Quantum fluctuations of a Coulomb potential

Quantum fluctuations of the electromagnetic field produced by an elementary particle are investigated. Using the Schwinger-Keldysh formalism, it is found that the two-point correlation function of the Coulomb potential has a nonvanishing value already at zeroth order in the Planck constant. It is proved that to this order, the correlation function is gauge independent and has a well-defined coincidence limit. In this limit, the leading term in the long-range expansion of the correlation function is calculated explicitly. Furthermore, the spectrum of the electromagnetic fluctuations is investigated. It is found, in particular, that in a wide range of practically important frequencies, the spectral density of the electromagnetic field fluctuations exhibits an inverse frequency dependence. It is shown also that in the case of a macroscopic body, the h(0) part of the correlation function is suppressed by a factor 1/N, where N is the number of particles in the body. Relation of the obtained results to the problem of measurability of the electromagnetic field is mentioned.