# Photon number density operator

## Abstract

A new operator is introduced to represent the density of photons in configuration space. It
has some features in common with operators previously introduced by Mandel and Cook but
has better transformation properties.
The operator is introduced first in the Coulomb gauge where only transverse photons are
necessary to describe physical states. It is the first component of the four vector obtained by
contracting the electromagnetic field tensor with the vector potential. It is also shown that in
the free field case the corresponding photon current and the photon number density operator
satisfy a continuity equation.
In the Lorentz gauge, longitudinal and scalar photons are allowed and the operators are
defined in respect to an indefinite metric as proposed by Gupta. The Coulomb gauge operator
expressed in the new metric cannot give the right number of ghost photons for arbitrary states
and has to be discarded as a valid photon number density operator in the Lorentz gauge. It is
shown that the photon number density operator in the Lorentz gauge differs from the one in
the Coulomb gauge by a divergence term. The total number of photons for physical states is
the same for both operators. The ghost states, which are the longitudinal and scalar photon
states, in respect to the old metric are different for the two operators.
The form of the photon number operator in the Lorentz gauge using the new metric can
be substantiated by symmetry arguments. The new operator is able to count the ghost state
photons in respect to the old metric in a free field and in the case of two fixed charges present.