Photon position observable

TL;DR

This paper applies biorthogonal quantum mechanics to define a covariant position observable for photons, linking photon detection probabilities with the Klein-Gordon wave function and Glauber correlation functions.

Contribution

It introduces a biorthogonal basis approach to the photon position operator, ensuring covariance and localization, and connects photon detection probabilities with established quantum optical measures.

Findings

Photon position operator is covariant and localized.

Transition probability matches the first order Glauber correlation.

Bridges photon counting with electromagnetic energy density detection.

Abstract

In biorthogonal quantum mechanics, the eigenvectors of a quasi-Hermitian operator and those of its adjoint are biorthogonal and complete and the probability for a transition from a quantum state to any one of these eigenvectors is positive definite. We apply this formalism to the long standing problem of the position observable in quantum field theory. The dual bases are positive and negative frequency one-particle states created by the field operator and its conjugate and biorthogonality is a consequence of their commutation relations. In these biorthogonal bases the position operator is covariant and the Klein-Gordon wave function is localized. We find that the invariant probability for a transition from a one-photon state to a position eigenvector is the first order Glauber correlation function, bridging the gap between photon counting and the sensitivity of light detectors to electromagnetic energy density.