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G. Barnich and M. Bonte
When all polarizations are quantized, it is important to understand how equivalence with reduced phase space quantization is achieved. Arguably the most
transparent implementation is through the quartet mechanism [3] which implies
the cancellation of the contributions from unphysical polarizations and ghost
variables when computing matrix elements of gauge invariant operators between
gauge invariant states in the context of Hamiltonian BRST operator quantization.
Furthermore, the associated path integral is simply related to the manifestly Lorentz
invariant Lagrangian BRST path integral by integrating out momenta (see, e.g. [4]
for a comprehensive review). More generally, as is well known in the context of
topological field theories, these cancellations no longer work perfectly when there
is non-trivial topology.
Whereas the quartet mechanism is relatively straightforward for the free electromagnetic field where the quartets are associated with temporal oscillators for the
scalar potential and to oscillators for the longitudinal part of the vector potential on
the one hand, and to oscillators for the ghost fields on the other, this is no longer
the case in the presence of charged sources, where gauge invariance becomes a nontrivial issue [5]. For the simplest source representing a charged point particle at rest,
it turns out that the BRST invariant vacuum state is a coherent state constructed
out of unphysical null oscillators that represents the quantum Coulomb solution
[6]. Some technical details and clarifications on this elementary construction are
provided in Appendix “Details on Quantum Coulomb Solution”.
The ultimate aim of this research is a better understanding of the degrees of
freedom responsible for black hole entropy. In this context, it is intriguing to note
that in one of the earliest papers on linearized quantum gravity by Bronstein [7]
(see [8] for perspective), the last part of the paper follows closely the derivation
by Dirac, Fock, and Podolski on the Coulomb law in order to obtain Newton’s law
between two test masses from the creation and destruction operators associated with
the metric fluctuations h 00 . How to extend the considerations below to the case of
linearized gravity will be discussed elsewhere.
In order to avoid facing the question of the detailed interaction of the quantized
electromagnetic field with charged dynamical matter, it is instructive to first
consider the case where these interactions can be idealized as boundary conditions
imposed on the free electromagnetic field. This naturally leads one to consider the
electromagnetic field in the presence of charged conducting plates. In the absence of
charge, this is precisely the set-up of the Casimir effect [9] at non-zero temperature
[10] (see also [11, 12] and e.g. [13–15] for reviews).
As we will discuss in details below, that there is an additional physical polarization at zero value for the discretized longitudinal momentum, besides the two
transverse ones at non-zero values, is well known in this context. We will focus on
how to determine the dynamics of these “edge” modes and isolate their contribution
to the partition function and the entropy, which scales with the area of the plates.
We then provide a microscopic understanding of the charged vacuum capacitor,
where there is an additional contribution to the partition function that comes from
the zero mode of the additional polarization and that also scales with the area but
does not contribute to the entropy [16].
G. Barnich and M. Bonte
When all polarizations are quantized, it is important to understand how equivalence with reduced phase space quantization is achieved. Arguably the most
transparent implementation is through the quartet mechanism [3] which implies
the cancellation of the contributions from unphysical polarizations and ghost
variables when computing matrix elements of gauge invariant operators between
gauge invariant states in the context of Hamiltonian BRST operator quantization.
Furthermore, the associated path integral is simply related to the manifestly Lorentz
invariant Lagrangian BRST path integral by integrating out momenta (see, e.g. [4]
for a comprehensive review). More generally, as is well known in the context of
topological field theories, these cancellations no longer work perfectly when there
is non-trivial topology.
Whereas the quartet mechanism is relatively straightforward for the free electromagnetic field where the quartets are associated with temporal oscillators for the
scalar potential and to oscillators for the longitudinal part of the vector potential on
the one hand, and to oscillators for the ghost fields on the other, this is no longer
the case in the presence of charged sources, where gauge invariance becomes a nontrivial issue [5]. For the simplest source representing a charged point particle at rest,
it turns out that the BRST invariant vacuum state is a coherent state constructed
out of unphysical null oscillators that represents the quantum Coulomb solution
[6]. Some technical details and clarifications on this elementary construction are
provided in Appendix “Details on Quantum Coulomb Solution”.
The ultimate aim of this research is a better understanding of the degrees of
freedom responsible for black hole entropy. In this context, it is intriguing to note
that in one of the earliest papers on linearized quantum gravity by Bronstein [7]
(see [8] for perspective), the last part of the paper follows closely the derivation
by Dirac, Fock, and Podolski on the Coulomb law in order to obtain Newton’s law
between two test masses from the creation and destruction operators associated with
the metric fluctuations h 00 . How to extend the considerations below to the case of
linearized gravity will be discussed elsewhere.
In order to avoid facing the question of the detailed interaction of the quantized
electromagnetic field with charged dynamical matter, it is instructive to first
consider the case where these interactions can be idealized as boundary conditions
imposed on the free electromagnetic field. This naturally leads one to consider the
electromagnetic field in the presence of charged conducting plates. In the absence of
charge, this is precisely the set-up of the Casimir effect [9] at non-zero temperature
[10] (see also [11, 12] and e.g. [13–15] for reviews).
As we will discuss in details below, that there is an additional physical polarization at zero value for the discretized longitudinal momentum, besides the two
transverse ones at non-zero values, is well known in this context. We will focus on
how to determine the dynamics of these “edge” modes and isolate their contribution
to the partition function and the entropy, which scales with the area of the plates.
We then provide a microscopic understanding of the charged vacuum capacitor,
where there is an additional contribution to the partition function that comes from
the zero mode of the additional polarization and that also scales with the area but
does not contribute to the entropy [16].