378
G. Barnich and M. Bonte
7 Extra Contributions to Partition Function
When naively decomposing the additional massless scalar field into its zero mode,
the particle, and the remaining bulk modes in two dimensions, their contributions to
the partition function is straightforward. In the charged case, the former is given by
Z
0
NPG (β, μ) = Tre
−β(H 0
NPG −μQ) .
(35)
This can be related to the well-known partition function of a free particle of unit
mass by completing the square. The result is
ln Z
0
NPG (β, μ) = ln Δq −
1
2
ln
2π ¯
h
2 β
+
A
2d
βμ
2 .
(36)
Here Δq denotes the divergent interval of integration of q, while the last term
reproduces the Gibbons–Hawking contribution −βF (β, μ) to the partition function
as discussed in (21).
The partition function of a massless scalar in two dimensions can be obtained
as usual after putting the field in a box with periodic boundary conditions and by
neglecting the zero mode. The standard result in the limit of large volume in two
dimensions, which is the area of the plates in the current context, is
ln Z
NPG =
A
2π
ζ(3)( ¯
hβ)
−2 .
(37)
A different discussion along the lines of [27, 28] gives instead
F
0
NPG (β, 0) = β
−1
[div + ln β + cte],
(38)
which differs by a factor of 2 in the ln β term from (36). Note however that this
difference will not matter for the considerations below as long as the μ dependent
part will still be given by −
A
2d μ 2 .
8 Charged Black Body Partition Function
In order to discuss the full, finite result, one may follow and adapt the discussion of
the finite temperature Casimir effect [10, 11] (see e.g. [13–15] for reviews).
One considers segments on the z-axis given by
I = [0, d], II = [d, L z ], I I I = [0, L z /η], IV = [L z /η, L z ].
(39)
The analog F C (β, μ) of the Casimir free energy is defined as
G. Barnich and M. Bonte
7 Extra Contributions to Partition Function
When naively decomposing the additional massless scalar field into its zero mode,
the particle, and the remaining bulk modes in two dimensions, their contributions to
the partition function is straightforward. In the charged case, the former is given by
Z
0
NPG (β, μ) = Tre
−β(H 0
NPG −μQ) .
(35)
This can be related to the well-known partition function of a free particle of unit
mass by completing the square. The result is
ln Z
0
NPG (β, μ) = ln Δq −
1
2
ln
2π ¯
h
2 β
+
A
2d
βμ
2 .
(36)
Here Δq denotes the divergent interval of integration of q, while the last term
reproduces the Gibbons–Hawking contribution −βF (β, μ) to the partition function
as discussed in (21).
The partition function of a massless scalar in two dimensions can be obtained
as usual after putting the field in a box with periodic boundary conditions and by
neglecting the zero mode. The standard result in the limit of large volume in two
dimensions, which is the area of the plates in the current context, is
ln Z
NPG =
A
2π
ζ(3)( ¯
hβ)
−2 .
(37)
A different discussion along the lines of [27, 28] gives instead
F
0
NPG (β, 0) = β
−1
[div + ln β + cte],
(38)
which differs by a factor of 2 in the ln β term from (36). Note however that this
difference will not matter for the considerations below as long as the μ dependent
part will still be given by −
A
2d μ 2 .
8 Charged Black Body Partition Function
In order to discuss the full, finite result, one may follow and adapt the discussion of
the finite temperature Casimir effect [10, 11] (see e.g. [13–15] for reviews).
One considers segments on the z-axis given by
I = [0, d], II = [d, L z ], I I I = [0, L z /η], IV = [L z /η, L z ].
(39)
The analog F C (β, μ) of the Casimir free energy is defined as