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arXiv:2107.03262 · Physical Review A

Photon quantum mechanics in real Hilbert space

Margaret Hawton

Published Nov 17, 2021• 15 citations
arXivHTMLPDF

Abstract

Classically, electromagnetic pulses are described by real fields that couple to charged matter and propagate causally. We will show here that real fields of the form used in standard classical electromagnetic theory have a quantum mechanical interpretation in which the probability density for a photon to be at $\mathbf{x}$ is positive definite and operators representing all of the standard physical observables exist. A covariant alternative to the ${\ensuremath{\omega}}_{k}^{1/2}$ dependence that appears in most (but not all) presentations of quantum optics and quantum field theory is presented and real Lorentz scalar one-photon advanced and retarded potentials are derived.

DOI
10.1103/physreva.104.052211

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Classically, electromagnetic pulses are described by real fields that couple to charged matter and propagate causally. We will show here that real fields of the form used in standard classical electromagnetic theory have a quantum mechanical interpretation in which the probability density for a photon to be at x\mathbf{x}x is positive definite and operators representing all of the standard physical observables exist. A covariant alternative to the \ensuremathωk1/2{\ensuremath{\omega}}_{k}^{1/2}\ensuremathωk1/2​ dependence that appears in most (but not all) presentations of quantum optics and quantum field theory is presented and real Lorentz scalar one-photon advanced and retarded potentials are derived.
We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive-definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive-energy photon. With invariant plane-wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.
Recently a photon position operator with commuting components and a particle density that is positive definite for both positive and negative frequency states has led to a first quantized theory of the photon. Here we extend this theory to a manifestly covariant form and show for the first time that, with invariant plane wave normalization, the position operator is Hermitian with real instantaneously localized eigenvectors. Reality of the field and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The position space wave function is the real part of the projection of the photon's state vector onto a basis of position eigenvectors. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Gupta-Bleuler gauge condition.
Still today the discussion about the foundations, physical interpretation, and real scope of quantum mechanics has never ceased. It would be wrong to dismiss these issues as mere philosophical problems, because questions of consistency and interpretation are not devoid of practical utility. We present the foundations and main properties of a positive definite phase space quantum mechanics. A new quantization procedure is proposed as well. This new interpretation/formulation eliminates conceptual and technical difficulties from quantum mechanics: (i) many paradoxes typical of the wave-particle duality, EPR experiments, macroscopic superpositions, and collapse of wavefunctions disappear; (ii) the elimination of the wavefunctions from quantum theory is in line with the procedure inaugurated by Einstein with the elimination of the ether in the theory of electromagnetism; (iii) it is useful in considering the classical limit, can treat mixed states with ease, and brings certain conceptual issues to the fore; (iv) confirms the ensemble interpretation of the wavefunctions, derives its statistical interpretation, corrects the temporal dependence of the old wavefunctions, and considers pure classical states --localizable states-- beyond the Hilbert space; (v) the quantum equation of motion is of the Liouville kind and star-products are not needed, simplifying the formalism; and (vi) eliminates the hypothetical external quantum field of the pilot wave interpretation, solving its problems on the status of probability, and correcting well-known inconsistencies of the Bohm potential. Finally, we offer some perspectives on future developments and research in progress.
One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev. A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of quantum mechanics, this wave function gives the probability to count a photon at any position in space.
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.
We revisit the quantum oscillator model of the electromagnetic field and conclude that, while the nonlocal positive and negative frequency ladder operators generate a photon Fock basis, the Hermitian field operators obtained by second quantization of real Maxwell fields describe photon-antiphoton pairs that couple locally to Fermionic matter and can be modeled classically. Their commutation relations define a scalar product that can be the basis of a first quantized theory of single photons. Since a one-photon state collapses to a zero-photon state when the photon is counted, the field describing it must be interpreted as a probability amplitude.