TL;DR
This paper derives a Lorentz invariant photon number density expression using the photon wave function, showing its independence from wave function form and proposing a practical choice involving the vector potential-electric field pair.
Contribution
It introduces a Lorentz invariant photon number density formula based on the scalar product of polarization-sensitive position eigenvectors and photon wave functions, advancing photon localization theory.
Findings
The photon number density is Lorentz invariant and positive definite.
The scalar product used is independent of wave function form.
The vector potential-electric field pair is a convenient wave function choice.
Abstract
A Lorentz invariant positive definite expression for photon number density is derived as the absolute square of the invariant scalar product of a polarization sensitive position eigenvector and the photon wave function. It is found that this scalar product is independent of the form chosen for the wave function and that the normalized positive frequency vector potential-electric field pair is a convenient choice of wave function in the presence of matter. The number amplitude describing a localized state is a delta-function at the instant at which localization and detection are seen as simultaneous.
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