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Home , John Stewart Bell, John von Neumann

Is the State Real in the Hilbert Formulation?

Mani L. Bhaumik

Department of and Astronomy, University of California, Los Angeles, USA. E-mail: [email protected]

Editors: Zvi Bern & Danko Georgiev Article history: Submitted on November 5, 2020; Accepted on December 18, 2020; Published on December 19, 2020.

he persistent debate about the reality of a quan- 1 Introduction tum state has recently come under limelight be- Tcause of its importance to quantum informa- The debate about the reality of quantum states is as old tion and the quantum community. as quantum physics itself. The objective reality underly- all of the deliberations are taking place using the ele- ing the manifestly bizarre behavior of quantum objects gant and powerful but abstract formal- is conspicuously at odds with our daily classical phys- ism of quantum developed with seminal ical reality. The scientific outlook of objective reality contributions from . Since it is commenced with the precepts of that rather difficult to get a direct perception of the events cemented our notion of reality for centuries. Discover- in an abstract , it is hard to trace the ies starting in the last decade of the nineteenth century progress of a phenomenon. Among the multitude of revealing the uncanny quantum world in the microscopic recent attempts to show the reality of the quantum domain shook that perception. state in Hilbert space, the Pusey–Barrett–Rudolph With the particular exception of Einstein, who was the theory gets most recognition for their proof. But some lone supporter of his postulated wave-particle for of its assumptions have been criticized, which are still photons for about two decades, continued to not considered to be entirely loophole free. A straight- think of the microscopic quantum world within the con- forward proof of the reality of the wave packet func- fines of the ingrained classical physics. Finally, with de tion of a single particle has been presented earlier Broglie’s proposed extension of the wave–particle dual- based on the currently recognized fundamental real- ity to matter particles like electrons and its experimental ity of the universal quantum fields. Quantum states verification, irrevocably opened the door ushering in the like the atomic levels comprising the wave bizarre new world of quantum physics. packets have been shown to be just as real. Here With some talented younger physicists like we show that an unambiguous proof of reality of the Schrodinger,¨ Heisenberg, Born, Dirac and others, quantum states gleaned from the reality of quantum the development of the quantum physics proceeded in a fields can also provide an explicit substantiation of break neck starting in early 1926. Although these the reality of quantum states in Hilbert space. efforts led to the most successful description of events in Quanta 2020; 9: 37–46. the atomic domain, the revelations of quantum physics were so weird that immediately a debate started about the significance of it all. This is an open access article distributed under the terms of the Creative Commons Attribution License CC-BY-3.0, which Soon the dispute came to a climax at the 1927 Solvay permits unrestricted use, distribution, and reproduction in any medium, Conference in Belgium with the famed Bohr–Einstein provided the original author and source are credited. debate. While Bohr insisting that there was no reality

Quanta | DOI: 10.12743/quanta.v9i1.142 December 2020 | Volume 9 | Issue 1 | Page 37 before a is measured, Einstein maintained the Standard Model provides a remarkably uni- there must be a reality even before a quantum state is fied view of all types of matter and (ex- observed. Almost a century later the debate is surprisingly cept for gravitation) that we encounter in our still thriving. A conspicuous example is the substantial laboratories, in a of equations that can fit account presented in the recent review article by Matthew on a single sheet of paper. We can be certain Leifer [1]. All of these contemporary considerations are that the Standard Model will appear as at least conducted using the abstract Hilbert space formulation of an approximate feature of any better future the- initiated by John von Neumann. ory. [7] After John Bell’s epochal paper [2] presented Bell’s inequality and its numerous experimental substantiations, Thus, it would be cogent to consider the space filling the reality of the quantum state is now more acceptable in universal Quantum Fields as the primary ingredients of contrast to the conclusion of earlier Bohr–Einstein debate. physical reality uncovered by us so far. An abundant The most prominent recent theory of reality is presented proof of this can be encountered all around us in several by Matthew Pusey, Jonathan Barrett, and Terry Rudolph different ways. The most direct convincing evidence [3]. Other theories are also advanced by Lucien Hardy comes from the fact that elementary particles like an [4] as well as Roger Colbeck and Renato Renner [5]. electron has exactly the same properties, such as - However, none of these latest advances is considered to energy, charge, etc., irrespective of when or where in be entirely loophole free. Leifer’s considerably extensive the universe it comes into existence—in the big bang, in review [1] provides a distinct example of the difficulties of astrophysical processes throughout the eons or anywhere reaching a definitive conclusion using the circuitous way in a lab in the world. of deliberations in the abstract Hilbert space. Here we A manifestation of the fluctuations of the quantum present a rather straightforward way to prove the reality fields in a phenomenon like the electron anomalous g- of the quantum state. factor agrees up to an unprecedented twelve decimal In order to avoid any possible confusion, it would be places when the experimental results are compared to prudent to agree upon the definition of reality. In this the theoretical computation. Observed phenomena like regard, we rely upon the generally acknowledged conno- Lamb shift, Casimir effect further assert the existence of tation of reality. We consider something to be physically the fluctuations of the quantum fields. A very dramatic real if it is independently observed by several people confirmation of the indispensable effects of the quantum and they agree with each other that the result of their field fluctuations comes from the mass of the compos- observations is the same. Accordingly, one could rely on ite particles like protons and neutrons. The mass of the the following notions of the distinguished contemporary three valence quarks in a proton provided by the Higgs physicists for our understanding of reality. boson is only about 9 Mev while the total proton mass is Referring to the outstanding developments in the a whopping 938 Mev. This magical “mass without mass” cutting-edge quantum field theory or QFT in short, the ascends from the endowment of quantum fluctuations. distinguished Physics Nobel Laureate Frank Wilczek as- Perhaps the most spectacular graphic evidence is pro- serts vided by the observed anisotropy in the cosmic mi- crowave background radiation with their presumed origin the standard model is very successful in de- in the cosmic inflation in the early universe when the scribing reality—the reality we find ourselves quantum fluctuations of the reputed inflaton field enor- inhabiting. [6, p. 96] mously expanded from the very microscopic to macro- scopic dimensions providing seeds for galaxy formation Wilczek additionally enumerates afterwards. Any reasonable concept of physical reality The primary ingredient of physical reality, from should then owe its eventual origin to the fundamental re- which all else is formed, fills all space and . ality of quantum fields and their characteristic attributes. Every fragment, each space-time , has The elementary particles like electrons, one of the the same basic properties as every other frag- members of the initial act of material formation from ment. The primary ingredient of reality is alive the abstract but physical quantum fields, are quanta of the with quantum activity. Quantum activity has fields. Each of them can be rendered as a wave packet con- special characteristics. It is spontaneous and sisting of an admixture of the various fields. Accordingly, unpredictable. [6, p. 74] the wave packet function of the elementary particle ought to be considered as real as the primary quantum fields. Another esteemed Physics Nobel Laureate Steven Wein- More generally the fields whose quantization produces berg confirms the 24 other observed elementary particles in Nature, such

Quanta | DOI: 10.12743/quanta.v9i1.142 December 2020 | Volume 9 | Issue 1 | Page 38 as the photons, W+, W−, Z0, quarks and gluons, embod- quently, the contribution of the different quantum fields ied in the Standard Model of should be to the single particle would comprise of irregular dis- just as real. Consequently so would be the composite turbances of the fields with energy off the mass shell, particles like protons and neutrons. We can then consider resulting in the electron ripple to be very highly distorted. the quantum states of atoms and molecules comprising It is well known that such a pulse, no matter how the elementary and composite particles real as well. deformed, can be expressed by a Fourier with This would empower us with confidence to extend the weighted linear combinations of simple periodic wave reality of the quantum states to those in Hilbert space forms like trigonometric functions [9]. The result would formalism including all its operations to be as real as the be a wave packet function to represent an electron that wave packet of a single quantum particle and the wave would embody a fundamental reality of the universe since functions of all quantum states of the Schrodinger¨ and all the amplitudes of the wave packet would consist of Heisenberg formulation. contributions of irregular disturbances of the various real primary quantum fields. The ψ(x), for simplicity in one dimen- 2 The wave function of an electron sion, will be given by the inverse

Z ∞ A particle like an electron arises as a quantized excitation 1 + ψ(x) = √ ψ˜ (k) eıkxdk (1) of the underlying electron quantum field. Such an energy- 2π −∞ eigenstate of the field can be expressed as a specific Lorentz covariant superposition of field shapes of where ψ˜(k) is a function that quantifies the amount of 2π the electron field along with all the other quantum fields each wave number component k = λ that gets added to of the Standard Model of particle physics. the combination. Superposition of field shapes in a one-particle state From Fourier analysis, we also know that the spatial evolves in a simple wavelike manner with time depen- wave function ψ(x) and the wave number function ψ˜(k) dence e−ıωt. The individual field shapes, each with their constitute a Fourier transform pair. Therefore, we can find own computable dynamic time evolution, are actually the the wave number function through the forward Fourier vacuum fluctuations comprising the very structure of the transform as energy-momentum eigenstate. The quantum fluctuations Z +∞ 1 −ıkx are evanescent in the sense that they pass away soon after ψ˜ (k) = √ ψ (x) e dx (2) 2π −∞ coming into being. But new ones are constantly boiling up to establish an equilibrium distribution so stable that Thus, the Fourier transform relationship between ψ(x) their contribution to the electron g-factor, as mentioned and ψ˜(k), where x and k are known as conjugate variables, earlier, results in a measurement accuracy of one part in a can help us determine the frequency or the wave number trillion [8]. content of any spatial wave function. The Lorentz covariant superposition of fluctuations of So far, we have considered only a fixed momentum. all the quantum fields in the one-particle quantum state Obviously in any , momentum would can be conveniently depicted leading to a well behaved be expected to change all the time. As the momentum smooth wave packet that is everywhere continuous and increases, the shape of the curve representing equation continuously differentiable. (2) in the momentum space would be taller and thinner However, the detailed quantitative calculations are while, being a conjugate variable, the curve in the position quite involved. For simplicity, we can get the result using space would be shorter and wider. This naturally leads a heuristic perspective. For this purpose, let us consider us to the postulate of famed , which an isolated single quantum of a non-interacting electron is evidently a natural consequence of the position and quantum field. Since no is acting on such an elec- momentum being conjugate variables in wave packets. tron, its momentum would be constant and therefore its Fortunately, a fairly rigorous underpinning of the wave position would be indefinite since a regular ripple from packet for a single particle QFT state in position space a free electron field with a very well-defined energy and for a scalar quantum field has been provided by Robert momentum is represented by a delocalized periodic func- Klauber [10, p. 275]. Since particles of all quantum fields tion. are invariably an admixture of contributions from essen- Recalling that the electron in reality is an admixture tially all the fields of the Standard Model, the wave packet of all the quantum fields of the standard model, it should function of a single particle of a scalar quantum field can be noted that in the non-relativistic regime, there would be considered to be qualitatively representative of those not be enough energy to create any new particle. Conse- of the vector fields like the electron quantum field.

Quanta | DOI: 10.12743/quanta.v9i1.142 December 2020 | Volume 9 | Issue 1 | Page 39 The mathematical form in position space of a wave Even a century later, quantum mechanics still perplexes packet function of a particle for a scalar field is [10] most people including many . This is consistent Z + with the fact that on an average at least one popular book − 3 0 −ık0 x 3 0 |φi = (2π) 2 A k e d k (3) on the riddles of quantum mechanics is still being pub- lished every year by notable authors. However, as has

0 been explained in detail [12], there appear to be plausible Using the more common convention of eık x in position answers to the enigmas of quantum mechanics derived space for an inverse Fourier transform, k instead of k0 from the discoveries of the quantum field theory of the for momentum, ψ(x) for |φi and ψ˜(k) for A(k0), equation standard model. (3) in one space dimension becomes exactly equation (1), thus confirming that the single particle wave packet For example, a wave packet function is localized and function given by (1) can be rigorously derived from QFT. therefore can represent a quantum particle, but just holisti- In order to determine the time evolution of the wave cally, since only the totality of the wave packet represents packet function, we need to incorporate the time term to all the conserved quantities of the energy-momentum the spatial function. Accordingly, eigenstate of a particle such as mass, charge, and spin. This particularly important fact requires the total wave 1 Z +∞ ψ(x, t) = √ ψ˜ (k) ei(kx−ωt)dk (4) function to collapse during measurement, which has been 2π −∞ one of the most puzzling aspects of quantum mechanics. where ω = 2πν is the . It can now be evident in fact as a requirement because of On a cursory glance, the wave packet in equation the particular factual nature of the wave packet. (4) looks very similar to the one that has always been used in quantum mechanics so far, for example, to solve Most of the topics in quantum mechanics, extensively Schrodinger¨ equation for the hydrogen atom. But in real- used in many diverse fields like chemistry, biology, mate- ity the wave function given by (4) is substantially different rial science, science, involve non- in character since the amplitudes are fundamentally real relativistic quantum mechanics since the particle physical entities owing their origin to the primary reality are lower than the speed of . The very extensively of different quantum fields. studied quantum objects are the hydrogen atom and the When quantum mechanics, with its essential require- quantum . ment of a wave packet function to represent a localized In the hydrogen atom, the electron revolves in the cen- particle, emerged in the atomic domain of physics and tral field of the proton producing discrete energy levels commenced explaining its mysterious accomplishments that correspond to characteristics standing wave patterns with uncanny consistency, it appeared totally contrary to called orbitals. These are calculated by using the time- our intuition developed from classical physics. Naturally, independent Schrodinger¨ equation using the real electron the wave packet of a particle was considered to be just wave function in equation (1). The normalized hydrogen a fictitious mathematical construct merely necessary for wave functions, using polar coordinates for convenience, carrying out calculations. can be found in textbooks on quantum mechanics for At best, the amplitudes of a wave packet are considered example in [13, § 4.2]. to be real probability amplitudes to find the particle dur- ing a measurement. This is consistent with the fact that The foremost aspect to underscore here is that the stand- the amplitudes of the wave packet represent distinctly dif- ing wave patterns representing the quantum states of the ferent properties of various quantum fields except energy. hydrogen atom are embodiments of the wave packet func- The energy represented by any amplitude of the wave tion in equation (1) that is based on the primary reality of packet has the same characteristics even though the other the quantum fields. These standing wave patterns of the properties of the amplitude may have different attributes orbitals representing the quantum states are thus objec- corresponding to their origin in the various fields. tively as real as the primary quantum fields. Although the amplitude of the wave function is complex-valued, its squared modulus is real-valued. Fol- In fact, reality of some of these orbitals has been estab- lowing Einstein, Born [11] had interpreted the square lished by experimental measurements [14]. Consequently of the modulus of the wave amplitude as probability for it would be compelling to infer that all quantum states of the occurrence of the particle. Thus probability p(x) of quantum mechanics are based on reality, which can be finding the particle at the position x in the interval x and extended to the quantum states described in the abstract x + dx is Hilbert space formalism, the child of p(x) = ψ∗ (x, t) ψ (x, t) dx (5) and John von Neumann.

Quanta | DOI: 10.12743/quanta.v9i1.142 December 2020 | Volume 9 | Issue 1 | Page 40 3 Emergence of Hilbert space a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of Although von Neumann was an exceptionally talented quantum mechanics. polymath, his contribution to the development of the His habilitation (qualification to conduct independent Hilbert space formalism is one of his outstanding lega- university teaching) was completed in December of 1927, cies to science. Following Hilbert’s preliminary thoughts, and he began lecturing as a Privatdozent at the Univer- he not only initiated the formulation but also coined the sity of Berlin in 1928 at the age of 25, the youngest name Hilbert space providing an advanced comprehen- Privatdozent ever elected in the university’s history in any sive mathematical foundation of quantum theory. subject. In 1929, he briefly became a Privatdozent at the , where the prospects of becoming 3.1 Von Neumann’s contribution a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Shortly after receiving his PhD in Physics from the Uni- . In 1933, he was offered a lifetime versity of in 1926, von Neumann decided to professorship at the Institute for Advanced Study in New under David Hilbert, likely the foremost mathemati- Jersey. He remained a mathematics professor there until cian of the time, for completing his habilitation program. his death, although he had announced his intention to At the University of Gottingen,¨ considered to be the pre- resign and become a professor at large at the University miere center for mathematics in the world for quite a of California, Los Angeles. while, Hilbert became interested in the indispensable ap- As a true polymath, he made significant contributions plication of intricate mathematics in the emerging topics to various diverse fields throughout his momentous career. like relativity and quantum physics. Perhaps with that It is fascinating to note that he himself considered his perception, he was delighted to attract this mathemati- pioneering contributions to the foundations of quantum cal genius to his group and secured for him a six-month mechanics especially the Hilbert state formalism as his fellowship from the Rockefeller financed International most important scientific contribution. Education Board with additional letters of recommenda- tion from two of his former students, Richard Courant In a recent article, Klaas Landsman [19] summarizes and . von Neumann’s main accomplishments in his pioneer- With all the exhilaration of the emergence of quantum ing contribution to the development of the Hilbert state physics in 1926, Hilbert opted to devote his long-standing formalism of quantum mechanics, with the admiring com- winter lectures on mathematics that year to the novel topic ment that any one of these would have been a significant and convinced von Neumann to join the effort under his achievement for a 23 year old. These are: guidance. These impressive lectures covered almost all the contemporary developments including Heisenberg’s 1. Axiomatization of the notion of a Hilbert mechanics, Schrodinger’s¨ wave mechanics, Born’s space (previously known only in examples). probability interpretation, and finally Jordan’s and Dirac’s transformation theory to unify them into a single formal- 2. Establishment of a for (pos- ism. These lectures were published [15], which was the sibly unbounded) self-adjoint operators. last publication by Hilbert on the subject possibly because of his failing health he could not keep up with the fast 3. Axiomatization of quantum mechanics in moving subject. However, he was thrilled by the mar- terms of Hilbert (and operators): velous coincidence that the mathematical formalism de- veloped by him before the advent of quantum mechanics (a) Identification of with (possibly ideally harmonized with the formulation of the mathe- unbounded) self-adjoint operators. matical apparatus necessary for an elegant and rigorous (b) Identification of pure states with one- treatment of the emerging new subject. dimensional projections (or rays). Von Neumann, as the most outstanding of Hilbert’s (c) Identification of transition amplitudes with heirs, continued to carry the torch and in 1927 published inner products. three brilliant papers [16–18] that placed quantum me- (d) A formula for the stating the prob- chanics on a meticulous mathematical foundation includ- ability of measurement outcomes. ing a rigorous proof of the equivalence of matrix and (e) Identification of general states with density wave mechanics. He also coined the name Hilbert space operators. that soon took off with every one starting to use it for (f) Identification of propositions with closed the new formalism. The significance of the concept of subspaces (or the projections thereon). [19]

Quanta | DOI: 10.12743/quanta.v9i1.142 December 2020 | Volume 9 | Issue 1 | Page 41 3.2 Dirac’s contribution system are vectors in a multidimensional Hilbert space containing an orthonormal basis set of eigenfunctions. In the meantime, another star shined in the rising quantum The observables are Hermitian operators on that space, firmament. received his PhD in Physics under and measurements are orthogonal projections. Unitary Ralph Fowler at Cambridge in 1926 just about the same operators are used for changing a vector from one basis time did von Neumann. It is quite interesting to note that set to another. For elegance, Dirac’s bra-ket notation is both of them acquired a PhD in engineering before veer- used to characterize the vectors and delta functions are ing off to theoretical physics. Dirac, however, was the first used to express of vectors. to ever get a PhD in quantum mechanics, more No physical property of a quantum system changes specifically in , which he improved by by going from wave or matrix mechanics rendering to formally characterizing it using non-commutative opera- the Hilbert state formalism. A very simple example il- tors and Poisson’s brackets. lustrates the essence. The normalization relation for a Schrodinger¨ initially revealed the similarity between single particle wave packet function in position space is the two seemingly different formulations of quantum presented in the wave mechanical description as mechanics, his own wave mechanics and Heisenberg’s matrix mechanics. In early 1927, Dirac [20] and Jor- Z +∞ ψ∗ (x) ψ (x) dx = 1 (6) dan [21, 22], independently of one another, published −∞ their versions of a general formalism tying the various forms of the new quantum theory together in full gen- Using Dirac’s bra-ket notation equation (6) becomes erality. This formalism has come to be known as the Z +∞ Dirac–Jordan transformation theory. ψ∗ (x) ψ (x) dx = hψ|ψi = 1 (7) A few months later, in response to these publications −∞ by Dirac and Jordan, John von Neumann published his where the bra vector hψ| is complex conjugate Hilbert space formalism for quantum mechanics. Jor- of the ket |ψi. dan’s work in time went into oblivion while Dirac firmly The orthogonality relation is given by embraced the Hilbert space formalism advanced by von Z +∞ Neumann. Additionally Dirac introduced the elegant bra- ∗ ψm (x) ψn (x) dx = hψm|ψni = δmn (8) ket notation as well as delta functions. Von Neumann did −∞ not considered use of the delta functions to be rigorous. However, a later version oddly dubbed the rigged Hilbert where δmn is the Kronecker delta space was constructed, which restored rigorousness to  Dirac’s approach. 0 if m , n δmn =  (9) The contributions of von Neumann and Dirac to the 1 if m = n foundations of quantum theory using Hilbert space are considered to be equal as reflected in the embraced phrase The quantum wave functions, for example, the solutions the Dirac–von Neumann in mathematical formu- of the Schrodinger¨ equation describing physical states lation of quantum mechanics. Compilations of their in- in wave mechanics are considered as the set of compo- novation were published as books by Paul Dirac [23] in nents ψ(x) of the abstract vector Ψ called the state vector. 1930 and John von Neumann [24] in 1932. In many ways However, the state vector does not depend upon any par- their contributions are mutually complementary. For ex- ticular choice of coordinates. The same state vector can ample, while von Neumann’s contributions often empha- be described in terms of the wave function in position sized mathematical rigor, Dirac emphasized pragmatic or momentum state or write as an expansion in wave concerns such as and intuitiveness. functions ψn(x) of definite energy X Ψ = cnψn(x) (10) 4 Fundamentals of Hilbert Space n Formalism suggesting that every linear combination of vectors in a Hilbert space is again a vector in the Hilbert space. 2 The Hilbert space, acknowledged as the most appropriate The normalized square moduli |cn| of the complex co- for mathematical formulations of quantum mechanics, is efficients are then interpreted as the probability for the a square integrable, complex, linear, abstract space of vec- system to be in the state ψn analogous to Born’s initial tors possessing a positive definite inner product assured proposal where |ψ(x)|2 is interpreted as the probability to be a number. The states of a quantum mechanical density for the particle to be at x.

Quanta | DOI: 10.12743/quanta.v9i1.142 December 2020 | Volume 9 | Issue 1 | Page 42 4.1 Projective measurement For a self-adjoint Hermitian , quantum states associated with different eigenvalues of Aˆ are orthogonal Every vector in the Hilbert space as a linear combination to one another of the basis vectors ψ with complex coefficients c can n n hψ |ψ i = δ (19) be expressed in Dirac’s notation as m n mn X The possible results of a measurement are the eigenvalues |ψi = cn|ψni. (11) of the operator, which explains the choice of self-adjoint n operators for all the eigenvalues to be real. The proba- Multiplying both sides of equation (11), by hψm| gives bility distribution of an in a given state can X be found by computing the spectral decomposition of the hψm|ψi = cnhψm|ψni. (12) corresponding operator. For a Hermitian operator Aˆ on n an n-dimensional Hilbert space, this can be expressed in Since hψm|ψni = δmn, hψn|ψni = 1 terms of its eigenvalues λn following equation (18) as X cn = hψn|ψi (13) Aˆ = λn|ψnihψn| (20) n which is the transition amplitude of state |ψi to state |ψni. Inserting equation (13) into equation (11) gives If the observable Aˆ, with eigenstates {|ψni} and spectrum X {λn} is measured on a system described by the state vector | i | ih | i ψ = ψn ψn ψ . (14) |ψi, the probability for the measurement to yield the n λn is given by 2 Defining a projection operator Pˆn = |ψnihψn|, equation p(λn) = |hψn|ψi| (21) (14) becomes X After the measurement the system is in the eigenstate |ψi = Pˆn|ψi. (15) |ψni corresponding to the eigenvalue λn found in the mea- n surement, which is called the reduction of state. This leading to X seemingly unphysical reduction of state is a shortcut for ˆ ˆ Pn = I (16) the description of the measurement process and the fact n that the system becomes entangled with the state of the signifying that the sum of all the projection operators is macroscopic measurement equipment. The entanglement unity. leads to the necessary decoherence of the superposition | ih | The outer product ψ ψ is called the projection op- of states of the measured system leading solely to the | i erator since it projects an input ket vector φ into a ray observed eigenvalue with its specific probability. defined by the ket |ψi, as follows

|ψihψ||φi = (hψ|φi) |ψi (17) 4.3 Unitary Operators with a probability |hψ|φi|2 as the inner product between If the inverse of an operator Uˆ is the adjoint operator two state vectors is a complex number known as prob- Uˆ −1 = Uˆ † (22) ability amplitude. This is usually known as projective measurement and we will notice that it is important for then this operator is called a unitary operator and the measurement of a mixed state consisting of an ensem- ble of pure states. Uˆ †Uˆ = Uˆ Uˆ † = Iˆ (23)

4.2 Operator Valued Observables Unitary operators play a significant role in quantum me- chanics representing transformation in the state space. In a quantum system, what can be measured in an experi- Time evolution is just one example since the evolution of ment are the eigenvalues of various observable physical state vectors with time is unitary. This means the state quantities like position, momentum, energy, etc. These vector changes smoothly preserving the total probability. observables are represented by linear, self-adjoint Hermi- tian operators acting on Hilbert space. 4.4 Each eigenstate of an observable corresponds to eigen- vector |ψni of the operator Aˆ, and the associated eigen- In all of the operations in the Hilbert space formalism, value λn corresponds to the value of the observable in that the reality of the quantum state is not altered. We now eigenstate consider the landmark paper [25], of Einstein–Podolsky– Aˆ|ψni = λn|ψni (18) Rosen (EPR) where they presented a thought experiment,

Quanta | DOI: 10.12743/quanta.v9i1.142 December 2020 | Volume 9 | Issue 1 | Page 43 which attempted to show that “the quantum-mechanical Thus as long as ρˆ remains the same, a change in the description of physical reality given by wave functions is wave function of particle 2 does not affect any observ- not complete”, indicating the possible existence of some able since all observable results can be predicted from hidden variables to explain the apparent violation of local- the , without needing to know the ensem- ity enshrined in Einstein’s theory of relativity. However, ble used to construct it. Consequently no useful signal in 1964, offered [2] his celebrated theo- can be sent using entanglement and nonlocality between retical explanation known as Bell’s inequality revealing two observers separated by an arbitrary distance thereby that one of the key assumptions, the , no violation of the sanctified tenets of special theory of as applied to the kind of hidden variables interpretation relativity ensues. hoped for by EPR, was mathematically inconsistent with the predictions of quantum theory. Among an overabundance of experimental efforts per- formed under the generic topic of quantum entanglement, a loophole-free Bell inequality violation has presumably 5 Conclusion claimed to have been conclusively demonstrated [26]. These experiments demonstrate that although there are some possible non-local correlation in quantum systems, It would be cogent to acknowledge the space filling, ever it does not violate causality since no information can be immutable, universal quantum fields to constitute the transferred faster than the speed of light consistent with primary ingredients of reality uncovered by us so far. Ele- the theory of relativity. mentary particles like electrons are quanta of these fields A detailed discussion has been provided by Bhau- and as such are as real as the fields themselves. And so mik [27] illustrating how the reality of the quantum state is the wave packet functions depicting elementary parti- is not violated by quantum entanglement. In brief, the cles constructed in terms of the attributes of the quantum expectation value of an overall pure quantum state of a fields. composite system does not change although in an entan- The quantum states of, for example, a hydrogen atom glement experiment, wave function of a constituent mixed portrayed in terms of the wave functions that are solutions subsystem can change violating locality. of the Schrodinger¨ equation using the wave packet func- This is due to the fact that the actions of an exper- tion ought to be real as well. In fact, recent experimental imentalist on a subsystem of an entangled state can be observations confirm this reality. It is now well known described as applying a unitary operator to that subsystem. that the same quantum states can also be described in Although this produces a change on the wave function terms of Heisenberg’s matrix mechanics. Eventually it of the complete system, such a unitary operator cannot was recognized that the most elegant and efficient way to change the density matrix describing the rest of the sys- treat the quantum states is by utilizing the abstract Hilbert tem. In brief, if distant particles 1 and 2 are in an entan- space formalism developed predominantly by John von gled state, nothing an experimentalist with access only to Neumann and Paul Dirac. particle 1 can do that would change the density matrix of There have been concerns about whether the quantum particle 2. states described by the abstract Hilbert space have any ρ |ni The density matrix ˆ of an ensemble of states with reality. Using appropriate assumptions several recent probabilities pn is given by X theoretical treatments suggest that the quantum states in ρˆ = pn|nihn| (24) Hilbert space are real. However, the somewhat round- n about ways of going about the proof leave rooms for where |nihn| are projection operators and the sum of the possible loopholes. We show here that the reality of the P probabilities are n pn = 1. Thus there can be various en- quantum states can be confirmed in a straightforward sembles of states with each one having its own probability manner relying on the primary reality of quantum fields. distribution that will give the same density matrix. The expectation value of an observable hAˆi is   hAˆi = Tr ρˆAˆ (25) Furthermore, the time evolution of ρˆ only depends upon Acknowledgements the commutator of ρˆ with the Hamiltonian Hˆ following the von Neumann equation d The author wishes to thank Zvi Bern, Per Kraus, and Eric ı~ ρˆ(t) = [Hˆ , ρˆ(t)] (26) dt D’Hoker for helpful discussions.

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