Soft Degrees of Freedom,
Gibbons–Hawking Contribution and
Entropy from Casimir Effect
Glenn Barnich and Martin Bonte
Abstract Recent work on non-proper gauge degrees of freedom in the context of
the Casimir effect is reviewed. In his original paper, Casimir starts by pointing out
that, when the electromagnetic field is confined between two perfectly conducting
parallel plates, there is an additional physical polarization of the electromagnetic
field at zero value for the discretized longitudinal momentum besides the standard
two transverse polarizations at non-zero values. In this review, the dynamics of
these additional modes is obtained from first principles. At finite temperature, their
contribution to the entropy is proportional to the area of the plates and corresponds
to the contribution of a massless scalar field in 2+1 dimensions. When the plates
are charged, there is a further contribution to the partition function from the zero
mode of this additional scalar that scales with the area but does not contribute to
the entropy. It reproduces the result obtained when the Gibbons–Hawking method is
applied to the vacuum capacitor. For completeness, a brief discussion of the classical
thermodynamics of such a capacitor is included.
Keywords Edge modes · Casimir effect · Gibbons–Hawking entropy · Black
hole micro-states
1 Introduction
That seemingly unphysical polarizations of the electromagnetic field have an
important role to play in the presence of charged particles is known since the work
by Dirac [1], and Fock and Podolski [2], where the Coulomb force between two nonrelativistic electrons is constructed in terms of creation and destruction operators
associated with the scalar potential A 0 .
G. Barnich () · M. Bonte
Physique Théorique et Mathématique, Université libre de Bruxelles and International Solvay
Institutes, Bruxelles, Belgium
e-mail: gbarnich@ulb.ac.be
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_35
369
Gibbons–Hawking Contribution and
Entropy from Casimir Effect
Glenn Barnich and Martin Bonte
Abstract Recent work on non-proper gauge degrees of freedom in the context of
the Casimir effect is reviewed. In his original paper, Casimir starts by pointing out
that, when the electromagnetic field is confined between two perfectly conducting
parallel plates, there is an additional physical polarization of the electromagnetic
field at zero value for the discretized longitudinal momentum besides the standard
two transverse polarizations at non-zero values. In this review, the dynamics of
these additional modes is obtained from first principles. At finite temperature, their
contribution to the entropy is proportional to the area of the plates and corresponds
to the contribution of a massless scalar field in 2+1 dimensions. When the plates
are charged, there is a further contribution to the partition function from the zero
mode of this additional scalar that scales with the area but does not contribute to
the entropy. It reproduces the result obtained when the Gibbons–Hawking method is
applied to the vacuum capacitor. For completeness, a brief discussion of the classical
thermodynamics of such a capacitor is included.
Keywords Edge modes · Casimir effect · Gibbons–Hawking entropy · Black
hole micro-states
1 Introduction
That seemingly unphysical polarizations of the electromagnetic field have an
important role to play in the presence of charged particles is known since the work
by Dirac [1], and Fock and Podolski [2], where the Coulomb force between two nonrelativistic electrons is constructed in terms of creation and destruction operators
associated with the scalar potential A 0 .
G. Barnich () · M. Bonte
Physique Théorique et Mathématique, Université libre de Bruxelles and International Solvay
Institutes, Bruxelles, Belgium
e-mail: gbarnich@ulb.ac.be
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_35
369