Photon wave functions, wave-packet quantization of light, and coherence theory

TL;DR

This paper develops a photon wave function framework that parallels electron quantum mechanics, linking it with quantum field theory, and demonstrates how to describe and convert photonic states and coherence using wave functions and modes.

Contribution

It introduces a photon wave mechanics theory derived from energy-momentum relations, establishing equivalence with quantum field theory and providing practical mode conversion methods.

Findings

Photon wave functions are derived from energy-momentum relations.

Photon wave mechanics and QFT mode functions are shown to be equivalent.

A practical spectral filtering mode converter is proposed.

Abstract

The monochromatic Dirac and polychromatic Titulaer-Glauber quantized field theories (QFTs) of electromagnetism are derived from a photon-energy wave function in much the same way that one derives QFT for electrons, that is, by quantization of a single-particle wave function. The photon wave function and its equation of motion are established from the Einstein energy-momentum-mass relation, assuming a local energy density. This yields a theory of photon wave mechanics (PWM). The proper Lorentz-invariant single-photon scalar product is found to be non-local in coordinate space, and is shown to correspond to orthogonalization of the Titulaer-Glauber wave-packet modes. The wave functions of PWM and mode functions of QFT are shown to be equivalent, evolving via identical equations of motion, and completely describe photonic states. We generalize PWM to two or more photons, and show how to switch between the PWM and QFT viewpoints. The second-order coherence tensors of classical coherence theory and the two-photon wave functions are shown to propagate equivalently. We give examples of beam-like states, which can be used as photon wave functions in PWM, or modes in QFT. We propose a practical mode converter based on spectral filtering to convert between wave packets and their corresponding biorthogonal dual wave packets.