions (or free field VEV). More generally, a k-argument state can be observed as consisting of n-particles, n 6= k, since k and n-argument states are not orthogonal in the constructed scalar product. Given kernel functionals Un,m that satisfy A.1-6, kernel functionals Wn,m that also satisfy A.7 result from ∗ 2 ˆ ∗ ˆ W(f , f) := |f0| + W2(f 1, f1) + U((f −f) , f −f) (21)

with fˆ := (f0, f1(x)κ, 0, 0,...). The decoupling of the contributions of the higher order functionals in U from the two-point functional W2 illustrates that in the unconstrained revision, a two-point functional equal to the free field two-point functional no longer implies that the quantum field is a free field. Without densely defined Hermitian field operators for the real, observable components of the field, that is, without VEV that satisfy the additional constraints of formal Hermiticity W.a and totality W.b described below, the Jost-Schroer theorem [6, 59] does not follow. The Jost-Schroer theorem does not follow from A.1-6 and does not generally apply in the unconstrained development. For the example constructions with a free field two-point functional, the Jost-Schroer theorem provides that quantum fields (14) that satisfy A.1-6 are not the anticipated densely defined Hermitian Hilbert space operators. Indeed, in cases with an explicit adjoint to the field operator and an exhibition of interaction, the field is not Hermitian [32, 34]. Assertion that the field in (1) is Hermitian, densely defined and satisfies A.1-5 precludes contributions to the VEV such as (4) that result in the exhibition of interaction. The revised quantum-classical correspondence removes W.a-b as conditions that must be satisfied to realize relativistic quantum physics. Wightman’s original axioms [6, 7, 59] are distinguished here by two additional conditions. Consistently with the canonical formalism’s conjectured correspondence of the elevation of classical fields to Hermitian quantum field operators, Wightman’s prospective axioms included assumptions that implied the real, observable field components in the VEV (1) were Hermitian Hilbert space field operators: W.a) Formal Hermiticity: the kernel functionals of the degenerate scalar product (8) satisfy

Wk,n−k((x)n))(κ)n = Wn((x)n))(κ)n independently of k. In this case, the degenerate scalar product is W(g∗, f) = W (g∗ × f) for a Wightman functional W 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 40

W.b) Totality: P = S. Satisfaction of the physical conditions apply for all sequences of Schwartz tempered functions.

Together with A.1-2, satisfaction of these two conditions implies that the real, observable components of a field are realized as Hermitian Hilbert space operators. Sequences of tempered functions S are a ∗-algebra for the involution (9). Satisfaction of formal Hermiticity W.a and totality W.b also suffices to analytically extend the kernel functionals to the moment functionals of a generalized random process [1, 39]. However, condition W.b precludes the example nontrivial realizations of relativistic quantum physics [32, 34, 38]. The unconstrained revision relaxes satisfaction of the physical conditions to a Poincarétransform-stable, non-∗-involutive subalgebra P ⊂ S and admits scalar products that do not satisfy formal Hermiticity. Both W.a and W.b can be satisfied by particular realizations of A.1-7 but W.b is not satisfied by the example realizations that exhibit interaction and W.a is not satisfied by the nontrivial example realizations with nonzero spin. The refined correspondence of classical and quantum state descriptions associates a classical field with the VEV without the necessity for the Hermitian field operators that correspond to classical fields to be exact elevations of the classical field. Satisfaction of formal Hermiticity W.a provides that

hf × g|hi = hg|f ∗ × hi,

the scalar product of states f × g with h is also the scalar product of g with f ∗ × h. In the unconstrained constructions, f ∗ × h does not describe a physical state. Except for the vacuum, f ∗ 6∈ P if f ∈ P. With the refined quantum-classical correspondence discussed in section 1, both W.a and W.b can be abandoned. There is no phenomenological basis to assert that there must be a state of nature for every sequence of tempered functions, and location in relativistic quantum physics illustrates that a classical dynamical variable can be associated with the quantum description of state without a Hermitian elevation of the classical dynamical variable. Classical dynamical variables are more approximately and conditionally associated with Hermitian Hilbert space operators in these instances. All observers perceive the same events. The Poincarécovariance of states provides that the scalar product is invariant and, as a consequence, likelihoods are invariant to Poincaré transformations. h(a, Λ)g|(a, Λ)fi = hg|fi, for Poincarétransformations n ˜ Y −ipa T ˜ −1 (a, Λ)fn((p)n)(κ)n := e (S(A) ·)nfn((Λ p)n)(κ)n (22) k=1

with S(A) an Nc × Nc realization of the proper orthochronous Lorentz group, A ∈ SL(2, C), Λ the proper orthochronous Lorentz transformation determined by A, a a spacetime translation, 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 41 and f˜(p) is the Fourier transform of f(x). The 4 × 4 Λ has components 1 Λ = Trace(σ Aσ A†) (23) jk 2 j k for the four Pauli spin matrices σj [6]. The energy-momenta p are ~ times wavenumbers with units of inverse length and the product of mass with the speed of light is an inverse length mκk c/~, the inverse of the reduced Compton wavelength (18). Energies are the zero component of cpk. The Fourier transform of tempered functions in four dimensional spacetime is the evident multiple argument extension of Z dx ψ˜(p) := e−ikxψ(x) (24) (2π)2 using wavenumber k, p := ~k, and the Lorentz invariants px = p0ct − p · x and spacetime volume dx = dx0dx1dx2dx3. A.4 limits the spectrum of the generators of translations to the closed forward cone and this limits the support of the Fourier transforms of the generalized functions

˜ ˜ ∗ ˜ Tn((p)n)(κ)n := W`,n−`((p)n)(κ)n ge` ((p)`)(κ)` fn−`((p)`+1,n)(κ)`+1,n (25)

˜ 4` 4n−4` + with W`,n−`((p)n)(κ)n the kernel functionals, g` ∈ P(R ) and fn−` ∈ P(R ) to En [7].

+ + + + En := {(p)n : pn ∈ V , pn−1 + pn ∈ V , . . . , p2 + . . . pn ∈ V , p1 + p2 + . . . pn = 0}.

+ 2 V := {x : x ≥ 0 and x0 ≥ 0} is the closed forward light cone. In Wightman’s original ˜ axioms, the support of the functionsg ˜`, fn−` were not constrained and as a consequence, the ˜ + support of the kernel functionals W`,n−`((p)n)(κ)n was limited to En . In the constructions, 4n limiting the support of VEV to mass shells and requiring that functions in P(R ) are zero on the negative energy mass shells satisfies A.4. This development enables joint satisfaction of locality, positive energy support and Poincarécovariance, an unmet and apparently unattain- able task within established developments when interaction is manifest [2, 32, 41]. The revision introduces P ⊂ S with P limited to functions that label positive energy states. Satisfaction of the physical conditions in a more constrained function space results in physically nontrivial, locally commutative, positive energy, Poincarécovariant tempered distributions. In the revision, not every sequence of tempered functions labels a physical state. Spectral support need not be satisfied as a function, that is, independently of the support of the test functions (and indeed, VEV can not be functions [6]). Satisfaction of A.4 as a functional in the Hilbert space completion for a selected basis function space results in the example nontrivial realizations of relativistic quantum physics. With the refined correspondence of field operator and classical field, motivation for local commutativity as the statement that values of the field are causally related is diminished. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 42

Nevertheless, motivation for local commutativity persists in the approximation Φ(b x)κ ≈ Φ(x)κ from the discussion in section 1.1 and in that A.5 implies the appropriate Bose-Einstein or Fermi-Dirac statistics of indistinguishable particles. It is known that a relaxation of local commutativity admits realizations of relativistic quantum physics. It was established in [2] that without the constraint of local commutativity, realizations of QFT are available. In the constructed examples, satisfaction of local commutativity A.5 follows naturally. Local commutativity is that linear combinations of the VEV with transpositions of arguments conditionally vanish.

hf|Φ1 ... (ΦkΦk+1 ± Φk+1Φk) ... ΦnΩi = 0

2 if (xk − xk+1) < 0, 1 ≤ k < n, with the sign determined by particle statistics to satisfy m normal commutation relations [6], and in this abbreviated notation Φk := Φ(xk)κk . Imple- mentation of a space-like support condition uses functions of bounded support, for example, tempered test functions f ∈ S are used to verify the local commutativity of free field VEV. For the function spaces P ⊂ S that lack functions of bounded support,n the space-like support condition does not apply but the limited energy-momentum support of the functions in P and the example constructed VEV provide satisfaction of local commutativity. The example VEV exhibit unconditional signed symmetry for functions in P, section 2.5 and [32, 34]. For the example unconstrained realizations, A.5 can be tightened to the unconditional signed symmetry of the VEV but to include the established free field developments, only conditional local commutativity is required [32]. As might be anticipated, revisions to the axioms for QFT result in substantial deviations from established results implied by satisfaction of the canonical formalism: without equality of the quantum field in the VEV (1) with densely defined Hermitian field operators for the real, observable components of the field, “no go” theorems such as the “no simple two-point function” results of Jost-Schroer and Greenberg [6] and the “no infinitely divisible models” of Rinke [47] no longer necessarily apply; there are no states of strictly bounded support within the basis space of energy support constrained functions P nor their Hilbert space limits although states that are arbitrarily dominantly supported in bounded volumes are included; the scalar product implements interaction while the Hamiltonian is not associated with a classical interaction, and more than one classical interaction, including both short range and long range interactions, are exhibited in distinct dynamical instances of a single example realization; the VEV are boundary values of analytic functions but not of a single analytic function (no single domain of holomorphy, demonstrations of PCT and spin-statistics must be revisited for the unconstrained development); and the constructed Hilbert spaces are distinct from Fock Hilbert spaces (states of a free field are not necessarily asymptotically complete in the constructed Hilbert spaces) when there is interaction. The lack of completeness of the states of a free field in the example

mThis discussion neglects the possibility of parastatistics [6]. nOr, in the equivalent development with modiﬁed VEV (29) and P = S. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 43 constructed Hilbert spaces is interesting both with regard to Rinke’s results and for bound states.

2.3 The basis function space P and equivalent development for S In an unconstrained development, the choice of a basis function space P admits nontrivial realizations that are unavailable to Wightman’s original development of relativistic quantum physics [59]. Wightman selected the Schwartz tempered test functions S and in an unconstrained development, constraints on the VEV are relaxed by placing constraints on the function space P. Wightman’s selection of functions treats time similarly to space, and enables a canonical quantization. In the unconstrained development, time is distinguished. The functions in P have Fourier transforms with zeros on negative energy mass shells. This choice of basis function space P achieves satisfaction of the positive energy (spectral support A.4) condition in realizations with interaction. Revisiting the discussion in section 1, anticipation that VEV satisfy axioms A.1-6 and W.a-b as classical functions, that is, without regard to the basis function space that labels states, is another illustration of an inappropriate appeal to classical concepts. The VEV define a rigged Hilbert space realization of quantum mechanics: VEV are generalized functions dual to a basis function space [17, 18, 19]. The states of nature have nonnegative energies and these states can be selected by the basis function space as well as by constraints on the VEV functionals. A Hilbert space construction based upon the tempered functions S and the example constructed VEV results in a Hilbert space with elements that exhibit the physical properties A.1-7 except that the positivity of energies within A.4 is violated. Selection of the basis function space P isolates the subspace of physical, positive energy states. In the unconstrained development, satisfaction of physical conditions is preserved without imposition of the technical constraints W.a-b. The revision to the correspondence of classical and quantum fields eliminates the need for formal Hermiticity W.a and totality W.b. W.a-b imply that quantum field operators (14) are Hermitian. W.a-b are satisfied by examples within the unconstrained development and these examples are consistent with a canonical quantization. These examples with Hermitian field operators are all physically trivial like free fields. P consists of those tempered functions with Fourier transforms that vanish on the appropriate negative energy mass shells. The appropriate mass shell is determined by the field components labeled κj. 4n fn((x)n)(κ)n ∈ P(R ) if ˜ fn((p)n)(κ)n = 0 when any Ej = −cωj with j = 1, . . . n and q 2 2 2 ~ωj := mκj c + pj . (26) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 44

Appropriate zeros are implemented using multiplier functions for tempered functions [18]. A suﬃcient characterization of the functions fn ∈ P is that every n ˜ Y fn((p)n)(κ)n = (p0,j + ωj)g ˜n((p)n)(κ)n (27) j=1

4n with gn ∈ S(R ) and ωj from (26). This is abstracted as

P˜ := (p0 + ω)S˜ using (27) and f0 := g0. The p0,k + ωk are multipliers of tempered test functions since they are infinitely differentiable and of polynomially bounded growth [18]. The remaining issue is whether A.3 is satisfied: whether P is closed under Poincarétransformations. The Lorentz 2 4n 4n invariance of p and that (a, Λ)˜gn ∈ S(R ) for everyg ñ ∈ S(R ) provides that P is stable under Lorentz transformations. Indeed, the Lorentz invariance of p2 provides that a zero of the appropriate form (27) is stable with Lorentz transformation,

0 0 (p0,` − ω`) p0,` + ω` = (0, Λ)(p0,` + ω`) = (p0,` + ω`) (0, Λ)(p0,` − ω`) for the Poincarétransformation (a.Λ). The final factor is regular in a neighborhood of the negative energy mass shell. A Poincarétransformation of the functions is implemented as

T −1 (a, Λ)fn((x)n)κ1...κn := (S(A) ·)nfn((Λ (x − a))n)κ1...κn with the Lorentz transform Λ determined by A ∈ SL(2, C) in (23), a real Lorentz vector translation a and S(A) a Nc × Nc a realization of the proper orthochronous Lorentz group paramerized by A [6]. HP includes states labeled by Cauchy sequences of functions in P that are convergent in the Hilbert space norm (6). These include functions

n ˜ Y fn((p)n)(κ)n = (p0,` + ω`)g ˜n((p)n)(κ)n ) `=1

3n with gn ∈ S(R ) and, as a consequence, HP includes states labeled by functions that include generalized functions of point support in time [34]. P includes the functions used by Lehmann, Symanzik and Zimmermann to isolate the creation component of a field operator in their developments [6]. As a consequence, LSZ expressions for scattering amplitudes readily adapt to the unconstrained development. Explicit scattering amplitudes are evaluated in section 2.15 and [32, 34, 36, 38]. In this development of relativistic physics, there are no strictly localized states but there are essentially localized states [34]. Functions of the form (27) include no functions of bounded 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 45 spatial support. The essentially localized states include states arbitrarily dominantly supported within bounded volumes yet with tails that do not vanish within any finite region. Functions of the form (27) are anti-local [46, 56]. Since subspaces of Hilbert space elements are in one- to-one correspondence with projection operators [40], there are no projection operators onto bounded spatial volumes. Such projections are generally incompatible with relativistic causality [69]. This is in contrast with L2, the Hilbert spaces appropriate for nonrelativistic quantum mechanics. The tails of the functions that describe states can be physically negligible since minor support corresponds to unlikely and essentially unrepeatable observations. Significantly, these states that cannot be exactly localized but are essentially localized have Fourier transforms with no support on negative energy mass shells. The lack of support on negative energies implements one of the properties of physically relevant realization of the Lorentz group, positive energies. If the scalar product (8) is defined using generalized functions Wk,n−k with support limited to 2 2 mass shells, every pj = (mκj c) , and gk, fn−k ∈ P, then the support of the generalized functions ˜ + + Tn((p)n)(κ)n in (25) are within En . En includes only nonnegative energies. Since P includes no functions of bounded support, there are no von Neumann algebras associated with subdomains 3 of R in the nontrivial unconstrained developments. An alternative development based on S: The zeroes in (27) that distinguish the subspace P from the tempered functions S are multiplier functions [18] that may alternatively be applied to the kernel functionals. Application of the multipliers to the VEV produces an equivalent development but from (27), the basis function space is the Schwartz tempered functions S. In this equivalent, alternative development: formal Hermiticity W.a is entirely abandoned and as a consequence there are no Hermitian field operators; the VEV of established models including free and monomial models are modified; and satisfaction of totality W.b is preserved. With the multipliers appropriately applied to the VEV functionals rather than to the test functions, every function argument of a kernel functional has a multiplier (p0,j + ωj)/(2ωj) and every argument for the ∗-morphism of a function has a multiplier (−p0,j + ωj)/(2ωj). The appropriate application of multipliers to the kernel functionals is

k n Y Y −p0,j + ωj p0,` + ω` Wk,n−k((x)n)(κ)n 7→ Wk,n−k((x)n)(κ)n . (28) 2ωj 2ω` j=1 `=k+1

2 2 2 From section 2.5, the example VEV have point support on each mass shell, δ(pj − mκj c ) for each j ∈ {1, n}, that produces the equivalences,

p0,j + ωj = θ(p0,j) 2ωj −p0,j + ωj = θ(−p0,j) 2ωj 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 46 with θ(x) the Heaviside step function. The resulting equivalence of developments is:

VEV Basis space W ((x) ) P k,n−k n (κ)n (29) k n Y Y θ(−p0,j)θ(p0,`) Wk,n−k((x)n)(κ)n S. j=1 `=k+1 For the constructions in section 2.5, the VEV reduce to established free field VEV evaluated for functions from P as the separations of the dominant supports of the arguments grow. This is a physically desirable property of the development based on the subspace P: states labeled by functions with localized and isolated supports for each argument are free particle-like states. For this conditional equality with the free field VEV, to include examples that satisfy formal Hermiticity W.a, and for commonality with [32, 34, 38], the development studied here does not modify the VEV of established models: the development in these notes is based on the subspace P of S. Satisfaction of formal Hermiticity is necessary to display Hermitian field operators. In the alternative development based on S, the extension of the free field development that includes Hermitian field operators restores terms to the modified VEV (28) to satisfy formal Hermiticity W.a. Beginning with established free field VEV for Wk,n−k((x)n)(κ)n , every 2 term in (28) from the free field VEV is either unmodified (θ(p0,j) = θ(p0,j)) or is zeroed (θ(p0,j)θ(−p0,j) = 0). The zeroed terms are replaced to restore the VEV that produce Hermitian field operators. The free field VEV are presented in section 2.4. The established free field VEV satisfy formal Hermiticity and totality W.a-b that provide that (1) results from Hermitian field operators. Nontrivial examples that satisfy formal Hermiticity W.a and A.1-5 are constructed in [32, 34, 38] and nontrivial examples that violate formal Hermiticity but satisfy the strong, causal cluster decomposition condition A.6 are constructed in section 2.5. Both constructions use the basis function space P. The Jost-Schroer theorem discussed in section 1.4 provides that there are no augmentations to the nontrivial example constructions that satisfy both the physical conditions of relativistic quantum physics A.1-5 and the technical conditions W.a-b, formal Hermiticity and totality, to produce Hermitian field operators. The equivalence of developments is interesting for an analogy with a result from Reeh and Schlieder [46, 69] that states expand in states that are supported far from the support of the state of interest. Although the theorem of Reeh and Schlieder is demonstrated for assumptions distinct from A.1-7, the equivalent developments in (29) display the same tension between consideration of states labeled by functions of bounded support and an inferred global support of states. The development with states labeled by the anti-local functions from P is equivalent to a development that includes states labeled by test functions of bounded support from S. In this sense, the characterization of states as anti-local is preserved in the equivalent alternative that includes states labeled by functions of bounded support. Anti-local functions have global support in the sense that they do not vanish over any finite region. The result of Reeh and 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 47

Schlieder is that states described by (2) with Hermitian field operators, states described by operating on the vacuum with powers of field operators and functions fn((x)n)(κ)n supported solely within a bounded region O, are dense in the entire Hilbert space and not just dense for states supported within the same bounded region O. The equality (29) demonstrates that the more evident anti-locality of the development based on P is also exhibited in the developments based on S: anti-locality is inherent to the example VEV functionals that exhibit the properties of relativistic quantum physics. Selection of the basis function space P overcomes the challenge of discovering VEV functionals within S0 that satisfy the physical conditions A.1-5 for all functions within S, that reduce to free field VEV as the separations of the supports of function arguments grows or interaction terms weaken, and are nontrivial realizations of relativistic quantum physics. This longstanding problem remains without a resolution [1, 2, 6, 31, 39, 41, 59, 61]. The example realizations of relativistic quantum physics discussed here are admitted by elimination of canonical quantization’s assertion that the elevations of the quantum field (14) must be Hermitian Hilbert space operators. The lack of real functions within P precludes Hermitian field operators but not a correspondence of classical and quantum dynamics.

2.4 Relativistic free fields There are well established, physically trivial relativistic realizations for the sequence of kernel functionals W in (13) as duals to the basis function space of tempered functions S [6, 8, 59]. Physically trivial relativistic realizations include free, generalized free, and monomial models. Satisfaction of the physical conditions suffices to determine a scalar two-point function W2(x1, x2) as a Källén-Lehmann form, summation over a mass spectrum of the Pauli-Jordan function [57]. In this unconstrained development, two-point functionals derive from extensions of the Pauli-Jordan function to nonzero spin. The two-point functionals used in the example realizations have Fourier transforms

˜ + W2(p1, p2)κ1κ2 = δ(p1 + p2) δ2 M(p2)κ1κ2 − + √ (30) = δ(p1 + p2) δ1 δ2 2 ω1ω2 M(p2)κ1κ2 with δ± := θ(±E )δ(p2 − m2 ). k k k κk ± The support of δk are on the positive or negative energy mass shells. There is a mass mκ associated with each field component Φ(x)κ. The mass shell support condition is that Ej = ±ωjc from (26). There is a consequent simplification in the expressions for the free field VEV Fk.n−k 4 ∗ from that W2(f1, g1)κ1κ2 = 0 if both f1, g1 ∈ P(R ). Only W2(f1 , g1)κ1κ2 is nonzero. Without interaction, particle number is conserved and as a consequence, for f ∈ P, only Fk,k contributes,

∗ Fk.n−k(fk , fn−k) = 0 unless n = 2k. (31) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 48

For free fields, every state has a classical particle correspondence. The free field VEV satisfy formal Hermiticity, indeed, the VEV are generated algebraically from a field

+ − Φ(f)κ = Φ (f)κ + Φ (f)κ with creation and annihilation components that have commutation relations

− + [Φ (f1)κ1 , Φ (f2)κ2 ]± = W2(f1, f2)κ1κ2 − and otherwise commute [23, 55, 63]. There is a cyclic vacuum state with hΩ| ... Φ (f)κΩi = 0 + 4 ∗ and hΩ|Φ (f)κ ... Ωi = 0. In this evaluation of the VEV, each fj ∈ S(R ) [32]. With fj ∈ P + − for j ≤ k and fj ∈ P for j > k,Φ (fj)κ = 0 for j ≤ k,Φ (fj)κ = 0 for j > k, n Y Fk,n−k = hΩ| Φ(xj)κj Ωi j=1 and the result is that k X Y Fk,k({1, 2k}) = σ(S) W2({j, ij}) (32) S j=1 P in the abbreviated notation σ(S) for signs developed below in section 2.5. The summation S includes the k! distinct pairings j, ij with j ∈ {1, k} and ij ∈ {k +1, 2k}. S = {1, . . . k, i1, . . . ik} and evaluation of the signs σ(S) = ±1 is discussed below (46) with σ(So) = 1 for So = {1, . . . k, k + 1,... 2k}. Due to the simplifications in evaluation of the free field VEV for function sequences from P, the F exhibit signed symmetry (41) that implies satisfaction of local commutativity A.5 for sequences from P. The expression for free field VEV that applies generally for sequences from S has contributions Fk,n−k for all even n = 2`, all k and includes all (2`)!/(2``!) appropriately signed, distinct pairings of the n arguments. Due to the limitation on energy support of function sequences from P ⊂ S, only k! of the (2k)!/(2kk!) terms for n = 2k contribute to the free field VEV that apply for the basis space P: terms that include argument pairings with a second argument from {1, k} or a first argument from {k + 1, 2k} do not contribute. The VEV that apply for tempered functions S satisfy local commutativity for elements of S with bounded support.

The elements of the Nc × Nc array M(p)κ1κ2 in (30) are multinomials in the energy- momentum components [6] and in matrix notation, the array satisﬁes

DM(p) = C†(p)C(p) (33) with D the Dirac conjugation matrix from (9) and C†(p) is the Hermitian transpose of C(p). Matrix nonnegativity [28] of DM(p) results in the degenerate scalar product (8) [38]. Since DM(p) is a nonnegative matrix, it is Hermitian and it follows from (12) that

M(p)† = DM(p)DT . (34) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 49

Local commutativity A.5 is satisﬁed if the M(p) are (real orthogonal) permutation matrix similar to a direct sum decomposition into two components Ms(p),

M (p) 0 OM(p)OT = 1 , (35) 0 M2(p) that satisfy locality conditions T Mj(p) = ±Mj(−p). (36)

The + sign applies for the boson constituent M1(p) and the − sign applies for the fermion constituent M2(p). Two additional conditions on M(p) and D provide Poincaréinvariance of the scalar product. S(A)M(p)S(A)T = M(Λ−1p) (37) S(A)D = DS(A) with S(A) an Nc × Nc realization of the proper orthochronous Lorentz subgroup (22). A ∈ SL(2, C). (32) results in scattering amplitudes with particle number conserved, and incoming and outgoing momenta equal in pairs, that is, no exhibition of scattering. Constructed from (30) and the basis function space S, the free field Φo(f) is a Hilbert space operator that realizes the Wightman-Borchers, G˚arding-Wightman and Haag-Kastler axioms [6, 7, 59]. Constructed from (30) and the basis function space P, the free field Φo(f) can not be a Hermitian operator but an available extension of the basis function space achieves Hermitian field operators for the real, observable components of the field [32]. This extension is due to the particular form of free field VEV and is unavailable to the example constructions if interaction is manifest. Hermitian field operators appear to be peculiar to physically trivial realizations.

2.5 Nontrivial realizations of relativistic fields In this section, the example VEV that exhibit the physical properties of relativistic physics are displayed. Realization of the quantum mechanical description of state in section 2.1 is determined by a selection of basis function space P, completed in section 2.3, and VEV functionals W that satisfy A.1-7 from section 2.2. The VEV of interest exhibit interaction. The displayed constructions of kernel functionals W (13) dual to P suffice to satisfy conditions A.1-7. Connected functionals: From the definition of field (14), the VEV are identified with the kernel functionals

hΦ(xk)κk ... Φ(x1)κ1 Ω|Φ(xk+1)κk+1 ... Φ(xn)κn Ωi = (D·)kWk,n−k((x)n)(κ)n in the matrix notation (10) with the linear transformation D, Dirac conjugation, from the ∗- morphism of function sequences (9). These kernel functionals are constructed from connected functionals. The connected functionals include two-point functionals (30) from known free ﬁeld 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 50 examples and n argument connected functions with n ≥ 4. The VEV are a link-cluster-like expansion [6, 50] of these connected functionals. The expansion expresses the kernel functionals C C Wk,n−k as sums of products of connected functionals Wk,n−k. Wk,n−k((x)n)(κ)n is a connected C functional if Wk,n−k(fn)(κ)n does not contribute to scalar products when the support of the C fn((x)n) consists of two arbitrarily greatly space-like separated clusters. That is, Wk,n−k is connected if C lim Wk,n−k(fn(ρ))(κ) = 0 ρ→∞ n with 4n fn(ρ) := fn(x1, . . . xj, xj+1 − ρa, . . . , xn − ρa) ∈ S(R ), 2 a < 0, ρ ∈ R, and for 1 ≤ j < n for all components κj. In evaluation of scalar products (8), ∗ 4k fn((x)n) = hk(((x)k)gn−k((x)n−k) with hk, gk ∈ P(R ) for the appropriate masses labeled by 4n the κj and as a consequence, fn ∈ S(R ). The example constructions are based on connected functionals

C W1,1(x1, x2)κ1κ2 := W2(x1, x2)κ1κ2 2n C ˜ Y 2 2 Wk,2n−k((p)2n)(κ)2n := c2nδ(p1 + . . . p2n) δ(pj − mκj ) Qk,2n−k((p)2n)(κ)2n (38) j=1 C Wk,2n−1−k((x)2n−1)(κ)2n−1 := 0 with 2n ≥ 4, W0,0 := 1, W2 the free ﬁeld two-point functional from (30), and the elements of the Lorentz covariant array Qk,2n−k are explicit functions of the energy-momenta vectors. If kinematic constraints do not suﬃce to satisfy A.7,

W0,k = Wk,0 = W1,k = Wk,1 = 0 for k ≥ 2 and all selections of the κj’s. The development is simpliﬁed by setting the means (vacuum polarizations) of ﬁeld components to zero,

W1,0 = W0,1 = 0, for all components κj without loss of generality since means are independently assigned as discussed below. Odd orders of the kernel functionals vanish. In these examples, the elementary particles, those associated with the field components, are created or annihilated in pairs although more generally, the odd order kernel functionals need not vanish for developments that include scalar fields [32, 34]. The link-cluster-like expansion of connected functionals provides satisfaction of the strong cluster decomposition condition A.6. Satisfaction of the cluster decomposition condition A.6 also provides that isolated particles behave as nearly free particles. The kernel functionals (38) are selected with a free field two- point functional. hΦ(f)Ω|Φ(g)Ωi = hΦo(f)Ω|Φo(g)Ωi (39) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 51

with Φ(f) the constructed quantum field (14) and Φo(f) the free field of the same elementary masses mκ. Then cluster decomposition A.6 and elemental stability A.7 provide that a function argument with support that is distantly space-like separated from the support of all other arguments contributes to the scalar product (8) only if the argument appears in a two-point functional. If the isolated support is interpreted as the description of a particle (and that particle can be isolated, that is, is not confined) and all supports are isolated, then that state describes nearly free particles. Organization of connected functionals into VEV: The remainder of the effort is to C describe the expansion of W in terms of these W, specify the Qk,2n−k((p)2n)(κ)2n , verify that the resulting VEV satisfy A.1-7, and understand the implied physics. A convenient shorthand notation for kernel functionals is to designate

Wk,n−k(A) := Wk,n−k((x)A)(κ)A (40) for a set of n argument indices A = {i1, . . . in} with (x)A := xi1 , . . . xin and similarly for (κ)A. To develop the construction of VEV from the connected functionals, a few useful results for the degenerate scalar products of interest are collected. In the notation (40), a sequence of functionals V is designated as signed symmetric if the elements of the sequences satisfy

Vk,n−k(π2(π1({1, n}))) = σ(π1)σ(π2)Vk,n−k({1, n}) (41) for every k, n. In a notation with permutations

πa(A) := {πa(i1), πa(i2), . . . πa(in)}, for a = 1, 2, speciﬁed k, n and a set of indices A := {i1, i2, . . . in} with each ij ∈ {1, n}, permutations π1 and π2 are

π1({1, n}) = {π1(1), π1(2) . . . π1(k), k + 1, k + 2, . . . n}

π2({1, n}) = {1, 2, . . . k, π2(k + 1), π2(k + 2) . . . π2(n)} with π1 representing one of the k! permutations of {1, k} and π2 representing one of the (n−k)! permutations of {k +1, n}. Generally, π1(ij) = ij if ij 6∈ {1, k} and π2(ij) = ij if ij 6∈ {k +1, n}. Signs σ(πa) = ±1 are assigned to each permutation.

σ(π1) = 1 if π1({1, k}) = {1, k} σ(π2) = 1 if π2({k + 1, n}) = {k + 1, n}, and then σ(π1)σ(π2) = 1 if π2(π1({1, n})) = {1, n}.

0 σ(πa) = −σ(πa) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 52

0 if the permutations πa and πa diﬀer by one transposition of two adjacent fermion indices. a = 1, 2. 0 σ(πa) = σ(πa), if the transposition is either of two adjacent boson indices or transposition of a boson with an adjacent fermion index. These signs are selected to satisfy local commutativity A.5 for normal commutation relations [6]. Arguments are assigned as boson or fermion by the value of κj. This assignment is in (35). The sign associated with an ordering of indices A = {i1, i2, . . . in} = π2(π1(Ao)) with respect to a reference order Ao := {j1, j2, . . . jn} is σ(π1)σ(π2) achieved by a sequence of adjacent index transpositions. The reference order Ao above is {1, n}.A signed symmetrization V of an array of kernel functionals v is ! X X Vk,n−k({1, n}) := σ(π1) σ(π2) vk,n−k({1, n}) . (42) π1 π2 The summations P and P include all k! permutations of the indices labeled {1, k} and all π1 π2 (n − k)! permutations of the indices labeled {k + 1, n}, respectively. The V constructed from a degenerate scalar product v in (42) is signed symmetric. If w provides a degenerate scalar product, then its signed symmetrization W also provides a degenerate scalar product since

W(f ∗, f) = w(g∗, g) ≥ 0 (43) with X g = (f0, f1,... σ(π1)fk(π1({1, k})),...) π1 from (42) and (8). If V is a degenerate scalar product, then the scaled sequence of functionals Vc generated from V as

2 Vc := (|c0| , c1c0V1,0,... ckcn−kVk,n−k,...), (44)

∗ also provides a degenerate scalar product. Each ck ∈ C and the ck 6= 0. That Vc(f , f) ≥ 0 follows from V(g∗, g) ≥ 0 by appropriately assigning the elements of the sequence g ∈ P as scaled elements from f ∈ P in (8). The selected organization of the connected functionals C W in (38) into W can be considered a product of the sequence of kernel functionals for a free ﬁeld F with a sequence of kernel functionals U constructed below. This construction of W is designated

W := F ◦ U (45) and is described in (46) below. U derives from the higher order connected functionals in (38) and F from (32) is determined by the two-point functional (30). The free ﬁeld contribution 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 53

F implements classical particle correspondences and the higher order connected functionals U introduce interaction. It is demonstrated below that if both F and U are signed symmetric (41) and satisfy the conditions A.1-7, then the constructed W satisfies A.1-7. More generally, if both W and W0 are signed symmetric (41) and realize A.1-7, then W◦W0 realizes relativistic quantum physics and this leads to consideration of prime and divisible realizations [27]. Compatible interactions add, W = F ◦ U ◦ U 0 is a realization that exhibits both the interactions provided by U and U 0. After definition of the product (45), the sufficiency of F ◦ U to satisfy A.1-7 is developed. Then, since realizations of free fields are well established, a demonstration that U is signed symmetric and satisfies A.1-7 suffices to demonstrate that the constructed W realize relativistic quantum physics. The components of W = F ◦ U in (45) expand in products of the components (13) of F and U, X 0 Wk,n−k({1, n}) := σ(π1)σ(π2) F`,j(A) Uk−`,n−k−j(A ). (46) A P kn−k The summation A includes all ` ∈ {0, k}, all j ∈ {0, n − k}, and all ` j partitions without regard to order of the indices {1, n} into two subsets, a subset A and its set complement A0 = {1, n}\ A. In each term, ` arguments are selected from the first k indices to assign to F, and j arguments are selected from the last n − k indices to assign to F. The first k indices are ∗ associated with the ∗-morphism fk of a function and the last n − k indices are associated with ∗ the function gn−k in an evaluation of each Wk,n−k(fk , gn−k). The arguments of F are labeled 0 A = {i1, i2 . . . i`}, ordered with i1 < i2 . . . < i`, and the arguments of U are labeled A , the similarly ordered complementary set to A within {1, n}. In (46), ` designates the number of elements in A ∩ {1, k} and j designates the number of elements in A ∩ {k + 1, n}. Then k − ` is the number of elements in A0 ∩ {1, k} and n − k − j is the number of elements in A0 ∩ {k + 1, n}. Signs σ(π1)σ(π2) are defined by the permutations π1 of the indices {1, k} and π2 of the indices {k + 1, n} as described below (41). The signs are selected so that π1, π2 reorder indices from the reference order

0 Ao ∪ Ao := {1, `} ∪ {k+1, k+j} ∪ {`+1, k} ∪ {k+j+1, n}

0 0 to the particular order A ∪ A = π2(π1(Ao ∩ Ao)) in each term of (46). This ◦-product implements a link-cluster-like expansion [6, 50] with factors of connected functionals that include no arguments in common and all arguments are present in each term of the expansion. Developed in section 2.7, this organization implies satisfaction of the strong form of cluster decomposition A.7. An equivalent form of (46) simpliﬁes demonstration of positive deﬁniteness A.2 and local 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 54 commutativity A.5. The functionals in (46) are a signed symmetrization of functionals w,

! X X Wk,n−k({1, n}) = σ(π1) σ(π2) wk,n−k(π2(π1({1, n}))) π1 π2 (47) k n−k 0 X X F`,j(Ao) Uk−`,n−k−j(A ) w ({1, n}) := o k,n−k `!j! (k−`)!(n−k−j)! `=0 j=0 with

Ao := {1, `} ∪ {k+1, k+j} (48) 0 Ao = {1, n}\ Ao = {`+1, k} ∪ {k+j+1, n}. The equality of (47) with (46) follows by inspection from the combinatorial identity

X T`({1, `})Vk−`({` + 1, k}) X σ(π ) = σ(π )T (A)V (A0) (49) 1 `!(k − `)! 1 ` k−` π1 A that applies for signed symmetric sequences of generalized functions T and V . The summation P k A includes all ` partitions of {1, k} without regard to order into a subset A and its set 0 0 complement A = {1, k}\ A. A = {i1, i2 . . . i`} is ordered with i1 < i2 . . . < i` and A is the similarly ordered complementary set to A within {1, k}. There are k! permutations π1 of k argument indices. The permutations π1 and associated signs σ(π1) are discussed below (42). k The k! permutations can be organized into the ` partitions without regard to order A and the `!(k − `)! permutations of indices within T` and Vk−`. Then, the signed symmetry of the Tj and Vj combined with the signs σ(π1) from (41) result in the identity. The identity of (46) with (47) follows from applications of (49) to the sum of products of signed symmetric kernel 0 functionals F`,j(C) Uk−`,n−k−j(C ) with argument indices

C = π2(π1((Ao)) 0 0 C = π2(π1(Ao)) using the reference order (48). The signed symmetry (41) of Fk,n−k and Uk,n−k enables a reordering of the indices to satisfy iα < iβ if α < β within each of the sets of indices C and C0. With these designations, W from (46) is a linear combination of ordinary multiplication and addition, and the summation includes all appropriate partitions A, A0 without regard to order of the indices {1, n}. As a consequence, the ◦-product (45) is commutative, associative and distributive with addition of sequences, and deﬁnes a sequence of generalized functions if F and U are sequences of generalized functions. These results are a variation for kernel functionals that do not satisfy formal Hermiticity of identities for an s-product [22]. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 55

The identity (47) displays W as the signed symmetrization (42) of w. From (43), if w provides a degenerate scalar product then W provides a degenerate scalar product. The demonstration that w provides a degenerate scalar product is achieved using the tensor product of two linear vector spaces of function sequences with degenerate scalar products that result for scaled F and U respectively. Linear vector spaces with degenerate scalar products are also designated pre-Hilbert spaces. Positive deﬁniteness of the composition: The demonstration that W = F ◦ U is a degenerate scalar product given that F and U are signed symmetric degenerate scalar products derives from (47). w(f ∗, g) equals the degenerate scalar product of particular elements in the tensor product of spaces with degenerate scalar products F c and U c, respectively. The consequent nonnegativity of w(f ∗, f) provides that W is a degenerate scalar product by the arguments above for signed symmetrized and scaled functionals. If U provides a degenerate scalar product, then the tensor product of the pre-Hilbert spaces that result for scaled U and F has elements labeled f ⊗ g and a degenerate scalar product w((f ⊗ f )∗, g ⊗ g ) = F (f ∗, g ) U (f ∗, g ). (50) 1 2 1 2 c 1 1 c 2 2

The scales (44) on the degenerate scalar products are c` := 1/`! in both instances. In these pre-Hilbert spaces of function sequences based on P for kernel functionals Fc and Uc, ﬁelds ΦF and ΦU are deﬁned by (14). These operations extend to the tensor product space. The elevations ΦF ⊗ I and I ⊗ ΦU in the tensor product space are

(ΦF ⊗ I)f ⊗ g = (ΦF f) ⊗ g and (I ⊗ ΦU )f ⊗ g = f ⊗ (ΦU g).

The composite vacuum state is Ω = ΩF ⊗ΩU . To compact notation, the arguments f1(xν)κν ∈ P of the ﬁelds are understood. This construction is a variation of a construction due to Borchers and Uhlmann [27, 47]. (47) is recognized as

∗ X w(f , g) = hO(fn)Ω|O(gm)Ωi (51) n,m with Z n ` n−` ! X X (ΦF ⊗ ) ( ⊗ ΦU ) O(f ) := d(x) I I f ((x) ) . n n `!(n − `)! n n (κ)n (κ)n `=0

The summations associated with the jth operator factor, ΦF (xj)κj ⊗ I or I ⊗ ΦU (xj)κj , are xj and κj. From (8), Z ∗ X X w(f , g) = d(x)n+m wn,m((x)n+m)(κ)n+m n,m (κ)n+m ×f ((x) )∗ g ((x) ) n n (κ)n m n+1,n+m (κ)n+1,n+m 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 56 and then substitution of (47) results in

Z n m 0 X X X X F`,j(Ao) Un−`,m−j(A ) w(f ∗, g) = d(x) o n+m `!j! (n−`)!(m−j)! n,m (κ)n+m `=0 j=0 ×f ((x) )∗ g ((x) ) . n n (κ)n m n+1,n+m (κ)n+1,n+m

The degenerate scalar product (50) for w, the relation of VEV with kernel functionals (15), the ∗-morphism (9), elevations of the field operators, and the indicated associations of arguments result in Z ` n−` j m−j X X X (ΦF ⊗ ) ( ⊗ ΦU ) (ΦF ⊗ ) ( ⊗ ΦU ) w(f ∗, g) = d(x) h I I Ω| I I Ωi n+m `!(n − `)! j!(m − j)! n,m `,j (κ)n+m ×f ((x) ) g ((x) ) n n (κ)n m n+1,n+m (κ)n+1,n+m that demonstrates (51). As a consequence. w(f ∗, f) ≥ 0 since it is the squared norm of a vector in the tensor product space. Satisfaction of positive definiteness A.2 follows from (43), (44) and (46) if both F and U are signed symmetric and satisfy A.1-2 for function sequences in P. The equality of (47) with (46) rests on the signed symmetry of both F and U. U is constructed as signed symmetric and from (32), F is signed symmetric. From the established realization of the free field and the sesquilinearity of (8), the construction (46) reduces the demonstration that W satisfies positive definiteness A.2 to whether U provides a degenerate scalar product and is signed symmetric. For the example constructions based on P, satisfaction of local commutativity A.5 is equivalent to the signed symmetry (41) of W . Signed symmetry satisfies local commutativity without a condition for space-like support. A consequence of the zeroes in the energy support of functions in P, local commutativity reduces to signed symmetry even for free fields in this development. From the equivalences developed for (29), scalar products of symmetric Pauli-Jordan functions equal the scalar products of the positive energy support constrained functionals (30) for functions from P [34]. Discussed in section 2.3 and [34], the functions in P are anti-local and do not have bounded spatial support. The construction (47) displays W as signed symmetric. The kernel functionals F are the VEV of a free field (32) characterized by the two-point functional W2 in (30). Realizations of free fields [23, 55, 63] provide matrices M(p),D,S(A) that satisfy (12), (33), (36) and (37). Free fields of any spin have been constructed [6, 20] and example M(p),D,S(A) are provided in section 2.9. The higher order connected functionals U are constructed from these matrices and elementary generalized functions. Nonnegative, signed symmetric, high order connected functionals U: The final step in the construction is specification of the kernel functionals U that introduce interaction.

This construction provides the arrays Qk,2n−k((p)2n)(κ)2n that complete definition of the connected functionals (38). The Uk,n−k are constructed from elementary generalized functions and 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 57 the arrays M(p),D from free field realizations (30), and Lorentz transforms of the VEV use the free field S(A). U is constructed as a signed symmetric sequence of generalized functions that provides a degenerate scalar product (8) for function sequences from P. U provides a degenerate scalar product for function sequences from S but satisfaction of spectral support A.4 is limited to P. The Fourier transforms of the Uk,n−k are Faàdi Bruno formula expansions of generator functionals. With generators described by n sets of arguments (α, p, κ)n with each αj ∈ R, 4 pj ∈ R and κj ∈ N, the Fourier transforms of the Uk,n−k are

  2n   ˜ X X Y d (D·)k Uk,2n−k((p)2n)(κ)2n := σ(π1)  σ(π2)   GU;k,2n−k((α, p, κ)2n) (52) dαj π1 π2 j=1 for even orders and ˜ Uk,2n−1−k((p)2n−1)(κ)2n−1 := 0 for odd orders. π2 applies to arguments labeled {k + 1, 2n} and π1 to {1, k}. The matrix notation (10) applies to the k factors of D. The generator is evaluated at (α)n = 0 after diﬀerentiation. The logarithms of the generators are multinomials of the αj with coeﬃcients C that are the connected functionals Wk,n−k from (38).

n Z Z du Y d ln (G ((α, p, κ) )) := dσ(λ) (1+λα e−ipj u δ(p2 − m2 ) ) U;k,n−k n 4 j j κj (2π) dρj j=1 (53) X ×(D·)k exp( ρaρbHκaκb ). a,b∈Jk,n

4 evaluated at (ρ)n = 0 after diﬀerentiation. The summation of u is over R and results in energy-momentum conserving Dirac delta functions that produce translation invariance of the scalar product. The dσ(λ) is a nonnegative measure on R with ﬁnite moments. Z n cn := dσ(λ) λ . (54)

The summation over integers a, b, 1 ≤ a, b ∈ Jk,n ≤ n, is described below (58) in section 2.6. The Nc × Nc matrices H(p) are functions of the energy-momenta.  B(p + p ) if a, b ∈ {1, k} or a, b ∈ {k + 1, n}  a b κaκb  H := (55) κaκb Υ(−pa + pb)κaκb if a ∈ {1, k}, b ∈ {k + 1, n}   0 if a ∈ {k + 1, n}, b ∈ {1, k}. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 58

Addressed in section 2.8, this form for H(p) is selected to satisfy regularity A.1 for non-constant H(p). The matrices B(p) and Υ(p) derive from the array M(p) in the two-point functional (30). Z isp B(p) := dµB(s) e M(s) Z (56) −sp Υ(p) := dµΥ(s) e M(s)

4 with s ∈ R , dµB(s) and dµΥ(s) are real-valued Lorentz invariant measures, dµΥ(s) is a nonnegative measure, and sp = s0p0 − s · p is a Lorentz invariant. The elements M(p)κaκb are multinomials in the energy-momentum components and greater spins result in higher order multinomials [6]. From the limitation of the energy-momentum support of the degenerate + 2 scalar product (8) to En , the singularities of H(p) at (pa ± pb) = 0 excluded from the summations that evaluate scalar products. The selected support of the VEV and functions in P 2 satisfy regularity A.1 by appropriate exclusion of (pa ± pb) = 0. For nontrivial constructions, signed symmetry with transposition of the arguments is considered to preclude cancellation of terms in the signed symmetrization (42) in (52). The indicated dependence on the M(p)κ1κ2 and vanishing of the odd orders Uk,2n−1−k implements Poincaréinvariance of the scalar product [38] for nonzero spins. Odd orders can be implemented for spin zero fields [34]. The specification (52) of U completes realization of nontrivial relativistic quantum physics. Rather than a Feynman series for scattering amplitudes, explicit VEV with a Hilbert space realization of the states labeled by elements of P are displayed. This unconstrained construction realizes both quantum mechanics and relativity. A realization follows for every realization of free fields (a selection of a M(p),D,S(A)), nonnegative measure dσ (that determines the relative contributions of connected functionals cn), and Lorentz invariant measures dµB(p) and dµΥ(p) (that describe interaction). Relation to free fields: The connectedness of the Uk,n−k provides that the expression (47) for Wk,n−k reduces to Fk,n−k if the support of functions f, g that describe states are sufficiently widely space-like separated and localized. In these instances, the states are readily interpreted as nearly free particles. In these instances, connectivity provides that the contributions of Uk,n−k ≈ 0 except that U0,0 = 1. F is signed symmetric and as a consequence, ! X X σ(π1) σ(π2)Fk,n−k(π2(π1({1, n}))) = k!(n − k)! Fk,n−k({1, n}) π1 π2 and then W ≈ F from (47). Vacuum polarizations: The means of the field are set to zero in this development but nonzero means hΩ|ΦκΩi are independently specified for the boson field components. Finite means can be included without impact on the satisfaction of A.1-7 and as a consequence, a development with zero means is general. The addition of a constant to a quantum field Φ(ˆ x)κ with vanishing mean implements fields Φ(x)κ with finite means except, if κ is a fermion index, 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 59 then addition of a constant to the field violates anticommutation. Nonzero means result if another generator factor G1;k,n−k is included in (52).

k n X X G1;k,n−k((α, κ)n) := exp( α`W0,1,κ` + α`W0,1,κ` ). `=1 `=k+1

The Nc dimensional array W0,1 = DW1,0 from (8). If there is a classical correspondence, then the W0,1 are real. Summary of VEV constructions: Appropriate selections of a basis function space P and VEV functionals W determine a realization of nontrivial relativistic quantum physics. The construction of W, (45) with (46), (30), (32), (52) and (53), suffices to satisfy the conditions A.1-7. Necessary conditions have not been characterized. Generalizations to these constructions include odd order scalar field VEV, compositions that add connected functionals F ◦U ◦U 0, multiplication of the arrays B(pa + pb) and Υ(−pa + pb) by Lorentz scalar, Lorentz invariant functions fB(pa +pb) and fΥ(−pa +pb) with fΥ(−pa +pb) factorable using nonnegative measure similarly to Υ(−pa + pb), and extensions of dµB(s) to complex-valued measures [34, 36]. Re- maining tasks for the unconstrained development include a fuller characterization of the physics modeled by constructions. Limited demonstrations that the evolution of appropriate states is well approximated by classical Newtonian dynamics and that the scattering cross sections coincide with low order Feynman series expansions are in [32, 33, 34, 36, 38] but, for example, a characterization of implied bound states remains. The constructions realize quantum mechanics as described in sections 2.1 and 2.2. Discussed in section 2.8, the components of W 0 4n are generalized functions within S (R ) in three or more spacetime dimensions if the development is limited to finite masses, and four or more spacetime dimensions includes massless particles [37]. Regularity implemented by (55) generally violates formal Hermiticity W.a and totality W.b is inconsistent with satisfaction of spectral support A.4 for the example nontrivial constructions. The refined quantum-classical correspondence, an approximate correspondence of state descriptions in appropriate instances, eliminates the motivation to impose Hermiticity. The free field, monomials of free fields and generalized free fields satisfy formal Hermiticity W.a and totality W.b, and there are scalar field cases with constant H(p) that exhibit interaction and satisfy formal Hermiticity W.a [34]. Satisfaction of A.1-6, and both formal Hermiticity W.a and totality W.b appears to be peculiar to physically trivial VEV. As a consequence of the limitation of the momentum support of the VEV to mass shells and the selection of positive energies in the intersection of the supports of the VEV with functions from P, the Hamiltonian, the generator of time translations of the jth argument of a function in the description of the states, is

U(λ) = e−ik0,j λ = e−iωj λ (57) from (26), translation by λ = ct, and in the selected units [34]. In the n argument subspace of 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 60

HP , n Y U(λ) = e−ik0,j λ. j=1 The constructed VEV W satisfy A.1-7 for states described by function sequences from the basis function space P and their Hilbert space limits. Satisfaction of spectral support, relativistic invariance, local commutativity and elemental stability was discussed above or in [34, 38], and positive deﬁniteness, cluster decomposition and regularity are demonstrated below in sections 2.6, 2.7 and 2.8 respectively.

2.6 Positive definiteness The construction in section 2.5 reduced demonstration of positive definiteness A.2 to demonstration that U defined by (52) provides a degenerate scalar product. From (52), U is the signed symmetrization (42) of a sequence of kernel functionals, designated u in (60) below, and as a consequence of (43), a demonstration that u provides a degenerate scalar product suffices. The sesquilinearity of u(f ∗, g) is evident in the definition (8) and the remaining task is to demonstrate nonnegativity. A factorization of the generator (53) results in display that u(f ∗, f) ≥ 0 [38]. Considering the prefactors of D in (53), three sets of factors appear in the generator (53).

k X X (D·) exp( ρ ρ H ) = exp( ρ ρ (DBDT ) ) k a b κaκb a1 b1 κa1 κb1 a,b∈Jk,n a1>b1=1 (58) n k n X X X × exp( ρ ρ B ) exp( ρ ρ (DΥ) ). a2 b2 κa2 κb2 a3 b3 κa3 κb3 b2>a2=k+1 a3=1 b3=k+1

Pk Pk Pk b>a=1 designates a=1 b=a+1. This deﬁnes the summation over a, b ∈ Jk,n for (53). Sub- stitution of the identity (34) into (56) provides that

DB(p)DT = B(−p)†.

A factorization results from substitution of this identity into the generator (53), use of the three factors distinguished in (58), and the factorization (33) of DM(p).

Z Z du ln (G ((α, −p, κ) , (α, p, κ) )) = dσ(λ) 0A nA U;n,m n,1 n+1,n+m (2π)4 n m   Nc Z (59) X 0 n × exp  dµΥ(s) cn,j1 cm,j1  j1=1 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 61 with

+ + ! k d k k Y −ipj u 2 2 X A := 1+λαje δ(pj −mκj ) exp ρaρbB(pa + pb)κaκb dρj j=k+1 b>a=k+1 k+ X −sp kc := ρ e k+a C(s) . ,j1 k+a j1κk+a a=k+1

k The arguments λ, u and (p, α, ρ)k+1,k+ of A are concealed and the arguments s, ` and k (p, ρ)k+1,k+ of c,j1 are concealed in this condensed notation. The dependence on masses k k mκj is also concealed. The A are linear differential operators and precede the factors c,j1 . C(s) is the factored decomposition of the nonnegative matrix DM(s) = C†(s)C(s) from (33). The first n of the n + m energy-momenta pj and component indices κj in (59) are relabeled pj, κj 7→ pn+1−j, κn+1−j and the energy-momenta are reflected pj 7→ −pj in anticipation of transfer of the effect of the ∗-morphism from the functions to the VEV in (8). The nonnegativity of U(f ∗, f) follows from the nonnegativity of u(f ∗, f) and (43). From (52),  n  Y d (D·) u˜ ((p) ) := G ((α, p, κ) ). (60) k k,n−k n (κ)n  dα  U;k,n−k n j=1 j The Fourier transform (24) of the ∗-morphism (9) of functions is Z d(x) n e−ip1x1 . . . e−ipnxn f ∗((x) ) = f˜ ((−p) ) (2π)2n n (κ)n n n,1 (κ)n,1 with the Fourier transformation preceding the complex conjugation. The Fourier transforms of the generalized functions provide that u(f ∗, f) is nonnegative if   Z n ∗ X X ˜ Y d u(f , f) = d(p)n+m fn((p)n)(κ)   n dαj n,m≥2 (κ)n+m j=1  n+m  ˜ Y d ×fm(pn+1, . . . pn+m)(κ)   n+1,n+m dαj j=n+1 (61)   N  Z Z du Xc Z × exp dσ(λ) 0A nA exp dµ (s) 0c nc  (2π)4 n m  Υ n,j1 m,j1  j1=1 ≥ 0

from the substitutions of (59) for the generator of the kernel functionals uk,n−k((x)n)(κ)n , and relabeling summation variables. The nonnegativity of u(f ∗, f) follows from an elaboration on 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 62

Schur’s product theorem that the Hadarmard products of nonnegative matrices are nonnegative [28]. Substitution of the absolutely convergent power series approximation for the exponential functions, x exp(x) ≈ (1 + )N N for suﬃciently large N, and the binomial expansion displays (61) as a summation with nonnegative weights of terms

 k k2  N  3 Z Z du Xc Z  dσ(λ) 0A nA dµ (s) 0c nc  .  (2π)4 n m  Υ n,j1 m,j1   j1=1

Reordering ﬁnite summations expresses (61) as a series of manifestly nonnegative terms. With 4n+4m the summations over n, m, (κ)n+m and (p)n+m ∈ R from (61) given precedence, a relabeling of summation variables displays u(f ∗, f) as a summation of nonnegatively weighted terms   2 Z n d X X ˜ Y 0 k4 0 k5 d(p)n fn((p)n)(κ)n   ( An) ( cn,j1 ) ≥ 0. dαj n≥2 (κ)n j=1 ˜ One factor includes the functions fm,(κ)n+1,n+m and summation over m and the (κ)n+1,n+m, and ˜ the other factor includes the complex conjugates of the functions fn,(κ)n with summation over 0 0 n and the (κ)n. In the notation of (59), dependence on (p, κ)n is isolated within An, cn,j1 and n m n dependence on (p, κ)n+1,n+m is isolated within Am, cm,j1 . The n that appears in Am and n cm,j1 is eliminated by relabeling summation variables. There are summations over distinct 0 0 λ, u, s or j1 for the appropriately paired factors An and cn,j1 in the terms from (61). The measure dµΥ(s) is nonnegative. The remaining summations are binomial expansions of the exponential functions. As a consequence, the u(f ∗, f) generated by (52) and (53) is nonnegative for function sequences from S and U provides a degenerate scalar product for function sequences in P ⊂ S. Finally, the example constructions of W, (45) with (46), (30), (32), (52) and (53), satisfy the positive deﬁniteness condition A.2.

2.7 Cluster decomposition Satisfaction of the strong form of cluster decomposition (19) provides that the truncated functionals [6] are connected functionals. This is a stronger condition than the uniqueness of the vacuum condition used in [32, 34, 38] and the development in section 2.13 illustrates that this condition implies that the local observables of non-entangled, spatially distant bodies are essentially independent. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 63

The strong form of cluster decomposition (19) is equivalent to

∗ ∗ ∗ ∗ Wk,n−k(fk1 gk−k1 , fk2 gn−k−k2 ) → Wk1,k2 (fk1 , fk2 )Wk−k1,n−k−k2 (gk−k1 , gn−k−k2 ) for the kernel functionals W if the support of fk1 and fk2 are arbitrarily greatly space-like separated from the support of gk−k1 and gn−k−k2 . This equivalence follows from the form of the degenerate scalar product (8) and the ×-product of function sequences (11). The demonstration that cluster decomposition A.6 is satisfied first establishes that W satisfies cluster decomposition if F and U both satisfy cluster decomposition. Then, satisfaction of cluster decomposition by the free field provides that it suffices to demonstrate that U satisfies cluster decomposition. It is then observed that U satisfies cluster decomposition if u from (60), satisfies cluster decomposition. Finally, satisfaction of cluster decomposition by u is demonstrated to establish that W satisfies the cluster decomposition condition A.6. A partition of the indices {1, n} into two subsets P and P 0 = {1, n}\ P suitable to test satisfaction of cluster decomposition (19) has 1 ≤ k1 ≤ k, 1 ≤ k2 ≤ n − k, and

P := {1, k1} ∪ {k + 1, k + k2}. (62)

0 Then P := {k1 + 1, k} ∪ {k + k2 + 1, n}. The notation is (40). If F and U satisfy the cluster decomposition condition A.6, then the arbitrarily great space-like separation of any (x)P from every (x)P 0 provides that

0 F`,j(A) = F`1,j1 (A ∩ P )F`−`1,j−j1 (A ∩ P ) and 0 0 0 0 Uk−`,n−k−j(A ) = U`2,j2 (A ∩ P )Uk−`−`2,n−k−j−j2 (A ∩ P ).

A is from (46), `1 is the number or elements in A ∩ {1, k1}, and j1 is the number or elements in A ∩ {k + 1, k + k2}. From the complementary sets, `2 = k1 − `1 is the number or elements 0 0 in A ∪ {1, k1}, and j2 = k2 − j1 is the number or elements in A ∪ {k + 1, k + k2}. Substitution into the deﬁntion of kernel functionals (46) provides that

X Wk,n−k({1, n}) = σ(A)F`1,j1 Uk1−`1,k2−j1 F`−`1,j−j1 Uk−k1−`+`1,n−k−k2−j+j1 A X X 0 (63) = ( σ(A ∩ P )F`1,j1 Uk1−`1,k2−j1 )( σ(A ∩ P )F`−`1,j−j1 Uk−k1−`+`1,n−k−k2−j+j1 ) A∩P A∩P 0 0 = Wk1,k2 ({1, n} ∩ P ) Wk−k1,n−k−k2 ({1, n} ∩ P ) in an abbreviated notation with the intermediate dependencies on arguments suppressed. No 0 partitions that include an xj with j ∈ P with x` with ` ∈ P contribute, and the remaining 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 64 summations over partitions include only those partitions that do not mix indices from within P with indices within P 0. In (63),

σ(A) := σ(π1)σ(π2) = σ(A ∩ P )σ(A ∩ P 0) P P σ(A ∩ P ) := σ( π1)σ( π2) 0 P 0 P 0 σ(A ∩ P ) := σ( π1)σ( π2)

P P 0 P with πa and πa designating subgroups of permutations. π1 are the permutations of indices P 0 0 P {1, k1} = {1, k} ∩ P , π1 are the permutations of {k1 + 1, k} = {1, k} ∩ P , π2 are the P 0 permutations of {k+1, k+k2} = {k+1, n}∩P and π2 are the permutations of {k+k2 +1, n} = 0 {k + 1, n} ∩ P . Since there are no contributions from partitions that transpose an xj with 0 j ∈ P with an x` with ` ∈ P , these permutations are independent and a sign σ(π1)σ(π2) is P P 0 P P 0 unambiguously decomposed as σ( π1)σ( π1)σ( π2)σ( π2) in the contributing terms. The free ﬁeld F from (32) satisﬁes cluster decomposition. From (52),

  n   ˜ X X Y d (D·)k Uk,n−k((p)n)(κ)n := σ(π1)  σ(π2)   GU;k,n−k((α, p, κ)n) dαj π1 π2 j=1

P P 0 if n is even. Then the independence of the permutations πa and πa for a = 1, 2 and the lack of contributions from permutations of (x)P with (x)P 0 provides that     ˜ X P X P Y d (D·)k Uk,n−k((p)n)(κ) = σ( π1)  σ( π2)   GU;k1,k2 ((α, p, κ)P ) n dαj P P π1 π2 j∈P     X P 0 X P 0 Y d 0 × σ( π1)  σ( π2)   GU;k−k1,n−k−k2 ((α, p, κ)P ) dαj P 0 P 0 0 π1 π2 j∈P if

0 ln (GU;k,n−k((α, p, κ)n)) = ln (GU;k1,k2 ((α, p, κ)P )) + ln (GU;k−k1,n−k−k2 ((α, p, κ)P )) (64) when any (x)P is arbitrarily greatly space-like separated from every (x)P 0 and with k1, k2 deﬁned in (62). The logarithm of the generator (53) is a linear combination of connected functionals 0 and any connected functional that includes an xj with j ∈ P and an xi with i ∈ P does not contribute. As a consequence, (64) applies. The k1 + k2 spacetime arguments of GU;k1,k2 are 0 labeled as (x)P and similarly for the n − k1 − k2 arguments (x)P of GU;k−k1,n−k−k2 . In these 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 65

instances, exp(x + y) = exp(x) exp(y) provides that the generator GU;k,n−k factors with the (p, κ)P dependence in one factor and the (p, κ)P 0 dependence in the other. As a consequence, the construction of W, (45) with (46), (30), (32), (52) and (53), satisﬁes the cluster decomposition condition A.6. Discussed in sections 2.12 and 2.13, this strong form of the cluster decomposition condition suﬃces to imply a causal property: the local observables of non-entangled, distantly space-like separated bodies are essentially independent.

2.8 Regularity 0 4n If the kernel functionals Wk,n−k((x)n)(κ)n are generalized functions within S (R ), then the constructed W satisfies regularity A.1. The construction (45) with (46), (30), (32), (52) and (53) displays the VEV as finite sums of products of connected functionals (38) with no arguments in common among factors. Demonstration of regularity reduces to a demonstration that the 0 4n higher order connected functionals are within S (R ) since the two-point functional (30) is 0 8 a Källén-Lehmannform within S (R ). From (52), the higher order connected functionals, C Wk,n−k((x)n)(κ)n with n ≥ 3 from (38), are products of elementary generalized functions and since products are not generally defined, these particular products must be validated. To satisfy regularity A.1 in the example realizations that exhibit interaction, elemental stability A.7 is adopted. Elemental stability enables the introduction of interaction with generators GU;k,n−k in (53) that satisfy positive definiteness A.2 without singular contributions [34, 38]. To satisfy positive definiteness and neglect the singular n = 2 contributions from the generators GU;k,n−k, the W1,n−1 can not contribute to the scalar product (8). Positive definiteness A.2 ∗ ∗ implies that Wn−k,k(fn−k, fk) = 0 if Wk,n−k(fk , fn−k) = 0. For functions from P, conservation of energy sets all contributions from W0,n to zero if n ≥ 2. If the mass-spin combinations preclude decay of isolated elementary particles, then satisfaction of formal Hermiticity W.a is ∗ consistent with nonnegativity, W1,n−1(f1 , fn−1) = 0 with W1,n−1((x)n) 6= 0. More generally, ∗ W1,n−1(f1 , fn−1) = 0 is inconsistent with formal Hermiticity and elemental stability is adopted as an axiom for the examples in these notes. C Whether the connected functionals Wk,n−k((x)n)(κ)n are continuous linear functionals 0 4n 4n within S (R ) rests on whether the indicated summations over the submanifolds of (p)n ∈ R to evaluate (38) are regular. From (38) and (52), the submanifolds are determined by the mass shell and energy-momentum conserving Dirac delta functions, and any singularities must not coincide with singularities of H(p) to produce non-summable singularities. Contrasted with the neutral scalar field examples discussed in [34], multiple component fields introduce distinct elementary masses mκ and singularities from non-constant H(p). In this section, the number of spacetime dimensions is considered and is designated d. There are no nontrivial finite Lorentz invariant measures [6] and as a consequence, B(p) 4 and Υ(p) are either constant or singular within R . From (56), B(p) is at least as singular as 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 66 the Pauli-Jordan function [57]. From the singularities of the Pauli-Jordan function [6],

dk b B(p) ≈ a δ(p2) + (65) dpk p2

2 in a neighborhood of p = 0 and k > 0 for nonzero spins. A constant H(p) results from dµH (s) = cδ(s)ds for H = B, Υ. For functions in P and from the limitation of the support of U to mass shells, the singularity of B(pj + pk) is encountered only when pj is the argument of a ∗-morphism of a function (Ej = −~cωj) and pk is the argument of a function (Ek = ~cωk). In 2 these instances, the points (pj + pk) = 0 would be within the support of the summations that deﬁne the scalar product (8). However, the selected form (55) includes only energy-momentum support p , p with (p + p )2 > m2 c2 + m2 c2. Similarly for Υ(−p + p ), the singularities are j k j k κj κk j k encountered only if both arguments are associated with a function or both are associated with the ∗-morphism of a function. In the regular case (55), the singularities of H(p) are not within the support of the summations that evaluate the scalar products (8) [38]. Each connected functional from (53) is supported solely on submanifolds with energy and momentum conserved. After evaluation of the mass shell delta functionals and with consideration of the zeros in the energy support of functions in P and P∗, the Fourier transforms of the n ≥ 4 connected functionals include only momenta on the manifold indicated by

n Y 1 δ(ω ... + ω − ω ... − ω ) δ(p +p ...+p ) (66) 2ω 1 k k+1 n 1 2 n j=1 j with q 2 2 2 ωj = mκj c + pj from (26). Factors of 1/(2ωj) are multipliers of tempered functions and are not considered 3n further in the examination of regularity for ﬁnite masses. Within the submanifold of R with momentum conserved, pn = −p1 ... − pn−1, (67) energy conservation δ(Ek((p)n)) with

k n n X X X Ek((p)n) := ωj − ωj := sjωj (68) j=1 j=k+1 j=1 deﬁnes a generalized function except possibly for points on the surface Ek((p)n) = 0 with a vanishing gradient, ∇Ek((p)n) = 0 [17].

−1 if j ∈ {k + 1, n} s = j 1 otherwise. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 67

Any singularities of δ(Ek((p)n)) are now studied. The components of the gradient ∇Ek((p)n) within the submanifold with momentum conserved are dE ((p) ) dω dω dp k n = s j − n n dp j dp dp dp j j n j (69) pj pn = sj + ωj ωn from (68) with d − 1 dimensional momentum vectors pj, the shorthand (67, and j ∈ {1, n − 1}. 2 2 2 2 Summing squares provides that when the gradient vanishes, pj /ωj = pn/ωn. Then, substitution of ωj provides that the gradient vanishes if and only if

pj pn sj = − mκj mκn for each j ∈ {1, n − 1} and using (67). Then both the function and its gradient vanish,

Ek((p)n) = ∇Ek((p)n) = 0, only if n X sjmκj = 0 (70) j=1 from pj = −sjmκj pn/mκn at the zeros. A neighborhood of those points with a zero energy and vanishing gradient is given by

pj p1 = sj + ej mκj mκ1 with kejk < 1, 0 < kp1k and j ∈ {2, n − 1}. In this neighborhood,

n−1 n−1 X mκ X p = − p = − n p − e n j m 1 j j=1 κ1 j=2 using (70) and

2 2 mκj p1 · ej ej (p1 · ej) ωj ≈ ω1 + sjmκj + mκ1 mκj − mκ1 mκj 3 mκ1 ω1 2ω1 2ω1 Pn−1 to second order in small quantities and with en := − j=2 ej and for j ∈ {2, n}. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 68

On the submanifold with momentum and energy conserved, and within a neighborhood of the points where the gradient ∇Ek((p)n) equals zero,

n mκ X E ((p) ) ≈ 1 s m ω2 e2 − (p · e )2 k n 2ω3 j κj 1 j 1 j 1 j=2 R2 := (αp2 + βm2 ) 3 1 κ1 2ω1 from (70), polar coordinates for the n − 2 ej, and the deﬁnitions

n−1 2 X 2 R := ej , j=2 u1 := p1/kp1k and

n n 2 X 2 2 2 X 2 αR := mκ1 sjmκj ej − (u1 · ej) , βR := mκ1 sjmκj ej . j=2 j=2

The result is that energy conservation (66) deﬁnes a generalized function except possibly for the points with the gradient ∇Ek((p)n) vanishing when Ek((p)n) = 0, the points with R = 0.

2 R 2 2 δ(Ek((p)n)) = δ( 3 (αp1 + βmκ1 )) 2ω1 3 3 2ω1 2 2 2ω1 2 = 2 δ(αp1 + βmκ1 ) + 2 2 δ(R ). R αp1 + βmκ1

2 2 Since (66) is a generalized function for R > 0, δ(αp1 + βmκ1 ) is regular. For n ≥ 4 and a suﬃcient number of dimensions d, the singularities of the energy-momentum conserving delta functions (66) are locally summable for the regular selection (55) of H(p). The (d−1)(n−2)−1 Jacobian for polar coordinates for (e)2,n−1 contributes R in d spacetime dimensions for the nth order connected functionals. Then, the summations in evaluation of the degenerate scalar product (8) include

R(d−1)(n−2)−1 dR = R(d−1)(n−2)−3 dR R2 and d ≥ 3 suﬃces for the summations to converge and the contributions from the generator (53) to be continuous linear functionals for the tempered functions. Terms

R(d−1)(n−2)−1δ(R2) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 69

do not contribute for n ≥ 4, d ≥ 3 [17]. d ≥ 3 suffices for finite masses mκj and d ≥ 4 includes massless stable particles [37]. If the singularities (65) of non-constant H(p) were included in the evaluations of the degenerate scalar product (8), then three is not the sufficient number of dimensions. From (55), the 2 singularities of H(p) occur for (pi ± pj) = 0 and these can coincide with the singularities of the energy-momentum conserving delta functions (66) in the case of equal masses, mκi = mκj . Designate these coincident singularities as at least 1/(R)α with α ≥ 2. α = 0 for constant H(p). If the singularities of H(p) were included in the evaluation of scalar product, then the worst case would be n/2 factors of H(p) in an nth order connected functional and the factors of H(p) would contribute a coinciding singularity of up to 1/Rnα/2 in the case of equal masses. The summations in evaluation of the degenerate scalar product would include

(d−1)(n−2)−1 R (d−1)(n−2)− α n−3 nα dR = R 2 dR R 2 R2 and α n + 2 d > 1 + 2 n − 2 suﬃces for the summations to converge. For d = 4 and n = 4, α is limited to less than 2 necessitating selection of a regular case such as (55) to include non-constant H(p). Finite masses, the regular selection (55) for the arrays H(p) and three or more spacetime dimensions d suﬃce for the construction of W in section 2.5 to satisfy regularity A.1. Inclusion of massless elementary particles requires four or more dimensions [37].

2.9 Example M(p), S(A) and D The example realizations [38] are constructed from realizations of free fields [23, 55, 63]. Here, consideration is limited to finite masses; massless cases require additional regularization [37]. The constructions of W, (45) with (46), (30), (32), (52) and (53), derive from a Dirac conjugation D, a realization for a two-point function for the elementary particles M(p), and the corresponding realization of the Lorentz subgroup S(A), A ∈ SL(2, C). These Nc × Nc matrices satisfy (12), (33), (36) and (37). A neutral scalar field is realized by M(p) = S(A) = D = 1, the scalar realization of the Lorentz subgroup. A charged scalar field is realized by two component fields. This two component (Nc = 2) scalar field has S(A) := I, the 2 × 2 identity, and 0 1 D := 1 0 0 1 M(p) := = M(−p)T . 1 0 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 70

T This ﬁeld has a symmetry associated with a charge, SφM(p)Sφ = M(p).

eiφ 0 S := φ 0 e−iφ and DSφ = SφD. A massive vector boson field is realized with four component fields, Nc = 4. This spin-1 boson field uses a real 4 × 4 representation of the Lorentz subgroup.

D := I S(A) := Λ p p M(p) := (j) (k) − g jk m2 jk = C†(p)C(p). p := (p(0), p(1), p(2), p(3)) and parentheses were used to distinguish energy-momentum vector components from energy-momentum vectors pj. g is the Minkowski signature matrix and Λ T is a Lorentz transformation determined by A ∈ SL(2, C) in (23). M(p) = M(−p) . In four dimensions with p := (p0, px, py, pz) and cp0 = E, √  p2−m2c2  0 0 0 0  mc r  p2 +p2  p0 √ px y z 0 0  †  mc 2 2 2 p2−m2c2  C (p) =  p0−m c 0  .  p −p p  p0 √ y √ x y √ pz 0  mc 2 2 2 2 2 2 2 2 2 2   p0−m c (py+pz)(p0−m c ) py+pz   p0 √ pz √ −pxpz √−py 0  mc 2 2 2 2 2 2 2 2 2 2 p0−m c (py+pz)(p0−m c ) py+pz

A massive spinor fermion field is realized by four component fields, Nc = 4. This spin-1/2 fermion field uses a 4 × 4 reducible, complex representation of the Lorentz subgroup. ! 0 D := I I 0 ! A 0 S(A) := 0 A ! 0 P M(p) := P T 0 ! p + p p + ip P = P † := 0 z x y px − ipy p0 − pz 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 71

T with 2 × 2 P , I and A ∈ SL(2, C). M(−p) = −M(p) . cT (p) 0 c(p) 0 DM(p) = C†(p)C(p) = 0 c†(p) 0 c(p) is a nonnegative matrix for p within the forward cone and

 √ p +ip  √x y p0 + pz p +p c(p) =  0 z  . 0 √ mc p0+pz

T This ﬁeld also has a symmetry SφM(p)Sφ = M(p) associated with a charge,

φ 0 eiφ 0 S := φ := φ 0 φ 0 eiφ and DSφ = SφD. The mixed product property of the direct sum (U ⊕ V )(U 0 ⊕ V 0) = (UU 0 ⊕ VV 0) and Kronecker product (U ⊗V )(U 0⊗V 0) = (UU 0⊗VV 0) [29] provides that direct sums and Kronecker (tensor) products of M(p),S(A),D that satisfy (12), (33), (36), (37) are also solutions. Real orthogonal similarity transforms OM(p)OT ,OS(A)OT ,ODOT of solutions are also solutions. These compositions of example representations of M(p),S(A),D and their subrepresentations provide a rich class of realizations of relativistic quantum physics with interaction.

2.10 Hermitian Hilbert space operators, states and classical observables The properties of operators used in quantum mechanics are discussed in this section. Every subspace of states corresponds with a projection operator. Projection operators onto subspaces of a Hilbert space are self-adjoint Hilbert space operators. Projection operators E are bounded operators, kEψk ≤ kψk, and the domains of bounded operators always extend to the entire Hilbert space, DE = H. In the case of projections, the range RE = EH ⊂ H, a proper subset unless E = I, the identity. Every (closed) subspace of a Hilbert space corresponds with a projection operator, and every projection operator E corresponds to a subspace of a Hilbert space EH [40, 49]. For a projection E, any state |ψi ∈ H may be decomposed as an element |Eψi within the subspace EH and an element |(I − E)ψi in the orthogonal complement of the subspace, Riesz’s theorem. Every element in the orthogonal complement is orthogonal to every element in the subspace. Projections are self-adjoint and idempotent, E = E∗ = E2. The states of nature are described by nonnegative (hψ|ρψi ≥ 0), self-adjoint (ρ = ρ∗, Dρ = Dρ∗ dense in H), trace-class, normalized operators ρ (Trace[ρ] = 1). X Trace[ρ] := hek|ρeki (71) k 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 72

if the |eki are an orthonormal basis for the separable Hilbert space H. This basis is not unique 0 ∗ −1 and every unitary operator maps a basis to another basis, |Ueki = |eki. U = U for a unitary operator and unitary operators are bounded. These operators ρ are designated state density operators [62] and generalize the vector states discussed above. Vector states have state density operators ρ = E for E the projection onto a single element |ψi (RE = {|aψi : a ∈ C}). Born’s rule includes: if a state is described by the state density operator ρ, then the likelihood of observing a state in the subspace EH corresponding to a projection E given an initial state described as ρ is Trace[Eρ] ≤ 1. The operators associated with observables have mean values

E[A] := Trace[Aρ] ∈ R for Hermitian operators A and state density operators ρ. Measurements are real numbers and the expectation values of Hermitian operators are real. If A is Hermitian, A = A∗ on the domain of A, then for every f ∈ DA,

∗ and A = A for f ∈ DA. As a consequence, observables are limited to Hermitian operators. Signiﬁcant examples of observables from nonrelativistic quantum mechanics are the Hermi- tian location operators Xν. In L2, Z hψ|ψi = dx |ψ(x)|2 is ﬁnite at each time t with x = ct, x. The summation is over three dimensional space. Three location operators Xν, one for each of the three spatial dimensions, are given by

|Xνψi := xν|ψi and then the range of Xν is labeled by functions xνψ(x) ∈ L2 with the xν one of the three components of the spatial vector x = (x1, x2, x3) = (x, y, z). ψ(x) ∈ L2 from |ψi ∈ H. The 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 73 location operator is an example of an unbounded operator since there is no bound on the values assumed by location in a Euclidean space. xνψ(x) is not necessarily square-integrable for square-integrable ψ(x) and as a consequence, DXν ⊂ H, a proper subset. Functions ψ(x) ∈ L2 such that xνψ(x) ∈ L2 are dense in L2 but elements of slow decline are excluded from DXν 2 −1+δ (e.g., ψ(x) = (1 + x ) ∈ L2 for 1 δ > 0 in three dimensions but xνψ(x) 6∈ L2). If the dominant support of the function ψ(x) is in a neighborhood of a single location y(t), that is, if the spread of that peak is small compared to the precision of our location measurements, then for considerations of location, the state described by ψ(x) is essentially the same as a classically described state located at y(t). These states that are dominantly supported in small volumes approximate eigenfunctions of the location operator,

|Xνψi ≈ yν(t)|ψi. An eigenfunction of location would be labeled by a function equivalent to a Dirac delta function, that is, a function that is not square-summable using Lebesgue measure. Functions with point support are defined as linear functionals dual to functions and can be considered as limits of sequences of essentially localized functions that describe states. For functions ψ(x) with broad spatial support, likelihood is spread over a wide volume and the state description is not well approximated by a classical state, that is, one representative point of the support does not adequately describe the state. The support of the description of state, location in this case, is of more fundamental physical significance than the properties of the associated operators Xν. The domain of the operator Xν is not the entire Hilbert space but the location support is described for all states, and similarly, there are states arbitrarily near each state with a finite mean value for location that have divergent means hψ|Xνψi. For functions dominantly supported in small volumes, it is overwhelmingly likely, compared with any other finite volume in space, that an observation will result in a location indistinguishable from y(t) even if the expectation value diverges. We have no ability to measure the likelihood of locations arbitrarily far from our location. Our ability to observe is limited to a localized region in spacetime: our observed likelihoods are relative to finite volumes. These considerations are developed further in the section on technical concerns with canonical quantization, section 5. The Adjoint and real, observable fields: In Wightman’s original development and for unconstrained developments that satisfy formal Hermiticity and totality, an adjoint of the field operator Φ(f) follows from the scalar product (8), product of function sequences (11), and the ∗-morphism of sequences (9). The definitions of the field operator (14) and of an adjoint operator result in the identification hh|Φ(f)gi = hΦ(f)∗h|gi = W (h∗ × f × gi (72) = W ((f ∗ × h)∗ × gi = hΦ(f ∗)h|gi 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 74

with f = (0, . . . , f(x1)`, 0, 0,...), only one, one argument function is nonzero, and the nonzero component of f was designated `. From formal Hermiticity, W(f ∗, g) = W (f ∗ × g). The ﬁeld ∗ ∗ is Hermitian, Φ(f) = Φ(f), if Φ(f) = Φ(f ) on a domain DΦ(f) ⊂ HP .

f = (0,..., 0, f(x)`, 0,...) = f ∗

= (0,D`1f(x)1,...D`Nc f(x)Nc , 0,...) with a component for each κ ∈ {1,Nc} from (9). Real field components Φ(x)` with D`κ = δ`,κ using the Kronecker delta are designated real, observable. Real, observable field components occur for the boson field components in the examples of section 2.9. In this event, the field is Hermitian if f(x)` is real. From [34] and section 2.3, any real function of a single argument in P is in the equivalence class of zero. The domain of the adjoint of the field Φ(f)∗ includes only the zero element of the Hilbert space. For f ∈ P and if formal Hermiticity applies,

Φ(f)∗g = f ∗ × g 6∈ P

∗ unless g = 0 since P ∩ P = {(c, 0, 0 ...)} for c ∈ R [34]. For a nontrivial f(x)`, the adjoint maps elements |gi out of the Hilbert space unless |gi = 0. The fermion field components are not Hermitian, and Hermitian operators associated with the field, the real part Φ(f) + Φ(f)∗ and magnitude Φ(f)∗Φ(f), are also precluded by the lack of real functions in P. The constructed Φ(f) is also not a normal operator; the common domain of Φ(f)∗Φ(f) and Φ(f)Φ(f)∗ includes only zero. Unless satisfaction of A.1-7 extends from P to a ∗-involutive algebra such as S, the real, observable components of the field Φ(f) defined in (14) are not realized as Hermitian operators, and there are no Hermitian operators corresponding to the real part or magnitude of other field components. An extension to include real functions is precluded by the example implementations of interaction in section 2.5. Observables: In quantum mechanics, the operators that correspond to observable classical dynamical variables are Hermitian. The real eigenvalues of the Hermitian operator are the observable values. If the state is an eigenstate of the Hermitian operator, only one real value of an observable is associated with the state, similarly to a classical description of the selected observable attribute. However, it is more useful to think of the functions as the descriptions of states than it is to think of states as distributed over the eigenstates of an observable. The latter puts an emphasis on classical state descriptions that never fully apply. No state is simultaneously an eigenstate of location and momenta in the rigged Hilbert spaces of physical interest. Although useful as approximations for selected instances and attributes, position and velocity demonstrate that a classical description never applies to a state of nature. From the Heisenberg uncertainty principle, knowledge of velocity is lost for a position eigenstate and as a consequence, these eigenstates are not nearly classical descriptions of bodies. And, (generalized) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 75 eigenstates are often not elements of the Hilbert space although they are approximated by elements within the Hilbert space (the association of classical dynamical variable with quantum states is approximate, not ideal). As a consequence, what observer’s interpret as position measurements do not result in state descriptions with point support but rather result in a measured value associated approximately with an element of the Hilbert space. Eigenstates of momenta, like the eigenstates of position, are not elements of the rigged Hilbert spaces of interest in quantum mechanics but are generalized eigenstates and are approximated by elements of the rigged Hilbert spaces. The functions that most closely associate with observations of location of a body are the relativistically invariant localized functions, [44], sections 1.5 and 2.1, and appendix 7.2. Functions from an appropriate delta sequence such as the Gaussian functions (74) with ko = 0 for σo → 0 are good approximations for functions with point support and are elements of the L2 Hilbert spaces appropriate in nonrelativistic physics. The eigenfunctions of energy and momentum are harmonic functions,

h(p) = e−ikx, with p = ~k. The harmonic functions are also not square-summable. Functions such as the Gaussian functions (74) with ~ko = po are good approximations for momentum eigenfunctions for σo → ∞ and are elements of the Hilbert spaces. A substantial difference between nonrelativistic and relativistic quantum mechanics is that the kernel functions Wk,n−k((x)n) that define the scalar product of the nonrelativistic L2 Hilbert spaces have point support (delta functions) in position differences but to realize relativistic quantum physics, the kernel functionals have spatially distributed support. The spatially distributed support is necessary to satisfy Lorentz covariance with nonnegative energy support. Then, for example, Z d(x)n Wk,n−k((x)n)gk(x1, . . . xk) xjfn−k(xk+1, . . . xn) Z ≈ y(t) d(x)n Wk,n−k((x)n)gk(x1, . . . xk)fn−k(xk+1, . . . xn) for functions appropriately supported near y(t) in the jth argument but as demonstrated in section 5, multiplication by xj does not correspond to Hermitian Hilbert space operators. Sim- ilarly with momentum and the Fourier transforms except that in the case of momenta, the pν do define Hermitian Hilbert space operators as they must since the Pν generate the unitarily realized translations. Both nonrelativistic and relativistic scalar products are translation invariant. Of three prominent examples of classical dynamic variables, none fully satisfy conjectures associated with measurement as projection onto eigenstates of the Hermitian elevation of the classical dynamical variable. 1. Momentum. The operators associated with momentum are densely defined Hermitian operators implemented as elevations of pν, the arguments of the Fourier transforms of the 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 76

functions that describe states. The momentum operators possess dense sets of orthonormal generalized eigenfunctions labeled by the harmonic functions. These eigenfunctions are labeled by functions that are limits of functions that label elements of the Hilbert space realization. With the exception that these eigenfunctions are not elements and are only approximated by elements of the Hilbert space realizations, these associations satisfy conjecture for the quantum-classical correspondences of the canonical formalism. Nevertheless, a measurement can never cause “collapse” to a momentum eigenstate.

2. Location. In relativistic physics, the operator associated with location is not the elevation of xν, and the orthonormal functions associated with observations of locations for a free ﬁeld are the relativistically invariant localized functions of Newton and Wigner [44]. The relativistically invariant localized functions have support at all locations although the support is dominantly within a small volume near the associated location. Dirac delta functions are not orthogonal for the relativistically invariant scalar product, and only become approximately orthogonal in the nonrelativistic limit. The Hermitian operators Xbν associated with observations of location are described in (5) and are developed by associating states with observations of the location of bodies.

3. Fields. There are no eigenstates of the field operators (14). An association of the propagation of bodies in the quantum descriptions of states with trajectories of classical bodies from classical field theory establishes an association of quantum and classical field that is not based on elevation of the field to Hermitian Hilbert space operators. This correspondence of trajectories associates quantum dynamics with classical dynamics without a necessity for an elevation of classical fields to quantum field operators. This alternative association applies if the quantum descriptions are well approximated by classical descriptions. This alternative association was introduced in section 1.5.

The quantum-classical correspondence of the canonical formalism is that a quantum description is implemented by Hermitian Hilbert space operators that are elevations of classical dynamical variables. That is, location is the elevation of xν to Hilbert space operator and VEV functionals hΩ|Φ(f) ... Φ(h)Ωi are the scalar products of states that result from the action of Hermitian field operators Φ(g) on the vacuum state Ω in the case of the observable field components. Here, these correspondences are taken as examples of the inappropriate reliance on classical concepts. Extrapolation of the classical dynamical variables to Hermitian Hilbert space operators applies a classical correspondence to states that may not have classical correspondences, and approximate correspondences suffice physically. An example of an approximate correspondence is location for a relativistic free field. As discussed in section 1.5, section 2 and appendix 7.2, location is not the elevation of the classical dynamical variable xν in relativistic physics. Indeed, such correspondences are not generally available. The Hermitian product of corresponding Hermitian operators does not generally correspond to the product of the classical dynamical variables. This is demonstrated by counterexample in section 5. The classical corre- 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 77 spondences of a quantum description are richer than consideration of the elevations of classical dynamical variables to Hermitian Hilbert space operators. Physical considerations allow more approximate and conditional quantum-classical correspondences.

2.11 Heisenberg uncertainty One of the great insights in the development of quantum mechanics is that location and momentum are not independently specified descriptions of bodies. Both location and momenta derive from the function ψ(x) that describes a state of a single body. The support of the functions ψ(x) are associated with the likelihoods of locations and the support of the Fourier transforms ψ˜(p) are associated with the likelihoods of momenta. Likelihoods are provided by Born’s rule using the Hilbert space scalar product. If the scalar product is L2, appropriate in nonrelativistic physics, then the location operator corresponds to the Hermitian operator that derives from multiplication of the function ψ(x) by the value of an argument xν and the momentum operator corresponds to the Hermitian operator that is the elevation of pν. Using the Fourier transform relations (100) and linearity, in L2 location corresponds to −i~d/dpν and momentum to i~d/dxν. This is discussed further in appendix 7.2. If at a time t the support of ψ(x) is dominantly supported in the neighborhood of a point y(t), then Z Z 2 2 hψ|Xνψi := dx xν|ψ(x)| ≈ yν(t) dx |ψ(x)| = yν(t) for a normalized ψ(x). y(t) is any suitable representative of the neighborhood of support. From Parseval’s equality (99), the definition of Fourier transform (24) and implied identities, if at a time t the support of the Fourier transform ψ˜(p) is dominantly supported near a momentum q(t) = ~k(t), then with ν = 1, 2, 3, Z Z ˜ 2 dψ(x) hψ|Pνψi := ~ dk kν|ψ(p)| = i~ dxν ψ(x) ≈ qν(t). dxν There are no states within the quantum mechanical description that approximate a classical state arbitrarily well. There is no knowledge of momentum for a location eigenfunction, and no knowledge of location for a momentum eigenfunction. Are there most nearly classical selections for functions that describe quantum mechanical states? Here, a nearly classical state is one with precisely known location and velocity. The time derivative of location is velocity v and in the nonrelativistic approximation p ≈ mv. There is no limit on the precision of knowledge of the mass. Are there states that approximate both location and momentum simultaneously and arbitrarily well? The result is that there is an optimal accuracy that can be simultaneousally obtained for knowledge of location and momentum, and a choice of functions that provides the optimal simultaneous knowledge. This result is the Heisenberg uncertainty principle. In quantum mechanics, not only is dynamics no longer described by a smooth trajectory specified by an initial position and velocity, but we cannot know both the location and the time derivative 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 78 of the location of a body sufficiently well to specify a trajectory. That is, our description of state does not support the concept of Newtonian mechanics, that we solve for the future of a body from the current location and time derivatives of the location. The Heisenberg uncertainty principle is a lower bound on the simultaneous accuracy in location and momentum knowledge in a quantum mechanical description. We can approximate the location and time derivative of location for a body, and this approximation becomes better the heavier the body, but the concept of Newtonian mechanics is not supported by the quantum description of state. Also, in quantum mechanics, we cannot necessarily identify a particular body to propagate forward in time. In quantum mechanics, similarly described bodies are indistinguishable and cannot be labeled to identify a trajectory except while a single body is sufficiently isolated from other bodies. While isolated, a dominant peak in the support of a function that describes a state can be identified and followed with a high likelihood until it approaches other similar bodies. At this point, its identity becomes lost in ambiguity. The Heisenberg uncertainty principle is derived in ordinary, nonrelativistic quantum mechanics. It is a result that follows from the nonrelativistic L2 scalar product. In L2, three location operators Xν = xν and three momentum operators Pν = i~d/dxν, one for each spatial dimension, are self-adjoint in L2 and satisfy the Born-Heisenberg-Jordan relation

A := Xν − hψ|Xνψi, and B := Pν − hψ|Pνψi.

ν = 1, 2 or 3 and the intersection of the domains includes states labeled by the Schwartz tempered functions that are dense in L2. From the commutation of Xν and Pν it follows that [A, B] = −i~ and that A, B are mean zero for the state |ψi. From the interpretation of hψ|Aψi as the mean value of the quantity associated with the operator A for normalized states hψ|ψi = 1, identify variances of position and momenta as

2 2 2 2 σx = hψ|A ψi, and σp = hψ|B ψi for the normalized state labeled by ψ(x). The self-adjointness of A and B follows from the self-adjointness of Xν and Pν on L2. Self-adjointness and the Cauchy-Schwarz-Bunyakovsky inequality then provide that

2 2 2 2 2 2 2 |hψ|[A, B]ψi| = |i~hψ|ψi| = ~ = |2i=m(hψ|ABψi)| ≤ 4|hψ|ABψi| ≤ 4σxσp.

This is Heisenberg’s uncertainty principle. For states in L2,

σxσp ≥ ~/2, a lower limit on the geometric mean of the variances of location and momentum. This limits the simultaneous accuracy of location and momentum descriptions in each dimension ν. The lower bound is achieved if hψ|ABψi is imaginary and |Aψi = c|Bψi for c ∈ C. The labels ψ(x) for the states that meet both lower bounds are

2 2 e−(x−xo) /(4σo)+ipox/~ ψ(x) = 1 . (74) 2 (2πσo) 4 These Gaussian functions satisfy

Z 2 (x − xo) 2 hψ|ABψi = i dx 2 |ψ(x)| 2σo that is imaginary and

dψ(x) i~(x − xo) Aψ(x) = (x − xo)ψ(x) = cBψ(x) = c(−i~ − poψ(x)) = c 2 ψ(x) dx 2σo

2 for real xo := hψ|Xνψi and po := hψ|Pνψi, c = 2σo/(i~) and hψ|ψi = 1. The parameter σo = σx and σp = ~/(2σo). This is the function shape in each dimension. The spatial function that labels a minimum uncertainty state is the product of three factors, one for each spatial dimension. These Gaussian functions are denoted minimum packets and are the states most like classical states in the sense that the geometric mean of the uncertainties in simultaneous knowledge of the location and momentum of the state is minimized. The Heisenberg uncertainty principle applies in relativistic physics with some revision. In a relativistic development, the operators Xν (|Xνψi := |xνψi) are not self-adjoint and therefore are not associated with an observable quantity. As a consequence, the development above of the Heisenberg uncertainty relation does not apply for the operator pairs Xν,Pν. Nevertheless, the Heisenberg uncertainty principle applies when classical approximations to the relativistic 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 80

1/2 −1/2 physics apply, and with Xν replaced by Xbν := −i~ω d/dpνω for the Hermitian operator associated with the location of a single body. ω is from (26). Classical particle approximations apply if the states are described by functions with localized, isolated concentrations of support [33]. The eigenfunctions of Xbν are the relativistically invariant localized functions [44] and are not strictly localized. If the Xbν is used in the development of the Heisenberg uncertainty relation, then the minimum uncertainty packets are not Gaussian functions. The relativistic minimum uncertainty packets are normalized inverse Fourier transforms of

2 2 g˜(p) = ω1/2e−σo(p−po) eip·xo/~.

If the state description consists of sufficiently localized and isolated concentrations in the support of each argument, then the state can be described as classical particles and the Heisen- berg uncertainty principle applies for functions ψ(x) used to define particular functions in P. Recall (p + ω)ψ˜(p) u˜(p) := 0 √ ∈ P. (75) 2ω If the state is described by localized, isolated concentrations of support, then the contributions to the kernel functionals of the n-argument connected functionals are negligible. Due to cluster decomposition, the contributions to scalar products of the high order connected functionals are negligible when a state consists solely of widely spatially isolated bodies. In these cases, the Wk,n−k are well approximated by the free field contribution (32) to the kernel functionals. From (32), the significant contributions to scalar products (8) for product states fn ∈ P with

n Y fn((x)n) = uj(xj) ∈ P j=1 and functions uj with Fourier transforms of the Gaussian form consists of sums of products of ∗ factors W2(uj u`). Substitution results in Z ∗ ˜ W2(uj u`) = dk1dk2 W2(p1, p2)u˜j(−p1)˜u`(p2) Z δ(E + ω ) δ(E − ω ) = dk dk δ(k + k ) √1 1 √2 2 u˜ (−p )˜u (p ) 1 2 1 2 2ω 2ω j 1 ` 2 Z 1 2 (76) ˜ ˜ = dk2 ψj(p2)ψ`(p2) Z = dx ψj(x)ψ`(x), the L2 scalar product of the two spatial functions ψj(x), ψ`(x). In these instances, the relativistic scalar product reduces to L2 scalar products of states described by functions ψj(x) from (75). ∗ ∗ W2(uju`) = 0 and W2(uj u` ) = 0 when uj ∈ P. Indeed, the equivalence of the contribution to 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 81

the scalar product of uj and uk with the L2 scalar product of ψj and ψk demonstrates that the scalar product is small if the overlap of the functions described by ψj and ψk is small. This jus- tiﬁes analyzing the support of functions described by uj using the support of the ψj if classical particle approximations apply. The overlap of states described by uj and uk is small when the overlap of the functions ψj and ψk is small and the contributions of the high order connected functionals is negligible. The uj ∈ P in (75) have essentially the same support as the ψj in these cases. From this demonstration it also follows that the Heisenberg uncertainty principle that applies to the variances of location and momentum measurements in the Hilbert space L2 with state descriptions ψj, ψk applies approximately also to the descriptions of isolated bodies labeled by uj, uk ∈ P in relativistic quantum physics if classical particle approximations apply. Otherwise, the numbers of bodies in the state description is uncertain and the interpretation of functions as describing uncertainties in body locations and momenta does not apply. The Heisenberg uncertainty principle states that there is a lower limit on the simultaneous knowledge of body positions and momenta. As a consequence, the classical concept of propagating a trajectory into the future from initial conditions is not supported by the quantum mechanical description of state. This provides that the classical approximation to quantum mechanics is necessarily conditional and limited.

2.12 Observation The process of observation is fundamental and central in quantum mechanics. This process explains how the perception of nature can differ from the description. One of the concepts of classical physics is that the state can be observed without disruption: an observer is external to the systems under consideration and is essentially omnipotent. However, in quantum mechanics, observation is an interaction within the quantum description. The quantum description evolves a state that describes the composite of system under observation and observer. Re- gardless of the initial states, states evolve to a linear combination over states that describe the distinct alternatives for results from an observation: for example, an initially localized states spreads over space with time. Within the quantum mechanical description, it is inconsistent with our experience that there can be only one observer state: observers will record one result from among the possibilities. To be included in the quantum description, a distinct state of the observer must be entangled with each of the possibilities for observation. States that describe different observations are distinct. Within the linear expansion of state, we are described by one of those observer states. This relative state or “many worlds” description [9] is naturally derived from the quantum mechanical description of state although it was described several decades after the initial interpretations. Initial interpretations maintained more classical descriptions for observers. Even the description as “many worlds” reveals a classical predisposition. Consid- eration of the state as a superposition over many classical worlds, each with a state described by classical dynamical variables, rather than as a single quantum mechanical description is an example of the persistent classical predisposition in the study of physics. The relative state 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 82 interpretation uses the natural quantum mechanical description of states by functions with entangled states leading to a decomposition into relative states. An observer never has knowledge of the systems beyond what is determined in the processes of interaction. The description that we perceive results from the process used to observe. Nevertheless, the quantum mechanical descriptions of observation explains our comfort with the classical view: often, within the precision of our observations, the quantum mechanical description appears to replicate the classical description. A richer range of descriptions is included among the descriptions of state in quantum mechanics. A localized interaction results in observation of a localized state; two slits in a screen followed by localizing detectors results in observation of wave-like interference when the initial states are widely supported and propagate toward the screen. In common instances and to great likelihood, the perception of a quantum description is indistinguishable from a classical description although in general, in instances such as the uniformly illuminated double slit, a quantum description of state may deviate substantially from a classical description. Quantum mechanics supecedes the concepts of classical mechanics. In a modern development of quantum mechanics [9], an observation is any interaction that results in entanglement of the descriptions of bodies with descriptions of observers. Below, the states of an observer are labeled by their perceptions. The relevant final state of an observed body is determined by the entangled observer state. The possibilities are described by the scalar product (that determines likelihoods), the (not directly observed, but prepared) initial state, and chance (the entangled final state of the observer that describes us). Obser- vations modify the relevant function that describes a body. To achieve an observation, the support of the description of observer state must overlap with the support of observed body’s state description at the time of observation. The final state description is relative to our, the observer’s, particular state description. Our state is also described by a function; quantum mechanics consistently describes all of nature. That all natural processes are described consistently dictates that quantum mechanics also describes observers. In a quantum mechanical description, the descriptions of an observer are included in the descriptions of states as elements in a Hilbert space, and the states of an observer are generally expressed as linear combinations of other state descriptions. Unlike a classical concept of state as determined and immutable except predictably from classical forces, observation generally results in a random selection of observed state from within a linear expansion of the temporal propagation of the initial state. The initial states expands in descriptions associated with each of the possible results of the observation. This description of observation follows the quantum mechanical descriptions of state as elements in a Hilbert space. Interactions generally entangle the states of bodies, and observations entangle states of bodies with states of systems described as observers. Introduced in section 2.1, if |ψi describes the initial state of a body and one history of perceived observations, then an interaction of the observer and a body will result in an expansion of the evolved 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 83 state into states labeled by updated histories of the observer’s perceptions, X X |U(t)ψi = |E U(t)ψi = hg |U(t)ψi |g i. (77) θ θ θ θ θ For convenience, the assumption is that our observations are to finite resolution and the terms in this summation are denumerably indexed. The last equation displayed is a simplification that applies if the subspaces associated with the projections Eθ are spanned by the single element |g i. The θ label observer perception histories, the Eθ are orthogonal projection operators, θ P EθEϑ = 0 if θ 6= ϑ, and θ Eθ = I at least within an orthogonal subspace that includes the initial state |ψi and the states |g i of interest. Whether this summation includes all pos- θ sible states in the Hilbert space is not of issue. Demonstrated below, our only concern is a particular subspace |EθU(t)ψi that includes our history of perceptions, us. The histories of observer perceptions are updated for the last interaction with the body. It is characteristic of a Hilbert space description that any state can be expanded as a linear combination of a dense set of elements. Well designed measurements entangle descriptions that approximate eigenstates for the measured quantityo with the observer’s states. Observers often interpret the body’s state as that eigenstate, as a classically described state. That is, for example, in a well designed measurement, the observer interprets the perception of a body supported within a small neighborhood of xo as a point-like body located at the point xo. The likelihood of adding the observable associated with the state included in the description g to the history of the observer is given by Born’s rule, θ Likelihood=Trace(E ρ) = |hg |ψi|2 θ θ in the case of a normalized vector state ρ := |ψihψ| and with Eθ the projection onto the normalized vector state |gθi. “The observer” is the description of an observer relative to one of the possibilities for perceived state, in this case relative to the body in the description |g i. The θ description of the observer state is included in |g i. The last observation adds the perception θ labeled by θ to the history of observations. An observer is any sufficiently complex system capable of transitioning to distinct states in response to interactions with external bodies. We never observe that recorded observations change, that further observations or com- munications change the record of past measurements or that the current state is inconsistent with past perceptions. This experience indicates that states labeled by distinct perceptions are orthogonal: the Hilbert space decomposes into orthogonal subspaces labeled by the distinct histories of observations. This decomposition provides no likelihood of transitions from our history of perceptions to a distinct history. The observer states are entangled with the corresponding states of bodies and these joint states can be labeled by a history of perceived observations. The observation may be in error or imperfect, that is, the actual state entangled

oDiscussed in section 1.1, these eigenstates may be distinct from the elevations of a classically conceived measured quantity. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 84 with an observer may not be the perceived state, but distinct histories of perceptions lie in orthogonal subspaces. There is no likelihood of a recollection of a history distinct from the recorded history of observations and observations are necessarily repeatable in sequence, for example, the locations along a track of a body. Entanglement was introduced in section 2.1 and entanglement develops in the time evolution of a state that initially describes an observer and a body that may be initially independent. An element of the Hilbert space, for example, a vector state |ψi generally includes descriptions of an observer and a body. After interaction, that is, for sufficiently large t, the time translation of the vector state, X |U(t)ψi = c |g i θ θ θ from (77), is a linear combination of vector states |g i that each describe a distinct state of the θ observer entangled with the corresponding descriptions of the body. Each term is labeled by θ, the record of observation, that is, the state of the observer, and the terms are orthogonal, hg |g i = 0 if θ 6= ϑ. The result of the time evolution implemented by U(t) results in a ϑ θ perception added to the initial knowledge of the observer. An initially non-entangled object approaches and its support significantly overlaps with the observer’s support. The state then expands as distinct descriptions of the possibilities for final observer states entangled with the corresponding object states. For example, a distinct state of the observer may correspond to the final momentum of the object as the object and observer separate. Descriptions of observers are typically localized. The observables of the object include features of the function sequence describing the object as discussed in section 1. Unlike a classical description with each body conceived to have a state described independently of the states of other bodies, in quantum mechanics, constituent descriptions of states become entangled. This “many worlds” description of nature as a superposition over elements in a Hilbert space is another source of discomfort with quantum mechanics. This superposition will include states that describe alternatives to our particular history of perceptions. However, further study of this initially disconcerting concept illustrates that the idea results in no weirdness and indeed, is completely consistent with our experience as well as consistent with the quantum mechanical description of nature, bizarre images summoned by “many worlds” notwithstanding. The development that describes observation from the premise that the “mathematical formalism of quantum mechanics is sufficient as it stands” is the Everett-Wheeler-Graham (EWG) relative state interpretation named for Hugh Everett III, John Archibald Wheeler and Neill Graham [9]. This development is natural to the quantum mechanical description of nature and follows from that states are described by elements of Hilbert spaces that expand as linear combinations of other states, and that observers must also be described within quantum mechanics. The bizarre images conjured by this description emerge from the predisposition for classical description. Consistency with nature requires the quantum mechanical descriptions. In the EWG development, there is a virtual collapse of the description of nature to the 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 85 relative state for each particular description of an observer, and we are described by one of those terms. Predictions based solely on this relative state agree with predictions using the full description of state. This agrees with our experience; we need only know what we have observed and the other possibilities are of no consequence to our ability to predict future outcomes. The alternative “worlds” have no reality for us. The alternatives do not affect our predictions although our experience is one from among many possibilities. This development resolves one of the mysteries of quantum mechanics. Why do we never perceive a state in a superposition over contradictory states like a superposition of ‘live cat’ and ‘dead cat’ in Schrödinger’scat thought experiment [54]? Our observations entangle us with one or the other state. For the Schrödinger cat thought experiment, the descriptions of final states would be a linear combination of a g θ that describes observation of a live cat entangled with a live cat and an undecayed isotope, and an orthogonal g that describes observation of a dead cat entangled with a dead cat and a ϑ decayed isotope. The formalism of quantum mechanics is adequate to describe our experience. Early interpretations of quantum mechanics included ad hoc assertions like “collapse of the wave packet” or “hidden variables” in attempts to preserve a classical description of all or part of nature. Further development has demonstrated that our observations of nature are adequately described by quantum mechanics without such augmentations. This description of measurement is in violent conflict with classical concepts; this Hilbert space development of observation was not the first description that arose in physics. Never- theless, this description using linear combinations over the elements of the Hilbert space, the approximate association of elements with classical states, and reliance only on unitary time evolution is the natural quantum mechanical description of measurement. Concepts that preserve a “classical domain” are less disruptive to established classical concepts but employ ad hoc assumptions and incur the famous measurement paradoxes that reinforce the need for a quantum mechanical treatment of measurement. These classical measurement paradoxes include the Einstein-Podolsky-Rosen (EPR) [14], Schrödinger’scat [54], and Wigner’s friend [67] paradoxes. The EPR paradox is discussed in section 1.1. Schrödinger’scat paradox illustrates that the quantum description can not be relegated to a microscopic world and Wigner’s friend illustrates the ad hoc nature of the assertions necessary to descriptions of measurement as “collapse of the wave packet.” The EWG relative state interpretation [9] avoids imposition of classical idealizations into quantum mechanics and considers that quantum mechanics is a complete and consistent model of nature. The EWG interpretation of quantum mechanics develops the concept of relative (entangled) state. A result of the Hilbert space development and our established physical principles is that there is no perceptible physical distinction between: considering oneself as essentially a classical system; and considering observers as included in an entangled superposition over the various possibilities for observers with differing perceptions, one for each of the myriad possible outcomes for each measurement. This result, demonstrated below, uses the physically motivated condition that distinct observer histories lie in orthogonal subspaces. Physical predictions conditioned on our knowledge are indifferent to whether a “wave function collapse” 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 86 law or the exclusively unitary time translation law for observations is adopted. The less ad hoc description of interaction is that a trace (likelihood) preserving, unitary similarity transform that determines the evolution of the state density operator. A Hermitian operator A has a real conditional expectation value corresponding to an observable quantity. The conditional expectation is the expected value with states limited to a subspace of the Hilbert space. A Hermitian operator A is representedp as X A = akEk k is Hermitian with orthogonal projections Ek and ak ∈ R [19, 40]. The ak are the values of the observable that represents the states in the subspaces EkH. The conditional expectation value of the observable associated with the Hermitian operator A and conditioned upon one particular description θ of the observer is then

E[E AE ] Eθ[A] := θ θ E[E ] θ (78) Trace(E AE ρ) = θ θ Trace(Eθρ) with ρ the nonnegative, unit trace, state density operator [62] introduced in section 2.10. The Eθ are the orthogonal projections from (77) that project onto the subspaces of states that represent the state of the observer labeled by perception history θ. The Eθ project onto a composite body plus observer state. Two example constructions of composite states with essentially independently speciﬁed descriptions of the observer and body are developed below. Eθ projects onto the subspace of the composite states that include the speciﬁed history of observer perceptions and the entangled state of the body under observation. From the idempotence of projections and transposition invariance of the trace (71) [28], Trace(AB)=Trace(BA), Trace(Eθρ) = Trace(EθρEθ) with EθρEθ the state density operator projected into the subspace labeled by θ. The relative state density operator [9], relative to the observer history labeled θ, is

θ ρ := Nθ EθρEθ (79) normalized to unit trace.

θ Trace(ρ ) = Nθ Trace(Eθρ) = Nθ E[Eθ] := 1.

pAppropriate limits of these sums are included. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 87

The relative state density operators are orthogonal, ρθρϑ = 0 from θ 6= ϑ, and remain orthogonal at equal times with time evolution implemented by unitary time translation, U(t)∗U(t) = 1. In terms of this relative state density operator, the conditional expectation is

Eθ[A] = Trace(Aρθ).

The conditional expectation is relative to the subspace labeled by the history of perceived observations θ and this history also labels the entangled descriptions of the body . It was argued on physical grounds that states labeled by diﬀerent histories of perceptions were orthogonal since it is our experience that there is no likelihood of a change to history. As a consequence, there will be a decomposition of unity into projection operators Eθ labeled by the perception histories θ. X Eθ = 1 θ with the θ labeling states that describe the history of possible perceptions, plus one additional orthogonal subspace of all remaining state descriptions. Distinguish this projection as E0. Decomposition of the Hilbert space into orthogonal complements follows from the orthogonality of states with distinct histories, and Riesz’s theorem on orthogonal subspaces in a Hilbert space [49]. If the operator A corresponding to an observable quantity is in the commutant of the Eθ from (78), that is, if [A, Eθ] = 0 for each Eθ except possibly with an exception for E0, then a mixture of the relative state density operators ρθ is equivalent to the complete state density operator ρ. This equivalence applies in the subspace that includes observers and then consideration is limited to the subspace P (I−E0)H. The resolution of unity Eθ = 1, the idempotence of projections, the commutation of A and the Eθ, the transposition invariance of the trace (71) and the deﬁnition of relative 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 88 state results in the equivalence,

E[A] = Trace(Aρ) X = Trace(EθAρ) θ X 2 = Trace(Eθ Aρ) θ X = Trace(EθAEθρ) θ X = Trace(AEθρEθ) θ X θ = Trace(Aρ )/Nθ θ X θ = E [A]/Nθ. θ Then, using the deﬁnition of relative state density operator (79), if an operator A is in the commutant of the Eθ, the mixture of relative states,

eq X θ X θ ρ = ρ /Nθ = E[Eθ] ρ θ θ has the same observable properties as the complete state description ρ.

Trace(Aρ) = Trace(Aρeq ). (80)

If E[Eθ] = 0, the term does not contribute to the summation. And, each of the relative state density operators ρθ provide the expectation values conditioned on an observer history. The subspace E0H is of no interest. This equivalence of a mixture of the mutually orthogonal relative state density operators with the complete description of the state ρ explains why the quantum mechanical description of state is not in contradiction with a classical concept of an observer. Each term in a mixture evolves independently in time: knowledge of the entire state description is not required to propagate the relative states forward in time. Conditional expectations are equivalent to distinguishing one particular history, our history, as a classical observer. All our future observations evolve from our current relative state description ρθ. We need not know or account for the “other branches” to predict the future that is relevant to us, the future conditioned on our history. This suggests a “collapse of the wave function” upon observation described in early developments of quantum mechanics [62], but there is no physical distinction between a virtual or an actual collapse of the state description ρ to ρθ upon observation given the assumptions of orthogonal subspaces labeled by histories and the commutation of A with the Eθ. An actual collapse would result in a classical description of the observer but the universality of likelihood conservation, that is, a unitary implementation 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 89 of time translation, is preserved if the collapse is considered as virtual. The equivalence applies for any future time, ∗ X θ ∗ U(t)ρ U(t) = E[Eθ] U(t)ρ U(t) . θ

The ﬁrst example of the commutation of observables A with projections Eθ is a tensor product of two Hilbert spaces. Two Hilbert spaces H1 and H2 describe the states of the body and observer, respectively. A second example of this commutation is developed in section 2.13 and is based on a strong cluster decomposition property of the kernel functions in a scalar product (8). The second example describes a body that is initially distantly spatially separated from the observer and the commutation is with arbitrarily great likelihood but inexact. We anticipate on physical grounds that observable features of bodies are (nearly) independent of the states of the observer when the bodies are distantly spatially separated from the observer, and unentangled. In this case as well as the tensor product case, the observables A of the body commute with the projections Eθ onto the subspaces labeled by observer histories. A tensor product is the composition of Hilbert spaces H1 and H2 into a composite

H := H1 ⊗ H2.

From descriptions of the body ψ, g ∈ H1 with scalar product hψ|gi1, and states of the observer described u, v ∈ H2 with scalar product hu|vi2, the tensor product composite Hilbert space H has states labeled ψ ⊗ u, g ⊗ v with scalar product

hψ ⊗ u|g ⊗ vi := hψ|gi1 hu|vi2.

1 2 If ek and ek are orthonormal bases for separable H1 and H2 respectively, then an orthonormal basis for the composite Hilbert space has labels

1 2 ekj = ek ⊗ ej and dim(H) = dim(H1) × dim(H2). Operators deﬁned in H1 ⊗ H2 include extensions of the operators for H1 and H2. Hilbert space operators in the tensor product include C := A ⊗ B,

hψ ⊗ u|C g ⊗ vi := hψ|Agi1 hu|Bvi2.

Then any operators A ⊗ I commute with any I ⊗ B. If H1 includes the descriptions of the body and H2 includes the descriptions of the observer, then this example satisfies the assertions above in discussion of the physical equivalence of the mixture of relative states ρθ to ρ. For an observation to occur, the time translation U(t) from (77) must couple the constituent Hilbert spaces H1 and H2: U(t) is not in a form U1⊗U2. Tensor products demonstrate that the assumed concept of body observables within the commutant of the projections onto the observer states is realizable. B(H1) ⊗ I and I ⊗ B(H2) demonstrate that there are commuting sets of operators 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 90 of interest. B(H) is the set of all bounded operators for the Hilbert space H. This notation is discussed further in appendix 7.4 and that this composition suffices to define a Hilbert space is discussed further in appendix 7.6. With the observer as well as the body under observation included in the quantum mechanical description, time evolution is continuous, unitary time translation. The equivalence of ρeq and ρ provides that there is no need for a collapse of the state upon measurement, nor a need to distinguish what interactions constitute measurements and cause collapse. Measurement is distinguished as any interaction that results in an entanglement of our history of perceived observations with state descriptions. The equivalence of descriptions ρeq and ρ explains why only the history of one observer, one history of perceptions, needs to be maintained. An observer need not carry the entire history of possibilities, the state density matrix ρ, to describe the natural world. Any future observations are relative to the current state of the observer, that is, future observations are conditioned upon a particular history θ, within a particular orthogonal subspace. A description of that relative state is all that must be maintained to properly describe the future evolution of relevant observables. With attention focused on one term in the mixture of relative states, this term is interpreted as the result of an effective, virtual collapse to a more classically described state upon measurement. But, the state density matrix continues to evolve unitarily. Since the mixture of relative states is physically equivalent to the complete description of the natural world, limiting knowledge to the relative state does not modify predictions for observables. There is no consequence to considering the alternative states since their inclusion does not effect predictions of later observations. Whether the other “branches” ρϑ with ϑ not equal to our history θ are real is a metaphysical issue of no empirical consequence. There is no observable effect to considering other branches or not. Our physical predictions are indifferent to whether the branching or collapsing is considered. There is a physical equivalence of: the complete description of state that includes the observer distributed over possibilities ρ; a mixture of independent states labeled by the possible results recorded by the observer ρeq ; and a description that collapses to the state relative to one classically described observer upon each measurement ρθ. Results are indifferent to the approach considered. The three are physically equivalent within quantum mechanics considered as a complete description of nature. The alternative “branches” are never observed and if you consider yourself a classically described body then there will never be a contradiction to that sense of self, that is, there will never be an instance when the observer perceives his alternative states. Nevertheless, logical contradictions to a classically described state are evident in the EPR, Schrödinger’scat and Wigner’s friend measurement paradoxes. The EWG interpretation is inherent to quantum mechanics and requires no ad hoc adjustments.

2.13 Localized observables, separation and independence The intuitive notion that separation implies independence provides the second example of Her- mitian operators A associated with observables of a body that (nearly) commute with the 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 91

projections Eθ onto states that include a particular description for an observer. In addition to the tensor product model of independent body and observer states discussed in section 2.12, a body that is initially distantly spatially separated from the observer provides another example of independent descriptions for observer and system. If the strong form of cluster decomposition (19) applies, scalar products factor when the support of the functions that describe bodies and observers are greatly space-like separated. This is suggestive of the tensor product decomposition of Hilbert spaces but applies approximately in likelihood since there are no functions of bounded support in P and applies only if states are appropriately separated at some time. In these general cases, the EWG interpretation of quantum mechanics applies. If localized observations are considered, then the strong form (19) of the cluster decomposition property provides that the observables of a body will nearly commute with the projections onto the various states of an observer when the body and observer state are initially distantly spatially separated and not entangled. This intuitive, satisfying property is enabled by consideration of localized observations. As discussed in section 2.10, observables always correspond to Hermitian operators although the form of the operator may not be the canonical formalism-anticipated elevation of a classical dynamical variable. Observables essentially limited to bounded spatial volumes and isolated bodies associates observables with bodies. And, if the body is initially greatly spatially isolated from the observer, then these observables (arbitrarily nearly) commute with projections onto states that describe the observer. This commutation is the property used to develop the equivalence of the mixture of relative states with general state descriptions in the EWG development of observation discussed in section 2.12. For the example realizations of relativistic quantum physics, the independence of the body and observer states is approximate since there are no functions of bounded support within P. As a consequence, the equivalence (80) of the mixture of relative states ρθ with ρ is to arbitrarily great likelihood. The approximate independence results from spatial separation, and localization of the observation,

0 ≈ k[Eθ,A]k kAk with A a Hermitian operator representing the localized observable and for the projection, kEθk = 1. The commutation is only approximate because the support of a body description can never be ideally spatially isolated from the state of an observer. The indistinguishability of bodies of similar descriptions in quantum mechanics implies that bodies are not identified with function arguments. State descriptions are symmetric with interchange of arguments in the example of a scalar field, boson, case. But, if the support of an argument is spatially isolated from the support of the other arguments, evolution of that support can be identified as the evolution of an identifiable body’s state to arbitrarily great likelihood until the support significantly overlaps the support of other arguments. Observables of a spatially isolated body can be associated with that body independently of the remainder of the state description. With isolation, the association is a body with a region of support to arbitrarily great likelihood. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 92

The strong form of the cluster decomposition condition (19) provides that the truncated functions deﬁned by link-cluster expansion (cf. eqn. 10.29 [6], [50]) are connected functionals,

C T Wk,n−k = Wk,n−k.

C The vanishing of connected functionals Wk,n−k as the supports of arguments are greatly space-like separated, and satisfaction of the strong form of cluster decomposition (19) provides that Wn,m((y)n+m) = Wj,`((x)P )Wn−j,m−`((x){1,n+m}/P ) as λ → ∞ if yik = xk for ik ∈ P and yik = xk + (0, λa) for ik ∈ {1, n + m}/P , the set complement of the j + `-element set of integers P in the set of integers {1, n + m}, with the evident embellishment of the (x)n notation, and kak > 0. P := {1, j} ∪ {n + 1, n + `}. The demonstration is for a selected case of an elementary particle described by ψ(x), a function of a single spacetime argument with observable features derived from ψ(x). The support of ψ(x) is spatially isolated from the support of the descriptions of observer’s states.

Designate by Vxo a bounded spherical volume centered on xo that includes all but a negligible amount of the support of ψ(x). Assert that there is a choice of functions {e`(x)} such that: the support of every e`(x) outside of Vxo is uniformly negligible; the observables of interest associated with the body described by ψ(x) within the volume Vxo can be expressed as single particle operators, n X |Afˆ i := (0,A1f(x1),..., Ajfn((x)n),...) (81) j=1 with X Z Ajfn((x)n) := a`e`(xj) dy1dy2 W2(y1, y2)e`(y1)fn(x1, . . . xj−1, y2, xj+1, . . . xn) `

4 if j ∈ [1, n], W2 is the two-point functional, y1, y2 ∈ R are spacetime vectors; the functions e`(x) are orthonormal, he`|e`0 i = δ`,`0 ; (82) a` are the values of the observable associated with states described by e`(x); and X ψ(x) ≈ he`|ψi e`(x). (83) `

The a` are eigenvalues of the single particle operator X A = a`|e`ihe`|. ` 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 93

The extension Aˆ of the operator A to multiple arguments in (81) is suggested by the indistinguishability of similarly described bodies. While this extension deﬁnes a Hilbert space operator, the interpretation of the observable may be less evident. As discussed in sections 1 and 2.1, the extension is most readily interpreted when a classical particle interpretation applies to the quantum state description. The operator A is Hermitian and here A is required to be associated with the volume Vxo . Only bounded operators are considered for localized observable operators

A. A is designated essentially localized within Vxo if the strong form of cluster decomposition (19) implies that hf|Afˆ i → 0 (84) as the dominant support of the vector state |fi is arbitrarily greatly spatially separated from

Vxo . The connected property of W2 provides that an A defined by (81) is essentially localized. The Pauli-Jordan function, W2(x1, x2) declines faster than exp(−r/λc) in space-like directions for large space-like separations r = kx1 − x2k. This rapid decline ensures that the expectations of an A of the form (81) for states widely separated from Vxo are negligible given the conditions satisfied by the e`(x). There are no strictly localized functions in P and the equalities are to arbitrarily great likelihood. The e`(x) have nonzero but negligible support outside of Vxo . These properties restrict the class of observables. For example, the linear harmonic oscillator energy eigenfunctions are localized but not uniformly negligible outside of a bounded volume: the greater the energy, the broader the support. The volume is selected so that the contribution of any ψ or e` of interest beyond Vxo is arbitrarily negligible. Examples of localized, one particle observables include location with the e`(x) ≈ δ(x − x`) and x` ∈ Vxo . Location of a photon within a charged coupled device is an example of a localized observable. The second set of operators of interest are the projections Eθ onto states that include descriptions of the observer (nearly) independently of the descriptions of the body. Eθ projects a joint description of observer and body onto a description with the observer in a state described by the particular history of perceptions θ. Initial observer states are labeled f and final observer o states are labeled hθ. These states do not include a description for the body to be observed, that is, the support of f and the h within V is arbitrarily negligible. Initial and final correspond o θ xo to before and after the observation. The projections onto the final observer states labeled hθ are Eθ := orthogonal projection onto the union of the ranges of Eg (85) `θ defined from the projections onto product states defined

|g i := |e × h i (86) `θ ` θ with e` := (0, e`(x1), 0,...) for the basis set of descriptions of the body essentially localized to within the volume Vxo and the hθ describe the observer states associated with a history of perceptions θ. It is assumed on physical grounds that these hθ are invariant to translations. That is, both the translation of |hθi by a and |hθi represent that same history of perceptions θ. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 94

Memory depends on internals and not location of the description of state. Below, it is assumed that the dominant support of the hθ is spatially distant from Vxo . Estimates for the commutation of the body’s essentially localized observables A with the projections Eθ onto observer states result from the assumptions of great initial spatial separation, lack of entanglement, and the strong form of cluster decomposition (19). The deﬁnitions for the observables (81), (82), (83), and (84), projections (85), and joint observer-body states (86) in the rigged Hilbert spaces of relativistic quantum physics provide demonstrations that

The results follow for the product (11) and the choice of {e`(x)}. And then ﬁnally, the near commutations of the A and Eθ are evaluated. The scalar product (8) and the decomposition of kernel functionals into connected functionals provide that hg |ψ × f i = he × h |ψ × f i `θ o ` θ o

≈ he`|ψi hhθ|foi using the result from strong cluster decomposition (19) that

∗ ∗ ∗ ∗ Wk,n−k(gn−k−1e` ψfk−1) ≈ W2(e` ψ)Wn−k−1,k−1(gn−k−1fk−1) (87) as a consequence of the distant space-like separation of supports. The separation of the volume

Vxo from the support of the initial observer also provides that

hg |g i = he × h |e × h i `θ `θ ` θ ` θ

≈ he`|e`ihhθ|hθi = 1 by normalizing kh k = 1. Similarly, hg |g i ≈ 0 if ` 6= µ. This approximate orthogonality θ `θ µθ then provides that X Eθ ≈ Eg (88) `θ ` 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 95 from (85). As a consequence,

= hhθ|foi |ψ × hθi from substitution of the expansion (83). In rigged Hilbert spaces of sequences of functions,

hg|fi |E fi := |gi g hg|gi

2 ∗ are orthogonal projections onto the vector state described by g, Eg = Eg = Eg . The expression for Eθ on the appropriate body plus observer states f = ψ ×fo demonstrates that the deﬁnition (85) approximates the projection operator with the desired property of projecting onto the observer state of interest independently of the descriptions of the body. The form (81) of the local observable Aˆ results in

n X |Aˆ ψ × foi = (0, Aψ(x1)fo,0,... Ajψ(x1)fo,n−1(x2, . . . xn),...) j=1

≈ (0, Aψ(x1)fo,0,... (Aψ(x1))fo,n−1(x2, . . . xn),...)

= |(A ψ) × foi as a consequence of the wide spatial separations of the support of the e`(x) and fo, and the shorthand that Aj applies to the jth argument. The commutators of the Aˆ and Eθ are

ˆ ˆ |AEθ ψ × foi ≈ hhθ|foi |A ψ × hθi

≈ hhθ|foi |(Aψ) × hθi 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 96 and

|EθAˆ ψ × foi ≈ |Eθ (Aψ) × foi

≈ hhθ|foi |(Aψ) × hθi. Then, the unitary implementation of time translation provides that

[Eθ, Aˆ] ≈ 0 for states that describe a body and observer that are initially unentangled and distantly space- like separated. Commutators are independent of unitary similarity transformations and as a consequence, independent of the unitarily implemented temporal translations. This development is not general but demonstrates an alternative example of observables Aˆ that (nearly) commute with all the projections Eθ onto states that include a particular description of an observer. That spatially isolated systems are essentially independent satisﬁes our intuition and experience. This sensible result is enabled within relativistic quantum physics by the consideration of observations that are essentially localized, likely association of locations with bodies, and satisfaction of the strong form of cluster decomposition (19).

2.14 Why does the flawed classical interpretation suffice in everyday life? Although quantum and classical predictions are consistent as approximations in appropriate instances, there is no necessity for an equivalence of the classical and quantum descriptions as reality. Quantum mechanics provides our current best understanding of nature and classical mechanics is an idealized and flawed description of nature that applies conditionally for appropriate states. However, this begs the issue of why classical physics provides such an excellent description of our everyday experience. Why were we able to describe so much before consideration of quantum mechanics early in the twentieth century? The comfort we have with a classical description of nature is explained as a confirmation bias built into the process of observation, and that we commonly deal with slow and isolated massive bodies. The conditions for which classical predictions apply are commonly satisfied. Due to the large masses of most bodies dealt with in our everyday experience, contradictions to the classical model are essentially cloaked from our view. Indeed, it was not until the accumulation of evidence for the atomic structure of matter, encounters with difficulties in the study of statistical physics, and contrary results in spectroscopy that the exceptions to classical concepts emerged. There are many quantum states not well approximated by classical descriptions, but these exceptions occur for situations that are not our usual experience. If one anticipates that the classical description applies, then the confirmation bias is that classical mechanics suffices for our perceptions using typical processes of observation and our experience with heavy, slow bodies. Slow is with respect to the speed of light, 3 × 108 meters/second. Examples that quantify this result are evaluated below. Quantum mechanics is quite different from classical 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 97 mechanics at small distances, distances on the order of the Compton wavelength (18), but our typical observations do not probe these small distances. For heavier bodies, the Compton wavelength is proportionally smaller. Entanglement of states is accessible to our observation, for example, as described in the Einstein-Podolsky-Rosen paradox discussed in section 1.1, but entanglement is not typically noticed. Particle creation and annihilation are quantum mechanical but again, are not encountered in everyday experience. For quantum states well approximated by classical descriptions, the observable features of interest appear and behave nearly like classical dynamic variables. This approximation applies for quantum mechanics with −34 the measured value of ~ = 1.054 × 10 joule-seconds and whether or in what sense the ~ → 0 limit of quantum mechanics should be classical mechanics is a distinct concern of no interest to this argument. Whether ~ is small is entirely a consideration of the units used, but ~ is “small” in the sense that classical approximations suffice for much daily experience, for example, ~/mc is small with respect that distances in our common experience. Classical mechanics describes identifiable bodies that follow smooth trajectories and quantum mechanics describes indistinguishable bodies that are variously wave-like or body-like but are always described by functions that do not vanish over any finite spatial volume in relativistic physics. Two significant qualifications for application of classical descriptions to quantum states are that a classical model does not necessarily apply to every feature of a quantum state simultaneously, and classical descriptions apply only in appropriate instances, that is, for appropriate states. From Heisenberg’s principle in section 2.11, products of the standard errors in location and velocity for a body are lower bounded by the reduced Planck’s constant divided by twice the body mass. As a consequence, classical descriptions apply more accurately for massive bodies than for light mass ones. Heavy is with respect to masses determined by Planck’s constant and location −33 accuracy, m ~/(2cσx) introduced in section 1. Heavy is m 1.8 × 10 kilograms for σx = 10−10 meters. An electron mass is 9.1 × 10−31 kilograms. Particle creation and annihilation, indistinguishability, and discrete energy levels are purely quantum mechanical effects. However, and for example, propagation of isolated, localized bodies is well approximated by classical models as long as entanglement is negligible and the support of the function that describes the body is not sufficiently large that overlap with other bodies is significant. While the descriptions of quantum states develop wide support over time, apparently localized bodies are typically observed. Localization typically derives from the scattering or emission of radiation from the body. State descriptions with sufficiently small spatial spreads are perceived as classical bodies. There is no contradiction to the classical concept of location as a point apparent in observation of states with spatial spreads small with respect to measurement precision or body size. Radiation typically originates from near the surface of an extended body. Observing the direction of arrival and passive range for this radiation localizes the body. An observer “sees” bodies with an accuracy no better than comparable to the radiation’s wavelength. The sizes of atoms are small contrasted with typical macroscopic bodies that consist of very large numbers of atoms. The size of atomic bodies is characterized by a radius associated with the ground state energy 2 2 −10 −34 of the hydrogen atom, the Bohr radius ~ /(me ) = 0.5×10 meters. ~ = 1.054 ×10 joule- 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 98 seconds is the reduced Planck constant, e is the electron charge, e = 1.6 × 10−14 kg1/2m3/2/sec, and m is the electron mass, m = 9.1 × 10−31 kilograms. When a body is observed, the accuracy of the location measurement typically greatly exceeds this characteristic size. The wavelength of visible light is greater than 3 × 10−7 meters. Typical observations result in essentially localized descriptions of bodies, that is, states described by a function with support localized to within a neighborhood of any representative point. The localization is to a great likelihood, and unlikely deviations from this localization are not significant. For appropriate quantum states and many observable properties, the description provided by classical mechanics approximates the observable features of the quantum states. A question of interest is: what would it take to observe the diffraction and interference predicted by quantum mechanics with baseballs? The method used is to contrast the diffraction and interference of spatially distributed states of electrons and baseballs that propagate through appropriate slotted screens. Because the position and momentum support of the functions that describe states are related by Fourier transformation, there are wavelengths naturally associated with a body, the de Broglie wavelength. For a body described by a function with momentum support dominantly within a small neighborhood of po = ~ko, the body is associated with a single wavelength. For a square-integrable function that is nearly a plane wave eiko·x, the de Broglie wavelength λ is determined by ko · x = 2π and Planck’s constant. That is, λ := 2π/kkok = h/kpok from po = ~ko and 2π~ = h, Planck’s constant. If the motion is nonrelativistic, then po ≈ mvo with vo the body’s velocity and h λ ≈ . (89) mkvok Wave-like effects are more evident for slow and light mass bodies since the wavelengths are larger. But then, for sufficiently small kvok, why isn’t wave-like motion observed for heavy bodies? The discussion below addresses the observation of diffraction and interference, particularly as vo → 0 and λ → ∞. Relativistic relative velocities are typically not observed in common experience. The nonrelativistic approximations in quantum mechanics are the common experience. For application of the classical concept that bodies are identifiable and follow trajectories, nonrelativistic relative motions and isolated, localized bodies suffice. Otherwise, particle creation and annihilation become likely or particles become indistinguishable. In nonrelativistic motion, the energy

p p2 E = ck = (mc2)2 + (pc)2 ≈ mc2 + (90) ~ 0 2m since the magnitudes of the relative velocities are much less than the speed of light. As a consequence, 2 1 λckj ωj ≈ + (91) λc 2 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 99 expressed using the reduced Compton wavelength (18). The time evolution of states for the example realizations in section 2.5 reveals the quantum mechanical behavior of body motion and identifies when that motion deviates significantly from a classical approximation. Free motion of a body describes the evolution of an isolated, localized body between localized overlaps with the descriptions of other bodies. The localiza- tions considered here are propagation of a free body through a slit. Propagation through a slit can be considered as preparation of the state in a known spatial configuration with minimal disruption of the momentum. In this example, the evolution of a body is free propagation before the screen, and after it has passed through a slotted screen but before it is detected. Slotted screens are used to examine the visibility of the diffraction and interference predicted in quantum mechanics. For this demonstration of diffraction and interference, states with n-arguments (arguments equal particles in free propagation of isolated bodies) are described by a function fn ∈ P with

n Y fn((x)n) = uj(xj) j=1 and the particles are labeled by functions with Fourier transforms of a form.

(p0 + ω)ψ˜j(p) u˜j(p) = √ 2ω with tempered functions ψj(x) that describe localized wave packets. The selected functions are Gaussian (74), the optimally classical-like states. These Gaussian functions

2 2 −σ (k−koj ) i(k−koj )·xoj ψ˜j(p) = e oj e have a spatial spread of σoj centered on a location xoj with momenta centered on ~koj. The spatial spreads σoj are generally dependent on the particular spatial dimension as well but this generality is not maintained in the notation here. The Gaussian summation

∞ √ Z 2 √ 2 α ds e−αs +βs = π eβ /(4α) (92) −∞ provides the inverse Fourier transform and the associated spatial packets are

3 2 1 −(x−x )2/(4σ2 ) ψ (x) = e oj oj e−i·koj ·x. j 2 2σoj The development of (76) in section 2.11 on the Heisenberg uncertainty principle establishes that these uj describe nearly isolated, freely propagating packets in the case that the separation of the support of functions labeled by the ψj(x) are sufficient to neglect high order connected 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 100 function contributions to the kernel functionals from section 2.5. With attention now isolated on a single function, the notation is simplified by dropping the particle identifying subscript j. Relativistic momenta are unlikely if the most likely momenta po = ~ko and the spread in momenta ~/(2σo) are sufficiently small with respect to mc. That is, for the function labeled by ψj, the mean momentum kkok kmax and the spread 1/σo kmax with ~kmax /m c from ~ko = po and po/m ≈ vo when velocities are much less than speed of light c. Then, to ensure that velocities are nonrelativistic to the desired likelihood, the spatial spread σo must be much larger than the reduced Compton wavelength (18).

σ ~ = λ . o mc c If the support of the function does not include relativistic momenta and if the support is suﬃciently small with respect to separations, then the support of the function is nonrelativistic and isolated and the support moves like a classical body [33]. Quantum mechanically, a time translation is generated by a Hamiltonian that in the example realizations of relativistic quantum physics derives from (57),

−ik λ −i λ −i λc k2λ U(λ) = e 0 ≈ e λc 2 from (91) for sufficiently short times with respect to kkk and for each field component (elementary mass) labeled κ. This unitary realization of time translation applies for each argument of the function fn that describes the n-particle state in the nonrelativistic approximation, and this translation is exact before the nonrelativistic approximation. This Hamiltonian that is the “free field Hamiltonian” for canonical quantization is the exact Hamiltonian for these example nontrivial realizations of relativistic quantum physics [34]. Interaction derives from the form of scalar product. In the nonrelativistic approximation, the time translated function U(t)uj(x) is approximated by the inverse Fourier transform of

−i c t−i λcc k2t −σ2(k−k )2 ψ˜(p, t) = e λc 2 e o o with t an independent parameter and in the instance that xo = 0. With a convenient deﬁnition of a time-dependent length, λ ct t `2 := c = ~ , c 2 2m the inverse Fourier transform is the result of three Gaussian summations.

Z −ix·k ψ(x, t) = dk e ψ˜(p, t) (2π)3/2

2 2 −σ2k2−i c t (−ix+2σoko) o o λc 2 2 e 4(σo+i`c ) = 2 2 3/2 e . (2(σo+i`c )) 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 101

From the demonstration of (76) in section 2.11 above, the support of uj approximately coincides with the support of the corresponding ψj. Simplification of the magnitude of this result provides that 2 (x−vot) − 2 4 2 e 2(σo+`c /σo) |ψ(x, t)|2 = , 4 4 3/2 (4(σo+`c )) 2 a function centered on the anticipated 2`c ko = λcctko = (~ko)t/m ≈ vot using that ~ko = po and po/m ≈ vo, the velocity in the nonrelativistic approximation. The spatial spread of the function grows with time, 4 2 2 2 `c 2 (λcct) σ(t) := σo + 2 = σo + 2 . (93) σo 4σo Since there is a spread in initial velocities, this result is consistent with an interpretation that the function uj describes a statistical distribution over classical states, a statistical mix of bodies with determined locations and momenta. However, a closer examination, for example to explain interference, illustrates that the interpretation as a statistical distribution over classical states fails without supplementation by additional concepts such as “pilot waves.” With a more elaborate notation, this development is applied with a distinct spread for each of the three spatial dimensions. Distinct sizes σok would apply for each dimension labeled by k. The result is (x −v t)2 − k k 3 2(σ2 +`4/σ2 ) Y e ok c ok |ψ(x, t)|2 = (94) (4(σ4 +`4))1/2 k=1 ok c with k labeling spatial dimensions and not spacetime arguments. Combined with typical sizes and masses for subatomic and macroscopic bodies, this description (94) for the free propagation of bodies explains why wave-particle duality is not evident in everyday experience. Two effects, diffraction from propagation through a slit and the interference of the emissions from multiple slits are addressed to assess the visibility of the wave-particle duality. One significant observation is that the wave effects persist at very low rates of body flux, that is, with a very small likelihood that more than one body travels through the slits at a time. The diffraction and interference result from the description of states of single bodies and are not due to the interaction of multiple bodies. The description of each body “interferes with itself.” The propagation of a microscopic and a macroscopic body are contrasted. For this contrast, bodies are characterized by their mass m and a size S. The contrast is between electrons, m = 9.1 × 10−34 kilograms with a classical electron radius of 2.82 × 10−15 meters, and a baseball, m = 0.145 kilograms and a radius of 0.0375 meters. In this characterization, the electron is 33 orders of magnitude lighter and 14 orders of magnitude smaller than the baseball. Consider a screen with two slits of width ds and a separation between the centers of the slits of D. To apply the available expression for evolution of Gaussian functions, the Gaussian pulse 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 102 shape will be taken as an approximation for the transmission characteristic of the slit. The illumination of the two slits is assumed to be nearly uniform in amplitude and the illumination is assumed to be from a single, distant source. The illumination is assumed to arrive at the screen as a nearly plane wave, with the direction of propagation collinear with a normal vector for the screen and any interactions of the bodies during their free transit are asserted to be negligible. Several constraints must be met to view diffraction and interference effects. The body, an electron or a baseball, must pass through the slit. For this it is assumed that

ds > S suffices. S is the size, twice the radius, of the body. To observe the quantum mechanical effects, the deflection to the first interference maximum must be less than 90 degrees and for the calculations derived below to apply, the observations must be in the far field, far behind the screen. To observe diffraction, the assumption here is that the detection is made sufficiently far behind the screen that the width of the spread beam is at least twice the slot width. Given the assumed transmission characteristic of the slit, the description of diffraction is based on propagation of the Gaussian functions. The slit prepares the state with an initial uncertainty in the dimension within the screen perpendicular to the slit of σok = ds. Then from (93) with σok = ds, the width of the function subsequent to the screen in this dimension is s 2 2 λcct λcct ~t σ(t) = ds + ≈ = 2ds 2ds 2mds

2 in the far field of the screen, λcct ds. The remaining two spatial dimensions are assumed to be unaffected by the slit. In this far field, the angular half width of the beam behind the screen is 1 σ(t) sin φ = 2 ≈ ~ . vt 4mdsv The criterion used to consider the diffraction significant is if the spreading is twice the initial σok = ds. Then, s 2 2 λcct ds + > 2ds 2ds that requires a minimum propagation time of √ 12 md2 t > s ~ and that this time t results in a body in the far field,

vt D. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 103

The visibility of diffraction at this minimal time required for the diffraction to double the support size plus satisfaction of the far field condition requires that D v √ ~ . (95) 2 12 mds This is a condition on the maximum λ = h/mv to observe quantum mechanical effects at the minimal time. The wavelength is determined from the de Broglie wavelength relation (89). The deflection angle θ of the first interference maximum with respect to the normal to the screen is determined by the slit separation. 2π D sin θ = λ = ~ mv from the de Boglie wavelength (89) for the mass m body. To be visible, θ < π/2 implied by 2π v > ~ . (96) mD There is a maximum wavelength for the interference to be visible. To observe the first interference pattern maximum, the spreading of the illumination due to diffraction by the slits must be sufficiently broad that the first off-axis interference maximum is illuminated. That is, φ ≥ θ. Smaller φ with respect to θ result in an attenuated interference pattern. In the far field, this condition for the visibility of the double slit interference pattern is that

~ 2π~ sin φ = ≥ sin θ = or, D ≥ 8πds. 4mdsv mvD

This condition is unaffected by ~ or m but is affected by 2S < ds. Note that both the minimum propagation time t and the minimum velocity v for visibility of the quantum mechanical effects depend only on the ratio m/~ and the other conditions are independent of m and ~ (neglecting an implied dependence of the sizes S of bodies on m). (95) also requires a minimum v and is a stronger condition than (96) if D ≥ 8πds. The bodies must also be undisturbed during the illumination of the slits, and before detection. That is, the initial illumination of the slits must be nearly uniform and coherence must not be destroyed. Undisturbed implies that not scatter, absorption nor emission of radiation that shifts the phase of the waveform significantly occurs. State changes during the propagation of the illumination from the source to the detector via the slits corrupt the anticipated pattern of detections. State changes include localizing observations such as the scatter of radiation. Discussed in sections 1 and 2.12, many interactions cause the state of the body to be localized and then the illumination may be of a single slit rather than uniformly across the two, or the illumination from one slit may be phase shifted with respect to the illumination from the other slit. As discussed in section 2.12, if the interaction causes perception of a localized state relative to the observer, then all future observations follow for the localized description of the body 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 104 rather than for the initially prepared, more broadly supported state. These conditions pose a far greater challenge at the scales demanded by baseballs than for electrons. Illumination in this case is the flow of bodies, either electrons or baseballs. In summary, to view the quantum mechanical effects in a minimal time, four conditions must be satisfied by the properties of the bodies and the screen, √ 12 d2m D d > 2S, D ≥ 8πd , t > s , v √ ~ . s s 2 ~ 12 mds

Setting D = 8πds and a reasonable ds, the results below follow. In these results, distances are in meters, time is in seconds, and the velocity is in meters per second. For the electron, ~/m = 0.12 meters squared per second and −8 −15 −6 −14 7 ds = 4.0 × 10 > 5.6 × 10 ,D = 1.0 × 10 , t > 4.7 × 10 , v 2.1 × 10 .

−34 For the baseball, ~/m = 7.3 × 10 meters squared per second and 32 −32 ds = 2S = 0.15,D = 4.0, t > 1.2 × 10 , v 3.3 × 10 .

The electron velocity is nearly relativistic, v2/c2 = 0.005, but this velocity corresponds to only about 1.3 eV of energy. With slower electrons, the diﬀraction from the slits would be visible but not the interference from the double slit. Due to the great elapsed time required, we cannot anticipate viewing the quantum mechanical eﬀects with baseballs even if we could prevent localization and maintain coherence of the baseball states. Further insight is gained by repeating the calculations with hypothetical baseball-sized electrons and electron-sized baseballs. For the baseball-sized electrons,

ds = 2S = 0.15,D = 4.0, t > 0.76, v 5.3 and the quantum mechanical eﬀects are visible. For the electron-sized baseballs,

−8 −15 −6 18 −25 ds = 4.0 × 10 > 5.6 × 10 ,D = 1.0 × 10 , t > 7.5 × 10 , v 1.3 × 10 and viewing the quantum mechanical effects with baseballs is again unfeasible. In these examples, the large mass mattered more than a large size. Since we typically deal with bodies of masses more comparable to baseballs than electrons, the classical description applies in our everyday experience. The observed interference and diffraction patterns are difficult to explain for a classical model that describes a body by a single, smooth trajectory. A quantum mechanical concept is that the description of the state of each body is a function with spatially distributed support. This description of state more naturally explains the observed patterns particularly when flux densities are sufficiently low that the likelihood of more than one body passing through the slotted screens coincidentally is very small. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 105

2.15 Scattering amplitudes, the geometry of spacetime, and units Scattering amplitudes are large time diﬀerence transition amplitudes evaluated in a limit with incoming and outgoing states described as plane waves [4, 23, 55, 63]. For a scalar ﬁeld and states Q described by product functions with additional parameters τj, qj, fn((x)n) = j `(xj; τj, qj),

Sn,m = lim hU(t)`(t, qn+1) . . . `(t, qn+m)|U(−t)`(−t, q1) . . . `(−t, qn)i t→∞ with U(t) the unitary operator that translates the states in time. The LSZ (Lehmann-Symanzik- Zimmermann) expressions [6] for scattering amplitudes use functions with Fourier transforms

iEj τj `˜(pj; τj, qj) := e (ωj + Ej)f˜(pj − qj) ∈ P (97)

˜ 3 with τj a real parameter, qj a momentum vector and f(p) ∈ S(R ) is a Schwartz tempered test function. From section 2.3, `˜(pj; τ, qj) is a limit within the Hilbert space of functions in P. A convenient choice of test function is the Gaussian function (74)

2 3/2 σ 2 2 f˜(p) = o e−σop > 0. π f˜(p) is a point-wise nonnegative delta sequence heavily weighted near zero momentum in the plane wave limit σ → ∞ and o Z dp f˜(p − q) = 1.

The LSZ scattering amplitudes are VEV of products of fields Z Φ(`(t, q)) := dp (ω + E)eiEtf˜(p − q) Φ(˜ p) in this scalar field example. The VEV are evaluated for the constructed kernel functionals W˜ k,n−k((p)n) from (15) and section 2.5. Temporal translations of the field evaluated with (97) are independent of time with the choice of τ = t. Z U(t)Φ(`(t, q))U(t)−1 = dp (ω + E)e−i(ω−E)tf˜(p − q) Φ(˜ p)

= Φ(`(0, q)) due to the limitation of the spectral support of the example VEV to mass shells. The VEV Q of states described by functions j `(xj; τj, qj) are independent of time, and as a consequence, these states never converge to states of a free ﬁeld if the VEV include terms that exhibit interaction. These states are a basis for states that do not asymptotically converge to states of a free ﬁeld. 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 106

In a more familiar notation,

Z ↔ −1 U(t)Φ(`(t, q))U(t) = i dx uˆ(t, x) ∂ o Φ(t, x) with ↔ f(x) ∂ o g(x) := f(x)˚g(x) − f˚(x)g(x), f˚(x) the ﬁrst time derivative of f(x) and

1 Z uˆ(x) := dp eiωte−ip·xf˜(p − q) 2π a smooth solution of the Klein-Gordon equation. For the example of connected functionals for a scalar ﬁeld, the nonzero contribution to the plane wave, non-forward, elastic scattering amplitude is

S2,2 = lim hU(t)`(t, q3)`(t, q4)|U(−t)`(−t, q1)`(−t, q2)i t→∞ (98)

= c4δ(Q1 + Q2 − Q3 − Q4)δ(q1 + q2 − q3 − q4) for q 2 2 2 Qj := m c + qj , and this coincides up to a factor of i with a first order perturbation expansion of a Feynman series in box normalization with an interaction Hamiltonian of :Φ4 : [32, 34, 36] and corresponds with a Yukawa-like equivalent potential in first Born approximation. The incoming momenta are q1, q2 and the outgoing momenta are q3, q4. The higher order scattering amplitudes correspond P ` to the first contributing order for an interaction Hamiltonian density Hint (x) = ` a` : Φ(x) : 2`−4 with ` ≥ 4 and a` = c` (2π) /`!. c` is from (54). Asymptotic equality of scattering cross sections for the example constructions that realize quantum mechanics with first order, weak coupling contributions from Feynman series associates the constructions with familiar models for relativistic quantum physics. In cases with nonzero spin, for example, Compton scattering, the cross sections also deviate from Feynman series results for extremely relativistic exchange momenta (at small distances) [36]. In the example realizations of nontrivial relativistic physics, a single quantum model exhibits both long and short range interactions depending on properties of the interacting states, on whether the spatial extents of interacting states are spatially isolated or overlapping. Short range forces dominate for overlapping supports while the nonrelativistic evolution of localized, isolated support is described by long range (g/r) potentials, section 3 and [33]. The states that correspond most closely with classical descriptions, states with localized, isolated support, propagate along trajectories that correspond generally with the long range forces, gravity and 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 107 electrodynamics. When functions overlap, in cases such as plane waves, short range interactions more similar to the weak and strong nuclear forces dominate. In these example unconstrained constructions, the scattering amplitudes are independent of time in the plane wave limit [34]. In this development, the propagation of the support of states can be followed through intermediate times and this evolution is similarly nontrivial. The lack of time dependence in finite time, plane wave limits is explained by a Hamiltonian that contributes only a phase in the plane wave limit and that plane waves cover all space. To describe momentum, functions in the basis function space P are required to have Fourier ˜ transforms. fn((~k)n) denotes the scaled argument Fourier transform of fn((x)n). The description of states as functions over spacetime is replicated in the Fourier transform functions. The Fourier transform adopted here applies in four dimensional spacetime and is the evident multiple argument extension of (24). The arguments of the Fourier transforms are scaled to momentum from wavenumber, p := ~k with ~ equal to Planck’s constant h divided by 2π. To describe relativity, spacetime coordinates in four dimensions are designated x := (ct, x) from (158) and energy-momentum vectors are p := (E/c, p) and p := (px, py, pz). c is the speed of light and kx is without units. px = ~kx := 2 2 4 Et − p · x, x and p use the Minkowski signature (159). x, p ∈ R are Lorentz four-vectors 3 2 2 2 2 2 2 and x, p ∈ R are three dimensional Euclidean vectors. x := (ct) − x , p := (E/c) − p . p · x is the dot product and x2 := x · x is the squared Euclidean length. The units of spacetime coordinates are length, and wavenumbers k have the units of inverse length. Mass m is in natural units, e.g., kilograms, and a convenient length associated with a mass m is the reduced Compton wavelength λc from (18). The Fourier transforms of generalized functions are defined to satisfy Parseval’s equality T˜(ψ˜) := T (ψ). (99) As a consequence and when applicable, the Fourier transforms of generalized functions are Z dx T˜( k) = eikxT (x) ~ (2π)2 with the sign change of the exponent in the exponential function. The Fourier transform is invertible [49]. Z dk ψ(x) = eikxψ˜( k). (2π)2 ~ 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 108

Among the properties of Fourier transforms are that

−ika ψ(x − a) ↔ e ψ˜(~k)

ψ(Λx) ↔ ψ˜(Λp) (100) dψ(x) ↔ igkψ˜(p) dx 2 1 2 e−αkxk ↔ e−kkk /(4α) 4α2 with kxk2 the Euclidean (sum of squares) length squared of the Lorentz four-vector x, for a Lorentz transformation Λ, spacetime translation a, and complex α with 0. g is the Minkowski signature matrix (159) and gk = (k0, −k). The fourth property suggests that functions ψ(x) with broad spacetime support have Fourier transforms with concentrated energy- momentum support, and vice versa. Indeed, in general, the standard deviations of |ψ(x)|2 in 2 spacetime and of |ψ˜(~k)| in energy-momenta considered as probability distributions satisfy 1 1 σxσk ≥ 2 in each dimension. σxσk ≥ 2 is the origin of the Heisenberg uncertainty principle discussed in section 2.11. More precise knowledge of the location of a body implies degraded knowledge of the time rate of change of the position, and this eﬀect is more pronounced for low mass bodies than for heavy bodies due to ~k = p ≈ mv at nonrelativistic velocities v. Then

σ σ ≥ ~ x ˚x 2m in each of the three dimensions. The squared magnitudes of scalar products are likelihoods and as a consequence, the scalar ∗ products are independent of the units selected by observers. The scalar products Wn+m(gm, fn) are independent of units and with the convention that functions fn do not scale with units, the −4n units of the VEV Wk,n−k are (length) and the W˜ k,n−k are without units. The units of the ˜ ˜ ∗ ˜ ∗ Fourier transforms fn are set by Parseval’s equality Wn,m(gfm, fn) = Wn,m(gm, fn). That is, from the convention for Fourier transforms (24), the units of

Z dx ψ˜( k) := e−ikxψ(x) ~ (2π)2 are (length)4 since ψ(x) is dimensionless and the Lebesgue measure dx is (length)4. The definition of the kernel functionals is completed by setting the scale of the two-point functional. Without vacuum polarization, the two-point functional for the example nontrivial realizations 2 QUANTUM MECHANICAL DESCRIPTIONS OF NATURE 109 coincides with the two-point functional of a free ﬁeld, a Pauli-Jordan function. Z W2(f, g) = dx1dx2 W2(x1, x2)f(x1)g(x2) Z ˜ ˜ = dk1dk2 W2(k1, k2)f(−~k1)˜g(~k2)

˜ ˜ using Parseval’s equality (99) and that f(−~k1) = fe(~k1). A convenient scaling of W2(p1, p2) uses the Compton wavelength (18).

˜ 2 2 W2(p1, p2) = δ(λc(k1 + k2)) θ(E2)δ(λc k2 − 1) (101) p + p p 2 = δ( 1 2 ) θ(E )δ( 2 − 1). mc 2 mc

The units of wavenumber are inverse length and pj = ~kj. Gaussian functions provide an explicit example of an expansion of a state description as a linear combination of more localized elements in the Hilbert space. The identity

2 2 x2 Z s (x−s) − q − 2 2 − e 4σ2 = 1 ds e 4(1− )σ e 42σ2 2(1−2)4σ2π with 0 < 1 is an expansion of the more broadly supported Gaussian function as a linear combination of arbitrarily narrower Gaussian functions centered on s. This expansion is in one dimension. The likelihood of a transition from the normalized state labeled by

x2 − 2 ψ(x) = Nψe 4σ to the more localized, normalized state labeled by

2 − (x−s) gs(x) = Nse 42σ2 is 2 2 − s |hg |ψi|2 = e 4(1+2)σ2 s 1 + 2 with hψ|ψi = hgs|gsi = 1. As anticipated, the likelihood of transition to a function centered on s declines rapidly the farther s is from the center of support at x = 0. And, the likelihood of transition to a function of width σ declines slowly with reductions in relative function width. 4 A geometry of the spacetime of arguments x ∈ R of the functions fn((x)n) that describe states is not of evident significance. Coordinates are selected to simplify the description of the propagation of isolated, localized bodies when classical interpretations apply to particular states. That is, isolated, localized bodies are described approximately by free field theory, 3 CLASSICAL CORRESPONDENCES 110 and this follows from the organization and the properties of the connected functionals. The associated classical dynamics is determined by the correspondence of the time translations of features of the states with the classical dynamic quantities, and the time translations are determined by the kernel functionals Wk,n−k((x)n). Any geometric interpretation of the interaction is captured in the kernel functionals, not in an independent geometry assigned to the arguments of the functions. The spacetime coordinates x are the arguments of the functions that describe states and the scalar products determine the interaction. Function arguments are interpreted as spacetime coordinates and the arguments of the Fourier transforms of the functions are interpreted as energy-momenta. These arguments are summation variables in the descriptions of generalized functions as summations [18]. A physically relevant geometry derives from the observed dynamics of states. The free field Wk,n−k((x)n) in (32) use (30) and describe non-interacting states. The relevant classical spacetime geometry, flat in this example without interaction, is associated with the dynamics of the states. In the case of gravity, classical trajectories are geodesics of a curved Riemannian spacetime for particles with negligible interactions other than gravity. This curved spacetime will locally coincide with a spacetime of the arguments but due to the classical model of interaction as curvature of the spacetime, on larger scales the function arguments will deviate significantly from the inferred classical spacetime coordinates. And, the classical dynamical models apply to the quantum descriptions of states only if classical descriptions apply. Classical descriptions apply when the state descriptions of bodies are strongly peaked in small volumes and the peaks are well isolated. In these instances, the classical dynamics provide approximations for the evolution of the states. Concentrations in the support of the functions are considered geometric bodies in the classical approximation. For example, a classical Riemannian configuration space is then inferred from the quantum dynamics of appropriate states described by classical trajectories. The quantum dynamics associated with the evolution of more general states will not necessarily have an interpretation in terms of classical bodies, nor a geometrical interpretation. The problematic development of quantum gravity is a result of the notion of “quantizing” classical geometrodynamics using methods and correspondences of the canonical formalism.

3 Classical correspondences

In this section, the correspondences of quantum mechanical descriptions of nature with classical mechanical descriptions are derived for the realizations of relativistic quantum physics in section 2. The quantum-classical correspondence follows from an identiﬁcation of classical dynamical variables as representatives for the support of functions. This correspondence, introduced in section 1, substitutes for the canonical formalism’s assertions for elevations and operator identities. Issues include: whether the realizations of relativistic quantum physics are associated with classical dynamical models; when the associations apply; and what particular models are associated. 3 CLASSICAL CORRESPONDENCES 111

The constructions in section 2 realize quantum mechanics. As a consequence, the propagation of general states over any interval is available. In particular, the propagation of states with classical interpretations match with classical developments. For appropriate states, classical dynamics provides insight into quantum dynamics. If the functions that describe states are predominantly supported on localized and isolated spatial volumes, then these volumes propagate like bodies described in classical dynamics. To conform with our daily experience, classical dynamics must provide an accurate representation of the quantum dynamics in these appropriate instances. A classical perception of the quantum state applies in these instances. The quantum mechanical evolution of states with localized and isolated support is described by a classical evolution of representatives for the location and momenta support [33]. This correspondence is suggested by Schrödinger’sanalysis of the linear harmonic oscillator [52] and Ehrenfest’s development of classical correspondences: quantum mechanical expectation values satisfy classical equations of motion [12, 42]. In this note, representatives of the support of states are used to approximate expectation values for appropriately supported states. In these appropriate instances, the support of the functions that describe nature are represented by classical dynamical variables and this correspondence provides an evident, technically sound classical correspondence for relativistic quantum physics. This classical correspondence is studied in this section. Exemplified by relativistic location discussed in [44] and section 1, an exact correspondence of classical dynamical variables with eigenstates of Hermitian operators, a correspondence hypothesized in the canonical formalism, is not necessarily realizable in relativistic quantum physics. But, an association of features is realizable and intuitive. Exact elevation as- serts an extrapolation of classical dynamical models to small distances and overlapping supports where classical correspondences have not been observed to apply. To simplify development, classical correspondences for only one example realization of a neutral scalar field, Nc = 1, are studied in this section. A development of classical correspondences is followed by two studies. The first study describes n bodies in the most classical-like quantum descriptions of state: products of independently described Gaussian functions with localized and isolated support. Gaussian functions exhibit the optimal simultaneous knowledge of position and momentum consistent with Heisenberg uncertainty. In this sense, products of Gaussian functions are the most classical-like state descriptions. In this instance, results rest entirely on the description of state: there is no need to evaluate the scalar product beyond a verification that interaction is manifest. These results apply generally for VEV that include interaction. Free field VEV (32) constrain corresponding classical trajectories to straight lines but the n-point connected VEV introduced in section 2.5 enable a range of corresponding trajectories that exhibit interaction. A nonrelativistic approximation of the Hamiltonian is used to associate the most classical-like states with the trajectories of −g/r pair potentials. Non- relativistic approximation limits applicability of the results. For the Gaussian product state descriptions, the results apply to infinitesimally brief periods and an undetermined number of iterations of infinitesimal time translation. The second study evaluates scalar products for entangled pairs of bodies. By Born’s rule, these scalar products determine state transition 3 CLASSICAL CORRESPONDENCES 112 likelihoods. Results include that the realizations of relativistic quantum physics from section 2 include the classical correspondences introduced in section 1, but the nonrelativistic approximation to the Hamiltonian precludes resolution of a unique, corresponding classical pair potential. For the scalar products that exhibit interaction and with the analysis limited to brief duration intervals, many functions are nearly equivalent to the propagation of the initial function: discrimination of one classical correspondence requires additional methods that avoid approximations to the Hamiltonian and achieve estimates for the propagation of states over intervals sufficient to resolve trajectories. The dynamics of the constructions is described in the scalar product of state descriptions hg|fi. Scalar products that realize nontrivial relativistic quantum physics derive from elementary generalized functions constructed in section 2. Well known scalar products include the free field VEV and variants such as monomials and generalized free fields that exhibit no interaction [6]. The new examples result in nontrivial scattering amplitudes, for example (98) for a selected example neutral scalar field. Example scattering amplitudes include the low order contributions of Feynman series. There are nontrivial constructions associated with every realization for a free field in four dimensional spacetime. An unfamiliar aspect of these constructions is that the Hamiltonian (57) is a “free field Hamiltonian” yet scattering is nontrivial. The scattering amplitudes derive from the nontrivial VEV. And, since quantum mechanics is fully realized, the propagation of states over any interval is defined: results are not limited to scattering amplitudes over infinite intervals. Analysis of the finite time evolution of states is unavailable in the canonical quantization-based developments [5, 63]: the Feynman rules evaluate asymptotic time, plane wave limit transition amplitudes. In this section, the more general propagation of states with classical correspondences is described. Results include that Newtonian gravity, the observed long range classical model applicable to a neutral scalar field, is represented in the constructions. However, while included by the constructions, the analysis methods are not yet capable of resolving whether a unique classical model corresponds to the VEV from section 2.5. The analysis at this stage suggests a correspondence of nonrelativistic, localized and isolated states with the classical dynamics of −g/r pair potentials. The nontrivial scattering amplitudes imply a correspondence with non-straight trajectories.

3.1 A nonrelativistic classical correspondence A correspondence of Newtonian mechanics with the unconstrained development of relativistic quantum physics was introduced in section 1. This correspondence is that the classical dynamical variables are representatives for the dominant support of the functions that describe states. This support includes both the spatial support and the support of the Fourier transform over momenta. A correspondence with Newtonian mechanics applies when localized states are 3 centered on suﬃciently isolated classical trajectories ξk(λ) ∈ R with nonrelativistic relative 3 CLASSICAL CORRESPONDENCES 113 velocities. These states are described by functions

ϕn(λ) := ϕn(x1 − ξ1(λ), . . . , xn − ξn(λ); λ) n Y −iq ·(x −ξ (λ)) := e j j j fn(x1 − ξ1(λ), . . . , xn − ξn(λ); λ) (102) j=1 n Y −iq ·(x −ξ (λ)) := e j j j fn((x − ξ(λ))n; λ) j=1

4n with fn((x)n, λ) ∈ P(R ) that are dominantly supported within a neighborhood of (x)n = 0 and with Fourier transforms supported in a neighborhood of (p)n = 0. From the translations of Fourier transforms (100), the Fourier transforms of ϕn(λ) are dominantly supported in a neighborhood of (λcq)n = (ξ˙)n. Derivatives with respect to the temporal parameter λ are ˙ designated ξk(λ) and λ := cτ for a time τ. The breadth of the support of fn((x)n; λ) varies with λ. Spreading of the support is implied by the Heisenberg uncertainty in location and momenta. For appropriately isolated and localized states, the label on arguments corresponds to labels of classical bodies, k ∈ {1, n}. For a classical correspondence to apply, each argument is associated to great likelihood with a classical body designated k with trajectory ξk(λ). In a nonrelativistic ﬁrst approximation, the trajectories ξk(λ) lack a temporal contribution,

ξk(λ) = (0, ξk,x(λ), ξk,y(λ), ξk,z(λ)), and satisfy Newton’s equations of motion

¨ dV ξk = − . (103) dξk No distinction is made in the notation of (102) between the four component Lorentz vector and the three component spatial vector ξj(λ). The dimensionless potential energy is

n n 1 X X V = Φ (kξ (λ) − ξ (λ)k) (104) mc2 jk j k j=1 k>j for n massive bodies interacting with a pair potential Φjk(r). The dimensionless kinetic energy T of n identical bodies is n 1 X T := ξ˙ (λ)2 2 k k=1 in nonrelativistic instances and the rest mass energy is n in these units. The trajectories ξk(λ) are the temporal evolution of the representatives of the volumes of predominant support of the quantum description of state (102). More generally, trajectories ξk(λ) satisfy Einstein’s geometrodynamics or Maxwell’s equations with the Lorentz force. 3 CLASSICAL CORRESPONDENCES 114

In nonrelativistic instances, momentum support near the representative classical momenta provides that energy is conserved only if the numbers of incoming and outgoing particles agree. In nonrelativistic instances, a scalar product based on (8) simpliﬁes,

X hf|gi = hfn|gmi n,m X ≈ hfn|gni n with fn representing a sequence f described in (7) but with only one fn((x)n) nonzero. In nonrelativistic cases, only hfn|gmi with n = m contribute in an evaluation of state transition amplitudes: particle number is conserved. A classical correspondence applies only as long as the support of the state description remains localized and isolated, and particle number is conserved. Classical correspondences necessarily apply only for functions of suﬃciently nonrelativistic, isolated and localized support that a likely association of the support of functions with classical bodies is achievable. The quantum mechanical evolution of states is a unitary mapping of the functions

ϕn(x1 − ξ1(0), . . . , xn − ξn(0); 0).

A classical correspondence is established if the representatives of the isolated and localized support of the functions behave like classical bodies, if

|U(−λ)ϕn(0)i ≈ |ϕn(λ)i. (105)

That is, the representatives for the support of the function that describes the state evolve along the trajectories of classical bodies. The representatives of both the location and momentum support follow classical trajectories. The sign results from that negative translation of a function argument corresponds to positive translation of the arguments of the ﬁeld. The argument λ of ϕn(λ) is the parameter for the classical trajectories ξk(λ) and a spread of support of the functions. The temporal translation of the functions ϕ(λ) is

U(−λ)ϕn(0) = ϕn(x1 −ξ1(0)−(λ, 0, 0, 0), . . . , xn −ξn(0)−(λ, 0, 0, 0); 0) and ϕ(λ) := ϕn(x1 −ξ1(λ), . . . , xn −ξn(λ); λ) from (102). (105) states that a translation in temporal support is nearly equivalent to a propagation along the classical trajectories and a spreading of the support. The approximation (105) 3 CLASSICAL CORRESPONDENCES 115 is in the norm (6).

2 0 ≈ k|U(−λ)ϕn(0)i − |ϕn(λ)ik

= hU(−λ)ϕn(0)|U(−λ)ϕn(0)i − hU(−λ)ϕn(0)|ϕn(λ)i

−hϕn(λ)|U(−λ)ϕn(0)i + hϕn(λ)|ϕn(λ)i 2 2 = kU(−λ)ϕn(0)k − 2

|hU(−λ)ϕn(0)|ϕn(λ)i| ≈ kϕn(0)k kϕn(λ)k (107) suffices to satisfy the classical correspondence (105). The Cauchy-Schwarz-Bunyakovski inequality provides that |hU(−λ)ϕn(0)|ϕn(λ)i| ≤ kϕn(0)k kϕn(λ)k with equality occurring if |U(−λ)ϕn(0)i = c|ϕn(λ)i for a finite constant c ∈ C. If the classical correspondence (105) applies approximately for brief increments of time with a sufficiently controlled error, then it applies as long as the classical correspondences and any approximations apply: as long as nonrelativistic approximations apply, and the support of the functions remains sufficiently isolated and localized to reliably associate the support with classical bodies. If |U(−λ)f(0)i ≈ |f(λ)i for sufficiently small , then |U(−λ)f(0)i ≈ |f(λ)i as long as classical correspondences apply and, for example, k |U(−λ)f(0)i − |f(λ)ik = O(2λ2). From the unitary realization of time translation, an iteration of (105) provides that

|(λ)i := |U(−δ)ϕn(λ)i − |ϕn(λ + δ)i, then

λ n kU(− n ) f(0) − f(λ)k λ n λ (n−1)λ (n−1)λ = kU(− n ) f(0) − U(− n )f( n ) + ( n )k λ λ n−1 (n−1)λ (n−1)λ = kU(− n ) U(− n ) f(0) − |f( n ) + ( n )k λ λ n−1 λ (n−2)λ (n−2)λ (n−1)λ = kU(− n ) U(− n ) f(0) − U(− n )f( n ) + ( n ) + ( n )k n−1 X λ n−1−k kλ = k U(− n ) ( n )k k=0 n−1 X kλ ≤ k( n )k k=0 from the triangle inequality and the unitary realization of time translation. Then the error in kλ µ the iteration is less than n times the maximum over k of the errors k( n )k. A maxk k(δ)k ≈ cδ for µ > 1 suffices for convergence at small λ to imply convergence of the iteration (108) for as long as the classical correspondences apply. If the error improves sufficiently with an increased number of steps n in the iteration (108), then a demonstration of (105) over brief intervals provides that (105) applies as long as the classical correspondences apply. The intervals λ of particular interest result in significant propagation of the support with ˙ respect to the spread of support. When λkξk(λ)k is much greater than the spread of the support of ϕn(λ), the trajectories of the volumes of support resolve from the spread. For smoothly differentiable trajectories ξk(λ), (105) is necessarily satisfied for sufficiently small λ. But, if an approximation to the Hamiltonian is applied, then the error in the classical correspondence (105) is O(λ). An O(λ) error bound is not sufficient to extrapolate to long duration intervals from brief intervals. e−iHλ = e−i(Hnr +)λ results in an O(λ) error. Hnr is the nonrelativistic approximation to H, the Hamiltonian (57). The greatest duration λ supported by nonrelativistic approximation (116) to the Hamiltonian is examined in section 3.4. To better estimate likelihoods over long intervals, evaluation of the scalar products with the exact Hamiltonian (57) is indicated but requires more capable methods: the nonrelativistic approximation to the Hamiltonian results in tractable Gaussian summations to evaluate the scalar products of interest but limits the duration of intervals. One question addressed in section 3.4 is whether the nonrelativistic approximation to the Hamiltonian (57) is sufficient to resolve the corresponding trajectory. With the nonrelativistic approximation, 3 CLASSICAL CORRESPONDENCES 117 there is always a sufficiently long duration λ that the approximation (116) deteriorates, a λ such that λ & π. A result of section 3.4 is that nonrelativistic approximations for the Hamiltonian do not suffice to resolve the classical correspondences of the selected example VEV from section 2.5. The two Hamiltonian operators, the exact Hamiltonian based upon√ ωk and nonrelativistic approx- 2 imations (116) and (91) to ωk, are qualitatively different: m − ∆ is anti-local [56] and the Laplacian ∆ is local. Demonstrated in section 3.4, analyses of the quantum-classical correspondences based on the Taylor series approximations to the Hamiltonian are insufficient. Indeed, the Heisenberg uncertainty principle does not permit a unique association of classical trajectory with the observed evolution of the support of functions over brief intervals: such an association is an effective near simultaneous measurement of both momentum and location that exceeds the uncertainty bound. Another perspective is suggested by the development of the Riemann- 2 Lebesgue lemma [49]. The likelihood |hU(−λ)ϕn(0)|ϕn(λ)i| will asymptotically converge to zero unless the phases exp(iωkλ) are compensated by exp(−iωkλ) from ϕn(λ). Approximation to ωk will leave an uncompensated phase and an asymptotically vanishing scalar product, a large interval contradiction to the classical correspondence (105). Physically, the small overlap of the spacetime functions implied by the lack of functions in P with bounded spatial support precludes perfect isolation and localization, and eventually the cumulative effect of these small overlaps violate a reliable classical correspondence. Eventually the assumptions underlying a classical correspondence (105) fail and (108) no longer applies. An approximation to the Hamiltonian decreases the duration of the longest interval λ with an accurate estimate for likelihoods. Classical correspondences apply when details of the evolution of a quantum state can be neglected, including: if the support of the state description is sufficiently isolated and localized that it can be reliably segregated as a particular body over an interval of time, that is, one body can be discriminated to great likelihood from other bodies; if entanglement is negligible; and a nonrelativistic approximation includes that functions are not so localized that their Fourier transforms include a significant likelihood of relativistic momenta. Representatives of the support of the functions (111) do not necessarily follow the classical trajectories ξk(λ) when satisfaction of these conditions weakens. The approximations weaken when: the velocities become relativistic; or the trajectories come sufficiently close that accelerations ξ¨(λ) diverge; or the likely association of classical bodies with the support of functions (110) becomes ambiguous. Indeed, the first Born approximation potential for the plane wave scattering in section 2.15 is Yukawa-like and not a −g/r potential. Close approach occurs and classical associations are lost for the necessarily overlapping plane waves used in the evaluation of scattering amplitudes like (98). Particle creation and annihilation provide counterexamples to a representative classical evolution for all states. Superposition does not apply to classical correspondences (105). Approximation of the evolution of the quantum state by a single classical trajectory per body 3 CLASSICAL CORRESPONDENCES 118

(105) deteriorates with increased durations λ. With interaction, the description of trajectories is more diverse. Sets of initially nearby trajectories diverge, for example, initially nearby bound orbits and unbound trajectories. Covering the range of trajectories with a single spatial spread may contradict the asserted distinguishability of corresponding bodies. Accommodation of the uncertainty in positions and velocities as initial conditions for the classical trajectories results in a spread of associated classical trajectories. In free field theory, this uncertainty is reflected in the growth in the spatial variance of the support of the Gaussian functions with time. In a 2 2 2 2 2 nonrelativistic approximation to free field theory, an initial variance σo 7→ σo +λc λ /4σo with λc the reduced Compton wavelength (18) [24]. An imaginary contribution to the spread parameter implements a broadening of spatial support without decreases in momentum support. This use of classical dynamical variables as representatives for the support of the quantum mechanical description of appropriate states appeared in Schrödinger’s1926 analysis of the nonrelativistic linear harmonic oscillator [52]. For a linear harmonic oscillator, a particular Gaussian function ψ(x, λ) of spread σ satisfies Schrödinger’s equation.

(x − A cos wt)2 βx sin wt ψ(x, t) = exp − − i − iφ(t) 4σ2 ~ satisfies Schrödinger’sequation if the spread and frequency are given by the mass m and spring constant k, r k σ2 = √~ , w = 2 mk m √ w kA2 β = mk A, φ(t) = t − sin 2wt 2 4~w with A the oscillation amplitude. The support of this function is centered on the classical trajectory ξ1(λ) = A cos wλ/c of a linear harmonic motion. The classical limit is evident when the spread of the support satisfies σ2 A2. As m → ∞, 2E σ2 = √~ A2 = 2 mk k

1 2 with the total energy 2 kA a constant of the motion. The functions ψ(x, t) that persist along the classical trajectory have a particular spread σ and are not eigenstates of the energy.

3.2 The most classical-like state descriptions and −g/r potentials In this section, a nonrelativistic classical correspondence is established for representatives of the support of the most classical-like states. The most classical-like states have optimally known 3 CLASSICAL CORRESPONDENCES 119 locations and momenta consistent with the Heisenberg uncertainty principle; the descriptions of classical bodies include both a location and a momentum. It is established in this section that over infinitesimal intervals and extensions to greater duration intervals, the corresponding classical bodies are described by Newtonian dynamics with −g/r pair potentials regardless of the form of the VEV beyond that interaction is exhibited. However, the study in this section does not determine the strength of the potential, g. Propagation along non-straight trajectories of −g/r potentials is enabled, but the particular g corresponding to a selection of VEV from section 2.5 is not determined in this infinitesimal interval development. Interaction strength, the cn in the description of the VEV, determine the scattering amplitudes, for example (98), but not the infinitesimal duration propagation of states when classical correspondences apply. Another classical correspondence, the first Born approximation potential that applies to plane wave scattering, is distinct from the classical correspondence exhibited in the motion of localized states. The first Born approximation equivalent potential is Yukawa-like and associated with a mass m [32, 33, 38]. The classical correspondence is not established as operator identities independent of the description of state, but rather, the classical correspondence varies with the properties of state descriptions. The classical dynamics of the nonrelativistic, localized and isolated states is not fully determined by (105) for these infinitesimal duration intervals λ, but for consistency with observation, only a single trajectory can be associated with the evolution of each state as long as the classical correspondence applies. The most classical-like states possess optimally determined, simultaneous locations and momenta. A sufficiently isolated and localized spatial support corresponds with classical bodies. A result of this section is that for these most classical-like states and for sufficiently brief duration intervals λ, i(n+eC )λ/λc |U(−λ)ϕn(0)i ≈ e |ϕn(λ)i (109) with U(λ) the unitary time translation operator (57), eC the most likely representative, nonrelativistic classical energy of the n corresponding bodies, and eC = T + V with V the potential energy for n particles interacting with a −g/r pair potential [33]. T is the kinetic energy. The trajectories for −g/r pair potentials naturally correspond with these most classical-like states. The effects of the potential V are introduced by a phase (106) permitted in physically equivalent descriptions of states in quantum mechanics. The most classical-like states are described by products of Gaussian functions (74). From section 2.11, Gaussian functions achieve the optimal simultaneous knowledge of position and momentum consistent with Heisenberg uncertainty. The product form implements an independent specification of the locations and momenta for each corresponding body. A classical description includes an independently determined location and momentum for each body. The 4n states ϕn((x−ξ(λ))n; λ) ∈ P(R ) selected for this study are described by products of Gaussian functions with Fourier transforms n Y (p0,j + ωj) ϕñ((p)n; λ) = ψ˜(pj; ξj(λ), λ) (110) p2ω j=1 j 3 CLASSICAL CORRESPONDENCES 120 with 2 2 ψ˜(p; ξj(λ), λ) := exp −L0(λ) (p − qj(λ)) + i(p − qj(λ))·ξj(λ) . (111)

The ψ˜(p; ξj(λ), λ) are Gaussian functions with spatial support described by a complex distance L0(λ). Momentum support is centered on qj(λ) and the inverse Fourier transforms of the Gaussian functions,

3 2 1 2 (x − ξj(λ)) ψ(x; ξj(λ), λ) = 2 exp − 2 − iqj(λ)·x , (112) 2L0(λ) 4L0(λ) have support centered on the trajectories ξj(λ). The support of the inverse Fourier transform ofϕ ˜n((p)n; λ) is determined similarly to (155) in appendix 7.2 from the support of the factors 2 1 ψ(x; ξj(λ), λ), the result of the anti-local diﬀerential operators (−∆j + m ) 2 , and with point 3 support in time. ∆j is the Laplacian for xj ∈ R . The support spread parameter may be complex. λ L (λ)2 = L (0)2 − i c λ (113) 0 0 2 in the nonrelativistic free propagation of a Gaussian function [24]. ~qk are the momenta of the trajectories, ˙ λcqk(λ) := γk ξk(λ) (114) ˙ ≈ ξk(λ) in a nonrelativistic instance with λc the reduced Compton wavelength (18) and 1 γk := q . ˙2 1 − ξk

The energy corresponding to the classical body that follows a trajectory ξk(λ) is ~cω(qk) from 2 −2 (26) and the associated energy-momentum Lorentz vector is qk := ω(qk), qk and qk = λc . In 2 2 the nonrelativistic limit in an appropriate coordinate frame, (~qk) (mc) , relative velocities ˙ 2 ˙ are much less than the speed of light and ξk(λ) 1. To ﬁrst contributing order in ξk, (91) is the nonrelativistic approximation for ωk. The −g/r potentials have a property that naturally associates them with the propagation of products of Gaussian functions. For a −g/r pair potential, the potential energy (104) is described by −gmc2 Φkj(kξk − ξjk) = kξk − ξjk and the potential energy is related to the acceleration of the trajectories, n X¨ V = ξk · ξk. (115) k=1 3 CLASSICAL CORRESPONDENCES 121

The representation (115) follows from a chain rule,

1 1 ∂ 1 ∂r ∂r ∂r = − ∇ · x = − x + y + z r r ∂r r ∂x ∂y ∂z with r := kxk, and from the deﬁnition of V (104),

n n X X∂Φkj(kξk − ξjk) V = − · (ξk − ξj) ∂(ξk − ξj) k=1 j>k n  n n  X ∂ X X = −  Φkj(kξk − ξjk) · ξ` ∂ξ` `=1 k=1 j>k n XdV = − ·ξ`. dξ` `=1 Substitution of Newton’s equation of motion (103) results in the desired identity (115). The remaining task is to establish that the classical correspondence (105) is satisﬁed for suﬃciently small λ for localized and isolated product states (110) with Gaussian functions (111). From exp(a + b) = exp(a) exp(b) and the realization of time translation (57), the energy- momentum dependence of time translations of (110) can be isolated in the exponent of the function U(−λ)ϕn(0).

This exponent of U(−λ)ϕn(0) is

n X 2 2 iωkλ − L0(0) (pk − qk(0)) + i(pk − qk(0))·ξk(0) k=1 n X −1 1 −1 ˙ 2 1 2 ≈ i(λc + 2 λc ξk(0) + λcqk(0) · (pk − qk(0)) + 2 λc(pk − qk(0)) )λ k=1

2 2 −L0(0) (pk − qk(0)) + i(pk − qk(0))·ξk(0) n X −1 1 −1 ˙ 2 = i(λc + 2 λc ξk(0) )λ k=1 λ −(L (0)2 + i c λ)(p − q (0))2 + i(p − q (0))·(ξ (0) + ξ˙ (0)λ) 0 2 k k k k k k n X −1 1 −1 ˙ 2 2 2 = i(λc + 2 λc ξk(0) )λ − L0(λ) (pk − qk(0)) + i(pk − qk(0))·ξk(λ) . k=1 3 CLASSICAL CORRESPONDENCES 122

A nonrelativistic approximation of the Taylor series for ωk (26) expanded near qk(0),

2 qk · (pk − qk) (pk − qk) ωk ≈ ω(qk) + + (116) ω(qk) 2ω(qk) applies for λckpk − qk(0)k 1 and is substituted for the neutral scalar ﬁeld Hamiltonian (57) applied in the n-argument subspace, X H = ωk. k

The nonrelativistic, λckqk(0)k 1, approximation (91) for ω(qk), the relationship of the ˙ classical momenta and trajectories (114) λcqk(λ) = ξk(λ), the Taylor series ˙ ξk(λ) ≈ ξk(0) + λξk(0),

2 and the definition (113) for the spreading support of functions L0(λ) produce the final expression for the exponent. The approximation (116) is developed below. The result at this point is a special case of the classical correspondence (105). If the VEV are free field VEV (32), then qk(λ) = qk(0) and (105) is demonstrated since the last line is the exponent of ϕn(λ) with the addition of a phase. For the free field VEV, the incoming and outgoing momenta must be equal in pairs. The phase φ(λ) is independent of the momenta pk and φ(0) = 0. The question addressed now is whether the infinitesimal λ expansion of the Gaussian functions generalizes to include accelerating qk(λ). But, even without interaction, the spreading of the support, LS(λ), is determined in (113) to implement the classical correspondence (105). For infinitesimal λ, a Taylor series for qk(λ) introduces acceleration. The approximation ˙ ˙ ¨ ξk(λ) ≈ ξk(0) + λξk(0) ¨ introduces the acceleration ξk(0) to the trajectories. To first order in λ,

2 2 (pk − qk(0)) − (pk − qk(λ)) ≈ 2λ (pk − qk(0)) · q˙ k(0). (117)

Then, with a correction to the term linear in (pk − qk(0)), the exponent of the function 3 CLASSICAL CORRESPONDENCES 123

U(−λ)ϕn(0) becomes n X −1 1 −1 ˙ 2 2 2 i(λc + 2 λc ξk(0) )λ + i(pk − qk(0))·ξk(λ) − L0(λ) (pk − qk(0)) k=1 n X −1 1 −1 ˙ 2 ≈ i(λc + 2 λc ξk(0) + q˙ k(0) · ξk(0)) λ + i(pk − qk(λ))·ξk(λ) k=1

2 2 2 −L0(λ) (pk − qk(λ)) + 2λL0(0) (pk − qk(0)) · q˙ k(0) n X −1 1 −1 ˙ 2 −1 ¨ ≈ i(λc + 2 λc ξk(0) + λc ξk(0) · ξk(0)) λ k=1

2 2 +i(pk − qk(λ))·ξk(λ) − L0(λ) (pk − qk(λ)) .

−1 ¨ The term iλc ξk(0) · ξk(0) was added to the phase and subtracted from the exponent to correct (pk − qk(0))ξk(λ) to (pk − qk(λ))ξk(λ) using the Taylor expansion. The 2λ (pk − qk(0)) · q˙ k(0) 2 term that corrects the breadth of the Gaussian function is negligible with respect to (pk−qk(λ)) for sufficiently isolated trajectories and small λ. These approximations are addressed below. Finally, this last line is the exponent ofϕ ñ(λ) with the addition of a phase. For sufficiently small λ and with substitution of the nonrelativistic approximation (116) for ωk, n X 2 2 iωkλ − L0(0) (pk − qk(0)) + i(pk − qk(0))·ξk(0) k=1 1 ˙2 ¨ λ 2 2 − i(1 + 2 ξk + ξk · ξk) − L0(λ) (pk − qk(λ)) + i(pk − qk(λ))·ξk(λ) = (λ). λc This demonstrates the classical correspondence (105), that

iφ(λ) |U(−λ)ϕn(0)i ≈ e |ϕn(λ)i for product states (110) with Gaussian functions (111). The phase φ(λ) is independent of the momenta pk and φ(0) = 0. However, as discussed in section 3.1, the error due to substitution ˙ of the nonrelativistic approximation (116) for ωk in the Hamiltonian (57) is O(kξkkλ) preclud- ing extrapolation of this demonstration of a classical correspondence from inﬁnitesimal λ to arbitrarily greater duration intervals. The phase introduced in this development of (105) is n X 1 ˙ 2 ¨ λ φ(λ) := 1 + ξk(0) + ξk(0) · ξk(0) . (118) 2 λc k=1

At the peak of likelihood, each pk = qk(λ), the function (110) with Gaussian factors (111) is real and φ(λ) is the phase at the peak likelihood of the support of the state. The identity 3 CLASSICAL CORRESPONDENCES 124

(115) provides that this phase is the classical energy of n particles interacting with a −g/r pair potential. λ φ(λ) ≈ (n + eC ) λc with eC the classical nonrelativistic energy of n bodies following the trajectories ξj(λ) given by Newton’s equation (103) with a −g/r pair potential. The energy includes the contribution of the rest mass of n identical mass m bodies. There is a single elementary mass m in this P ¨ example of a neutral, scalar field. An interpretation of k ξk ·ξk as the potential V of a classical interaction associates the most classical-like states with the trajectories of −g/r pair potentials. P 2 This associates an effective Hamiltonian k pk/2m+V (r) with evolution of the peak likelihood: neglecting the derivative of the function ϕn(λ), the time derivative at the peak of likelihood is the classical energy. This suggest that the phase corresponds to the nontrivial dynamics of the peak of likelihood and introduces the Hamiltonian from the nonrelativistic canonical formalism. The more comprehensive scalar product introduced in section 2.5 enables motion of the support of U(−λ)ϕn(0) along a classical trajectory. The nontrivial motion of the peaks in likelihood follow the trajectories ξk(λ) and this motion is generated approximately and in appropriate instances by the exact relativistic Hamiltonian (57). The motion is also associated with the P 2 nonrelativistic Hamiltonian k pk/2m + V (r) that includes an explicit interaction. This is suggestive of Ehrenfest’s nonrelativistic classical correspondences [12, 42]. These associations apply for the most classical-like state descriptions, those with a reliable association with classical bodies. The expectation of the Hamiltonian (57) in the n argument subspace remains E[H] = Pn E[id/dλ] = E[ k=1 ωk]. A significant outstanding question remaining in this development is whether a unique set of trajectories ξk(λ) correspond with the VEV by likelihood. Nevertheless, the association of the quantum dynamics described by the VEV from section 2.5 with the long range −g/r potentials such as electrostatics and Newtonian gravity is the anticipated correspondence. The approximation for the energies (116) derives from Taylor’s theorem. To second order in pk − qk, 2 2 qk · (pk − qk) (pk − qk) (qk · (pk − qk)) ωk ≈ ω(qk) + + − 3 ω(qk) 2ω(qk) 2ω(qk) and the Cauchy-Schwarz-Bunyakovski inequality provides that

2 2 2 (qk · (pk − qk)) ≤ qk (pk − qk) .

For nonrelativistic velocities 2 2 qk ω(qk) and then 2 2 (qk · (pk − qk)) (pk − qk) 3 . 2ω(qk) 2ω(qk) 3 CLASSICAL CORRESPONDENCES 125

This justiﬁes neglect of the last term from the Taylor series in (116) if the support of functions excludes relativistic momenta. The dominant support of the function (111) is in a neighborhood of qj of scale proportional to 1/|L0| and the nonrelativistic approximation is accurate when the dominant momentum support of the functions does not include relativistic momenta, if |L0| is bounded from below [33]. K 1 kpk − qkk ≤ |L0| λc for an integer K of order 3-5. If λc |L0| (119) then the error in (116) is negligible with respect to ωk. Similarly, the term from (117) that corrects for acceleration is negligible. The correction derives from 2 2 (p − qk(λ)) ≈ (p − qk(0) − λq˙ k(0))

2 2 2 = (p − qk(0)) − 2(p − qk(0))λq˙ k(0) + λ q˙ k(0) .

The Cauchy-Schwarz-Bunyakovski inequality provides that

2 2 2 2 ((pk − qk(0)) · q˙ k(0)λ) ≤ (pk − qk(0)) q˙ k(0) λ

2 and the correction may be neglected relative to (pk − qk) for suﬃciently small λ,

2 2 2 q˙ k(0) λ (pk − qk(0)) over all but a negligible volume of the support of pk. The remaining correction is second order in the small quantity λ. This suggests that for states described by Gaussian functions with sufficiently isolated, localized and nonrelativistic support, the classical dynamical variables of bodies described by Newton’s equation of motion with a −g/r pair potential are representative of the evolution of the support of the quantum description of state over infinitesimal intervals λ. In these instances, (105) provides a correspondence of classical and quantum descriptions of states. The strength of the potential is undetermined by this development, although this might be anticipated since the results depend only on the state description without any involvement of the scalar product that determines state transition likelihoods. The only role of the VEV in this development is to enable nontrivial motion; the free field VEV (32) require that incoming and outgoing momenta are equal in pairs and the VEV from section 2.5 enable interaction. Nontrivial motion is included, but not revealed in this development of the evolution of localized states. Interaction is manifest in the scattering amplitudes (98). The possibility that only straight trajectories appear is eliminated by the nontrivial scattering amplitudes. 3 CLASSICAL CORRESPONDENCES 126

3.3 Two body classical trajectories

Two body problems provide explicit examples of classical trajectories ξk(λ). The motion of two classical bodies is executed within a plane if the forces depend only on the distance between bodies. The location of two bodies in a plane is speciﬁed by a location for the center of mass and a separation x of the bodies,  r cos θ  x := ξ1 − ξ2 :=  r sin θ  (120) 0 with r := kξ1 − ξ2k, the distance between the two bodies, and θ the angle of rotation about the center of mass in the plane of motion. The notation is abbreviated, r = r(λ) and θ = θ(λ). Selection of an inertial coordinate frame collocated with the center of mass results in ξ2 = −ξ1 and 1 ξ = x. (121) 1 2 For the pair of bodies separated by r and rotating at an angular rate of θ˙ about the center of mass, the angular momentum and energy are L := r2θ˙ (122) eC = T + V (r).

In this development, angular momentum L is a distance and the classical energy eC is dimensionless. From (120), in the center of mass coordinates and with the notation ri := r(0), rf := r(θ), θif = −θfi = θ, θii = θff = 0, quantities of interest include

r r ξ · ξ = a b cos θ a b 4 ab

˙ r˙arb rbL (123) ξa · ξb = cos θab + sin θab 4 4ra 2 ˙ ˙ r˙ar˙b L r˙aL r˙bL ξa · ξb = + cos θab − − sin θab. 4 4rarb 4rb 4ra 1 ˙2 ˙2 a and b := i or f. From T = 2 (ξ1 + ξ2) and ξ1 = −ξ2, (123) provides that r˙2 L2 e = + + V (r). C 4 4r2

T ≥ 0 and for nonrelativistic motion, 1 T . A derivative of eC provides a useful relation between angular momentum, the separation r and the potential energy for circular orbits (r ˙ = 0) dV (r) L2 = . dr 2r3 3 CLASSICAL CORRESPONDENCES 127

Also for circular orbits, L θ = λ r2 from summation of the angular momentum when r is constant. The angular momentum L and energy eC are constants of the motion along trajectories ξj(λ) that satisfy Newton’s equation (103). A −g/r pair potential is g V = − , r and Newton’s equation of motion in Jacobi coordinates results in 2g x¨ = − x r3 with x from (120) and the factor of two is from the reduced mass of two equal mass m particles. The interaction is characterized by a length g. For gravity, g = Gm/c2 and for electrostatics g = Ke2/mc2. For a mass of one a.m.u. and an elementary charge of one, g = 1.23 × 10−53 m for gravity and g = 1.54 × 10−18 m for electrostatics. The trajectories ξ1, ξ2 for a −g/r potential are conveniently parameterized by θ from (120). Kepler’s solution of Newton’s equation of motion (103) provides L2/2g r = 1 − r cos θ and 1 Z θ λ(θ) = dφ r(φ)2 L θ0 with λ(θ0) = 0 and p 2 2 r := 1 + eC L /g . Then, L2 cos θ L2 sin θ ξ1(λ) = , , 0 (124) 4g (1 − r cos θ) 4g (1 − r cos θ) with the inversion θ(λ). For eC > 0, the solution is unbound and diverges when r cos θ = 1. −1 −1 For these unbound trajectories, θ is constrained to the interval (β, 2π−β) with β = cos (r ). Bound states have eC ≤ 0 and g2 e ≥ − . C L2 2 2 eC = −g /L are the circular orbits. For the circular orbits of −g/r potentials,

3 g r 2 e = − ,L = p2gr, λ(θ) = √ θ C 2r 2g with λ(0) = 0. 3 CLASSICAL CORRESPONDENCES 128

3.4 VEV and nonrelativistic approximation of likelihoods In this section, the study of the classical correspondence (105) is extended to the state transition likelihoods for entangled pairs of bodies. Scalar products (8) are evaluated for two argument functions that correspond with classical two body trajectories. If a classical correspondence applies, that is, if, to a great likelihood, particle number is conserved and the support of each function argument is well represented by the classical dynamical variables of classical bodies, then a state of two bodies is described by a two argument function. This provides a second example, in addition to the independently specified Gaussian functions in section 3.2, of classical correspondences (105). This example describes bodies with correlated positions. From section 3.3, two body trajectories ξk(λ) are described by m, g, eC ,L and include bound and unbound trajectories. In this section it is demonstrated that the classical correspondence (105) includes two body states but the duration of the interval that a nonrelativistic approximation to the Hamiltonian supplies accurate likelihood estimates is insufficient to resolve a unique classical correspondence. Another result of note is that for the entangled descriptions of state, the evolution of the center of mass is independent of the motion of the coupled bodies. The center of mass is described like a single body. Cluster decomposition implies stability of bound states as long as the composite remains isolated from the support of additional state constituents, and the classical correspondence applies. The scalar products hU(−λ)ϕ2(0)|ϕ2(λ)i establish that (107) is satisfied for the indicated state descriptions. Then, the classical correspondence (105) follows from (107). In this two body example, condition (107) is

|hϕ2(0)|U(λ)ϕ2(λ)i| ≈ kϕ2(0)k kϕ2(λ)k.

First, a suitable description ϕ2(λ) of entangled two body states is developed. A function ϕ2((x)2; λ) with Fourier transform

2 (p + ω ) Y 0,j j ip1·ξ1(λ)+ip2·ξ2(λ) ϕ˜2(p1, p2; λ) := e f˜2(p1, p2; λ), (125) p2ω j=1 j is in the Hilbert space HP of section 2 if

˜ 6 f2((p)2; λ) ∈ S(R ).

This ϕ2((x)2; λ) corresponds with two classical bodies if the support over (p)2 is nonrelativistic and near a classical momentum qj(λ), and the support over (x)2 is localized and isolated. The generalized functions U(λ)ϕ2((x)2; λ) have point support over time centered on the temporal parameter λ [34]. ξ˙j = λcqj in nonrelativistic instances, and from the properties of Fourier transforms (100), the spatial support of ϕ2((x)2, λ) is centered on trajectories (x)2 = (ξ(λ))2 3 CLASSICAL CORRESPONDENCES 129

if the support of f2((x)2; λ) is centered on (x)2 = 0. If the separation of the trajectories is suﬃcient with respect to the breadth of the support, then these functions have localized and isolated concentrations of support that is well represented by classical dynamical variables. For the functions (125), Parseval’s equality, the dual function (9), and the temporal translation operator (57) provide the scalar products of interest.

hU(−λ)ϕ2(0)|ϕ2(λ)i = hϕ2(0)|U(λ)ϕ2(λ)i Z −i(p +p )λ = d(p)4 W˜ 2,2((p)4) ϕ˜2(−p2, −p1; 0) e 0.3 0,4 ϕ˜2(p3, p4; λ) (126) Z √ = d(p)4 W˜ 2,2(−p2, −p1, p3, p4) 4 ω1ω2ω3ω4×

−ip ·ξ (0)−ip ·ξ (0) −i(ω +ω )λ+ip ·ξ (λ)+ip ·ξ (λ) e 1 1 2 2 f˜2(p1, p2; 0) e 3 4 3 1 4 2 f˜2(p3, p4; λ)

2 2 after evaluation of the mass shell Dirac delta functions δ(pj − m ) in the VEV described in sections 2.4 and 2.5. W˜ k,n−k((p)n) designates the kernel functional that remains from the VEV W˜ k,n−k((p)n) after evaluation of the mass shell delta functions. In the two body example, Jacobi coordinates achieve a separation of variables in New- ton’s equations of motion. Convenient functions for Jacobi coordinates and with the indicated support are 2 2 2 2 −Lc(λ) (p1+p2) −LS (λ) (p1−p2−2q1(λ)) f˜2(p1, p2; λ) := e e . (127) Poincar´einvariance is exploited to express scalar products in the center of mass coordinate frame with ξ2 = −ξ1.

The spatial support of these functions is characterized by distances Lc(λ),LS(λ) ∈ C with Lc(λ) characterizing the center of mass and LS(λ) completing the characterization of the state of the two bodies. In this representation, the support of the function describing the center of mass is centered on zero with momentum support centered on zero, and the supports of the descriptions of the two bodies are centered on ξ1 and −ξ1 with the the momenta centered on q1 and −q1. The locations of the bodies are correlated in this example. Independent, product-form descriptions of two body states were studied in section 3.2 and [33]. In the notation of section 2.5, (45) expresses the VEV as

W2,2(1234) = U2,2(1234) + W2(13)W2(24) + W2(14)W2(23), (128) 3 CLASSICAL CORRESPONDENCES 130

with connected functionals W2 and U2,2 from (38). The Fourier transforms of the connected functionals for the selected example neutral scalar ﬁeld are

2 ˜ √ Y 2 2 W2(12) = 2 ω1ω2 δ(p1 + p2) δ(pj − m )θ(−p10)θ(p20) j=1 4 ˜ Y 2 2 U2,2(1234) = c4 δ(p1 + p2 + p3 + p4) δ(pj − m ) j=1 from (30), (52) and (53). In this example of a neutral scalar ﬁeld, Q2,2 = 1. After evaluation 2 2 of the mass shell delta functions δ(pj − m ), the VEV are generalized functions solely of the momenta, √ 2 ωjωk W˜ 2(−pj, pk) = δ(pj − pk) 4 Y (129) 2ωj U˜2,2(−p1, −p2, p3, p4) = c4 δ(ω1 +ω2 −ω3 −ω4)δ(p1 + p2 −p3 −p4). j=1 The development now digresses to evaluate the scalar product (126) for the functions (125) with (127) and the VEV (129). A convenient change in summation variables simpliﬁes calculation of the scalar product.

0 0 p1 := p1 + p2 and p2 := p1 − p2 and then 02 02 2 2 p1 + p2 = 2p1 + 2p2. 0 0 Similarly for p3, p4. The inverse is p0 + p0 p0 − p0 p := 1 2 and p := 1 2 . 1 2 2 2

In these convenient coordinates, the VEV (129) result from the indicated substitutions for (p)4 and

1 0 1 0 0 0 0 1 0 0 0 0 d(p)4 δ(p1 −p3) δ(p2 −p4) = 4 d(p )4 δ( 2 (p1 +p2 −p3 −p4)) δ( 2 (p1 −p2 −p3 +p4))

1 0 0 0 0 0 = 2 d(p )4 δ(p1 − p3) δ(p2 − p4)

1 0 1 0 0 0 0 1 0 0 0 0 (130) d(p)4 δ(p1 −p4) δ(p2 −p3) = 4 d(p )4 δ( 2 (p1 +p2 −p3 +p4)) δ( 2 (p1 −p2 −p3 −p4))

1 0 0 0 0 0 = 2 d(p )4 δ(p1 − p3) δ(p2 + p4)

1 0 0 0 d(p)4 δ(p1 +p2 −p3 −p4) = 4 d(p )4 δ(p1 − p3). 3 CLASSICAL CORRESPONDENCES 131

The nonrelativistic approximation of energies ω3 + ω4 for momenta near the corresponding 0 0 classical values of p3 near zero and p4 near the nonrelativistic 2q1 follows the Taylor expansion of ωk (116). This nonrelativistic approximation results in

0 02 2 02 q1 · (p4 − 2q1) p3 q1p3 ω3 + ω4 ≈ 2ω(q1) + + − 3 ω(q1) 4ω(q1) 4ω(q1) (131) 2 2 0 1 02 ≈ + λcq1 + λcq1 · (p4 − 2q1) + 4 λcp3 λc

02 0 2 2 to leading contributing order in the small quantities. p3 ,(p4 − 2q1) and q1 are small with re- −2 ˙ spect to λc . The nonrelativistic approximation (91) for ω(q1) and the relation (114), λcq1 = ξ1, produce the ﬁnal approximation. Expansion about 2q1(λ) produces a less accurate representative for the Hamiltonian (57) for larger accelerations and over longer duration intervals: the nonrelativistic approximation for the energy signiﬁcantly deteriorates for large λ in the scalar product hϕ2(0)|U(λ)ϕ2(λ)i when interaction is exhibited. The designations

ξi := ξ1(0), ξf := ξ1(λ)

qi := q1(0), qf := q1(λ)

Li := LS(0),Lf := LS(λ) conveniently condense notation. In this notation, for the functions (127) and if the nonrelativistic approximation (131) for time translation (57) applies, the scalar products (126) become

2 2 Z −i( +λcq )λ 0 √ hϕ2(0)|U(λ)ϕ2(λ)i = e λc 1 d(p )4 16 ω1ω2ω3ω4 W˜ 2,2(−p2, −p1, p3, p4, )×

2 −L (0)2p02 −ip0 ·ξ −L (p0 −2q )2 e c 1 e 2 i e i 2 i × (132)

λc 02 0 2 02 0 2 0 2 −i( p +λ q ·(p −2q ))λ −L (λ) p ip ·ξ −L (p −2qf ) e 4 3 c f 4 f e c 3 e 4 f e f 4 .

There are three terms in the evaluation of (132), one each for U2,2(1234), W2(13)W2(24) and 2 W2(14)W2(23). Evaluated at λ = 0, this scalar product (132) is the squared norm kϕ2(0)k . 2 The squared norm kϕ2(λ)k equals (132) evaluated at λ = 0 but with Lc, LS and the classical dynamical variables ξ1, q1 evaluated at λ rather than zero. The ωj are the functions (26) of momenta pj. From the form (129) of the VEV after evaluation of the mass shell Dirac delta functions, no factors of ωj appear in the contribution of the free ﬁeld VEV and a factor of Q4 −1/2 j=1 ωj appears in the contribution of the four point connected function. The approximation 1 − 2 p ωj ≈ λc 3 CLASSICAL CORRESPONDENCES 132

˙2 2 2 suffices as a nonrelativistic approximation: ξ1 = λc q1 1 and the variation of ωj is dominated by the variation of the exponent within the dominant support of f˜2. After the indicated substitutions, only Gaussian summations (92) remain in the evaluation of the scalar product (132). Substitution of the connected functionals (129) into the VEV (128) results in elementary Gaussian summations for the scalar products of interest. These summations are evaluated in appendix 7.8. Of the three terms in the scalar product, only two of these terms contribute significantly in cases with classical correspondences. 2 Common factors among terms within the scalar products hϕ2(0)|U(λ)ϕ2(λ)i, kϕ2(0)k and 2 kϕ2(λ)k , and common phases among the terms within hϕ2(0)|U(λ)ϕ2(λ)i are extraneous to satisfaction of the condition (107) that provides the classical correspondence (105). Demon- strated in appendix 7.8, if the support of the center of mass spreads like a free particle (113) of mass 2m, λ L (λ)2 = L (0)2 − i c λ, (133) c c 4 then the description of the center of mass is a common factor. As a consequence, the description of the function characterizing the center of mass of the two bodies does not contribute to whether a classical correspondence applies: if a classical correspondence applies, then any description for the center of mass suffices. The characterization of the center of mass is decoupled from description of the two interacting bodies. With this entanglement of the states of the constituent particles, the center of mass behaves like a single body with internal structure. Similarly, and also evaluated in appendix 7.8, 2 2 −i( +λcq )λ e λc f is a common phase among the terms within hϕ2(0)|U(λ)ϕ2(λ)i. With these approximations and simplifications, the scalar products (132) are evaluated in appendix 7.8. With the common factor and phase neglected, the contribution of U2,2(1234) to the scalar product hϕ2(0)|U(λ)ϕ2(λ)i is

3 1 2 ! 2 2 2 2 ! 2 c4 π 4πλc Li Lf exp(−i2qi · ξi + i2qf · ξf )× 2π 2 2 2 2 2 2 Li Lf (qf Li + qi Lf ) −2i (q2 − q2 ) ! i f 2 2 ˙ exp L qi · ξi + Li qf · (ξf − ξf λ) × (134) 2 2 2 2 f qf Li + qi Lf ! ! −4L2L2 2 −(q · (ξ + ξ˙ λ) − q · ξ )2 exp i f q2 − q2 exp f i i i f . 2 2 2 2 i f 2 2 2 2 qf Li + qi Lf 4(qf Li + qi Lf ) 2 The contribution of U2,2(1234) to the norms kϕ2(λ)k are

1 3 2 4 ! 2 c4 π 4πλc |Lf | 2 . (135) 2π |Lf | 2 2 2 qf (Lf + Lf ) 3 CLASSICAL CORRESPONDENCES 133

2 and similarly with Li and qi for kϕ2(0)k . With the common factor and phase neglected, the contribution of W2(13)W2(24) to the scalar product hϕ2(0)|U(λ)ϕ2(λ)i is

3 ! 2 2 ! 2 2 2 ! 1 π (ξ − ξ + ξ˙ λ) 4L L (qi − qf ) exp − i f f exp − i f × 2 L2 + L2 4(L2 + L2 ) L2 + L2 i f i f i f (136) L22q + L2 2q ! 2 ˙ i i f f exp(i2q λcλ) exp i(ξi − ξf + ξf λ) · f 2 2 Li + Lf

2 and the contribution of W2(13)W2(24) to the norms kϕ2(λ)k are

! 3 1 π 2 (137) 2 2 2 Lf + Lf

2 and similarly with Li for kϕ2(0)k . If the trajectories of the two corresponding classical bodies are suﬃciently distinct, then the contributions of W2(14)W2(23) to the scalar products are negligible. Isolated and localized support implies that the classical bodies reliably associate with designated function arguments. Consideration now turns to satisfaction of the condition (107) that implies the classical correspondence (105) for these example two body cases. Substitution of the scalar products (132) from appendix 7.8 evaluate (107). The transcendental relations that result from substitution of (134), (135), (136) and (137) into the condition (107) are simpliﬁed by considering satisfaction of (107) in parts: the exponential functions in the contributions of U2,2(1234) and the W2(13)W2(24) to the scalar product hϕ2(0)|U(λ)ϕ2(λ)i approximate one; and the remaining magnitudes are equal. The condition for the exponential functions in the contribution of U2,2(1234) is

−2i (q2 − q2 ) ! i f 2 2 ˙ exp(−i2qi · ξi + i2qf · ξf ) exp Lf qi · ξi + Li qf · (ξf − ξf λ) × q2 L2 + q2L2 f i i f (138) ! ! −4L2L2 2 −(q · (ξ + ξ˙ λ) − q · ξ )2 exp i f q2 − q2 exp f i i i f ≈ 1 2 2 2 2 i f 2 2 2 2 qf Li + qi Lf 4(qf Li + qi Lf ) from (134). The condition for the exponential functions in the the contribution of W2(13)W2(24) is 2 ! 2 2 2 ! (ξ − ξ + ξ˙ λ) 4L L (qi − qf ) exp − i f f exp − i f ≈ 1 (139) 2 2 2 2 4(Li + Lf ) Li + Lf 3 CLASSICAL CORRESPONDENCES 134 from (136). If these exponential functions approximate unity, then the remaining magnitudes from substitution of (134), (135), (136) and (137) into (107) are

3 1 ! 2 2 ! 2 1 π 2 πλc + π c4 2 2 2 4 4 2 2 2 2 Li + Lf Li Lf (qf Li + qi Lf ) 1  3 1  2 ! 2 2 ! 2 1 π 2 πλc ≈  + π c4  × (140) 2 2 2 8 2 2 2 Li + Li |Li| qi (Li + Li ) 1  3 1  2 ! 2 2 ! 2 1 π 2 πλc  + π c4  . 2 2 2 8 2 2 2 Lf + Lf |Lf | qf (Lf + Lf ) The classical correspondence (105) applies if (138), (139) and (140) are satisﬁed. Instances with no interaction, c4 = 0, are illustrative of the classical correspondence (105). In these instances, the classical correspondence (105) is demonstrably satisﬁed for all λ. The term contributing the condition (138) does not apply if c4 = 0. The condition (139),

2 2 2 2 (ξ − ξ + ξ˙ λ) 4L L (qi − qf ) i f f + i f = 0, 2 2 2 2 4(Li + Lf ) Li + Lf is satisﬁed by straight trajectories, ˙ ξi + ξiλ = ξf and qi = qf if the spread of the constituent bodies remains constant,

2 2 Li = Lf . In the absence of interaction, a consequence of the free field VEV and the two body classical correspondence (105) is the observed straight line motion of classical bodies. The anticipated spread of the support of the bodies derives from the spreading of the center of mass (133). (139) is satisfied either: for sufficiently small λ and any continuous trajectory: or for straight, trivial trajectories and any λ. The straight trajectories are resolved from the spread of support at large λ. A constant ξ˙1(λ) is the only possibility for trajectories that correspond with the free field VEV (32). The contribution of W2(13)W2(24) attenuates exponentially with increasing λ for trajectories that deviate from straight lines. Next, it is demonstrated that both (138) and (139) are satisfied for sufficiently small λ by smooth trajectories. (138) is expressed

2 2 2 2 a0Lf Li + ia1Lf + ia2Li + a3 − i2qi · ξi + i2qf · ξf ≈ 0 (141) 2 2 2 2 qf Li + qi Lf 3 CLASSICAL CORRESPONDENCES 135

2 with real ak and the complex LS(λ) . In (141),

2 2 2 a0 = 4(qi − qf ) 2 2 a1 = 2(q − q ) qi · ξi i f (142) 2 2 ˙ a2 = 2 (qi − qf ) qf · (ξf − ξf λ) 1 ˙ 2 a3 = 4 (qf · (ξi + ξiλ) − qi · ξf ) . a0 ≥ 0 and a3 ≥ 0. For diﬀerentiable trajectories, the Taylor series to leading order in small λ results in 2 2 a0 ≈ 4(2qi · q˙ i) λ

a1 ≈ −4(qi · q˙ i) qi · ξiλ (143) a2 ≈ −4(qi · q˙ i) qi · ξiλ 1 2 2 a3 ≈ 4 (q˙ i · ξi) λ . 2 As λ → 0, both a0 and a3 approach zero as λ while a1 and a2 approach zero as λ. a1 − a2 is order λ2. Finally, ˙ −i2qi · ξi + i2qf · ξf ≈ 2i(q˙ i · ξ1 + qi · ξi) λ. 2 2 2 2 Then, for a finite |qf Li + qi Lf |, a sufficiently small λ satisfies (138). Similarly for (139), the Taylor series results in ˙ 1 ¨ 2 ξi − ξf + ξf λ = 2 ξiλ

qi − qf = −q˙ iλ,

2 2 Then as λ → 0, (139) is also satisfied for smooth trajectories ξ1(λ) and finite |Li + Lf |. Both (138) and (138) are satisfied for sufficiently small λ by smooth trajectories ξ1(λ). 2 The real part of (138) is exactly zero if Lf is the root of the numerator polynomial in (141),

ia L2 + a L2 = − 2 i 3 . f 2 a0Li + ia1 However, as λ → 0, this root is 2 2 LS(0) = −LS(0) , 2 satisﬁed by an imaginary LS(0) . But,

2 Re(LS(λ) ) > 0 for the functions (125) described by Gaussian functions (127) to be elements of HP . The magnitude of the exponential function in the contribution of U2,2(1234) is exactly one for a generalized 3 CLASSICAL CORRESPONDENCES 136

state, a divergent limit of elements from HP as the spatial support approaches a point. How- ever, these generalized states include support on relativistic momenta and as a consequence, lack classical correspondences. Satisfaction of (138) is approximate as a consequence. Circular orbits provide an illustrative example to explore the maximal duration periods λ consistent with the nonrelativistic approximations used to evaluate hϕ2(0)|U(λ)ϕ2(λ)i. For this study, only the contribution of U2,2(1234) to hϕ2(0)|U(λ)ϕ2(λ)i is considered. Nevertheless, the resulting estimate for the maximal λ applies generally since the nonnegative contribution of W2(13)W2(24) only reduces any upper bound on the period λ that satisﬁes (107). 2 2 2 2 ˙2 2 2 For circular orbits,r ˙ = 0, and then from (123) in section 3.3, λc qf = λc qi = ξi = L /4r using (114). The condition (138) that the magnitude of the exponential function approximates one simpliﬁes to 2 ˙2 2 2 |λc a3| ξi |Li + Lf |. From substitution of (123) and (142) for circular orbits,

2 1 ˙ ˙ ˙ 2 λc a3 = 4 (ξf · (ξi + ξiλ) − ξi · ξf ) 1 rf L riL = 4 (rf r˙i − rir˙f ) cos θ − ( + ) sin θ 4ri 4rf r˙ r˙ L2 r˙ L r˙ L 2 +( i f + ) λ cos θ − ( i − f ) λ sin θ 4 4rirf 4rf 4ri L sin θ L2 2 = 1 − λ cos θ 4 2 4r2 L2 |L2 + L2 | 4r2 i f with r = ri = rf . Simplifying,

r sin θ L 2 − λ cos θ |L2 + L2 |. (144) 2 4r i f

2 2 1 In (144), the function spread can not dominate r: if r sin θ 2|Li + Lf | 2 then the orbit is cloaked within the spread of function support and this instance does not have a classical correspondence. As a consequence, (144) can be satisﬁed only for small θ. For suﬃciently small θ, sin θ ≈ θ, cos θ ≈ 1, and the classical correspondence (105) implied by satisfaction of (138) and (140) applies if L2 λ2 |L2 + L2| 16r2 i i 3 CLASSICAL CORRESPONDENCES 137 from θ = (L/r2) λ for circular orbits with θ(0) = 0. Substitution of (123) results in 1 ξ˙2λ2 |L2 + L2|. 4 i i i

This bound provides that the approximations in the evaluation of hϕ2(0)|U(λ)ϕ2(λ)i limit λ to durations that are insufficient to resolve the corresponding trajectory. From the discussion in section 3.1, propagation of the support of a function must be significant with respect to the spread of support, 2 ˙ 2 2 2 λ ξk(λ) |Li + Li |, to resolve trajectories by likelihood. Then, (144) contradicts a resolution of the corresponding classical trajectory. The nonrelativistic approximations (116) and (91) result in estimates for the scalar product (126) that are insufficient to resolve the trajectories associated by (105) with the VEV, (128) with (129). The duration of the periods that the nonrelativistic approximations to the Hamiltonian provide accurate estimates for the scalar product is too brief to resolve trajectories from the spread in the support of states. Section 3.1 developed that approximations for the Hamiltonian apply only over finite intervals, and the estimate of this interval for circular orbits is too brief to resolve the corresponding trajectory. The estimate for largest interval does extend the classical correspondence established in section 3.2 from infinitesimal to greater periods, but, the most likely trajectory associated with VEV (128) and (129) has not been resolved. For the brief duration intervals, all continuous trajectories are included in a near- equivalence class of the propagation of the initial state. This breadth of the near-equivalence class of the propagation of the initial state over brief intervals is consistent with the Heisenberg uncertainty principle. The limitation on periods (144) is a surprising result: only the relativistic Hamiltonian provides insight into the nonrelativistic classical correspondence (105). Nonrelativistic approximations to the Hamiltonian do not suffice when interaction is exhibited. The bound (144) applies for general potentials and g λ2 |L2 + L2| 8r i i results from substitution for L for the circular orbits of −g/r potentials. Greater periods λ result from greater radius orbits or weaker potential strengths. Function spreads LS(λ) are both upper and lower bounded to satisfy the conditions in this section and a classical correspondence: the support of states has a characteristic size proportional to |LS(λ)| and support must be sufficiently spread to be nonrelativistic, Re(LS(λ)) λc; the sizes of the support of states in circular orbits are also lower bounded by (144) if the nonrelativistic evaluation of the scalar products satisfies (107); and to comply with a classical correspondence, states with non-overlapping dominant support have |LS(λ)| that are upper bounded to avoid a significant overlap of supports. 3 CLASSICAL CORRESPONDENCES 138

The ﬁnal condition (140) to satisfy a classical correspondence (105) is fulﬁlled by selection 2 of a real LS(λ) . Similarly to the classical correspondences of the linear harmonic oscillator, the function spread parameter LS(λ) is available to satisfy the classical correspondence. If

2 2 Lf = αLi (145)

2 2 2 2 2 2 2 for real functions α := α(Li , qi , qf ) with α(Li , qi , qi ) = 1 and real spread parameters Li = 2 Li , then condition (140) is a continuous function of α > 0 with at least one maximal value. Continuity, the nonnegativity of terms and (145) provide that (140) has a maximal value. Substitution of (145) for the evolution of LS(λ) into (140) results in the identiﬁcation of a function of α. 1  3  ! 2 2 2 2 b1  + 2 2 2  1 + α α (α + qf /qi ) b0

F (α) := 1 (146) 1  3  2 2 ! 2 b1 1 1 2 q b1 2 i (1 + )  + 5 2  b0 α α qf b0 with 3 1 π 2 b0 = 2 2 2Li 1 5 1 (147) 3 2 2 2 2 c4 π 4πλc π 1 b1 = 4 2 2 = λcc4 2 2 2π Li 2qi Li Li 2qi from simpliﬁcation of (140). Condition (140) is that α is determined to set

F (α) ≈ 1.

The Cauchy-Schwarz-Bunyakovsky inequality provides that

F (α) ≤ 1 if (138) and (139) are satisfied. A solution for α > 0 to maximize F (α) and satisfy F (α) ≈ 1 follows from the observation 2 2 that the continuous F (α) ≥ 0 converges to zero for both α → 0 and α → ∞ if both qf /qi and 2 2 b0/b1 remain finite. Indeed, if qf /qi = 1, then α = 1 satisfies (140) regardless of b1 and b0, and this is approximately the case for infinitesimal duration periods. If c4 is sufficiently small, then the terms proportional to b1/b0 are negligible with respect to the remaining terms and

α = 1 3 CLASSICAL CORRESPONDENCES 139

precisely satisfies (140). If c4 is sufficiently large that the other terms are negligible with respect to the terms proportional to b1/b0, then 2 2 α = qf /qi also precisely satisfies (140). From (114) for large c4, the resulting evolution of LS(λ) is ˙ 2 2 ˙ 2 2 ξ(0) LS(λ) = ξ(λ) LS(0) . (148) 2 2 2 For circular orbits, qf = qi and (145) is the assignment suggested by symmetry, LS(λ) = 2 2 LS(0) . The condition (140) associates LS(λ) with trajectories ξ1(λ). For weak or strong coupling, or brief duration periods, (140) is precisely satisfied. The worst instances for satisfaction of (140) correspond to infinite duration intervals,

2 2 2 2 qf /qi → 0, and qf /qi → ∞. The estimates for the scalar product derived from the nonrelativistic approximation for the 2 2 p Hamiltonian do not apply to infinite intervals. For qf /qi = α → 0, F (α) → b1/(b0 + b1) ≤ 1. 2 2 p For qf /qi → ∞ and α = 1, F (α) → b0/(b0 + b1) ≤ 1. This completes the demonstration that, for sufficiently brief intervals, the classical correspondences (105) apply to two body states described by functions (125) with Gaussian functions (127) for the VEV of a neutral scalar field given in (128) and (129). This demonstration applies for sufficiently localized, nonrelativistic, and isolated state descriptions, and nontrivial continuous trajectories. While the dynamics of appropriate localized states were associated with the trajectories of −g/r pair potentials in section 3.2, g has not been determined. The intervals supported by nonrelativistic approximations in the estimation of the scalar product hϕ2(0)|U(λ)ϕ2(λ)i are insufficient to resolve the particular trajectories associated with the VEV. It is remarkable that the exact relativistic Hamiltonian is required to determine the nonrelativistic correspondences of the VEV. To resolve whether a unique classical pair potential satisfies the classical correspondence (105), the nonrelativistic approximations must be removed to extend the estimates for the scalar products of interest (132) over longer duration intervals. (126) presented the scalar products without an approximation for the Hamiltonian. An evaluation of Z 0 √ hϕ2(0)|U(λ)ϕ2(λ)i = d(p )4 16 ω1ω2ω3ω4 W˜ 2,2(−p2, −p1, p3, p4, )×

−L (0)2p02 −ip0 ·ξ −L (0)2(p0 −2q )2 e c 1 e 2 i e S 2 i ×

−i(ω +ω )λ −L (λ)2p02 ip0 ·ξ (λ) −L (λ)2(p0 −2q (λ))2 e 3 4 e c 3 e 4 1 e S 4 1 requires methods beyond those of this note. Consideration of hU(−λ)ϕ2(0)|ϕ2(λ)i as a func- ˙ tional of the trajectories ξk(λ) and their first derivatives ξk(λ) leads to a calculus of variations 3 CLASSICAL CORRESPONDENCES 140 optimization problem [33] but the lack of methods to evaluate the summations and evaluate the Euler-Lagrange equations persists. Numerical quadratures include the concerns on the limitations of approximations and a lack of insight into a selection of LS(λ). In section 3.2, infinitesimal propagation of the most classical-like state descriptions indicates −g/r pair potentials. The association of the most likely value within isolated, localized volumes of dominant support with classical dynamic variables (105) complies with Ehrenfest’s nonrelativistic quantum-classical correspondences. But, nonrelativistic approximation of the Hamiltonian is generally insufficient to identify one classical correspondences for the VEV from section 2.5. The error in an approximation to the Hamiltonian is linear in λ and even though the error may be small, satisfaction of (105) can only be iterated forward over limited duration periods λ. The limited duration intervals cloak the association of classical trajectories with VEV as suggested by Heisenberg uncertainty.

3.5 Discussion of the classical correspondences The examples in sections 3.2 and 3.4 establish that the classical correspondence (105) applies with nontrivial trajectories to example descriptions of relativistic quantum physics over brief intervals. The correspondence of classical trajectories with the temporal evolution of the support of quantum states described by appropriate functions (105) is suggested by Schrödingerand Ehrenfest’s correspondences of quantum and classical mechanics [12, 42, 52]. The connected VEV from section 2.5) enable energy-momentum to be conserved without constrained momenta equal in pairs; these connected VEV realize nontrivial dynamics. If the free field VEV (32) are replaced with VEV that include appropriate n-point connected functionals, a Hamiltonian H that is the “free field Hamiltonian” in a canonical quantization results in an exhibition of interaction. In the example realizations of relativistic quantum physics from section 2, the scalar product of state descriptions determines the exhibition of interaction. For appropriate states, classical dynamical variables that satisfy Newtonian mechanics for −g/r pair potentials are representatives for the location and momentum support of the quantum mechanical descriptions of state. Trajectories are representatives of the evolution of this support for appropriate quantum mechanical descriptions of state. This correspondence is an alternative to the canonical formalism that describes quantum dynamics through elevation of classical dynamical variables to densely defined, Hermitian Hilbert space operators. Representation of the support of functions by classical dynamical variables is a more robust and conditional correspondence than the hypothesized operator elevations. This suggested correspondence avoids an over extrapolation of classical dynamical laws and does not suffer the counterexamples illustrated in sections 1 and 5. This revised classical correspondence deviates from the established canonical quantization descriptions of relativistic quantum physics. Rather than construct an elevation of a classical Hamiltonian, constructions of VEV satisfy the axioms of section 2.2: interaction is exhibited by the VEV without reference to an interaction Hamiltonian. The VEV determine the scalar products that determine state transition likelihoods. Nontrivial quantum dynamics are exhib- 3 CLASSICAL CORRESPONDENCES 141 ited but the classical correspondence for nonrelativistic, localized and isolated state descriptions has not been uniquely determined in the current state of development. It remains a goal of the unconstrained development to discover a quantum mechanical basis for the phenomenologically successful results of canonical quantization. Classical correspondences include: Newton’s mechanics at the first level of nonrelativistic approximation; and Maxwell’s equation with the Lorentz force and Einstein’s geometrodynamics in more general situations. Selections from the constructions of section 2 are anticipated to exhibit both the required classical correspondences and high energy phenomenology. Estimated in section 3.4 for brief intervals λ, the equivalence class of the evolution of one state includes correspondences with many smooth trajectories. This ambiguity in the correspondence is anticipated in the Heisenberg uncertainty principle. The class of functions nearly equivalent to the temporal propagation of an initial function includes functions centered on non-straight trajectories. One trajectory is not uniquely distinguished by the analysis of likelihood in section 3.4 over brief intervals λ. In [33], selection of trajectories of extremal, but not necessarily near certain, likelihood optimized the duration that (105) is satisfied and g is estimated from c4, and a choice for growth of the volume of support selects trajectories that satisfy Newton’s equation. However, these calculations also used nonrelativistic approximations. Trajectories must be distinguished by likelihood, that is, by the calculation of scalar products, or trajectories would not be distinguishable. If states centered on trajectories corresponding to a variety of classical pair potentials remain within equivalence classes, then the preserva- tion of equivalence classes by Hilbert space operators provides that trajectories will never be distinguishable no matter how distantly separated trajectories become. Plane wave scattering amplitudes are not associated with a −g/r potential: correspondence with classical trajectories is conditional and applies for state descriptions with isolated, localized and nonrelativistic support, and not plane waves. Indeed, the plane wave scattering amplitudes associate in first Born approximation with short range, Yukawa-like potentials [32, 33, 38] while the motion of localized states is governed by a long range −g/r pair potential. Significant outstanding questions include: 1. removal of nonrelativistic approximations (116) and (91) to evaluate likelihoods over longer duration intervals; 2. extension of classical correspondences to electrodynamics and gravity; 3. and an understanding of the role of eigenstates of H + V , how does the approximation (109) that applies for infinitesimal intervals and selected states with a phase at peak of likelihood equal to the classical energy extend to an equation of motion with an elevation of the classical potential V to Hilbert space operator, how do discrete energy spectra appear? Like Schrödinger’sdevelopment of the linear harmonic oscillator that includes states of any energy, a state of any classical energy can be matched in this development of a classical corre- 4 INCONSISTENCY OF THE CLASSICAL DESCRIPTION WITH NATURE 142 spondence. Energy eigenfunctions are not functions that follow classical trajectories even for the linear harmonic oscillator [52]. The revised classical correspondence, the representation of the support of functions that describe nature by classical dynamical variables, provides an evident and technically sound approach to quantum-classical correspondence. Together with the Hilbert space structure of quantum mechanics [11, 62], these considerations offer a realizable approach to relativistic quantum physics.

4 Inconsistency of the classical description with nature

Quantum mechanics is a striking change from classical mechanics. Several competing schools of thought persist on whether quantum mechanical descriptions are real depictions of nature. The diﬃculty for many physicists is that quantum mechanics forces us to abandon well-developed classical concepts that describe distinguishable point-like bodies evolving smoothly along trajectories. Quantum mechanics demands a change from well-developed classical concepts as the description of nature. The considerations listed below require a general abandonment of classical concepts. The conﬂict of classical concepts with nature is illustrated by many considerations including: 1. Planck’s calculation of the spectrum of black body radiation

2. the photoelectric eﬀect

3. Stokes ﬂuorescence and Compton scattering

4. Gibb’s paradox

5. the creation and annihilation of particles

6. interference patterns in a Michelson interferometer at very low “single quantum at a time” light intensity

7. discrete energy bands in the radiation spectra from atoms

8. the Einstein-Podolsky-Rosen paradox for conserved quantities such as angular momentum, discussed in section 1.1. Taken together, these considerations are very difficult to rectify with a classical world view. Each is a motivation for quantum mechanics, and together with principles of simplicity and universality, a quantum mechanical description for nature is strongly indicated. For consistency with observations, Max Planck’s calculation for the spectrum of black body radiation included that the energy in electromagnetic radiation is quantized in discrete particle- like amounts with an energy E proportional to photon frequency ν, E = hν. h became known 4 INCONSISTENCY OF THE CLASSICAL DESCRIPTION WITH NATURE 143 as Planck’s constant. Planck’s revelation and Albert Einstein’s insight that the photoelectric effect is also explained by a quantized photon energy produced agreement with observations. The observed photoelectric effect is strong evidence for the quantization of electromagnetic radiation as photons. In the photoelectric effect, electrons are not emitted from a surface until the light frequency is sufficient, that is, until the light quanta include sufficient energy that the dominant reaction, interaction of an electron with a single photon, results in an electron with sufficient energy to escape the surface. This observation is (nearly) independent of the amplitude of the incident radiation. The exception is nonlinear optics, if the electron absorbs the energy of multiple photons. These results indicate that the energy of photons is quantized, and contradicts the classical description of electromagnetism as waves and energy as a freely specified real parameter. If the illumination is a wave, we should anticipate that once the amplitude of the wave was sufficient that electrons would escape the surface. In his identification of the quantized energy E = hν of an electromagnetic field, Einstein was also motivated by the observed heat capacity of solids, in particular, contradictions to the equipartition of energy in classical statistical mechanics, and Stokes fluorescence [58], that the frequency of a photon emitted by a body was less than the illuminating frequency. (Stokes fluorescence has since been supplemented with observations of anti-Stokes radiation, when vibrational modes of an emitter contribute to the energy of the re-emitted photon.) Together with Joseph John Thomson’s observations of the electron, Robert Milliken and Harvey Fletcher’s measurement of a discrete electric charge, and Jean Perrin’s observations and Albert Einstein’s description of Brownian motion, these findings provided motivation for the descripion of matter as a composition of atoms. The indistinguishability of bodies of the same mass, spin, polarization and charges naturally resolves Gibb’s paradox. Gibb’s paradox arises if entropy is is not an extensive quantity. In statistical physics, an extensive quantity is proportional to the amount of substance, that is, for an extensive entropy, twice the volume of gas should have twice the entropy. Were entropy not to be extensive, then Gibbs paradox is a violation of the second law of thermodynamics, that entropy of isolated systems does not decrease. Other solutions have been suggested to resolve Gibb’s paradox, but the indistinguishability of bodies is a natural resolution consistent with the Bose-Einstein or Fermi-Dirac statistics of similarly described bodies in quantum mechanics. Note that this indistinguishability is in contradiction to the classical concept that the trajectories of individual bodies can be distinguished and followed. The creation and annihilation of particles is another contradiction to the classical concept of continuous trajectories. In these cases, trajectories disappear and trajectories with distinct descriptions appear. Such transformations of identifiable bodies are not described by classical physics. With creation and annihilation, one cannot follow the trajectory of a single identifiable entity, nor generally even determine the number of bodies in a relativistic state description. A Michelson interferometer consists of two light paths split and later recombined using a half-silvered mirror, two reflecting mirrors, and an optical path matching slab of clear glass. An interference pattern of light and dark intensity rings is visible on detectors at an end of 4 INCONSISTENCY OF THE CLASSICAL DESCRIPTION WITH NATURE 144 the optical path, for example, on photographic film. This pattern persists even as the intensity of the light is lowered. From the model of light quanta established in the photoelectric effect and Planck’s calculation of black body radiation spectra, it follows that the illumination can be reduced to less than a single photon at a time within the interferometer on average. This establishes that each photon interferes with itself, and since the arms of the interferometer are separated and can be of unequal length, that the photon takes both paths. This contradicts a classical concept that the photon is a classical particle with a definite trajectory. The interference pattern is consistent with the classical concept of electromagnetism as a wave, but that description is contradicted by the photoelectric effect. The interferometer is a strong argument for the reality of the quantum description of state. It is difficult to understand how at low intensities the interference pattern can be responsive to changes in arm path length without a physical presence in each of the two separate arms for each photon. Discrete line spectra are observed in the emissions of light, for example, from a gas of hydrogen or sodium atoms. That the spectrum is not continuous across frequencies contradicts the concept that energy is a real number parameter, a boundary condition for equations of motion. Line spectra contradict that any in a range of real numbers can provide the initial conditions that result in a hydrogen atom. There is no mechanism within classical mechanics that results in the observed atomic spectra (nor the observed heat capacities of solids at low temperatures). One of the predictions of quantum mechanics is line spectra, and observations of the Lamb shift of hydrogen atom energy levels is one of the most precise tests in physics. The Lamb shift is correctly estimated by the Feynman series rules for quantum electrodynamics. Quantum mechanics resolves many observed flaws of classical mechanics, from extensive entropy to conservation laws for quantized quantities to discrete atomic spectra to the consistency of observed interference patterns with particle-like descriptions of waves. Despite all this, classical descriptions of physics continue to be used as the underpinnings of quantum physics. The concept of a continuous evolution of distinguishable bodies traveling trajectories, the descriptions of Newtonian physics and Einstein’s geometrodynamics, are evidently useful approximations to quantum physics but in limited instances. The concept that quantum dynamics is the “quantization” of these classical dynamic models rests on expedience, experience with nonrelativistic quantum mechanics, and successful but limited phenomenology. Any correspondences of quantum with classical need only occur if appropriate classical approximations apply, and canonical quantization is an unjustified extrapolation otherwise. Indeed, fully quantum mechanical descriptions of relativistic dynamics of interest have not yet been identified although suggestive examples appear in section 2 and [32, 36, 38]. The examples demonstrate a union of the principles of relativity and quantum mechanics and the refined quantum-classical correspondences open a rich set of possibilities for quantum dynamics. 5 TECHNICAL CONCERNS WITH CANONICAL QUANTIZATION 145

5 Technical concerns with canonical quantization

In this section, concerns with the canonical formalism are developed. Inquiry into an appropriate mathematical development of quantum mechanics was advocated notably by John von Neumann. This inquiry has been extended by Léonvan Hove, Res Jost, Rudolf Haag, Arthur Wightman, Huzihiro Araki, Nikolay Bogolubov, Hans-JürgenBorchers, Franco Strocchi and many others [70]. The development uses the concept of Hilbert space advanced notably by David Hilbert, Erhard Schmidt, Frigyes Riesz, Marshall Stone and John von Neumann. The discussion here distinguishes a general development of quantum mechanics from the particular quantum-classical correspondences of the canonical formalism. Discussed in section 2, the general development is that: quantum mechanics describes the states of nature as elements of rigged Hilbert spaces; the evolution of the observable features of appropriate states is approximated by classical mechanics; energies are nonnegative; the temporal evolution of state descriptions is unitary (likelihood preserving) and causal; the likelihoods of observation are cal- culated from Born’s rule; the local observables of non-entangled, distantly space-like separated bodies are essentially independent; and likelihoods are relativistically invariant. The canonical formalism conjectures that the elevations of classical dynamical variables are Hermitian operators and specifies identities among operators to describe dynamics [11, 60, 62]. In section 1.4, the conjectured quantum-classical correspondence in question is that Φ(b x)κ = Φ(x)κ and the implementation of dynamics is (3). These conjectures appear not to be realizable in relativistic physics except for free fields and related developments. Quantum mechanics has provided a consistent and successful description of nature. Never- theless, as emphasized by von Neumann, unresolved issues remain even in the development of nonrelativistic quantum mechanics [62]. If classical dynamical variables elevate to Hermitian Hilbert space operators and there is a “quantization” of the corresponding classical dynamics, what determines the appropriate order for products of noncommuting operators? And, is it a concern that while functions of classical dynamical variables are classical dynamical variables (for example, generalized coordinates), products of Hermitian operators are not necessarily Hermitian operators? Are some classical quantities distinguished by having a quantum correspondence while other quantities do not? Or is it a concern that eigenfunctions of the operators associated with observables by the canonical formalism are not always elements of the Hilbert space, that is, do not describe states? In such cases, what state results from a “collapse to an eigenstate of the observable?” Introduced in section 1, the canonical formalism [11, 60, 63] includes assertions for the “quantization” of classical dynamics. Archetypes for the elevation of classical dynamical variables to densely defined Hermitian Hilbert space operators include location Xν and momentum Pν operators that satisfy the Born-Heisenberg-Jordan relation [Xν,Pµ] = −i~δν,µ in nonrelativistic physics. These Xν and Pν apply in the L2 Hilbert spaces suitable for nonrelativistic quantum mechanics. In the following few paragraphs, the discussion includes that: the locations and momenta associated with the functions that describe states are well defined as observable features even when expectations of the corresponding operators Xν 5 TECHNICAL CONCERNS WITH CANONICAL QUANTIZATION 146

and Pν diverge, and even for elements that are not in the domains of Xν and Pν; the operator Xν that is the elevation of location fails to be a Hermitian operator for the relativistic free field; and the elevation of products of locations x and momenta p are not necessarily Hermitian operators even in nonrelativistic quantum mechanics. That the eigenfunctions of the operators Xν and Pν are not elements of L2 Hilbert spaces was discussed in section 2. Each of these points is a difficulty for canonical quantization or illustrate better alternatives for quantum-classical correspondences. First, greater detail on Hilbert space operators is introduced. Study of Hilbert space operators, particularly unbounded operators in infinite dimensional Hilbert spaces, is a subtle and elegant subject [40] that illustrates many of the “paradoxes of infinity.” In section 2, a Hilbert space operator A was introduced as a mapping of elements from a Hilbert space H into H, A : H 7→ H. That is,

|Aψi = |gi if |ψi ∈ DA ⊆ H, the domain of A, and then |gi ∈ RA ⊆ H, the range of A. The domains of bounded operators can be extended to the entire H (A is bounded if kAψk ≤ ckψk for a c ∈ R independently of the element |ψi and the least upper bound deﬁnes kAk. The domain of an unbounded A is necessarily a proper subset of elements in the Hilbert space but the domain may be dense in H. If |Aui diverges, then |ui 6∈ DA. A set of elements |eni is dense in H if every |ψi ∈ H is in a neighborhood of a ﬁnite linear combination of the |eni labeled such that

N X kψ − cnenk < n=1 for N > N with the cn ∈ C. → 0 as N grows without bound. A separable Hilbert space has a denumerable (finite or infinite) dense set of elements |eni. An example is that functions in the perfect Schwartz function space S [18] are dense in the square-integrable but not necessarily differentiable functions in L2. L2 has a dense, denumerable basis. For an unbounded operator B, a subsequence from a diverging sequence |Buni can be selected and relabeled such that kBun|| > nkunk. Then, the neighborhood of each element |ψi with |Bψi ∈ H contains a sequence constructed in the following with kB (ψ − ψn)k > n kψ − ψnk for n growing without bound. That is, an unbounded operator is not continuous anywhere in the Hilbert space. Indeed, using the divergent sequence |uni selected above, set

|uni kBunk |vni := , and then kBvnk = > 1. nkunk nkunk

The sequence |ψni := |ψi + |vni is a convergent sequence, kψ − ψnk = kvnk = 1/n → 0, with a non-convergent kB (ψ −ψn)k = kBvnk > 1. Then generally, neighboring states within a Hilbert space do not necessarily have nearly the same observables if observables are associated with unbounded Hilbert space operators. 5 TECHNICAL CONCERNS WITH CANONICAL QUANTIZATION 147

In contrast, states exhibit observable features even when the expectation values of the corresponding operators diverge or when the operation is undeﬁned. For the example of location,

2 2 ψ(x) = e−x /(4σ ) + (1 + x2)1−δ is dominantly supported near x = 0 for 0 < , δ 1 and a small length σ. The σ, , δ can be selected to have an arbitrarily large likelihood that an observation of location is near x = 0 compared with any other disjoint, equal volume neighborhood. The likely location is near x = 0 for the state described by ψ(x) even though hψ|xψi diverges. With location xν 7→ Xν considered as the Hilbert space location operator, the expectation value hψ|Xνψi diverges but an evident perception for the body in the state described by ψ(x) is that the greatest likelihood is that if the body is observed, it will be observed near the origin. At large x2 (the unobservable “far side of the moon”), there is a small likelihood of detection within any finite volume but a sufficient volume for the mean value to diverge. There is no actual divergence since we not capable of detecting position over an infinite volume. Our measurements are always localized; location Xν and momentum Pν are idealized observables. Actual observations are limited to within finite volumes. The summations of |ψ(x)|2 are convergent while summations of x|ψ(x)|2 2 −x2/(4σ2) −x2/(4σ2) diverge at large x . kψ(x) − e k is small but kXνψ(x) − Xνe k diverges in the L2 norm. However, it is not possible to perform detection to verify state descriptions over this infinite volume. Without an omnipotent classical observer, no knowledge of infinite volumes is available. The likelihoods of physical interest are the relative likelihoods of detection within finite volumes. The unknowable, distant support of functions that might describe states only hypothetically affects our considerations. Correspondences with the classical concepts of location and momentum are provided by the support of the functions that describe states. Considering support, the likelihoods of observations of position and momenta are nearly the same if the descriptions of the states are nearly the same. The support of functions that describe states are more robust and consistent features than the results of unbounded, densely defined Hermitian Hilbert space operators Xν and Pν. The support of functions as appropriate features in the description of quantum states appeared in early developments of quantum mechanics. Erwin Schrödinger[35, 52] contrasted quantum mechanical and classical developments of the linear harmonic oscillator. The dynamics of the support of Gaussian states in the quantum mechanical development compare well with classically predicted dynamic values. Paul Ehrenfest [12, 42] presented a more general development for correspondences of expected values in quantum mechanical developments with classical predictions. For functions that are dominantly peaked in either location or momentum support, these developments are interpreted here as comparing representatives of the neighborhoods of support for the functions that describe quantum mechanical states with classical dynamical variables. The correspondence of classical dynamical variables with the descriptions of states is that concentrations of the support of the functions or their Fourier transforms compare with classical predictions. But, these quantum-classical correspondences apply only for appropriate 5 TECHNICAL CONCERNS WITH CANONICAL QUANTIZATION 148 functions. Extrapolation of the location operator Xν to relativistic physics does not succeed. With consideration of relativity, the Xν that results from elevation of the classical dynamical variable xν is not a Hermitian operator. As seen in section 2.10, only Hermitian operators are associated with observable quantities. Transition likelihoods are determined by the scalar product and these likelihoods are events independent of observer. As a consequence, the scalar product must be a Lorentz invariant in relativistic physics. The scalar product (8), and the Källén- Lehmann form for the two-point functional (30) as a nonnegative summation over masses of the Pauli-Jordan function result in consideration of

∗ hXν ψ|gi = hψ|Xνgi Z + = dxdy ∆ (y − x) ψ(y) xνg(x)

6= hXνψ|gi Z + = dxdy ∆ (y − x) yνψ(y) g(x)

+ + since the Pauli-Jordan function is not of point support, xν∆ (y − x) 6= yν∆ (y − x). Then ∗ Xν 6= Xν for ψ, g ∈ DXν . The elevation of xν to Hilbert space operator Xν is not Hermitian in relativistic quantum mechanics. This “localization problem” is one of many technical problems that develop in the extrapolation of the canonical formalism to describe relativistic quantum physics. In contrast, the energy-momentum operators Pν are generators of translations in a unitary realization of the translation group and are self-adjoint as a consequence of Stone’s theorem [40]. In the argument above, hψ|Pνgi = hPνψ|gi follows from

pνδ(p − q) = δ(p − q)qµ.

A unitary realization of the translation group results from translation invariance of the scalar product. Translation invariance reﬂects that the principles of physics do not vary with location. Location, an observable of evident importance in physics, does not correspond to the elevation of xν in relativistic quantum mechanics. Discussed in section 1.5, the Hermitian operator associated with location is not elevation of xν but rather is the Hermitian operator that is a linear combination of locations times projection operators onto subspaces of states associated with the most likely observation of a body at those locations. Suitable forms of location operators are determined by the relativistically invariant localized functions that describe those states most closely corresponding to a body at a particular location in relativistic quantum physics. These forms [44] are discussed in section 2.1 and appendix 7.2. Quantities that are dynamical variables in classical mechanics can be excluded as Hermitian operators even in ordinary (nonrelativistic) quantum mechanics. The product x3p is a classical dynamical quantity and in ordinary quantum mechanics with a single spatial dimension, the 5 TECHNICAL CONCERNS WITH CANONICAL QUANTIZATION 149

corresponding Hermitian operator in L2 should be

d x3 d d x3 i x3/2 x3/2 = i ( + ) ~ dx ~ 2 dx dx 2 since the classical dynamic variables x and p correspond with unbounded, self-adjoint operators x and i~d/dx in this one spatial dimensional L2 example. However, this densely deﬁned, formally Hermitian operator that corresponds with x3p has square-integrable eigenfunctions

√ exp (−λ/(2x2)) s (x) := 2λ λ x3/2

3/2 3/2 with imaginary eigenvalues −i~λ [6]. 0 < λ ∈ R. As a consequence, X PX is not a Her- mitian operator in the L2 Hilbert space. This sλ(x) ∈ L2 is defined for x > 0 and equals zero otherwise, or sλ(x) can be appropriately extended to negative x. The formally Hermitian op- 3/2 3/2 3 3 erator X PX that corresponds to x p is not Hermitian for L2 although x p is well defined in classical dynamics. Nevertheless, for the example of linear harmonic motion and minimum 3 uncertainty packet states st(x) with small spatial variances, the trajectory of x p given by 3/2 3/2 Newtonian mechanics approximates hst|X PX sti from quantum dynamics [35, 52]. This 3/2 3/2 establishes that there are particular states with real hst|X PX sti that agree with the classical approximations x3p even though X3/2PX3/2 is not a Hermitian operator. The Hermitian operator corresponding to x3p need not be the elevation of x3p. Once again, the support of functions for appropriate selections of states exhibit quantum-classical correspondences while a correspondence of operators fails. Assertion that classical dynamical variables elevate to Hermitian operators is unjustified without satisfaction of additional considerations. And, even when the operator that is the elevation of the classical dynamical variable is not Hermitian, the classical dynamics can approximate the quantum mechanics for appropriate states. There are no eigenstates of the quantum field (14). An eigenstate would be labeled by a generalized function that is a limit e of a sequence of states in the Hilbert space such that

Φ(f) e = λ e.

Then, and in the simpliﬁed notation of a scalar ﬁeld,

Φ(f)(e0, e1, e2, e3 ...) = f × (e0, e1, e2, e3 ...)

= (0, fe0, fe1, fe2 ...)

= (λe0, λe1, λe2, λe3 ...) from (11) and (14) that leads to the recursive λ en+1 = f en. Then e0 = 0 results in the conclusion that e = 0. 6 CONCLUSION 150

6 Conclusion

Quantum mechanics supersedes, not “quantizes,” classical descriptions. A description of the states of nature as elements of rigged Hilbert spaces manifests the wave-particle duality, and implements the discrete line spectra of atomic emissions, the quantized energy of photons, and the indistinguishability of similarly described bodies that results in an extensive entropy. This description is fundamentally contradictory to the classical physics intuition of an omnipotent observer following distinguishable bodies evolving smoothly along trajectories although isolated, essentially localized observable features correspond with classical bodies. Classical dynamics provides approximations for quantum dynamics over limited duration intervals for particular states. On a large scale for functions with isolated support concentrated in small areas, the quantum description is perceived as classical body-like. However, for significantly overlapping descriptions, or when entanglement applies, or when non-commuting observables are considered, the differences of the quantum and classical descriptions for bodies are not negligible. The scale for “small” is generally set by the Compton wavelength. Trajectories for distinguishable bodies are not a concept in quantum mechanics except when a correspondence with classical bodies applies. Observation affects the description of states and a sequence of observations does not record the unique evolution of an initial state but rather one possible evolution and includes the process of observation. These concepts of quantum mechanics contradict basic premises from classical mechanics and in this sense the concepts are disconcerting. The concepts of quantum mechanics are necessitated by consistency with nature. Within quantum mechanics and to accommodate relativity, the canonical formalism’s exact elevation of classical to quantum descriptions is relaxed to more approximate and conditional correspondences. An exact correspondence of classical dynamical variables with eigenstates of corresponding Hermitian Hilbert space operators is not necessary. More approximate and conditional correspondences are physically appropriate, equivalent within the limitations of observation, and do not excessively extrapolate the inherently limited classical models. While the canonical formalism succeeds in nonrelativistic physics, the change to scalar product necessitated by compliance with relativity is inconsistent with exact elevation of location and apparently with nontrivial fields. More appropriate quantum-classical correspondences admit a rich class of nontrivial realizations for relativistic quantum physics. The understanding of quantum mechanics expressed here is not a consensus view. Indeed, many physicists reject or at least minimize concepts discussed. The description emphasizes a consistent depiction of nature, consistent with observation and free of ad hoc accommodations to maintain a classical description for observers or to base quantum dynamics on classical dynamics. The argument is not that acceptance of the quantum mechanical description as reality is the only explanation for any one effect, but rather that the rigged Hilbert space realization of state descriptions and correspondence of features is the simple, comprehensive, and consistent explanation [45]. The state, an element of an appropriate Hilbert space, is the entire state description and the perception of these states by observers and the evolution of states 7 APPENDICES 151 are the concerns of mechanics. The perspective advanced here is that established assertions of the canonical formalism are productive but limited conveniences rather than foundational principles for physics. The phenomenological successes that result from canonical quantization in relativistic physics are compelling but limited, and the example realizations provide alternatives within a quantum mechanical description. The classical developments of Newton, Maxwell and Einstein must necessarily provide approximations for the evolution of quantum states in instances with classical correspondences, but the examples discussed here demonstrate advantages of alternatives to canonical quantization-extrapolations of classical developments.

It seems clear that the present quantum mechanics is not in its ﬁnal form. Some further changes will be needed, just about as drastic as the changes made in passing from Bohr’s orbit theory to quantum mechanics. Some day a new quantum mechanics, a relativistic one, will be discovered, in which we will not have these inﬁnities occurring at all. – Paul Dirac, in Albert Einstein : Historical and Cultural Perspectives : The Centennial Symposium in Jerusalem, edited by Gerald James Holton and Yehuda Elkana, 1979, p. 85.

The pressure for conformity is enormous. I have experienced it in editors’ rejection of submitted papers, based on venomous criticism of anonymous referees. The replacement of impartial reviewing by censorship will be the death of science. – Julian Schwinger.

7 Appendices

7.1 The Dirac-von Neumann axioms for quantum mechanics The Dirac-von Neumann axioms emerged from the uniﬁcation of developments from Heisenberg and Schr¨odinger. Here, the statement of the Dirac-von Neumann axioms is adapted from Strocchi’s discussion in [60].

I. Observables: The Hermitian operators corresponding to the observables of a quantum mechanical system are included in the algebra of bounded self-adjoint operators B(H) of a separable Hilbert space H.

II. States: The pure states of a quantum mechanical system are described by rays a|si, a 6= 0 ∈ C, identiﬁed with normalized elements |si ∈ H, ksk = 1. More generally, a state is described by a nonnegative, unit trace, state density operator ρ ∈ B(H).

III. Expectations: If a state is represented by a normalized pure state |si ∈ H, then, for the observable corresponding to A ∈ B(H), the experimental expectation is given by Born’s rule, hs|Asi. More generally, experimental expectations are Trace(Aρ). 7 APPENDICES 152

Axioms I-III describe a Hilbert space realization of quantum mechanics and are implicit in the development of section 2.1. Conditions A.1 and A.2 of section 2.2 imply a Hilbert space realization. State density operators ρ ∈ B(H) are discussed in section 2.10. Only elements of the Hilbert space are considered as states: limits of states such as the eigenstates of position and momentum are in the dual to the basis function space but depart from the rigged Hilbert spaces of interest [6]. Superselection sectors in the Hilbert spaces of interest illustrate that not all self-adjoint operators are observables. Hilbert spaces of interest can be represented as a direct sum of superselection sectors and the observables are associated with operators limited to within sectors. The limitation of observables to bounded operators departs from Dirac’s development but reflects that only finite values are observable, for example, we detect locations within a finite volume of interest. Limiting consideration to bounded operators provides several technical conveniences [24, 60]. Axioms I-III describe a general Hilbert space realization of quantum mechanics. A characterization of observables remains. The Dirac-von Neumann axioms IV-V discuss the properties of operators.

IV. Dirac canonical quantization: The Hermitian operators that describe canonical coordinates qi and momenta pj , i, j = 1,...N, obey canonical commutation relations

[qi, qj] = 0

[pi, pj] = 0

[qi, pj] = −i~δij.

V. Schr¨odingerrepresentation: The canonical commutation relations are realized in the Hilbert space Z 2 4 H := L2 = {ψ(x)| dx |ψ(t, x)| < ∞, x ∈ R , x = x1, x2, x3}

by:

|qiψi := xi |ψi ∂ψ |piψi := −i~ | i. ∂xi

In the Dirac-von Neumann axioms, canonical variables qi, pi are exact elevations of classical dynamical variables. An exact elevation is defined in section 1.1 and conjectures an exact association of classical dynamical variable with the eigenstates of the corresponding Hermitian operator qi. For a normalized eigenstate |ϕi of an observable qi, |qiϕi = λi|ϕi with λi ∈ R, and the experimental expectation is hϕ|qiϕi = λi. In the case that the canonical variables qi, pi are 7 APPENDICES 153 location and momentum, respectively, the canonical commutation relations are known as the Born-Heisenberg-Jordan relations. Discussed in section 1.1 and in the L2 Hilbert space applicable to nonrelativistic physics, these elevations result in Hermitian Hilbert space operators. This correspondence in axiom V can be characterized as an “elevation of c-number to q-number.” Axioms IV-V are generally taken to describe this particular correspondence of classical and quantum observables. In relativistic physics, the exact elevation of location is not a Hermitian operator. Relativis- tic location operators are discussed in [44], section 1 and appendix 7.2. For free field theories, the properties of position and momentum operators restricted to the one particle subspace extend to Fock space by second quantization but in the interpretation of state discussed in section 2.1, the extension of one-particle operators to multi-argument states is ambiguous when interaction is manifest. Some revision to axioms IV-V is necessary to describe quantum-classical correspondences in nontrivial relativistic quantum physics. In the unconstrained development, the physical characteristics of relativistic quantum physics are described by conditions A.1-7 of section 2.2. Conditions A.1-7 characterize the VEV of fields and consequently the scalar product (8) of the Hilbert space. In the unconstrained development, the functions that describe states are considered as depictions of reality and the classical correspondences derive from perception of these states. Discussed in section 2.1, the spatial supports of the functions describe the possibilities for observation of location and the supports of the Fourier transforms describe the possibilities for observation of momenta. When the support of functions is sufficiently isolated and localized, then the idealized position of a classical body is a representative for the likely locations. This revised correspondence is introduced in section 1.1. As discussed in sections 1.1 and 1.4, assumptions such as axiom V for observables as Hilbert space operators are not necessarily consistent with realization of relativistic quantum physics. A construction of the Hermitian operator corresponding to an observable classical dynamical variable is described in section 1.5. This construction is based on the perception of states and relaxes the conjectured correspondence in axiom V to a more approximate and conditional correspondence of classical with quantum state descriptions. The correspondence of operators qi with location is revised to representation of the spatial support of the descriptions of states, and the correspondence with fields is revised to representation of the evolution of localized, isolated concentrations in the spatial support of the functions that describe states. Localized, isolated concentrations in the spatial support correspond with classical particles that evolve according to the corresponding classical field theory. Conditions W.a-b in section 2.2 extrapolate the axiom V conjectured elevation of canonical variables to correspondence of a classical field with Hermitian field operators. The unconstrained development derives the quantum-classical correspondences from the representation of states. This derivation reproduces axioms IV-V for nonrelativistic physics but neither axioms IV nor V extrapolate to fields in the example constructions of relativistic quantum physics. The unconstrained development of relativistic quantum physics is free from the apparently unrealizable quantum-classical correspondences of axioms IV-V. Axiom V describes an elevation of classical dynamical variables to unbounded Hermitian 7 APPENDICES 154

Hilbert space operators. This unboundedness is in contradiction to axiom I. Unboundedness introduces the consideration that sums of observables are not necessarily observables (if necessarily limited domains are sufficiently disjoint). And, the (generalized) eigenstates of location and momentum are not elements of the Hilbert space. These and additional technical difficulties with the canonical formalism are discussed in section 5. The discussion here focuses on the physical differences of classical and quantum characterizations of nature.

The lack of a clear distinction between the role of the two sets of axioms, I, II, III and IV,V, is at the origin of the widespread point of view, adopted by many textbooks, by which all of them are characteristic of quantum systems. The distinction between classical and quantum systems is rather given by the mathematical structure of [the algebra of observables] and it will have diﬀerent realizations depending on the particular class of systems. – Franco Strocchi [60].

In the unconstrained development discussed here, the diﬀerences of state descriptions, classical from quantum, are emphasized over abstract similarities of classical and quantum dynamics. Introduced in section 1.1, understanding the formalisms of classical and quantum mechanics as depictions of reality is central to realizing relativistic quantum physics. The consideration that enables realization is that bodies are described by Lebesgue summable functions (7) rather than by classical dynamical variables. Then, the Heisenberg uncertainty principle is considered an intrinsic characteristic of the description of nature rather than a limitation of our measurement processes. The description of state precludes us from “seeing” a classically described body; the states of nature are not classically described despite the familiar and useful approximations provided by classical description. Described in sections 1.1 and 4, the classical and quantum descriptions of nature are fundamentally dissimilar, and the description of relativistic quantum dynamics is not necessarily an elevation of classical dynamics. The discussion in section 1 suggests that imposition of the conjecture to canonically quantize ﬁelds precludes realization of relativistic quantum physics.

7.2 Translations, position operators and relativistically invariant localized states Hermitian Hilbert space operators that correspond to position are studied in this appendix. Quantum-classical correspondences are richer than a canonical formalism-conjectured elevation of classical dynamical variables to Hermitian Hilbert space operators. Elevation was defined in section 1.1 as an exact representation of the classical dynamical variable by an eigenstate of the corresponding operator. In the case of location, the elevation of location xν with Hilbert space operators Xν that have Dirac delta functions as eigenstates. The revised correspondence emphasized in these notes is to consider that observers imperfectly interpret states in terms of classical concepts. The correspondence of classical spacetime with the arguments xν of 7 APPENDICES 155 the functions that describe states is preserved, as is the correspondence of energy-momentum with the scaled arguments pν = ~kν of their Fourier transforms. Introduced in section 1.5, a consistent association of location with the eigenstates of a Hilbert space operator that is Hermitian for the Hilbert space scalar product is enabled by consideration of approximate representation of the features of the quantum description of state. To describe the location of a classical body, a classical location is associated with the spacetime support of appropriate functions, and a function argument is associated with a classical body. Functions with localized and isolated support are appropriate to associate with classical bodies. Discussed below, multiplication of the Fourier transforms fñ((p)n) of functions by pν generates translations in location and multiplication of the functions fn((x)n) by xν generates translations in momenta. Translation invariance of the scalar product and Stone’s theorem leads to the conclusion that the momenta are densely defined Hermitian operators. However, depending on whether the development is relativistic or nonrelativistic, the scalar product is not invariant to translations in momenta. In a nonrelativistic development, the scalar product is invariant to translations in momenta but in a relativistic development, the independence of the speed of light from source velocity dictates a Lorentz invariant scalar product. As a consequence, the energy-momenta of bodies are limited to mass shells p2 = m2 and translation of the momenta is not a unitarily implemented symmetry in a development that is compliant with relativity. If multiplication by xν were a densely defined Hermitian operator, then translations in momentum would be unitarily implemented and the scalar product would be momentum translation invariant. Multiplication of functions by xν can not correspond to the location operator in relativistic physics since it cannot be Hermitian, and as developed in section 2.10, only Hermitian operators are associated with observables. Indeed, it is demonstrated in section 5 that the operator Xν that elevates xν is not Hermitian. The “position” operators Xν are not Hermitian due to the form of the relativistic scalar product (8). Xν is a Hermitian operation in L2, the Hilbert space appropriate in nonrelativistic physics, but in relativistic physics, the operator Xν no longer corresponds to location even though the supports of the functions that describe states still correspond to the location of bodies in appropriate instances. From the properties (100) of the Fourier transform, translation of a function corresponds to −ika multiplication of the Fourier transform by e . Recall that kx = k0ct − k · x, a Lorentz invariant. For both the free field and more general unconstrained developments, the Hamiltonian is H = ~ck0 [34]. From the scalar product (8), translation of the function to x − a corresponds to translation of a field to x + a. Fourier transforms correspond

−ika e ψ˜(~k) ↔ ψ(x − a) and then for suﬃciently small kak, Taylor series approximation of the exponential function and ψ(x) provide that dψ(x) (1 − ika)ψ˜( k) ↔ ψ(x) − a · . ~ dx 7 APPENDICES 156

Then the operator Pν that corresponds to multiplication of the Fourier transform by pν is d Pν = −i~gνν . dxν

The Minkowski signature is gνν = 1, −1, −1, −1 for ν = 0, 1, 2, 3 respectively from (159) below. From U(a)ψ(x) := ψ(x − a), the translation invariance of the scalar product provides that translations U(a) are unitary and the generators, U(a) = exp(−iaK), are the energy-momentum operators Pν = ~Kν. From Stone’s theorem, the energy-momentum operators are densely deﬁned and Hermitian. Similarly, multiplication of functions by xν generates translations in energy-momenta. Fourier transforms correspond

eiqx/~ψ(x) ↔ ψ˜(p − q) and then for suﬃciently small kqk,

dψ˜(p) (1 + iqx/ )ψ(x) ↔ ψ˜(p) − q · . ~ dp

Then the operator Xν that corresponds to multiplication of a function by xν is d Xν = i~gνν . dpν From T (q)ψ˜(p) := ψ˜(p − q), if the scalar product were energy-momentum translation invariant then the

T (q) = exp(iqX/~) would be unitary and its Hermitian generators would be the position operators Xν. The correspondence of Xν and Pν with position and momentum respectively follows from interpretation of the support of the arguments of functions as spacetime coordinates and of the Fourier transforms as energy-momenta. Considering only spatial coordinates, ν = 1, 2, 3, d d Xν = xν, −i~ and Pν = i~ , pν (149) dpν dxν that apply to functions or the Fourier transforms of functions, respectively. The signs are opposite for the derivatives with respect to the time coordinate and for the derivatives with 7 APPENDICES 157 respect to energy. Signs are determined by the convention for the Fourier transform (24). The arguments xν and pν correspond with location and momentum, respectively, and the operations on functions Xν and Pν are given by (149) and canonically commute. But the properties of Xν and Pν as Hilbert space operators depend on the Hilbert space realizations, that is, depend on the scalar product. In the Hilbert space L2 appropriate for nonrelativistic physics [55], the scalar product is both spacetime and energy-momentum translation invariant and Xν and Pν are both densely deﬁned Hermitian operators. In nonrelativistic physics, the operators Xν and Pν correspond with location and momentum, respectively. For the relativistic scalar product, only the Pν are Hermitian since in relativistic physics, there is translation invariance but energy-momentum translations are constrained to mass shells. Demonstrated in section 5, multiplication by xν is not realized as a Hermitian Hilbert space operator for the relativistically invariant scalar products of elementary particle states. The identiﬁcation of Xν as the location operator and Pν as the momentum operator in nonrelativistic physics results in the Born-Heisenberg-Jordan relation for their commutator.

[Xν,Pν] = −i~. This commutation implies the Heisenberg uncertainty relation as discussed in section 2.11 and a Baker–Campbell–Hausdorﬀ identity implies that similarity transforms of the location operators translate the eigenvalues. ia −iaν Pν / iaν Pν / ν e ~Xνe ~ = Xν + [Xν,Pν] ~ (150) = Xν + aν.

3 As a consequence, there is a simultaneous eigenfunction of the Xν for every a ∈ R given a function that is in the union of the null spaces of the Xν, ν = 1, 2, 3. From Xνψ0(x) = 0,

ia1P1/~ ia2P2/~ ia3P3/~ ψa(x) := e e e ψ0(x) satisﬁes Xνψa(x) = aνψa(x).

However, this result followed solely from the Born-Heisenberg-Jordan relation and there are many other diﬀerential operators Xbν that also canonically commute with the momentum operators Pν. From Clairaut’s theorem, the operations

d −1 Xbν = −i~u(p) u (p) (151) dpν mutually commute as long as u(p) is twice continuously diﬀerentiable. The Xbν canonically commute with the energy-momentum operators,

[Xbν,Pµ] = −i~δν,µ 7 APPENDICES 158

with ν, µ = 1, 2, 3. Then, there is a class of differential operators Xbν parametrized by twice continuously differentiable functions u(p) that satisfy the Born-Heisenberg-Jordan commutation relations [Xbν,Pν] = −i~ (73) and that mutually commute, [Pν,Pµ] = [Xbν, Xbµ] = 0. The physically relevant operations are Hermitian in the scalar product of the Hilbert space realization of quantum mechanics. Hermitian choices for the Xbν are prospective location operators for 3 the Hilbert spaces of interest: the Xbν have eigenstates for each location a ∈ R . The momentum operators Pν correspond to multiplication of the Fourier transform functions by pν and are generally self-adjoint (densely defined and Hermitian) due to the translation invariance of descriptions of physics. The form of the operators Xbν associated with location depends upon the Hilbert space realization of quantum mechanics. Three distinct sets of position operators are developed in this appendix: the elevation of xν that applies in a canonical quantization of nonrelativistic physics (149); one for the relativistic free field in Fock space; and one for the relativistic free field construction based upon the basis function space P used in the unconstrained development. 3 The eigenfunctions of these operators are labeled by locations a ∈ R , are orthogonal for distinct locations, and are dominantly supported near each location a although they are not of point support in the relativistic cases. Idealized classical locations are then representatives of the support of these eigenfunctions: the classical dynamic variables provide approximations to the support of the eigenfunctions. In relativistic physics, the lack of negative energy support and Lorentz covariance of the states implies that Dirac delta functions are not eigenfunctions nor generalized eigenfunctions of a Hermitian operator within the Hilbert space. The evident correspondence of particles with state descriptions provides that position operators apply for descriptions of free fields [32, 63]. A free field decomposes into canonically commuting particle creation and annihilation constituent operators; every state can be represented as a linear combination of particle creation operators applied to the vacuum state. When there is interaction, as discussed in section 2.1, these position operators apply to states in the one particle subspace and more generally provide an approximation for the locations of bodies described by states that are well described as classical particles. If high order connected functionals contribute significantly, then states are not necessarily interpretable as classical particles and the location of a particle as an observable is less evident. The extension of a single particle operator to multiple particle states (7) is a second quantization and follows (81) when interaction is lacking or in a Fock space. In Fock space, there is an underlying interpretation of states as particles [4, 23, 55, 63]. More generally, the interpretation of a location operator when there is interaction has not been determined. The discussion of section 2.1 illustrated that general state descriptions are not associated with determined numbers of particles. A particle interpretation is a classically inspired perception of the quantum description of state. In (151), u(p) = 1 is the established position operator Xν and this Xν is Hermitian in the Hilbert space L2 applicable in nonrelativistic physics. The eigenfunctions of the Xν are Dirac delta functions ideally associated with classical positions. More generally, in nonrelativistic limits, kpk mc, Xbν ≈ Xν if u(p) ≈ u(mc, 0, 0, 0). However, as discussed in section 1.1, 7 APPENDICES 159 eigenfunctions for a Hermitian operator are orthogonal in the Hilbert space scalar product: eigenfunctions of Hilbert space operators are not freely selected. The u(p) in (151) suffice to select Hermitian operators Xbν with orthogonal eigenfunctions for distinct eigenvalues for realizations of relativistic physics. The Hermitian operator associated with location is in the form (5) in section 1.5 derived from the association of classical bodies with states that are dominantly supported within small isolated volumes. From the discussion of Heisenberg’s uncertainty principle in section 2.11, product states with factors (p + ω)ϕ ˜(p) f˜(p) := 0 √ 2ω have scalar products that coincide with the L2 scalar product of theϕ ˜(p) in the one-particle subspace or more generally for states that are well described by classical particles. Using the translates (150), the selection of Dirac delta sequences (74) for ϕ(x) results in functions that describe the relativistically invariant localized states of [44]. For the relativistic free field in Fock space, the relativistically invariant localized functions are translates (150) of the limits of the Gaussian functions (74) after evaluation on the positive mass shell.

2 2 1/2 −σ (k−ko) 1/2 ψ˜0(p) := ω e o → ω .

For the relativistic free ﬁeld construction based upon the basis function space P used in the unconstrained development [32], the Hermitian operator that corresponds to location in a Hilbert space construction based upon P has the additional constraint that the eigenfunctions must not be supported on negative energy mass shells.

2 2 (p0 + ω) −σ (k−ko) (p0 + ω) ψ˜0(p) := √ e o → √ . 2ω 2ω

0 That is, Xbν : P 7→ P is required. States labeled by distinct locations a 6= a become orthogonal 3 as σo → 0 and then there is a (generalized) eigenfunction associated with every location a ∈ R . In both cases, the functions ψ˜a(p) are appropriate test functions that label elements of the constructed Hilbert space for ﬁnite σo. In both cases, elements in the null space of Xbν are limits σo → 0. 1/2 1/2 If either u(p) = (ω/mc) or u(p) = (p0 +ω)/(mc ω) with ω from (26), then the resulting Xbν is Hermitian in the one particle subspace for the relativistic scalar product. From (30) or 7 APPENDICES 160

(101), Z dp ˜ 1/2 d −1/2 hf1|Xbνg1i = δ(p0 − ω)f1(p) −i~ω ω g˜1(p) 2ω dpν Z dp −1/2 ˜ d −1/2 = −i~ ω f1(ω, p) ω g˜1(ω, p) 2 dpν (152) Z dp d −1/2 ˜ −1/2 = i~ ω f1(ω, p) ω g˜1(ω, p) 2 dpν

= hXbνf1|g1i from integration by parts. An alternative method to associate Xbν with location is to set Xbν equal to a Hermitian operator derived from Xν. 1 ∗ Xbν := (Xν + X ) 2 ν ∗ with the adjoint operators Xν defined for the free field scalar product (152). This derivation agrees with the result from the correspondence of states. That is, ˜ i~ d d −1 ˜ Xbνf1(p) = − + ω ω f1(p) 2 dpν dpν ω0 = −i f˜0 (p) − f˜ (p) ~ 1 2ω 1 1/2 d −1/2 ˜ = −i~ω ω f1(p) dpν ∗ from hXν f1|g1i := hf1|Xνg1i, (152), integration by parts, and with the prime designating differentiation with respect to pν. However, this association is not general. In this case of location, the adjoint shares a dense domain with Xν. The domain of the adjoint of the field Φ(f) from (14) is trivial, the zero element, for the construction based upon P and correspondence of the classical and quantum fields is established through association of observables and not 1 ∗ Φ(b f) = 2 (Φ(f) + Φ(f) ). The location eigenfunctions are orthogonal. The two-point functional for a free field (30) is expressed using the Fourier transforms, Z dk W (ψ∗g) = ψ˜(ω, p)g ˜(ω, p) 2 2ω and then the relativistically invariant localized functions centered on distinct locations a and a0 are orthogonal. Z ∗ 1 −ik·(a0−a) W (ψ ψ 0 ) = dk e 2 a a 2

1 3 0 = 2 (2π) δ(a − a ). 7 APPENDICES 161

To characterize the spacetime support of ψ˜a(p), [44] uses a transform with a Lorentz covariant measure supported solely on the mass shell,

Z dk p2 − (mc)2 f (x) := 2 θ(E)δ eikxψ˜ (p) a (2π)3/2 (mc)2 a Z dk mc = λ−1 eiωct/~−ik·xψ˜ (p). c (2π)3/2 ω a

In the limit σo → 0 and for t = 0, fa(x) = F (r/λc) with √ Z dk e−ik·(x−a) F (r/λ ) := mc λ3 √ = c (λ /r)5/4H(1) (ir/λ ), (153) c c (2π)3/2 ω F c 5/4 c (1) 2 2 a Hankel function H5/4(z), real constant cF [44], and r := (x − a) . The support of F (r/λc) is dominantly near r = 0 due to the divergence as r−5/2 at the origin and a rapid decline dominated by exp(−r/λc) at large r. λc is the reduced Compton wavelength. Similarly to the functions within P [34], the relativistically invariant localized functions fa(x) do not equal zero within any spatial neighborhood. In the unconstrained development, states that correspond to a location are labeled by functions from P. Functions within P with similar properties to the relativistically invariant localized functions are 2 2 (p + ω)e−σo(k−ko) eik·a ψ˜ (p) ≈ λ4 0 √ (154) a c mc ω from (p + ω) u(p) = √0 . mc ω The Hermitian operator associated with (5) is √ (p0 + ω) d ω Xbν := −i~ √ ω dpν (p0 + ω) and the functions (154) are eigenfunctions centered on a. The spacetime support of these functions agrees with the analysis of [44] and the spacetime support can also be characterized 3 using the inverse Fourier transform. For everyϕ ˜(p) ∈ S(R ), −iEτ f˜(p) = (p0 + ω)e ϕ˜(p) is a function within P. τ is real parameter that translates the temporal support of the function. The inverse Fourier transforms of these functions are generalized functions with point support in time. √ p 2 0 f(x) = 2π λc δ(x(o) − τ) −λc ∆ + 1 − iδ (x(o) − τ) ϕ(x) (155) 7 APPENDICES 162

3 0 3 with ∆ the Laplacian for R , δ (t) the ﬁrst derivative of δ(t), and ψ(x) ∈ S(R ). These functions f(x) do not vanish in any spatial neighborhood [32, 34, 56]. From (154), the inverse Fourier transforms ψa(x) are

2 2 Z dk (p + ω)e−σo(k−ko) ei(k−ko)·a ψ (x) ≈ λ4 eikx 0 √ . a c (2π)2 mcω

2 2 From (155) and for τ = ko = 0, σo → 0 and r := (x − a) ,

√ Z dk e−ik·(x−a) ω δ(ct/λ ) ψ (x) = mc λ3 √ c − iδ0(ct/λ ) a c 2π ω mc c

√ p 2 0 = 2π δ(ct/λc) −λc ∆ + 1 − iδ (ct/λc) F (r/λc).

F (r/λc) is from (153). This ψa(x) has point support in time, has no support on the negative energy mass shell, has spatial support concentrated in a neighborhood of a, and functions centered on distinct locations are orthogonal. These functions are generalized eigenfunctions of an operator that is Hermitian for the one particle subspace of the relativistic scalar product and canonically commutes with the momentum operators Pν. By the characterizations developed in section 1, the relativistically invariant localized states are dominantly localized within a volume of radius proportional to the Compton wavelength of the body, 3.0 × 10−13 m for an electron, and are orthogonal. As a consequence, states such as relativistically invariant localized states would typically be perceived as localized states despite their lack of bounded support. States that are well described by classical particles are described by functions with dominant spatial support that are mutually widely space-like separated and as a consequence of cluster decomposition, the scalar product is dominated by the contributions of the two-point connected functionals. In Fock space, all states have evident interpretations as particles while the modification of the scalar product in the unconstrained development changes the perception of states as particles. With interaction, the number of particles in the description of a state is generally undetermined. In an abbreviated notation with |fni designating a state described only by n-argument functions, generally hfn|gì= 6 0 for n, ` ≥ 2 but hfn|gì ≈ 0 if n 6= ` for states that are well approximated as classical particles, For these states, a fixed number of particles can be associated with the number of arguments of the functions that describe states. If the spatial support of fn((x)n) and g`((x)`) are widely and mutually space-like separated, then the contributions of the high order connected functionals are negligible when cluster decomposition is satisfied and then the scalar products are essentially those of a free field. This is a classical particle correspondence of the quantum description of state and for such states, the states are conventionally interpreted using the free field particle creation operator. More general states 7 APPENDICES 163 include components with any number of arguments and then the association of state description with classical particles is not evident. The form of the position operator is not determined by elevation of a classical dynamical variable. The form of the location operator in relativistic physics derives from Hilbert space realization and perception of the states. Single particle states |ψa(x)i are determined by the function class P and the relativistic scalar product (8). The established elevations of multiplication by xν are appropriate for the Schwartz functions S and L2 Hilbert spaces of nonrelativistic physics, but in relativistic physics, the two forms for Xbν developed above apply. The operators {Xν,Pν : ν = 1, 2, 3} and the two example {Xbν,Pν : ν = 1, 2, 3} are three Hermitian Hilbert space operator realizations of canonical commutation relations, [Xν,Xµ] = [Pν,Pµ] = 0 and [Xν,Pµ] = −i~δν,µ, in three distinct (isomorphic) Hilbert space realizations. The {Xν,Pν : ν = 1, 2, 3} are Hermitian operators associated with the observables of location xν and momentum pν respectively in the Hilbert space appropriate in nonrelativistic physics, L2, and the two example {Xbν,Pν : ν = 1, 2, 3} are Hermitian operators associated with the observables of location and momentum respectively in Hilbert space realizations of relativistic physics. The basis of the association is that there is an eigenfunction of the Xbν for every location a, the eigenfunctions are orthogonal in the appropriate scalar product, and the eigenstates are dominantly supported in small neighborhoods of the locations a. It is a peculiarity of the Hilbert space appropriate for nonrelativistic physics, the momentum translation independence of descriptions of nonrelativistic physics, that these eigenfunctions are Dirac delta functions. More generally, the elevation of classical location is not the Hermitian operator associated with location.

7.3 Classical descriptions of nature In classical physics, the description of state is considered independently of observation, or perhaps more precisely, the state is considered determined and observation need not disturb that state. The state and its evolution can be observed without the observer participating in the dynamics. The classical state is immutable except as predicted by classical dynamical laws. Bodies are described by locations and the rates of change of location of the points within the body. These points are the body in classical mechanics, and classical dynamics describes the smooth motion of these points. Isaac Newton developed a calculus to describe the motion of bodies as smooth trajectories that satisfy second order differential equations. Trajectories are paths within a three dimensional volume and locations are labeled with, for example, East- North-Up coordinates. Trajectories are the history of the locations of the points in a body. In Newton’s mechanics, bodies remain in motion traversing straight line trajectories unless pulled or pushed by forces such as gravity. A universal time provides the rates of these motions. This universal time and the action at a distance described as force was of concern to many physicists. These concerns were mitigated by Albert Einstein with relativity and geometrodynamical gravity (general relativity). Relativity describes a physical basis for time and geometrodynamics 7 APPENDICES 164 describes the trajectories of bodies as geodesic lines in a curved spacetime. The geodesics locally appear straight but ultimately curve relative to each other and the curvature of spacetime is determined by the bodies. Geodesics are paths of extremal distance through the curved spacetime and are the paths followed by bodies subject solely to gravitation. The effect of bodies on curvature propagates at up to the speed of light so the concern with action at a distance is mitigated. Bodies respond to the curvature at their locations. More generally than gravity, action at a distance was mitigated in field theories with the concept that forces are due to the value of fields at the locations of the bodies. Changes to these fields causally propagate at no more than the speed of light from the sources of the fields located on bodies. An early example of a classical field theory is James Clerk Maxwell’s electrodynamics. Einstein sought a unified geometrodynamical description of the forces, and implementation of this idea was pursued by Einstein and continues in contemporary physics. However, regrettably from the viewpoint of classical physics, inconsistencies between classical concepts and observations had accumulated in the early 20th century. Many observations are utterly inconsistent with the classical concept of identifiable points moving smoothly along trajectories specified by initial conditions. Selected examples of these observations are introduced in section 4. These observations were eventually described with the development of quantum mechanics. Classical dynamical variables are real numbers such as the location, momenta and energy of geometric bodies and field strengths at each location. In particular, location is described by 3 three coordinates that label points in a three dimensional space, R . In Newtonian dynamics, these coordinates are the labels for the points in a Euclidean space, and in Einstein’s gravity, the coordinates are labels for the points in a locally Minkowski, Riemannian spacetime. In classical physics, bodies are described as geometric bodies within the configuration space and the geometry of this space has physical significance. Classical states are descriptions of the locations of bodies and the temporal rates of change of the locations. Together with classical dynamical laws, these initial conditions describe the state for all future times. The evolution of the state is determined by the initial locations and their rates of change. This classical concept rests on two properties that are not consistent with nature: bodies are distinguishable and as a consequence their particular evolution can be followed; and bodies follow continuous trajectories through space as time elapses. Both properties are excellent approximations (sort of true) in common instances but fail when our observations more closely probe nature. “Sort of true” is, of course, false and contradictions are evident in the observations of nature discussed in section 4. Newtonian mechanics provides excellent approximations for heavy bodies with relative velocities slow with respect to the speed of light. The observation that the speed of light is a constant independent of source velocity leads to key relativistic concepts including a dynamic time, the Lorentz transform that relates the spacetime coordinates assigned to events by distinct observers in uniform relative motion, and Lorentz invariants that are dynamic quantities described identically by all such observers. Newtonian mechanics provides an accurate description of heavy bodies with relative velocities slow with respect to the speed of light. The extension of dynamics to greater velocities 7 APPENDICES 165 and to James Clerk Maxwell’s electrodynamics includes the concepts of special relativity. In relativity, time is recognized as a dynamic quantity that varies with the properties of observers. A dynamic time explains the observed independence of the speed of light from the velocity of the light source. This independence of propagation speed from source velocity is predicted in Maxwell’s equations for electrodynamics, and was observed in the Michelson-Morley experiments and has subsequently been validated in many other observations. With recognition of relativity, time is no longer a universal parameter in dynamics but becomes a dynamical quantity. To accommodate this invariance of the speed of light, the symmetries of physics include Lorentz covariance that mixes and dilates observed time and space intervals. In some senses, dynamic time is affected as if time were the fourth dimension in a four dimensional Minkowski spacetime (the metric in this spacetime is x2 +y2 +z2 −(ct)2, with a sign change for the temporal contribution and inclusion of negative squared “distances”). A way to conceive of dynamic time is sometimes designated the Chronos principle, “time does not provide the description of motion, motion provides the description of time.” One of Albert Einstein’s many seminal contributions to physics was to grasp the significance and implications of what has become known as relativity. Relativity can be derived from consideration of two observers in relative motion and that the speed of light is a universal constant. The velocity difference of the two observers is constant, and the two observers are not accelerating. That the two observers do not measure the same time intervals follows from the constancy of the speed of light and that both observers see the same events but label them with distinct coordinates [43]. If we consider ourselves the stationary observer, we can contrast the time intervals we observe for a clock carried with the moving observer to the time intervals this second, moving observer actually measures. If the observer that we consider the moving observer builds a clock that counts the reflections of a light pulse that is contained between two mirrors, then the time t0 for the pulse to transit once between the mirrors is D t0 = (156) c with D the distance between the mirrors and c the universal constant, the speed of light. These are the time intervals measured by the moving observer. If we watch a light pulse transit the moving clock, the moving observer’s coordinates for the two events are

(ct0, x0, y0, z0) = x0(bottom) = (0, 0, 0, 0) and x0(top) = (ct0, 0, 0,D) if the coordinates are set so that the pulse travels the z-axis and the relative motion of these two observers is along the x-axis (orthogonal to the z-axis). The two events are the pulse leaving the bottom mirror and then later hitting the top mirror. Our observation of these same events 7 APPENDICES 166 is for transits of the light pulse for mirrors that are in motion. Our coordinates for the two events of the light pulse reﬂecting oﬀ the moving mirrors are

(ct, x, y, z) = x(bottom) = (0, 0, 0, 0) and x(top) = (ct, vt, 0,D) with coordinates in the same orientation. Because the velocity of light is the same for every observer, the time interval we measure is

pD2 + (vt)2 t = c because the distance traveled by the pulse in the interval t is the hypotenuse of a right triangle with sides D, the distance between the mirrors, and vt, the distance traveled by the moving mirrors during the transit interval. D does not vary since there is no relative motion of the observers in that dimension, and the observers agree on their relative speed v and that the light pulse is contained between the mirrors. That is, both observers see the same events. The speed of light is a universal constant and Pythagorus’ theorem gives us the distance traveled by the light pulse in our frame of reference. These two equations are solved for the relationship between the time intervals measured by the two observers and the result is that

t0 t = . (157) p1 − (v/c)2

We could also carry such a light pulse clock and then our measured time intervals are also t0 and we note that this moving clock is running slow, t > t0. The time intervals we observe for the moving clock are longer than our measured time intervals. The light pulse transits take longer when the clock is moving due to the constancy of the speed of light. These clocks define the time intervals for the two observers as seen by us. We conclude that the moving clock is running slow. However, from the second, moving observers view, he can consider himself as stationary and our clock as in motion. How can we see his clock as slow and he see our clock as slow? The motion of the two observers is relative and the first observer considers the second observer to be in motion with the slowed time. Resolution of this twin paradox, both of the two observers cannot have clocks that run slow, requires consideration of acceleration [43]. Indeed, if the two observers are brought back together, the two must agree on the age of each. The resolution of this paradox rests on the observation that to return the two observers to a common location, one or both must accelerate, that is, change their velocity, and then resolution of the paradox rests on the theory of gravitation. The Einstein principle of equivalence observes that there is no distinction between undergoing an acceleration and being in a gravitational field [43], and then the observed gravitational red shift implies that accelerations affect the rates of 7 APPENDICES 167 clocks and provides a resolution of the paradox. The ages of the observers depend upon the accelerations and relative speeds of the two observers, and the resolution of the twin paradox is that the traveling twin (the twin who accelerated to return) returns younger. Acceleration is not relative. The observer that did not accelerate ages the fastest. Or, the two observers could be symmetrically accelerated to a return event and then the two would be the same age. This relation between the two time periods t and t0, between the time interval t attributed to the light pulse clock by the observer in relative motion and the interval t0 observed by the person traveling with the clock, is an example of a Lorentz transformation, named for Hendrik Lorentz. The Lorentz transformation relates the coordinates for two observers in relative motion without accelerations and provides the time and distance measurements attributed to events by the two observers. The Lorentz transform derives from the covariance of Maxwell’s equations for electrodynamics [13] and relates the dynamic time and distance observations of coordinate systems in relative motion. With our consideration attached to one of the observers, denote his coordinates for bodies and events as x and our understanding of the coordinates for the observer in relative motion as x0. Expressed in common units with the spatial coordinates, the fourth coordinate of spacetime is x0 = ct. Then, the coordinates of spacetime are

x = (ct, x, y, z) (158) with c the speed a light (and the meaning of x is derived from context, either a Lorentz four- vector of spacetime coordinates or one of the three spatial coordinates). In four dimensional spacetime, relativity provides that the two sets of coordinates are related

x0 = Λx with Λ a nonsingular 4x4 matrix with the property that ΛT gΛ = g for

 1 0 0 0   0 −1 0 0  g =   . (159)  0 0 −1 0  0 0 0 −1 g is the Minkowski signature matrix (a metric tensor) and ΛT is the matrix transpose of T the Lorentz transformation Λ, (Λ )jk = Λkj. The transformations Λ of interest are the proper, orthochronous transformations. Proper designates that the determinant of Λ is positive, det(Λ) > 0, and orthochronous designates that the transformation preserves the positivity of time intervals, Λ00 > 0. Proper, orthochronous Lorentz transformations exclude reﬂections; if the spatial coordinate axes are right-handed then the primed spatial coordinate axes are right-handed. Examples of Lorentz transformations include a rotation of coordinates in the x-y 7 APPENDICES 168 plane through θ,  1 0 0 0   0 cos θ − sin θ 0  Λ =    0 sin θ cos θ 0  0 0 0 1 and a boost along the x-axis by a velocity v,  γ −βγ 0 0   −βγ γ 0 0  Λ =    0 0 1 0  0 0 0 1 with v β = c 1 1 γ = = . p1 − β2 p1 − (v/c)2 Key properties are that cos2 θ + sin2 θ = 1 and γ2 − (βγ)2 = 1. In the case of two observers in relative motion with velocity v aligned with the x-axes, the relationship between the two coordinates systems is a boost in the x-direction. ct0 = γct − βγx x0 = γx − βγct y0 = y z0 = z. x are the coordinates of the reference observer and x0 are the coordinates attributed to the second, moving observer. The Lorentz transform includes a Lorentz-Fitzgerald length contraction. Lorentz invariants are quantities that do not change value under Lorentz transformation. For example, both position (ct, x, y, z) and energy-momentum (E/c, px, py, pz) are Lorentz vectors that transform under a change of reference frame as x0 = Λx and p0 = Λp. Then x2, p2 and px are each deﬁned using the Minkowski signature (159) from px := Et − p · x and are Lorentz invariants. X px := pνgνµxµ ν,µ X T = pν(Λ gΛ)νµxµ ν.µ . X = (Λp)νgνµ(Λx)µ ν,µ = p0x0. 7 APPENDICES 169

The result (157) of the light pulse clock described above is a special case of this Lorentz transformation. The clock is collocated with one observer, the observer in motion, and the clock’s coordinates from the stationary observer’s viewpoint are x = (ct, vt, 0, z). The coordinates attributed to the moving observer by this stationary observer are

x0 = (γct − γβvt, γvt − γβct, 0, z) = (γct (1 − v2/c2), 0, 0, z) = (ct/γ, 0, 0, z).

In particular, t = t0/p1 − (v/c)2. This result coincides with (157). Note that the clock is stationary, x0 = 0, and that the coordinate orthogonal to the motion is unaffected, z0 = z, in the moving observer’s frame. The Lorentz transformation can also be verified and understood as the resolution to the paradox of light spheres. The universality of the speed of light, that the speed of light is independent of the velocity of the light source, leads to this apparent paradox. If a flash bulb goes off just as two observers in relative motion pass, both observers would describe light spheres centered on themselves as the light propagates outward. The “paradox” is how to understand that one sphere can be centered on two distinct observers as the observers separate. The pulse of light is observable, and the two observers descriptions must coincide even though they continue to separate. The resolution is that distance and time are dynamic. A point on the light sphere is any point (x, y, z) of radius ct expressed in either observer’s coordinates.

x2 + y2 + z2 = (ct)2 x02 + y02 + z02 = (ct0)2

If both descriptions of the light sphere are valid, then time intervals and length intervals measured by the two observers cannot agree since the centers for the two descriptions of the spheres do not agree. If x0 = x+ct, t0 = t, y0 = y, and z0 = z, then one of the two equations for the light spheres would be in error and the speed of light would not be a universal constant. However, assuming both equations for the light spheres are valid and that the two observers coordinate systems are related by the Lorentz transform resolves the paradox since x2 = x02 = 0 are Lorentz invariants. x2 +y2 +z2 −(ct)2 = 0 together with the Lorentz transform relation x0 = Λx between the time and distance measurements of the two observers in relative motion implies that x02 + y02 + z02 = (ct0)2. Both observers describe the light sphere as centered on themselves: their coordinates for events and the coordinates they attribute to the moving observer do not agree. The Lorentz transformation maintains consistency with a universally constant speed of light. One result of these considerations is that time is determined by state. One determines elapsed time by measuring the change in the state description of bodies, for example, measuring 7 APPENDICES 170 the distance traveled by a light pulse. No change in state, no elapsed time. Then, a particular classical state of nature is associated with a time and all later times correspond to changes from that initial state. The direction to time is established by change in state. There is no physical sense in which time can be reversed. Any change is a positive increment of time. Time is not a dimension like the spatial dimensions. Time is a label for states. One is not free to move in time. One cannot choose to move backward in time even though time is involved similarly to distance in the Lorentz transformation. This distinction is sometimes made in descriptions of spacetime as 3+1 dimensional rather than as 4 dimensional. Not only is time distinguished in the Lorentz transform and involves boosts rather than rotations, but it is a label for states rather than a dimension. You cannot go back to a time when formerly you were not present. You were not present. Otherwise, it is not the same time since it is not the same state. Your presence makes it not the same state. A new state description would include you. The state of nature determines the time, and you cannot be present to kill your father before he conceives you. The time travel popularized in science fiction does not occur, and certainly a being from the future cannot show up in what would otherwise be a rerun of a past state of nature. There is no concern for causal loop paradoxes that would occur if it were possible to travel in time as if it were a dimension. Time travel is nonsensical. Time changes are due to configuration changes to the spatial description of state and change has a positive time lapse associated with it, from present to future. Can nature return to a configuration nearly identical to a prior state except that you are present? Yes, but that is not in the past and the likelihood of such an occurrence is vanishingly small. All the future that had occurred would have been forgotten other than your memory of it, and your memory would include the erasure of events to return to the prior state. And, that state that is nearly identical to a past state except that you are present with your knowledge of the future is necessarily farther into the future.

7.4 Polar decomposition and type III operators A Hilbert space structure is characteristic of quantum mechanics. In sections 2 and 5, characterizations of Hilbert spaces and Hilbert space operators were introduced. In this appendix, the characterization is extended to describe types of ∗-algebras of Hilbert space operators, a characterization due to Francis Murray and John von Neumann published beginning in 1936 and including studies by Alain Connes, Uffe Haagerup, Minoru Tomita, Masamichi Takesaki, Huzihiro Araki, E. J. Woods and Robert T. Powers [40]. From the Murray-von Neumann classification of the bounded linear Hilbert space operators that possess adjoints, “discrete” type I and “purely infinite” type III cases appear in descriptions of free fields [70]. Hilbert space operators are a remarkably rich study and include the Type III operators that model Hermi- tian free scalar fields based on the Schwartz tempered functions S. Type III operators exhibit properties achievable only in infinite dimensional Hilbert spaces. Given a Hilbert space, the von Neumann algebra of all bounded operators with adjoints is always the discrete type I case although in infinite dimensional Hilbert spaces, subsets of operators of types II and III appear. 7 APPENDICES 171

However, in the unconstrained development, the characterization of ∗-algebras of operators has less emphasis. Like free fields, the unconstrained models are constructed from explicit kernel functionals defined on an algebra of function sequences. But, in the example unconstrained developments based upon the not-∗-involutive P, the adjoint of the field has a trivial domain, the zero element, and as a consequence, the bounded operator derived from the field, the Weyl form exp(iΦ(f)), is not an element of a von Neumann algebra. A von Neumann algebra includes the operator adjoints. Appendix 7.2 discusses an extension of the Hilbert space that demonstrates that Φ(f) defined by (14) is a Hilbert space operator if formal Hermiticity is satisfied, but the field does not satisfy the physical conditions in this extended Hilbert space. The support of states is not limited to positive energies in the extension of nontrivial examples. The orthogonal projections onto subspaces of states generate von Neumann algebras, and the Hermitian operators associated with observables remain elements of von Neumann algebras, but the fields (1) are generally not elements of a von Neumann algebra. All separable Hilbert spaces are isomorphic to `2, the Hilbert space with n-tuple vector elements and the Hilbert space operators include matrices. However, there are operators in infinite dimensional `2 that are not generally representable as matrices. The isometry of a general separable Hilbert space with `2 is developed by construction. From any Hilbert space with n linearly independent elements labeled

{uk : k ∈ N, 1 ≤ k ≤ n} with scalar product huk|uji, the Gram-Schmidt construction results in an orthonormal basis {ek : k ∈ N, 1 ≤ k ≤ n, hek|eni = δk,n} with δk,n the Kronecker delta. The elements in the Pn Hilbert spaces of primary interest to physics expand as u = k=1 akek with ak ∈ C. The Gram-Schmidt construction follows from deﬁnition of projections hf|ui |E ui := |fi. f hf|fi

2 ∗ It follows that Ef = Ef = Ef . An orthonormal set {|eki : 1 ≤ k ≤ n} is constructed from a set of linearly independent elements |uki as

k−1 X |vki |v i := |u i and |v i := |(1 − E ) u i with |e i := 1 1 k vj j k kv k j=1 k and the norm kvkk is from the scalar product (6). There are n linearly independent elements if no |vki = 0. The Gram-Schmidt construction associates each element |uki in a general Hilbert space of n linearly independent elements with a linear expansion with complex coeﬃcients of n- tuple vectors |eki := (... 0, 1, 0 ...), all zeros except that the kth entry is unity. This association is an isometry (structure preserving mapping) of the elements of the particular Hilbert space Hn with the n-tuple elements of the n-dimensional `2 Hilbert space. Riesz’s theorem provides 7 APPENDICES 172

that a separable Hilbert space can be decomposed as the direct sum of Hk := EkH with Ek the projection associated with the basis element |eki.

|Ekeki = |eki

|Eke`i = 0, k 6= `.

PN The orthogonal complement of |eki can be decomposed and so forth to express H = k=1 ⊕Hk for any basis |eki with 1 ≤ k ≤ n. This decomposition is not unique, and an alternative basis is constructed given any unitary operator U not equal to the identity.

0 |eki := |Ueki.

A direct sum is a composition of Hilbert spaces H1 and H2 into an H := H1 ⊕ H2, and the direct sum generalizes to any number of constituent spaces (the direct integral). From elements labeled ψ, g ∈ H1 with scalar product hψ|gi1, and elements labeled u, v ∈ H2 with scalar product hu|vi2, the direct sum composite Hilbert space H1 ⊕ H2 has elements labeled ψ ⊕ u, g ⊕ v and scalar product

hψ ⊕ u|g ⊕ vi := hψ|gi1 + hu|vi2 and dim(H) = dim(H1)+dim(H2). There are operators deﬁned in H1 ⊕H2 that are extensions of the operators from H1 and H2. Deﬁne Hilbert space operators C := A ⊕ B by

hψ ⊕ u|C g ⊕ vi := hψ|Agi1 + hu|Bvi2.

Then all the operators A ⊕ 1 commute with all 1 ⊕ B and the two subspaces H1 and H2 are stable under the operations of the Hilbert space operators A ⊕ B. If all linear operators on H are considered, then the subspaces are not stable. More general operators on the Hilbert space of dimension dim(H1) + dim(H2) are distinguished from the operators A ⊕ B that are essentially block diagonal. The domain of bounded operators A can be extended to the entire Hilbert space, DA = H. And, if A is bounded and its adjoint A∗ is a densely deﬁned Hilbert space operator, then A∗ is also a bounded operator (kA∗k = kAk = kA∗Ak1/2). The null spaces and ranges of bounded operators are subspaces of the Hilbert space, NA = ENA H and RA := ERA H associated with two projection operators ENA ,ERA . The null space consists of elements zeroed by the operator, ∗ |Aψi = 0 if |ψi ∈ NA. An isometry is a bounded Hilbert space operator A with A A = I . For an isometry A, AA∗ = (AA∗)∗ and (AA∗)2 = AA∗AA∗ = AA∗ and as a consequence AA∗ is ∗ ∗ ∗ also a projection. If A A = AA = I then the isometry A is unitary. If AA = E < I , then q the isometry A is designated a partial isometry. Partial isometries include right shifts in `2. qA more general nomenclature for partial isometry is a Hilbert space operator A such that both A∗A and AA∗ are self-adjoint projections. 7 APPENDICES 173

The polar decomposition of a bounded Hilbert space operator A is

A = V (A∗A)1/2.

(A∗A)1/2 is a nonnegative, bounded operator and V is an isometry (inclusive of partial isometries and unitary V ) [40]. An operator R is nonnegative if hψ|Rψi ≥ 0 for all ψ ∈ H. If the null spaces of A and A∗ consist only of zero (Au = 0 and A∗u = 0 have only the trivial solution u = 0), then A and A∗ are invertible and V is unitary,

V := A(A∗A)−1/2.

Indeed, VV ∗ = A(A∗A)−1/2(A∗A)−1/2A∗ = AA−1(A∗)−1A∗ = 1 and V ∗V = (A∗A)−1/2A∗A(A∗A)−1/2 = 1. Or, if H is ﬁnite dimensional, then V is unitary. For example,

0 0 0 1 1 0 A = = V (A∗A)1/2 = 1 0 1 0 0 0 and 1 0 0 0 A∗A = and AA∗ = . 0 0 0 1 These example A∗A and AA∗ have the same dimensionality but distinct initial and final spaces. The initial space of an isometry A is the complement of the null space (the null space is the span of vectors |ui such that |Aui = 0) and the final space is the range of the isometry A. The initial space is also the range of A∗A, and the final space is the range of AA∗. Hilbert spaces with an unbounded number of dimensions introduce the possibility that the V in a polar decomposition is a partial isometry. For example, the shift with     v1 0  v2   v1  A   :=    v3   v2   .   .  . . and the observation that there is no last element in a infinite dimensional sequence provides ∗ ∗ that A A = 1 while AA = 1 − E1 with E1 the projection onto e1 = (1, 0, 0,...). This is one of the paradoxes of infinity, that there is no last room in Hilbert’s hotel. The initial space of this A is H∞, the complement of the null space, and the final space of A is (1 − E1)H∞, the range of A. 7 APPENDICES 174

The Hilbert hotel example illustrates that in infinite dimensional Hilbert spaces, there are additional considerations for the limits of finite dimensional operators. Considering the isometry between general Hilbert spaces and `2, the linear operators are matrices for finite numbers of dimensions but there are more possibilities for operators than considering only the limits of square matrices within the completions of Hn to H∞. In infinite dimensional completions of `2, the V in the polar decompositions are not necessarily unitary. Precursors of these possibilities in finite dimensional Hilbert spaces are isomorphisms between Hn and HN with n 6= N. The Hilbert hotel example is illustrated in maps from Hn to Hn+1. In this example of a right shift, A : H2 7→ H3 with  0 0  0 1 0 A = 1 0 A∗ = .   0 0 1 0 1 As a consequence,

 0 0 0  1 0 A∗A = = 1 AA∗ = 0 1 0 = 1 − E . 0 1   1 0 0 1

A plausible, if vague, description of the origin of the additional limits for operators on Hn → H∞ is that all countably infinite sets have the same number of elements (cardinality ℵ0). That is, as n, N grow without bound, Hn = HN = H∞ even for n 6= N and non-square matrices become effectively square. A development of these limits requires a context and a topology and this is included in the study of von Neumann algebras. In 1936, Murray and von Neumann published their development of an equivalence of subspaces in Hilbert spaces. Two subspaces are designated Murray-von Neumann equivalent if there is an isometry mapping the first isomorphically (preserving the scalar products) to the second. Since subspaces are one-to-one with the projections of a Hilbert space, the equivalence of subspaces is an equivalence of projections. The resulting equivalence is that if an A ∈ M, the von Neumann algebra, is a root of a projection E, E = A∗A, then Q = AA∗ is also a projection and Q is designated as Murray-von Neumann equivalent to E. This equivalence is designated E ∼ Q. The demonstration of the equivalence is first to show that Q = AA∗ is a projection if E = A∗A is a projection, and then that this root operator A is an isometry from EH 7→ QH. Since kEk = 1, Q is a bounded operator and its domain is the entire Hilbert space. kQk = ∗ ∗ kAk kA k = kEk = 1 and DQ = H. Since the domain is the entire Hilbert space, and Q = Q on its domain, then Q is self-adjoint. Q2 = AA∗AA∗ = AEA∗ and E = E2 = A∗AA∗A = A∗QA. 7 APPENDICES 175

Then Q2 = A(A∗QA)A∗ = Q3 and for every |ψi ∈ H,

2 kQ(I − Q)ψk = hQ(I − Q)ψ|Q(I − Q)ψi 2 2 = hψ|Q (I − Q) ψi = hψ|(Q2 − 2Q3 + Q4)ψi = 0 from the self-adjointness and that Q2 = Q3. The conclusion is that Q = Q2 and then Q is an orthogonal projection since it is self-adjoint and idempotent. From the discussion above, the initial space of A is the range of A∗A = E, and the ﬁnal space is the range of AA∗ = Q. That is, AE = QA = A. As a consequence, for every |ψi ∈ H,

hEψ|Eψi = hψ|Eψi = hψ|A∗QAψi = hAψ|QAψi = hQAEψ|QAEψi for every |ψi ∈ EH. That is |AEψi ∈ QH has the same scalar product as the corresponding |Eψi ∈ EH. This is the isomorphism, A : |Eψi ∈ EH 7→ |AEψi ∈ QH for the projections E,Q. An example of Murray-von Neumann equivalent projections that are not unitarily related results from the Hilbert hotel example. Consider the isometry of H with `2, with A the right shift within a subspace EH by k with E diagonal and then A∗A = E. The right shifted subspace is QH and the projection onto this subspace is Q < E. Murray-von Neumann equivalence establishes a comparison for projection operators. If E1 ∼ E and hψ|E1ψi ≤ hψ|Qψi for all ψ ∈ H, then Q is said to majorize E and this relation is designated E Q. That is, for two projections E and Q, if E = TT ∗ and hT ψ|ET ψi ≤ hψ|Qψi, then E Q. For the purposes of quantum mechanics, a von Neumann algebra is an algebra of bounded Hilbert space operators that includes the operator adjoints and the identity, and is closed in the weak topology. Weak topology neighborhoods of Ao are {A : A ∈ B(H); |h(A − Ao)ψ|gi| < , ∀ψ, g ∈ H} and B(H) is the set of bounded operators on the Hilbert space H. A von Neumann algebra is a subset of B(H) and B(H) itself is a von Neumann algebra. Results for von Neumann algebras include that the closures in the strong and weak topologies coincide, and von Neumann’s double commutant theorem applies [40]. The double commutant theorem provides that a von Neumann algebra consists of bounded operators in B(H) that commute with all the operators that commute with a set of bounded self-adjoint operators that include the identity. That is, a von Neumann algebra is equal to its double commutant and a von 7 APPENDICES 176

Neumann algebra is generated as the double commutant of a set of self-adjoint operators (for example, selected projection operators). The strong topology neighborhoods of an operator Ao are {A : A ∈ B(H); k(A − Ao)ψkk < , ψk ∈ H, 1 ≤ k ≤ n} with the norm from the scalar product (6). The commutant of a von Neumann algebra M operating in a Hilbert space H is the set of operators M0 := {T ∈ B(H): TA = AT ∀A ∈ M}. A von Neumann algebra F is designated a factor if F ∩ F 0 = {λI}, a scalar multiple of the identity. For a factor F, the double commutant theorem provides that F ∩ F 0 = (F ∪ F 0)0 and that F 0 is a factor if F is a factor. An example of a factor is B(H), the entire set of bounded operators on a Hilbert space H, and B(H) is a type I factor in the Murray-von Neumann classiﬁcation introduced below. These algebras of bounded Hilbert space operators are denoted factors because they factor the observables in quantum mechanics. If observable quantities correspond to Hilbert space operators, the observable quantities correspond to self-adjoint operators. In the unconstrained development above, the observables derive from interpretation of the spacetime and energy- momentum support of the functions that describe states and states are associated with self- adjoint projection (bounded) operators. Bodies are considered independent if their associated observables commute. For a factor F,

{AB : A ∈ F,B ∈ F 0}00 = B(H) and the self-adjoint operators are factored into commuting sets that generate all the bounded operators of H. The direct sum of Hilbert spaces is generalized to von Neumann algebras to decompose von Neumann algebras into more elementary von Neumann algebras. These elementary constituent von Neumann algebras are factors. A von Neumann algebra M of operators on a separable Hilbert space is isomorphic to a direct integral of factors [40]. For the example of a Hilbert space that is the direct sum of two constituent Hilbert spaces, an extension of the bounded operators within H1 to operators in H = H1 ⊕ H2 are A ⊕ I and these operators generate a von Neumann algebra that is the extension of B(H1) to H. In ﬁnite dimensional spaces, the products of A ⊕ I with more general operators in an H of the same dimension as H1 ⊕ H2 include A 0 UX AU AX = 0 I YV YV and in commuted order are UX A 0 UAX = . YV 0 I YAV

The von Neumann algebra M associated with operators in the subspace H1 ⊂ H is

A 0 M = { : A ∈ B(H1)} 0 λI 7 APPENDICES 177 and the commutant with respect to H is λ 0 M0 = { I : V ∈ B(H )}. 0 V 2 The intersection is λ 0 M ∩ M0 = { 1I } 0 λ2I and as a consequence M is not a factor considered within H but the {A} generate a von Neumann algebra that is a factor B(H1) considered in the subspace H1 of H. That is, B(H1) = EB(H)E with 0 E = { I 0 0 a central projection that projects B(H) into B(H1), EH = H1.A central projection is an element of M ∩ M0. This development is independent of basis, indeed, the commutator of ∗ ∗ ∗ ∗ UAU ∈ B(H) is [UAU ,UBU ] = U[A, B]U for any unitary U. In a factor, I is the only central projection. In this example, the central projections of H decompose the von Neumann algebra into two factors. In the example of a direct sum of three constituent Hilbert spaces 0 H1 ⊕H2 ⊕H3 with Fi = B(Hi), F1 6= F2 ∪F3 since the commutant is all operators in B(H) that 0 commute with F1. Indeed, F2 ∪ F3 ⊂ F1. A central projection E decomposes a Hilbert space H into orthogonal subspaces EH and (I − E)H (algebras EB(H)E and (I − E)B(H)(I − E)). The result described below is that central projections decompose a von Neumann algebra into a sum of subalgebras of operators of three major types. Murray and von Neumann used the properties of projection operators to characterize the possibilities for algebras of bounded linear operators realized in infinite dimensional separable Hilbert spaces. In the Murray-von Neumann classification, all von Neumann algebras of operators realized in finite dimensional Hilbert spaces are type I. The types of von Neumann algebras of bounded linear Hilbert space operators are classified by the properties of the projection operators in factors [40]. Type II and III algebras arise from the possibility of partial isometries. In the Murray-von Neumann classification, a projection E is finite if there is no projection Q and isometry T in the factor F such that E ∼ Q < E. That is, there is no projection Q = TT ∗ ∼ E = T ∗T such that hψ|(E − Q)ψi > 0 for all |ψi ∈ H. A projection E is abelian if the operators {TA = EAE : A ∈ F} all commute, TATB = TBTA. A projection E within a factor is minimal (one dimensional) if and only if it is abelian, Proposition 6.4.3 [40]. A factor F is classified:

Type I if there is a minimal (abelian) projection. The projections Ek onto basis elements |eki are ﬁnite, abelian projections onto one-dimensional subspaces EkH. Type II if there are ﬁnite projections but no minimal (abelian) projections. In a type II algebra, there are noncommuting operators within the set of operators {TA = EAE : A ∈ F} for every E ∈ F 7 APPENDICES 178

Type III if all nonzero projections are infinite. For an infinite projection E, there is a ∗ ∗ partial isometry T ∈ F such that E = T T and TT = I . That is, any subspace EH is isometric to the Hilbert space H and as a consequence, every subspace EH is infinite dimensional

These three types are comprehensive and mutually exclusive. Theorem 9.1.3 in [40] provides that the commutant of a factor F 0 is the same type as the factor F. The set of bounded operators on a Hilbert space, B(H), is a type I von Neumann algebra since it includes the n minimal projections associated with a basis set of elements {|eki : 1 ≤ k ≤ n}. The type II and type III algebras are subalgebras of B(H). Indeed, the traces (71) of any projections within a type III factor are either 0 or n = ∞, zero or infinite dimensional projections. Type I von Neumann algebras are classified further by dimensionality as In including the denumerably infinite I∞. Type II von Neumann algebras are classified further by whether the identity I is a finite (II1) or an infinite (II∞) projection [40] (whether there is a projection ∗ ∗ Q < I such that Q = T T and I = TT ). The demonstration of Corollary 6.1.9 in [40] that any two projections in a factor have Murray-von Neumann equivalent subprojections results in that every projection E ∼ I in a type III factor. Since B(H) are type I factors, then any type II or III factor acting on H is a proper subset of B(H), F ⊂ B(H). Indeed, if a minimal projection Ek 00 is appended to a type II or III factor F, then the resulting von Neumann algebra ((Ek ∪ F) , the double commutant) is type I∞. Infinite dimensional Hilbert spaces are isometric to the infinite dimensional `2 that has minimal projections onto the elements of a basis. Nevertheless, `2 includes all three types of algebras of bounded, linear operators as subalgebras of B(`2). Subspaces of infinite dimensional Hilbert spaces can differ by the operators considered, that is, by the types of von Neumann algebras considered. A particular von Neumann algebra realized in a Hilbert space is characterized by the number and types of the factors acting within orthogonal subspaces. Types I and III algebras of bounded operators are of interest in quantum physics [69, 70]. ∗ In type I Hilbert spaces H, the projection operators onto a basis factor Ek = Tk Tk for ∗ ∗ ∗ ∗ Tk = UEk with a unitary U ∈ B(H). Then Ek = EkEk = EkU UEk = Tk Tk and the Murray- ∗ ∗ ∗ ∗ von Neumann equivalent projections are Qk = TkTk = UEkEkU = UEkU , projections onto an alternative basis. Partial isometries appear in type II and III cases. In these cases, Qk = ∗ ∗ ∗ VEkV for a partial isometry V and V V = I within the subspace EkH. Ek = V QkV . And, in the example of L2, L∞ ⊂ B(L2) is type I even though the projections χa,b, the characteristic functions that are unity over intervals (a, b) of the real line, can be made arbitrarily smaller by choosing smaller intervals b − a but the EkAEk are one dimensional. Then, “minimal” projections refer to the properties of bases under isometric transformations rather than to existence of a smallest projection. In addition to B(H), another example of a factor is associated with the tensor product of Hilbert spaces introduced in section 2.12. The tensor product [40] of H1 and H2 extends to infinite dimensions. The tensor product is defined by the linear completion of elements labeled 7 APPENDICES 179

ψ ⊗ u in the Hilbert space norm (6). Operators A ⊗ B are defined hψ ⊗ u|A ⊗ B g ⊗ vi := 00 hψ|Agi1 hu|Bvi2 and (A ⊗ B) with respect to B(H1 ⊗ H2) is the tensor product von Neumann 00 algebra for A ∈ B(H1) and B ∈ B(H2). Then (A ⊗ B) = B(H1 ⊗ H2) as a consequence of that (λ1I) ⊗ (λ2I) = λ1λ2 (I ⊗ I) = λI and the commutation theorem for tensor products, 0 0 0 (F1 ⊗F2) = F1 ⊗F2. The operators A⊗I and I⊗B commute since the operators are defined by 0 operations on independent Hilbert spaces. As a consequence, (B(H1)⊗I) = I⊗B(H2) from the commutation theorem for tensor products. The tensor product is defined between distinct types of algebras and the type of the product is determined by the composite projections E ⊗ Q. For example, a type II∞ von Neumann algebra results from a tensor product of a type II1 factor and a type I∞ factor. There are no minimal projections due to the type II1 factor and the identity is infinite due to the type I∞ factor (e.g., the right shift). Precursors to type II and III behavior, behavior that does not occur in finite dimensional spaces, are suggested by the properties of morphisms between Hn ↔ HnN+M as n grows without bound. The properties of infinite dimensional spaces allow type II and type III operators on ∗ a Hilbert space. Consider Hilbert space operators T : Hn 7→ HnN and T : HnN 7→ Hn with n, N ∈ N, the natural numbers.   In  0n  T :=   and T ∗ := 0 ... 0 . (160)  .  In n n  .  0n with In the n × n identity matrix and 0n the n × n matrix of zeros. There are N n × n matrix entries in T and T ∗. Then the nN × nN product TT ∗ is   In 0n ... 0n  0n 0n ... 0n  E := TT ∗ =   (161)  . . .. .   . . . .  0n 0n ... 0n and ∗ T T = In (162) is n × n. Hn = HnN for an unbounded n → ∞ and N → ∞ and this suggests a type III case. In this limit, RTT ∗ = RE = EH and RT ∗T = H. With n and N growing without bound, both the projection E and H are infinite dimensional with the range of E a vanishingly small (1/N) subspace of H. This suggests the same relative contribution as a single basis vector ek in Hn as n grows without bound in the type I case. If we consider only the completion of the N orthogonal vectors that are N-tuples of sequences of n zeroes with one sequence of n ones, √ and normalize vectors by 1/ n then the minimal projections converge to zero and T remains 7 APPENDICES 180

ﬁnite. The result is an algebra of operators on a inﬁnite dimensional `2 that are not type I. A limit of this example T is an isometric map of H into EH.

hT u|T vi = hu|T ∗T vi = hu|vi and ET u = TT ∗T u = T u. As a consequence, T u ∈ EH and E is a projection, E = E∗ = E2. Type III von Neumann algebras of Hilbert space operators include partial isometries T such ∗ ∗ that T T = I , the unit operator in H, and TT = E for any nonzero projection E [40, 70]. T : EH 7→ H, an isometry between subspace EH and the entire Hilbert space H. The subspace EH has the same infinite dimensionality as the entire Hilbert space H [40] for any projection E included in the von Neumann algebra. Such partial isometries do not occur in finite dimensional Hilbert spaces. These arguments are only suggestive, and an appropriate development of von Neumann algebras is in [40] and its references. Despite the isometry with `2, not all operators in an infinite dimensional Hilbert space can be realized as square matrices. Linear operators are sesquilinear functionals A(ψ, u) = hψ|Aui ∈ C ∗ ∗ on pairs of elements from the Hilbert spaces. In the type III case with E = TT , T T = I and T : H 7→ EH isometrically. The isometry T is an operator that maps an element of H labeled by ek to an element T ek, an element of EH also labeled by ek and ET ek = T ek. That is, EH is infinite dimensional, not one dimensional. The Hilbert spaces H and EH are isomorphic, and isomorphic to the same ℵ0-dimensional `2 in this type III case. Then, although elements of H are expressed in an orthonormal basis as X |ψi = aj |eji j and the elements of the Hilbert space include |T eki for the isometry T , X |T eki= 6 tj,k |eji = t1,k |e1i j even though |ET ψi = |T ψi in the case that a projection E = TT ∗ is associated with a basis ∗ element labeled by e1 and T T = I . An element of EH could also be expressed X |Eψi = |ψi = aj |T eji j in terms of the linearly independent |T eji. The first matrix expression of the isometry T ∗ P results in a contradiction. The contradiction is that only t1,k 6= 0 and then TT = k tj,kt`,k = ∗ P δj,1δk,1 = E but T T = k tk,jtk,` = t1,jt1,` 6= δj,` = I . There are additional considerations in the representations of operators in the type II and III cases. These observations motivate a development of the EWG interpretation of quantum mechanics that parses assertions for the structure of the Hilbert space to those that are physically 7 APPENDICES 181 necessary, for example, to avoid the assumption that the observables for composite systems are necessarily realized in tensor products of constituent spaces. John von Neumann studied decompositions of the descriptions of states into independent bodies. Physics uses constructive mathematics, achieving arbitrarily accurate predictions in a specified amount of effort. The distinctions between the Murray-von Neumann types of Hilbert space operators necessarily occur only in the infinite dimensional Hilbert spaces. Nevertheless, impos- sibility proofs for quantum descriptions can inadvertently rely on assumptions for the structure of quantum mechanics and the concerns are rectified by fuller consideration of the types of Hilbert space operators permitted [70] and the basis function spaces that label states [34]. The possibility of type III operators and whether there are states labeled by functions of bounded support affects the logical consistency of the union of relativity and quantum mechanics.

7.5 Hilbert space field operators In this appendix, it is demonstrated that the field (1) defined by (14) for the example constructions based upon P is a Hilbert space operator if the VEV that satisfy formal Hermiticity W.a although the field is not Hermitian. For degenerate scalar products that satisfy formal Hermiticity, Wk,n−k = Wn independently of k. Examples that satisfy formal Hermiticity are the scalar fields in [32, 34]. That the field defined by (14) results in Hilbert space operators applies more generally, but the demonstration in this appendix is limited to the examples that satisfy formal Hermiticity. If there exists a ϕ ∈ S and an extension of the degenerate scalar product with

hΦ(f)h|Φ(f)hi = hϕ|hiS , (163) then the field preserves equivalence classes. hϕ|ψiS designates an extension of the degenerate scalar product to S such that hg|fi = hg|fiS if g, f ∈ P. In (163), f, h ∈ P and from (14), only one, one argument component of f is nonzero. The demonstration that equivalence classes of the Hilbert space norm (6) are preserved follows from the linearity of Φ(f), an extension to the ∗-involutive S of the demonstration of positive definiteness A.2, and the -Schwarz-Bunyakovsky inequality. For the example constructions of W in section 2.5, the extension of the degenerate scalar product to S does not satisfy formal Hermiticity W.a due to A.7 nor is support of the states constrained to positive energies. Restricted to function sequences from P, spectral support A.4 is satisfied and formal Hermiticity W.a is assumed satisfied for this demonstration. Formal Hermiticity is necessary for Φ(f) to be a Hermitian Hilbert space operator and for Φ(f) to correspond to a generalized random process [19]. If the extended scalar product satisfies

2 kf × hk = hϕ|hiS , 7 APPENDICES 182 then the Cauchy-Schwarz-Bunyakovsky inequality provides that

2 kf × hk = hϕ|hiS ≤ kϕkS khk using p khk = khkS := hh|hiS and as a consequence, Φ(f) preserves equivalence classes and is a Hilbert space operator. This demonstrates that kΦ(f)hk = kf × hk = 0 if khk = 0 for f, h ∈ P ⊂ S. The result is that the operation Φ(f) from (14) preserves equivalence classes in the norm (6) and extends from operation in the basis space of function sequences to Hilbert space operator. Φ(f) is a non-Hermitian Hilbert space operator in the Hilbert space based upon the non-∗-involutive function space P. A demonstration of the existence of a ϕ with the property (163) and extension of the constructed degenerate scalar product from P to S with the desired equivalence for sequences from P suffices to define the field (14) as a Hilbert space operator. By the construction in section 2.3, the ∗-involutive basis function space S includes P ∪ P∗. Kinematic constraints, introduced in section 2.2, that apply in P and imply satisfaction of elemental stability A.7 generally do not extend to S. (21) is an extension to S of a degenerate scalar product that satisfies A.7 for function sequences in P. This extension of the scalar product (8) is to define X ∗ hf|gi := Kn,m(fn, gm) n,m for function sequences f, g ∈ S with kernel functionals   0 if n < 2 or m < 2 and m 6= n Kn,m((x)n+m) := Wn,n((x)2n) if n = m (164)  Wn,m((x)n+m) otherwise using the kernel functionals Wk,n−k((x)n) in (8) with two-point functional

+ + W1,1(x1, x2) := ∆ (x2 − x1) + ∆ (x1 − x2).

These extended kernel functionals are symmetric with transpositions of the arguments within {1, k} or within {k + 1, n}. Recall that W0,1 = Wk,3−k = 0 and that the kernel functionals can be extended to W0,1 6= 0 by the method introduced above in section 2.5 and discussed in [34]. This selection of W1,1 is symmetric with interchange of the arguments and contributions + of the symmetric W1,1 equal those of ∆ for functions from P due to the limited energy support. ∆+ is the Pauli-Jordan function from (30). This choice of kernel functions implements a degenerate scalar product (8) for function sequences in S and implements the property A.7. It 7 APPENDICES 183

is demonstrated in [34, 38] and section 2.8 that these Wk,n−k define continuous linear functionals hf|gi for g, f ∈ S. The only use of fn, gm ∈ P in the demonstration of the nonnegativity of the degenerate scalar product in section 2.6 is validity of A.7 and the invariance of contributing terms in the kernel functionals under interchanges of pj and pk within (p)n, or pj and pk within (p)n+1,n+m. The kernel functions Kn,m explicitly implement A.7 and the symmetric W1,1 restores invariance with argument interchanges for free field contributions. The free field functional simplifies to (31) for function sequences from P. For the construction (164) of the kernel functions based on the example W , a degenerate scalar product hϕ|ψi with the desired equivalence extends to S. A demonstration that the extension of the scalar product (8) to S satisfies (163) for khk = 0 remains. It suffices to demonstrate that (163) defines an ϕ ∈ S. I suspect but have not provided a demonstration that this follows if the

Qk,2n−k((p)2n)(κ)2n from (38) evaluated on mass shells and with energy-momentum conserved are multipliers in S. The convenient case demonstrated here is if formal Hermiticity W.a is satisfied. If formal Hermiticity is satisfied, ϕ = f ∗ × f × h. By relabeling summation indices, the definition of Kn,m results in the difference ∞ Z ∗ X ∗ ∗ ∗ hg|f × hi − hf × g|hi = d(x)n+m+1 (Kn,m+1 gnf1hm − Kn+1,m (f1 gn) hm) n,m=0 ∞ Z X ∗ = d(x)n+m+1 (Kn,m+1 − Kn+1,m) gnf1hm n,m=0

= h0 W1,1(g1, f1) − g0 W1,1(f1, h1)

= 0.

This follows from (164) and formal Hermiticity, and from (8), h0 = h1(x) = 0 is necessary to khk = 0 for the constructions of interest. The product (11), ∗-morphism (9), and the definition of Kn,m (164) produce this result. For the example constructions, the fields (14) are densely defined non-Hermitian Hilbert space operators if there is always a ϕ ∈ S that satisfies (163). The field is not a Hermitian Hilbert space operator, Φ(f)∗ 6= Φ(f), if there are no ϕ ∈ P that satisfy (163) and formal Hermiticity is satisfied. From (163), ϕ := Φ(f)∗Φ(f)h = Φ(f)∗ (f × h) ∗ and if ϕ 6∈ P, then Φ(f) maps function sequences in the basis space out of P and Φ(f)h 6∈ DΦ∗ . The mapping Φ(f) defined in (14) remains within P. 7 APPENDICES 184

7.6 Tensor products of linear vector spaces

Introduced in section 2.12, a tensor product is the composition of linear vector spaces H1 and H2 into a composite H := H1 ⊗ H2.

From elements described ψ, g ∈ H1 with degenerate scalar product hψ|gi1, and elements described u, v ∈ H2 with degenerate scalar product hu|vi2, the tensor product composite Hilbert space H has states labeled ψ ⊗ u, g ⊗ v and degenerate scalar product

hψ ⊗ u|g ⊗ vi := hψ|gi1 hu|vi2. (165)

In this appendix, the suﬃciency of this assignment to specify a degenerate scalar product on the tensor product space is addressed. (165) is evidently a degenerate scalar product of product elements ψ ⊗ u, g ⊗ v, but does this extend to more general elements of H? First it is established that the Cauchy-Schwarz-Bunyakovsky inequality applies for all product states, and then this result is used to demonstrate that the degenerate scalar product applies to linear combinations of product states. Then the denseness of product functions in S provides that the degenerate scalar product applies to H. The Cauchy-Schwarz-Bunyakovsky inequality applies for all product states.

2 2 2 |hg1 ⊗ f1|g2 ⊗ f2i| = |hg1|g2i1| |hf1|f2i2|

≤ hg1|g1i1hg2|g2i1hf1|f1i2hf2|f2i2

= hg1 ⊗ f1|g1 ⊗ f1ihg2 ⊗ f2|g2 ⊗ f2i from (165) and the Cauchy-Schwarz-Bunyakovsky inequality in the constituent linear vector spaces. As a consequence, (165) extends to linear combinations of product functions.

|hu + v|u + vi|2 = hu|ui + hv|vi + hu|vi + hv|ui = hu|ui + hv|vi + 2

7.7 The degenerate scalar product for free field VEV In this appendix, the well known result [6, 8, 59] that the sesquilinear form (8) defines a degenerate scalar product for function sequences from S and the free field VEV is demonstrated in the notation and methods of these notes. It suffices to demonstrate that the sesquilinear form is nonnegative. Every state labeled by a sequence from S is a linear combination of states labeled by function sequences from P. Discussed in [32] and applying the development of section 2.4, this result follows from the commutation relations − + [Φ (f1)κ1 , Φ (f2)κ2 ]± = W2(f1, f2)κ1κ2 , the properties of the vacuum state, and the limitation of the energy support of the creation − component to positive energies. There is a cyclic vacuum state with hΩ| ... Φ (f)κΩi = 0 and + hΩ|Φ (f)κ ... Ωi = 0. The expansion of a state described by an f ∈ S into a description by an g ∈ P results from normal ordering the factors of the creation and annihilation constituents of − + the field, Φ(f)κ = Φ (f)κ + Φ (f)κ. As a consequence, a demonstration of the nonnegativity of (8) for function sequences from P suffices for the free field VEV. From (32) in section 2.4, the free field VEV for P are k X Y Fk,k({1, 2k}) = σ(S) W2({j, ij}) S j=1 k (166) 1 X X Y = σ(S)σ(S0) W ({i0 , i }) k! 2 j j S S0 j=1 7 APPENDICES 186

P with the abbreviated notation σ(S) for signs from section 2.5. The summation S includes the k! distinct orderings of {k + 1, 2k} and result in k! distinct pairings j, ij with j ∈ {1, k} P and ij ∈ {k + 1, 2k}. The summation S0 includes the k! distinct orderings of {1, k}. From the discussion of section 2.5, symmetry of Fk,k({1, 2k}) under the indicated transpositions of 2 arguments and that σ(S) = 1 results in equality of the two expressions. S = {1, . . . k, i1, . . . ik}, 0 0 0 S = {i1, . . . ik, k + 1,... 2k} and evaluation of the signs σ(S) = ±1 is discussed in (46) with σ(So) = 1 for So = {1, . . . k, k + 1,... 2k}. The nonnegativity of the matrix DM(p) results in a factorization (33), DM(p) = C†(p)C(p) with D the Dirac conjugation matrix from (9) and C†(p) is the Hermitian transpose of C(p). Substituted into (30), this factorization results in Z ip1x1+ip2x2 + DW2(x1, x2)κ1κ2 = dp1dp2 e δ(p1 + p2) δ2 DM(p2)κ1κ2 Nc Z (167) X −ip2(x1−x2) + = dp2 e δ2 C(p2)`κ1 C(p2))`κ2 . `=1 With the designation of symmetrized functions X hk((x)k)(κ)k := σ(S)fk({i1, . . . ik}) S and for the free ﬁeld VEV (166), the sesquilinear form (8) becomes Z ∗ X X F(f , f) = d(x)n+m (D·)nFn,m((x)n+m)(κ)n+m n,m (κ)n+m

×fn(xn, . . . x1)κn...κ1 fm(xn+1, . . . xn+m)(κ)n+1,n+m k (168) X 1 X Z Y = d(x) DW (k+1−j, k+j) k! 2k 2 k (κ)2k j=1

×hk(xk, . . . x1)κk...κ1 hk(xk+1, . . . x2k)(κ)k+1,2k . Relabeling summation variables, the factorization (167) and Fourier transforms (24) produce

k N X 1 X Z Y Xc Z F(f ∗, f) = d(x) ( dp eipj (xj −xk+j )δ+C (p )C (p )) k! 2k j j `j κj j `j κk+j j k (κ)2k j=1 `j =1 (169) ×hk(x1, . . . xk)κ1...κk hk(xk+1, . . . x2k)(κ)k+1,2k   2 Z k ! k X 1 X Y + X Y ˜ = d(p)k δ  C`j κj (pj) hk((p)k)(κ) k! i k k (`)k i=1 (κ)k j=1 that is manifestly nonnegative and completes the demonstration. 7 APPENDICES 187

7.8 Evaluation of the scalar product hϕ2(0)|U(λ)ϕ2(λ)i In this appendix, the Gaussian summations that evaluate the scalar products (132) for the two body example in section 3.4 are completed. Each of three terms are evaluated, The three terms are the three terms in the expansion of the VEV functional W2,2(1234) in hϕ2(0)|U(λ)ϕ2(λ)i. These three terms include one of U2,2(1234), W2(13)W2(24) or W2(14)W2(23) in the notation (40). The contribution of the four-point connected functional U2,2(1234) to the expansion of hϕ2(0)|U(λ)ϕ2(λ)i is

2 2 Z Z −i( +λcq )λ 0 0 0 2 du i(ω +ω −ω −ω )u c e λc f d(p ) δ(p − p ) λ e 1 2 3 4 × 4 4 1 3 c 2π

−L (0)2p02 −ip0 ·ξ −L2(p0 −2q )2 e c 1 e 2 i e i 2 i ×

λc 02 0 2 02 0 2 0 2 −i( p +λ q ·(p −2q ))λ −L (λ) p ip ·ξ −L (p −2qf ) e 4 3 c f 4 f e c 3 e 4 f e f 4

2 2 Z Z −i( +λcq )λ 0 0 0 2 du iλ q ·(p0 −p0 )u ≈ c e λc f dp dp dp λ e c f 2 4 × 4 1 2 4 c 2π

−i λc p02λ −(L (0)2+L (λ)2)p02 e 4 1 e c c 1 ×

0 2 0 2 0 0 2 0 2 −ip ·ξ −L (p −2q ) −iλ q ·(p −2q )λ ip ·ξ −L (p −2qf ) e 2 i e i 2 i e c f 4 f e 4 f e f 4 .

This expression results from substitution of a Fourier expansion for the energy conserving delta function Z du δ(ω) = eiuω, 2π and substitution from the VEV (128) with (129) in coordinates (130) into the scalar product (132). From the nonrelativistic approximation for energies (131), the leading order of the nonrelativistic Taylor expansion near the classically corresponding values produces

0 1 02 0 1 02 ω1 + ω2 − ω3 − ω4 ≈ λcqf · (p2 − 2q1) + 4 λcp1 − λcqf · (p4 − 2q1) − 4 λcp3

0 0 = λcqf · (p2 − p4).

0 0 Conservation of momentum provides that p1 = p3. From the form of the VEV after evaluation of the mass shell Dirac delta functions (129), no factors of ωj appear in the contribution of Q4 −1/2 the free ﬁeld VEV and a factor of j=1 ωj appears in the contribution of the four point −1 connected function. The nonrelativistic approximation (91) for ωj is λc and as discussed in section 3.4. 7 APPENDICES 188

The second contributing term to the scalar product hϕ2(0)|U(λ)ϕ2(λ)i results from substitution of W2(13)W2(24) into the expansion of the scalar product (132).

2 2 Z −i( +λcq )λ 1 λc f 0 0 0 0 0 2 e d(p )4 δ(p1 − p3)δ(p2 − p4)×

−L (0)2p02 −ip0 ·ξ −L2(p0 −2q )2 e c 1 e 2 i e i 2 i ×

λc 02 0 2 02 0 2 0 2 −i( p +λ q ·(p −2q ))λ −L (λ) p ip ·ξ −L (p −2qf ) e 4 3 c f 4 f e c 3 e 4 f e f 4 (170) 2 2 Z −i( +λcq )λ 1 λc f 0 0 ≈ 2 e dp1dp2 ×

−i λc p02λ −(L (0)2+L (λ)2)p02 e 4 1 e c c 1 ×

0 0 2 0 2 2 0 2 −iλ q ·(p −2q )λ −ip ·(ξ −ξ ) −L (p −2qi) +L (p −2qf ) e c f 2 f e 2 i f e i 2 f 2 .

The third and ﬁnal contribution to the expansion of the scalar product hϕ2(0)|U(λ)ϕ2(λ)i is the same as (170) except that p3 and p4 are interchanged in the VEV. This contribution of W2(14)W2(23) to the scalar product hϕ2(0)|U(λ)ϕ2(λ)i is

2 2 Z −i( +λcq )λ 1 λc f 0 0 0 0 0 2 e d(p )4 δ(p1 − p3)δ(p2 + p4)×

−L (0)2p02 −ip0 ·ξ −L2(p0 −2q )2 e c 1 e 2 i e i 2 i ×

λc 02 0 2 02 0 2 0 2 −i( p +λ q ·(p −2q ))λ −L (λ) p ip ·ξ −L (p −2qf ) e 4 3 c f 4 f e c 3 e 4 f e f 4

2 2 Z −i( +λcq )λ 1 λc f 0 0 ≈ 2 e dp1dp2 ×

−i λc p02λ −(L (0)2+L (λ)2)p02 e 4 1 e c c 1 ×

0 0 2 0 2 2 0 2 iλ q ·(p +2q )λ −ip ·(ξ +ξ ) −L (p −2qi) −L (p +2qf ) e c f 2 f e 2 i f e i 2 f 2 .

The demonstration that this third term is real and contributes to a nonnegative norm derives from the delicate positivity properties of the free ﬁeld VEV [6]. If λc 2 2 2 2 i 4 λ + Lc(0) + Lc(λ) = Lc(0) + Lc(0) 2 2 = Lc(λ) + Lc(λ) , then the description of the center of mass results in a common factor in the scalar products and squared norms hϕ2(0)|U(λ)ϕ2(λ)i, hϕ2(0)|ϕ2(0)i and hϕ2(λ)|ϕ2(λ)i. A deﬁnition similar to 7 APPENDICES 189

2 2 2 λc (113) for L0(λ) , (133), Lc(λ) = Lc(0) − i 4 λ, satisﬁes this condition. With this assignment, the center of mass spreads as a free body with mass 2m and

3 Z ! 2 0 −(L (0)2+L (0)2)p02 π dp e c c 1 = 1 2 2 Lc(0) + Lc(0)

2 2 is a common factor to each term in hϕ2(0)|U(λ)ϕ2(λ)i, kϕ2(0)k and kϕ2(λ)k . As a consequence, the description of the function characterizing the center of mass of the two bodies does not contribute to whether a classical correspondence applies: if a classical correspondence applies then any description for the center of mass suﬃces. Similarly, phases common to the three contributing terms within hϕ2(0)|U(λ)ϕ2(λ)i are extraneous to satisfaction of (107). 2 2 −i( +λcq )λ e λc f is a common phase. With the common factor and phase neglected, the contribution of U2,2(1234) to the scalar product hϕ2(0)|U(λ)ϕ2(λ)i is

Z Z 0 0 2 du iλ q ·(p0 −p0 )u c dp dp λ e c f 2 4 × 4 2 4 c 2π

0 2 0 2 0 0 2 0 2 −ip ·ξ −L (p −2q ) −iλ q ·(p −2q )λ ip ·ξ −L (p −2qf ) e 2 i e i 2 i e c f 4 f e 4 f e f 4 .

Substitution of the Gaussian summations Z ˙ 0 ˙ 0 0 2 0 2 0 −iξf ·p u −iξf ·(p −2qf )λ ip ·ξf −Lf (p4−2qf ) dp4 e 4 e 4 e 4 e 3 (ξ˙ (u+λ)−ξ )2 ! 2 − f f ˙ π 4L2 −i2qf ·(ξf u−ξf ) f = e 2 e . Lf and

Z 0 0 2 0 2 0 iξ˙i·p u −ip ·ξi −L (p −2qi) dp2 e 2 e 2 e i 2 3 ˙ 2 ! 2 − (ξiu−ξi) i2q ·(ξ˙ u−ξ ) π 4L2 = e i i i e i 2 Li leaves a single u summation to evaluate the contribution of U2,2(1234). This contribution to 7 APPENDICES 190

the scalar product hϕ2(0)|U(λ)ϕ2(λ)i for the example two body state descriptions is

3 2 2 ! 2 Z c4λc π 2 2 exp(−i2qi · ξi + i2qf · ξf ) du exp(i2q λcu − i2q λcu) × 2π 2 2 i f Li Lf (ξ˙ u − ξ )2 (ξ˙ (u + λ) − ξ )2 exp(− i i ) exp(− f f ) 2 4L2 4Li f 3 1 2 ! 2 2 2 2 ! 2 c4 π 4πλc Li Lf = exp(−i2qi · ξi + i2qf · ξf )× (171) 2π 2 2 2 2 2 2 Li Lf (qf Li + qi Lf ) −2i (q2 − q2 ) ! i f 2 2 ˙ exp L qi · ξi + Li qf · (ξf − ξf λ) × 2 2 2 2 f qf Li + qi Lf ! ! −4L2L2 2 −(q · (ξ + ξ˙ λ) − q · ξ )2 exp i f q2 − q2 exp f i i i f . 2 2 2 2 i f 2 2 2 2 qf Li + qi Lf 4(qf Li + qi Lf )

2 The contribution of U2,2(1234) to the norms kϕ2(λ)k are

1 2 3 Z (L2 + L2 )(ξ˙ u − ξ )2 3 2 4 ! 2 c4λc π f f f f c4 π 4πλc |Lf | 2 du exp(− 4 ) = 2 . (172) 2π |Lf | 4|Lf | 2π |Lf | 2 2 2 qf (Lf + Lf )

2 and similarly with Li and qi for kϕ2(0)k . With the common factor and phase neglected, the contribution of W2(13)W2(24) to the scalar product hϕ2(0)|U(λ)ϕ2(λ)i is

Z 0 0 2 0 2 2 0 2 1 0 −iλ q ·(p −2q )λ −ip ·(ξ −ξ ) −L (p −2qi) +L (p −2qf ) dp e c f 2 f e 2 i f e i 2 f 2 2 2 3 ! 2 ˙ 2 ! 2 2 2 ! 1 π (ξi − ξf + ξf λ) 4Li Lf (qi − qf ) = exp − exp − × (173) 2 2 2 2 2 2 2 Li + Lf 4(Li + Lf ) Li + Lf L22q + L2 2q ! 2 ˙ i i f f exp(i2q λcλ) exp i(ξi − ξf + ξf λ) · f 2 2 Li + Lf

2 and the contribution of W2(13)W2(24) to the norms kϕ2(λ)k are

3 Z ! 2 1 −(L2 +L2 )(p0 −2q )2 1 π dp0 e f f 2 f = (174) 2 2 2 2 2 Lf + Lf

2 and similarly with Li for kϕ2(0)k . 7 APPENDICES 191

With the common factor and phase neglected, the contribution of W2(14)W2(23) to the scalar product hϕ2(0)|U(λ)ϕ2(λ)i is

Z 0 0 2 0 2 2 0 2 1 0 iλ q ·(p +2q )λ −ip ·(ξ +ξ ) −L (p −2qi) +L (p +2qf ) dp e c f 2 f e 2 i f e i 2 f 2 2 2 3 ! 2 2 ! 2 2 2 ! 1 π (ξ + ξ − ξ˙ λ) 4L L (qi + qf ) = exp − i f f exp − i f × 2 2 2 2 2 2 2 Li + Lf 4(Li + Lf ) Li + Lf L22q − L2 2q ! 2 ˙ i i f f exp(i2q λcλ) exp −i(ξi + ξf − ξf λ) · . f 2 2 Li + Lf This term does not significantly contribute to the scalar product if a classical correspondence applies. If the bodies are reliably associated with trajectories, then then the support of the states is sufficiently isolated and localized that the real parts of 1 (ξ + ξ − ξ˙ λ)2 + 4L2L2 (q + q )2 L2 + L2 4 i f f i f i f i f because the trajectories of the two classically corresponding bodies are sufficiently distinct to attenuate the term. As a consequence, the contributions of W2(14)W2(23) to the scalar products are negligible when the classical bodies reliably associate with designated function arguments: support is isolated and localized.

References

1. S. Albeverio, H. Gottschalk, and J.-L. Wu, “Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions,” Rev. Math. Phys., Vol. 8, 1996, pg. 763-817. 2. H. Baumg¨artel,and M. Wollenberg, “A Class of Nontrivial Weakly Local Massive Wight- man Fields with Interpolating Properties,” Commun. Math. Phys., Vol. 94, 1984, p. 331. 3. J.S. Bell, “On the Einstein-Podolsky-Rosen Paradox,” Physics, Vol. 1, 1964, p. 195. 4. J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, New York, NY: McGraw-Hill, 1965. 5. A.S. Blum, “The state is not abolished, it withers away: How quantum ﬁeld theory became a theory of scattering,” Studies in History and Philosophy of Modern Physics, Vol. 60, 2017, p. 46. 6. N.N. Bogolubov, A.A. Logunov, and I.T. Todorov, Introduction to Axiomatic Quantum Field Theory, trans. by Stephen Fulling and Ludmilla Popova, Reading, MA: W.A. Ben- jamin, 1975. 7 APPENDICES 192

7. H.J. Borchers, “On the structure of the algebra of ﬁeld operators,” Nuovo Cimento, Vol. 24, 1962, p. 214.

8. J.M. Cook, “The Mathematics of Second Quantization,” Trans. Am. Math. Soc., Vol. 74, 1953, pg. 222.

9. B.S. DeWitt, H. Everett III, N. Graham, J.A. Wheeler in The Many-worlds Interpretation of Quantum Mechanics, ed. B.S. DeWitt, N. Graham, Princeton, NJ: Princeton University Press, 1973.

10. J. Dieudonn´e, Treatise on Analysis, Volume II, trans. by I.G. MacDonald, Vol. 10-II in Pure and Applied Mathematics, A Series of Monographs and Textbooks, New York, NY: Academic Press, 1970.

11. P.A.M. Dirac, The Principles of Quantum Mechanics, Fourth Edition, Oxford: Clarendon Press, 1958.

12. P. Ehrenfest, “Bemerkung über die angenäherteGültigkeit der klassichen Machanik in- nerhalb der Quanatenmechanik,” Z. Physik, Vol. 45, 1927, p. 455.

13. A. Einstein, “Zur Elektrodynamik bewegter K´orper,” Annalen der Physik Vol. 322(10), 1905, p. 891.

14. A. Einstein, B. Podolsky and N. Rosen, “Can Quantum-Mechanical Description of Phys- ical Reality be Considered Complete?,” Physical Review, Vol. 47(10), 1935, p. 777.

15. Richard Feynman, “Simulating Physics with Computers,” International Journal of The- oretical Physics, vol. 21, 1982, p. 467.

16. R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume III, Reading, MA: Addison-Wesley Publishing Co., 1965.

17. I.M. Gel’fand, and G.E. Shilov, Generalized Functions, Vol. 1, trans. E. Salatan, New York, NY: Academic Press, 1964.

18. I.M. Gelfand, and G.E. Shilov, Generalized Functions, Vol. 2, trans. by M.D. Friedman, A. Feinstein, and C.P. Peltzer, New York, NY: Academic Press, 1968.

19. I.M. Gel’fand, and N.Ya. Vilenkin, Generalized Functions, Vol. 4, trans. A. Feinstein, New York, NY: Academic Press, 1964.

20. I.M. Gel’fand, M.I. Graev, and N.Ya. Vilenkin, Generalized Functions, Vol. 5, trans. E. Sale- tan, New York, NY: Academic Press, 1966. 7 APPENDICES 193

21. N.I. Akhiezer and, I.M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. 1, trans. M. Nestell, New York, NY: Frederick Ungar Publishing Co., 1966.

22. H. Gottschalk, “Green’s functions for scattering in local relativistic quantum ﬁeld theory,” dissertation, Ruhr-Univerist¨atBochum, 1998.

23. F. Gross, Relativistic Quantum Mechanics and Field Theory, New York, NY: Wiley- Interscience Publication, 1993.

24. B. C. Hall, Quantum Theory for Mathematicians, New York: Springer, 2013.

25. M. Hardy, “Combinatorics of Partial Derivatives,” The Electronic Journal of Combina- torics, Vol. 13, #R1, 2006.

26. H. Georgi. “Unparticle Physics,” May 2007, arXiv:hep-ph/0703260.

27. G.C. Hegerfeldt, “Prime Field Decompositions and Inﬁnitely Divisible States on Borchers’ Tensor Algebra,” Commun. Math. Phys., Vol. 45, 1975, p. 137.

28. R.A. Horn, and C.R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press, 1985.

29. R.A. Horn, and C.R. Johnson, Topics in Matrix Analysis, Cambridge: Cambridge Uni- versity Press, 1991.

30. J.D. Jackson, Classical Electrodynamics, New York, NY: John Wiley and Sons, 1962.

31. A. Jaﬀe, “Constructive Quantum Field Theory,” in Mathematical Physics 2000, ed. A. S. Fokas, London: Imperial College Press, 2000.

32. G.E. Johnson, “Consistent Descriptions of Quantum Fields,” Reports on Mathematical Physics, Vol. 80(2), 2017, p. 193.

33. G.E. Johnson, “Classical approximations of relativistic quantum physics,” April 2016, arXiv:quant-ph/1604.07654.

34. G.E. Johnson, “Introduction to quantum ﬁeld theory exhibiting interaction,” Feb. 2015, arXiv:math-ph/1502.07727.

35. G.E. Johnson, “Measurement and self-adjoint operators,” May 2014, arXiv:quant-ph/- 1405.7224.

36. G.E. Johnson, “Fields and Quantum Mechanics,” Dec. 2013, arXiv:math-ph/1312.2608.

37. G.E. Johnson, “Massless Particles in QFT from Algebras without Involution,” May 2012, arXiv:math-ph/1205.4323. 7 APPENDICES 194

38. G.E. Johnson, “Algebras without Involution and Quantum Field Theories,” March 2012, arXiv:math-ph/1203.2705.

39. G.E. Johnson, “Interacting quantum fields,” Rev. Math. Phys., Vol. 11, 1999, p. 881 and errata, Rev. Math. Phys., Vol. 12, 2000, p. 687. Included in the 2000 errata and contrary to the assertion in the 1999 note, the constructed Euclidean region semi-norm is insufficient to result in Hilbert space field operators.

40. R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol- ume I: Elementary Theory, and Volume II: Advanced Theory, Graduate Studies in Math- ematics, Vol. 15: American Mathematical Society, 1997.

41. G. Lechner, “Deformations of quantum ﬁeld theories and integrable models,” Commun. Math. Phys. Vol. 312, 2012, p. 265.

42. A. Messiah, Quantum Mechanics, Vol. I, New York, NY: John Wiley and Sons, 1968.

43. C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation, San Francisco, CA: W.H. Freeman and Co., 1973.

44. T.D. Newton and E.P. Wigner, “Localized States for Elementary Systems,” Rev. Modern Phys., Vol. 21, 1949, p. 400.

45. M.F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state,” Nature Physics, Vol. 8, 2012, p. 475.

46. H. Reeh and S. Schlieder, “Bemerkungen zur Unit¨ar¨aquivalenz von Lorentzinvarianten Feldern”, Nuovo Cimento, Vol. 22, 1961, p. 1051.

47. H. Rinke, “A remark on the asymptotic completeness of local ﬁelds,” Commun. Math. Phys., Vol. 12, 1969, p. 324.

48. J. Rosenberg, “A selective history of the Stone-von Neumann theorem,” in Operator Alge- bras, Quantization, and Noncommutative Geometry: A Centennial Celebration Honoring John von Neumann and Marshall H. Stone, R.S. Doran and R.V. Kadison ed., American Mathematical Society, 2004.

49. Walter Rudin, Real & Complex Analysis, New York, NY: McGraw-Hill Book Co., 1966.

50. D. Ruelle, Statistical Mechanics, Rigorous Results, Reading, MA: W.A. Benjamin, 1969.

51. D. Ruelle, “On the Asymptotic Condition in Quantum Field Theory,” Helv. Phys. Acta., Vol. 35, 1962, p. 147. 7 APPENDICES 195

52. E. Schr¨odinger,“Der stetige Ubergang¨ von der Mikro-zur Makromechanik,” Die Naturwis- senschaften, Vol. 14, Issue 28, 1926, p. 664.

53. E. Schr¨odinger,“Discussion of probability relations between separated systems,” Mathe- matical Proceedings of the Cambridge Philosophical Society, vol. 31 (4), 1935, p. 555.

54. E. Schr¨odinger,“Die gegenw¨artigeSituation in der Quantenmechanik,” Die Naturwis- senschaften, Vol. 23, Issue 48, 1935, p. 807.

55. F. Schwabl, Advanced Quantum Mechanics, trans. R. Hilton and A. Lahee, Berlin: Springer- Verlag, 1999.

56. I.E. Segal and R.W. Goodman, “Anti-locality of certain Lorentz-invariant operators,” Journal of Mathematics and Mechanics, Vol. 14, 1965, p. 629.

57. O. Steinmann, “Structure of the Two-Point Function,” J. Math. Phys., Vol. 4, 1963, p. 583.

58. A. Douglas Stone, Einstein and the Quantum, Princeton, NJ: Princeton University Press, 2013.

59. R.F. Streater and A.S. Wightman, PCT, Spin and Statistics, and All That, Reading, MA: W.A. Benjamin, 1964.

60. F. Strocchi, “The Physical Principles of Quantum Mechanics. A critical review,” Jan. 2012, arXiv:quant-ph/1112.1507.

61. D. Tong, String Theory, http : //www.damtp.cam.ac.uk/user/tong/string/string.pdf, 2009.

62. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton, NJ: Princeton University Press, 1955.

63. S. Weinberg, The Quantum Theory of Fields, Volume I, Foundations, New York, NY: Cambridge University Press, 1995.

64. A.S. Wightman, “Quantum Field Theory in Terms of Vacuum Expectation Values,” Phys. Rev., vol. 101, 1956, p. 860.

65. A.S. Wightman and L. G˚arding,“Fields as operator-valued distributions in relativistic quantum theory,” Arkiv f¨orFysik, Vol. 28, 1965, p. 129.

66. A.S. Wightman,“Hilbert’s Sixth Problem: Mathematical Treatment of the Axioms of Physics,” Mathematical Development Arising from Hilbert Problems, ed. by F. E. Browder, Symposia in Pure Mathematics 28, Providence, RI: Amer. Math. Soc., 1976, p. 147. 7 APPENDICES 196

67. E.P. Wigner, “Remarks on the mind-body question,” in I.J. Good, The Scientist Specu- lates, Heinemann, London, 1961.

68. https : //en.wikipedia.org/wiki/T urtles all the way down.

69. J. Yngvason, “Localization and Entanglement in Relativistic Quantum Physics,” Jan. 2014, arXiv:quant-ph/1401.2652.

70. J. Yngvason, “The Role of Type III Factors on Quantum Field Theory,” Reports on Mathematical Physics, Vol. 55(1), 2005, p. 135.