Photon Wave Mechanics: Energy Wave Function and Four-Potential Theories

Ole Keller

Brian J. Smith, M. G. Raymer

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, i.e., 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.

Margaret Hawton

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.

Lajos Jánossy

Lajos Jánossy

Margaret Hawton

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.