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arXiv:2412.13218 · Universe

The Quantum Electromagnetic Field in the Weyl–Wigner Representation

Emilio Santos

Published Dec 9, 2024• 5 citations
arXivHTMLPDF

Abstract

The quantum electromagnetic (EM) field is formulated in the Weyl–Wigner representation (WW), which is equivalent to the standard Hilbert space one (HS). In principle, it is possible to interpret within WW all experiments involving the EM field interacting with macroscopic bodies, the latter treated classically. In the WW formalism, the essential difference between classical electrodynamics and the quantum theory of the EM field is just the assumption that there is a random EM field-filling space, i.e., the existence of a zero-point field with a Gaussian distribution for the field amplitudes. I analyze a typical optical test of a Bell inequality. The model admits an interpretation compatible with local realism, modulo a number of assumptions assumed plausible.

DOI
10.3390/universe10120452

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The quantum electromagnetic (EM) field is formulated in the Weyl–Wigner representation (WW), which is equivalent to the standard Hilbert space one (HS). In principle, it is possible to interpret within WW all experiments involving the EM field interacting with macroscopic bodies, the latter treated classically. In the WW formalism, the essential difference between classical electrodynamics and the quantum theory of the EM field is just the assumption that there is a random EM field-filling space, i.e., the existence of a zero-point field with a Gaussian distribution for the field amplitudes. I analyze a typical optical test of a Bell inequality. The model admits an interpretation compatible with local realism, modulo a number of assumptions assumed plausible.
Abstract I revisit the Wigner (or Weyl–Wigner, WW) representation of the quantum electromagnetic field. I show that, assuming that Fock states are just mathematical concepts devoid of physical reality, WW suggests a realistic interpretation which turns out to be (classical) Maxwell theory with the assumption that there is a random radiation filling space, the vacuum field. I elucidate why, in sharp contrast, non-relativistic quantum mechanics of particles does not admit a realistic interpretation via WW. I interpret experiments involving entangled light beams within WW, in particular optical tests of Bell inequalities. I show that WW provides clues in order to construct local models for those experiments. I give arguments why Bell definition of local realism is not general enough.
We characterize the electromagnetic vacuum as a stochastic field. Some consequences, like the particle behaviour of light, are studied. The stochastic approach is connected with the standard Hilbert space formalism via the Weyl transform. Several experiments involving spontaneous parametric down conversion are studied comparing Hilbert space and Weyl–Wigner formalisms. This allows an intuitive picture of entanglement to be obtained as a correlation between field fluctuations in distant places, involving the vacuum fields. The analysis shows that the Bell definition of local realism is not general enough, whence the reported violation of Bell inequalities does not refute local realism.
The Wigner representation for quantum mechanics of particles is generalized to Bose fields. The standard Hilbert space quantization becomes, via the Weyl transform, a quantization method that consists of adding a Gaussian zeropoint field distribution to the vacuum. I comment on the possible advantages of the method in order to study quantum fields in curved spaces. I study a unified formulation of non-relativistic quantum electrodynamics in the Weyl–Wigner formalism, in terms of (classical-like) c-numbers.
I contend that physics should provide a coherent account of reality, in addition to being an efficient algorithm for the prediction of empirical results. This article offers pictures of reality derived from theories of modern physics. In particular, it is shown that Bose quantum fields may be interpreted as pure wave fields via the Weyl–Wigner representation, the most relevant result being the existence of a stochastic vacuum field corresponding to the quantum vacuum fluctuations of the standard, canonical, formulation of field theory. That field provides explanations for the particle (photons) behavior of the electromagnetic field. Also, a realistic interpretation is offered for interference experiments with actual particles, like atoms. The interpretation of classical general relativity is standard but emphasis is given to the principle of equivalence. Problems like spacetime singularities (black holes) and the empirical violation of Bell inequalities are touched on but slightly. Asides from these problems, the main incompleteness of the article is the absence of a realistic interpretation of Fermi fields.
In this chapter, we will introduce different representations of the electromagnetic (EM) field. One may ask, why do we need different representations for the EMfield? The answer is that usually we do not know the state of the EM field which in quantum optics is represented by the annihilation and creation operators. However, results of measurements of the field are given in terms of real variables, for example, intensity. The representations allow us to recognize the state of the field from values of the measured quantities such as average amplitude, intensity, and correlation functions. The nature of the state is present in terms of the interpretation of the apparently classical (measured) variables. We will discuss two basic types of representations often used in quantum optics, Fock state (photon number) representation and coherent states representation, the later one introduced by Glauber.a We also discuss properties of fields with thermal and Poisson distribution of photons. The photon number states are very often used as a basis for quantum optics problems, and despite of many difficulties have recently been generated experimentally [11-13].
I assume that everywhere in space there is a real random electromagnetic radiation, or zeropoint field (ZPF), which looks similar for all inertial observers, so that the stochastic properties of the field should be Lorentz invariant. This fixes the spectrum except for a single adjustable parameter measuring the scale, which is identified with Planck's constant, so making the ZPF identical to the quantum electromagnetic vacuum. Photons are just fluctuations of the random field or, equivalently, wavepackets in the form of needles of radiation superimposed to the ZPF. Two photons are "classically correlated" if the correlation involves just the intensity above the average energy of the ZPF, but they are "entangled" if the ZP fields in the neighbourhood of the photons are also correlated. These assumptions may explain all quantum optical phenomena involving radiation and macroscopic bodies, provided the latter may be treated as classical. That is, we have an interpretation of quantization for light but not for matter. Detection of photons involves subtracting the ZPF, which cannot be made without a fundamental uncertainty. This explains why photon counters cannot be manufactured with 100% efficiency and no noise (dark rate), which prevents the violation of a genuine Bell inequality (this is the so-called detection loophole). The theory thus obtained agrees very closely with standard quantum optics if this is formulated in the Wigner representation.