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arXiv:2301.11434

The Field Structure of Free Photons

Anthony Rizzi

Published Jan 26, 2023• 0 citations
arXivHTMLPDF

Abstract

Using a quantum field theoretic description of the photon it is shown that, as intuitively expected but not before theoretically proven, the vector potential of a photon has a likely amplitude associated with a discrete frequency and therefore energy, and momentum. In particular, by finding the wave-functional for the vector potential, it is shown that the likely absolute amplitude spectrum has delta function at a given frequency. This analysis is extended to n-photon systems. It shows that such systems have a vector potential distribution whose most likely element has a strong sinusoidal component which has an amplitude corresponding to n-fold more energy than a single photon system. An analogous result for photons of different energy is also derived. Through the use of Parseval's theorem for stochastic systems, the calculations and associated analyses introduces a simple tool for exploring the nature of QFT Schrodinger wave-functional generally.

DOI
10.48550/arxiv.2301.11434

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Using a quantum field theoretic description of the photon it is shown that, as intuitively expected but not before theoretically proven, the vector potential of a photon has a likely amplitude associated with a discrete frequency and therefore energy, and momentum. In particular, by finding the wave-functional for the vector potential, it is shown that the likely absolute amplitude spectrum has delta function at a given frequency. This analysis is extended to n-photon systems. It shows that such systems have a vector potential distribution whose most likely element has a strong sinusoidal component which has an amplitude corresponding to n-fold more energy than a single photon system. An analogous result for photons of different energy is also derived. Through the use of Parseval's theorem for stochastic systems, the calculations and associated analyses introduces a simple tool for exploring the nature of QFT Schrodinger wave-functional generally.
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A photon structure is advanced based on the experimental evidence and the vector potential quantization at a single photon level. It is shown that the photon is neither a point particle nor an infinite wave but behaves rather like a local “wave-corpuscle” extended over a wavelength, occupying a minimum quantization volume and guided by a non-local vector potential real wave function. The quantized vector potential oscillates over a wavelength with circular left or right polarization giving birth to orthogonal magnetic and electric fields whose amplitudes are proportional to the square of the frequency. The energy and momentum are carried by the local wave-corpuscle guided by the non-local vector potential wave function suitably normalized.
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After reviewing the properties of the photon considered as a quantized particle of zero mass, positive energy, and unit spin, the expansion of the unquantized and quantized electromagnetic fields and vector and scalar potentials in terms of the photon wave functions and creation and destruction operators is reviewed and extended. The most general vector and scalar potentials are obtained through the use of the eigenfunctions of the curl operator. The dichotomy between the photon and wave picture of electromagnetic radiation is discussed and resolved. The results are applied to the calculation of the exact electromagnetic matrix elements and transition probabilities (i.e., with retardation taken into account exactly) for hydrogenic atoms. The exact matrix elements are very simple in form. The notion of multipole radiation of the usual treatments is irrelevant. However, it is shown how multipoles appear as an approximation.
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.