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arXiv:1502.03208 · Journal of Advanced Physics

The Unfinished Search for Wave-Particle and Classical-Quantum Harmony

Partha Ghose

Published Jun 3, 2015• 8 citations

Abstract

The main purpose of this paper is to review the progress that has taken place so far in the search for a single unifying principle that harmonizes (i) the wave and particle natures of matter and radiation, both at the quantum and the classical levels, on the one hand and (ii) the classical and quantum theories of matter and radiation on the other hand. The famous paradoxes of quantum theory, the mysterious nature of measurements in quantum theory and the principal no-go theorems for hidden variables are first briefly reviewed. The Koopman-von Neumann Hilbert space theory based on complex wave functions underlying particle trajectories in classical phase space, is an important step forward in that direction. It provides a clear and beautiful harmony of classical waves and particles. Sudarshan has given an alternative but equivalent formulation that shows that classical mechanics can be regarded as a quantum theory with essentially hidden non-commuting variables. An extension of KvNS theory to classical electrodynamics provides a sound Hilbert space foundation to it and satisfactorily accounts for entanglement and Bell-CHSH-like violations already observed in classical polarization optics. An important new insight that has been obtained through these developments is that entanglement and Bell-like inequality violations are neither unique signatures of quantumness nor of non-locality---they are rather signatures of non-separability. Finally, Sudarshan's proposed solution to the measurement problem using KvNS theory for the measuring apparatus is sketched to show to what extent wave and particles can be harmonized in quantum theory.

DOI: 10.1166/jap.2015.1197

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The main purpose of this paper is to review the progress that has taken place so far in the search for a single unifying principle that harmonizes (i) the wave and particle natures of matter and radiation, both at the quantum and the classical levels, on the one hand and (ii) the classical and quantum theories of matter and radiation on the other hand. The famous paradoxes of quantum theory, the mysterious nature of measurements in quantum theory and the principal no-go theorems for hidden variables are first briefly reviewed. The Koopman-von Neumann Hilbert space theory based on complex wave functions underlying particle trajectories in classical phase space, is an important step forward in that direction. It provides a clear and beautiful harmony of classical waves and particles. Sudarshan has given an alternative but equivalent formulation that shows that classical mechanics can be regarded as a quantum theory with essentially hidden non-commuting variables. An extension of KvNS theory to classical electrodynamics provides a sound Hilbert space foundation to it and satisfactorily accounts for entanglement and Bell-CHSH-like violations already observed in classical polarization optics. An important new insight that has been obtained through these developments is that entanglement and Bell-like inequality violations are neither unique signatures of quantumness nor of non-locality---they are rather signatures of non-separability. Finally, Sudarshan's proposed solution to the measurement problem using KvNS theory for the measuring apparatus is sketched to show to what extent wave and particles can be harmonized in quantum theory.

In this paper we continue the study started in [1] on the operatorial formulation of classical mechanics given by Koopman and von Neumann (KvN) in the Thirties. In particular we show that the introduction of the KvN Hilbert space of complex and square integrable "wave functions" requires an enlargement of the set of the observables of ordinary classical mechanics. The possible role and the meaning of these extra observables is briefly indicated in this work. We also analyze the similarities and differences between non selective measurements and two-slit experiments in classical and quantum mechanics.

In this paper, we continue the study started in Ref. 1, of the operatorial formulation of classical mechanics given by Koopman and von Neumann (KvN) in the 1930s. In particular, we show that the introduction of the KvN Hilbert space of complex and square integrable "wave functions" requires an enlargement of the set of the observables of ordinary classical mechanics. The possible role and the meaning of these extra observables is briefly indicated in this work. We also analyze the similarities and differences between non-selective measurements and two-slit experiments in classical and quantum mechanics.

Hilbert-space techniques are widely used not only for quantum theory, but also for classical physics. Two important examples are the Koopman-von Neumann (KvN) formulation and the method of ``classical'' wave functions. As this paper explains, these two approaches are conceptually distinct. In particular, the method of classical wave functions was not due to Bernard Koopman and John von Neumann, but was developed independently by a number of later researchers, perhaps first by Mario Schönberg, with key contributions from Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan. The primary goals of this paper are to explain these two approaches, describe the relevant history in detail, and give credit where credit is due.

We demonstrate the existence of a complex Hilbert Space with Hermitian operators for calculations in \textit{classical} electromagnetism that parallels the Hilbert Space of quantum mechanics. The axioms of this classical theory are the so-called Dirac-von Neumann axioms, however, with classical potentials in place of the wavefunction and the indeterministic collapse postulate removed. This approach lets us derive a variety of fundamental expressions for electromagnetism using minimal mathematics and a calculation sequence well-known for traditional quantum mechanics. We also demonstrate the existence of the wave commutation relationship

[\hat{x},\hat{k}]=i

, which is a unique classical analogue to the canonical commutator

[\hat{x},\hat{p}]=i\hbar

. The difference between classical and quantum mechanics lies in the presence of

\hbar

. The noncommutativity of observables for a classical theory simply reflects its wavenature. A classical analogue of the Heisenberg Uncertainty Principle is developed for electromagnetic waves, and its implications discussed. Further comparisons between electromagnetism, Koopman-von Neumann-Sudarshan (KvNS) classical mechanics (for point particles), and quantum mechanics are made. Finally, supplementing the analysis presented, we additionally demonstrate an elegant, completely relativistic version of Feynman's proof of Maxwell's equations \citep{Dyson}. Unlike what \citet{Dyson} indicated, there is no need for Galilean relativity for the proof to work. This fits parsimoniously with our usage of classical Lie commutators for electromagnetism.

The emerging field of entanglement or nonseparability in classical optics is reviewed, and its similarities with and differences from quantum entanglement clearly pointed out through a recapitulation of Hilbert spaces in general, the special restrictions on Hilbert spaces imposed in quantum mechanics and the role of Hilbert spaces in classical polarization optics. The production of Bell-like states in classical polarization optics is discussed, and new theorems are proved to discriminate between separable and nonseparable states in classical wave optics where no discreteness is involved. The influence of the Pancharatnam phase on a classical Bell-like state is deived. Finally, to what extent classical polarization optics can be used to simulate quantum information processing tasks is also discussed. This should be of great practical importance because coherence and entanglement are robust in classical optics but not in quantum systems.

The measurement problem in quantum mechanics is re-examined and it is shown that it cannot really be solved in a satisfactory way, within the framework of the usual interpretation of the theory. We then discuss von Neumann's attempts to prove that quantum mechanics is incompatible with the introduction of hidden variables, and develop a more detailed form of Bell's argument, showing that von Neumann's analysis is invalid. Using certain ideas that are implicit in the "differential-space" theory of Wiener and Siegel, we go on to propose a new deterministic equation of motion, describing a kind of coupling of the measuring instrument to the observed system that explains in detail how the wave packet is "reduced" during a measurement in a continuous and causally determined way. By averaging over the hidden parameters, we then recover the usual statistical results of quantum mechanics as a special case. However, a more detailed analysis of the theory shows that new experimental and theoretical questions can now be raised, which go outside the framework of the quantum theory as it is now formulated. These questions are examined briefly.