Abstract
The electromagnetic field inside a spherical cavity of large radius R is considered in the presence of stationary charge and current densities. R provides infra-red regularization while maintaining gauge invariance. The quantum ground state of physical photons forming the magnetic field is found to be a coherent state with a definite mean occupation number. The electric field, which is determined by the Gauss law constraint, is maintained by a minimum uncertainty coherent state, according to the projection operator approach to the quantization of constrained systems. The mean occupation number of this state is proportional to the square of the total charge. The results confirm formulae obtained previously from a calculation with a finite photon mass for infra-red regularization.
Highlights
The quantum N -portrait of black holes and the corpuscular nature of gravity recently developed by Dvali, Gomez and collaborators [1,2,3,4,5,6,7,8,9] are interesting approaches to tackling profound questions in quantum gravity, from ultra-violet finiteness over black hole entropy to the information paradox
They may help to shed light on dark energy: applying the same ideas to the observable universe gives an estimate of the dark energy density which is very close to the observed value [10]
An obvious approach to the above question is to view classical fields as coherent quantum states and to identify N with the mean occupation number. This approach is based upon an intuitive understanding of quantum field theory as an infinite collection of harmonic oscillators, for which the concept of coherent states is most straightforward
Summary
The quantum N -portrait of black holes and the corpuscular nature of gravity recently developed by Dvali, Gomez and collaborators [1,2,3,4,5,6,7,8,9] are interesting approaches to tackling profound questions in quantum gravity, from ultra-violet finiteness over black hole entropy to the information paradox. The long-range nature of gravity renders classical field configurations not square-integrable and, unsuitable as physical quantum states.1 Both challenges are shared by electrodynamics, and it has been argued before that understanding the analogous question – how many photons are bound by electric charges and currents – would yield answers that could be generalised to gravity [23,24,25]. The number of transverse, i.e., physical photons in the static magnetic field generated by a stationary current density j(x) was found to be This result is independent of the gauge parameter, but one may still wonder whether it is meaningful or just an artefact of the infra-red regularisation via a photon mass. The mean occupation number of this state is trivially given by (1)
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