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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 ac- ceptance of quantum mechanical state descriptions as physical reality. Schr¨odinger's1926 analy- sis 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 realiza- tions of relativistic quantum physics suggest a revision to the canonical formalism's anticipated quantum-classical correspondences. Schr¨odinger'sanalysis uses classical dynamical variables as representatives for the support of appropriate states, and Newton and Wigner's relativis- tic 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, Poincar´ecovariant, 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 con- dition 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 over- lapping 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 h'2(0)jU(λ)'2(λ)i . 187 1 Background and summary 1.1 Realization and canonical quantization Does a \quantization" procedure to elevate classical dynamical models to relativistically covari- ant scattering amplitudes suffice, or does quantum mechanics provide a comprehensive descrip- tion 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 ap- plication 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 me- chanics [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 approxi- mations 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 be- low 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 char- acteristics of relativistic physics. There are realizations of quantum mechanics that exhibit the physical characteristics
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