Soft Degrees of Freedom, Gibbons–Hawking Contribution and Entropy from. . .
385
e
d 3 x
i
Q
2
x
|| x | 3
ˆ
A
(−)
i ( x )−
1
2 j 0 ( x )(−Δ) −1/2 A
(−)
0 ( x )
,
(83)
where the subscript (−) denotes the creation part.
Acknowledgments G.B. is grateful to S. Theisen for pointing out reference [7] and to M. Henneaux for suggesting to explain the thermodynamics of the charged capacitor by standard methods.
The authors acknowledge useful discussions with F. Alessio, J. Crabbe, and S. Prohazka. This work
is supported by the F.R.S.-FNRS Belgium through a research fellowship for Martin Bonte and also
through conventions FRFC PDR T.1025.14 and IISN 4.4503.15.
Bibliography
1. P.A.M. Dirac, Proc. R. Soc. Lond. A 136(829), 453 (1932). doi:10.1098/rspa.1932.0094
2. V. Fock, B. Podolsky, Phys. Zs. Sowjetunion 1, 798 (1932)
3. T. Kugo, I. Ojima, Prog. Theor. Phys. Suppl. 66, 1 (1979)
4. M. Henneaux, C. Teitelboim, Quantization of Gauge Systems (Princeton University, Princeton,
1992)
5. P.A.M. Dirac, Can. J. Phys. 33, 650 (1955). doi:10.1139/p55-081
6. G. Barnich, Gen. Rel. Grav. 43, 2527 (2011). doi:10.1007/s10714-010-0984-6
7. M. Bronstein, Gen. Rel. Grav. 44, 267 (2012). doi:10.1007/s10714-011-1285-4
8. S. Deser, A.A. Starobinsky, Gen. Rel. Grav. 44, 263 (2012). doi:10.1007/s10714-011-1284-5
9. H.B.G. Casimir, Indag. Math. 10, 261 (1948). [Kon. Ned. Akad. Wetensch. Proc.100N3-4,61
(1997)]
10. J. Mehra, Physica 37, 145 (1967). doi:10.1016/0031-8914(67)90115-2
11. M. Fierz, Helv. Phys. Acta 33, 855 (1960)
12. L.S. Brown, G.J. Maclay, Phys. Rev. 184, 1272 (1969). doi:10.1103/PhysRev.184.1272
13. G.
Plunien,
B.
Muller,
W.
Greiner,
Phys.
Rept.
134,
87
(1986).
doi:10.1016/0370-1573(86)90020-7
14. B.E. Sernelius, Surface Modes in Physics (Wiley, New York, 2001). doi:10.1002/3527603166
15. M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, in Advances in the
Casimir effect. Integral Series Monographs Physics, vol. 145 (Oxford University, Oxford,
2009)
16. G. Barnich, Phys. Rev. D99(2), 026007 (2019). doi:10.1103/PhysRevD.99.026007
17. G.W. Gibbons, S.W. Hawking, Phys. Rev. D15, 2752 (1977)
18. S.W. Hawking, S.F. Ross, Phys. Rev. D52, 5865 (1995)
19. S. Deser, M. Henneaux, C. Teitelboim, Phys. Rev. D55, 826 (1997)
20. M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992)
21. J. Slater, N. Frank, Electromagnetism (Dover Publications, New York, 1969)
22. H.B. Callen, Thermodynamics (Wiley, New York, 1961)
23. F. Heinrich, Am. J. Phys. 54(8), 742 (1986)
24. H. Casimir, D. Polder, Phys. Rev. 73(4), 360 (1948)
25. J. Ambjorn, S. Wolfram, Annals Phys. 147, 1 (1983). doi:10.1016/0003-4916(83)90065-9
26. J.I. Kapusta, Phys.Rev. D24, 426 (1981). doi:10.1103/PhysRevD.24.426
27. J.S. Dowker, Phys. Rev. D37, 558 (1988). doi:10.1103/PhysRevD.37.558
28. J. Dowker, Class. Quant. Grav. 20, L105 (2003). doi:10.1088/0264-9381/20/8/102
385
e
d 3 x
i
Q
2
x
|| x | 3
ˆ
A
(−)
i ( x )−
1
2 j 0 ( x )(−Δ) −1/2 A
(−)
0 ( x )
,
(83)
where the subscript (−) denotes the creation part.
Acknowledgments G.B. is grateful to S. Theisen for pointing out reference [7] and to M. Henneaux for suggesting to explain the thermodynamics of the charged capacitor by standard methods.
The authors acknowledge useful discussions with F. Alessio, J. Crabbe, and S. Prohazka. This work
is supported by the F.R.S.-FNRS Belgium through a research fellowship for Martin Bonte and also
through conventions FRFC PDR T.1025.14 and IISN 4.4503.15.
Bibliography
1. P.A.M. Dirac, Proc. R. Soc. Lond. A 136(829), 453 (1932). doi:10.1098/rspa.1932.0094
2. V. Fock, B. Podolsky, Phys. Zs. Sowjetunion 1, 798 (1932)
3. T. Kugo, I. Ojima, Prog. Theor. Phys. Suppl. 66, 1 (1979)
4. M. Henneaux, C. Teitelboim, Quantization of Gauge Systems (Princeton University, Princeton,
1992)
5. P.A.M. Dirac, Can. J. Phys. 33, 650 (1955). doi:10.1139/p55-081
6. G. Barnich, Gen. Rel. Grav. 43, 2527 (2011). doi:10.1007/s10714-010-0984-6
7. M. Bronstein, Gen. Rel. Grav. 44, 267 (2012). doi:10.1007/s10714-011-1285-4
8. S. Deser, A.A. Starobinsky, Gen. Rel. Grav. 44, 263 (2012). doi:10.1007/s10714-011-1284-5
9. H.B.G. Casimir, Indag. Math. 10, 261 (1948). [Kon. Ned. Akad. Wetensch. Proc.100N3-4,61
(1997)]
10. J. Mehra, Physica 37, 145 (1967). doi:10.1016/0031-8914(67)90115-2
11. M. Fierz, Helv. Phys. Acta 33, 855 (1960)
12. L.S. Brown, G.J. Maclay, Phys. Rev. 184, 1272 (1969). doi:10.1103/PhysRev.184.1272
13. G.
Plunien,
B.
Muller,
W.
Greiner,
Phys.
Rept.
134,
87
(1986).
doi:10.1016/0370-1573(86)90020-7
14. B.E. Sernelius, Surface Modes in Physics (Wiley, New York, 2001). doi:10.1002/3527603166
15. M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, in Advances in the
Casimir effect. Integral Series Monographs Physics, vol. 145 (Oxford University, Oxford,
2009)
16. G. Barnich, Phys. Rev. D99(2), 026007 (2019). doi:10.1103/PhysRevD.99.026007
17. G.W. Gibbons, S.W. Hawking, Phys. Rev. D15, 2752 (1977)
18. S.W. Hawking, S.F. Ross, Phys. Rev. D52, 5865 (1995)
19. S. Deser, M. Henneaux, C. Teitelboim, Phys. Rev. D55, 826 (1997)
20. M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992)
21. J. Slater, N. Frank, Electromagnetism (Dover Publications, New York, 1969)
22. H.B. Callen, Thermodynamics (Wiley, New York, 1961)
23. F. Heinrich, Am. J. Phys. 54(8), 742 (1986)
24. H. Casimir, D. Polder, Phys. Rev. 73(4), 360 (1948)
25. J. Ambjorn, S. Wolfram, Annals Phys. 147, 1 (1983). doi:10.1016/0003-4916(83)90065-9
26. J.I. Kapusta, Phys.Rev. D24, 426 (1981). doi:10.1103/PhysRevD.24.426
27. J.S. Dowker, Phys. Rev. D37, 558 (1988). doi:10.1103/PhysRevD.37.558
28. J. Dowker, Class. Quant. Grav. 20, L105 (2003). doi:10.1088/0264-9381/20/8/102