Resolution of an apparent inconsistency in the electromagnetic Casimir effect

Abstract

The vacuum expectation value of the electromagnetic energy-momentum tensor between two parallel plates in spacetime dimensions D > 4 is calculated in the axial gauge. While the pressure between the plates agrees with the global Casimir force, the energy density is divergent at the plates and not compatible with the total energy which follows from the force. However, subtracting the divergent self-energies of the plates, the resulting energy is finite and consistent with the force. In analogy with the corresponding scalar case for spacetime dimensions D > 2, the divergent self-energy of a single plate can be related to the lack of conformal invariance of the electromagnetic Lagrangian for dimensions D > 4.

Full text

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arXiv:hep-ph/0610081v2 11 Apr 2007
Resolution of an apparent inconsistency in the
electromagnetic Casimir effect
H. Alnes1, K. Olaussen2, F. Ravndal1and I.K. Wehus1
1Department of Physics, University of Oslo, N-0316 Oslo, Norway.
2Department of Physics, NTNU, N-7491 Trondheim, Norway.
E-mail: i.k.wehus@fys.uio.no
Abstract.
The vacuum expectation value of the electromagnetic energy-momentum
tensor between two parallel plates in spacetime dimensions D > 4 is calculated in the
axial gauge. While the pressure between the plates agrees with the global Casimir
force, the energy density is divergent at the plates and not compatible with the total
energy which follows from the force. However, subtracting the divergent self-energies
of the plates, the resulting energy is finite and consistent with the force. In analogy
with the corresponding scalar case for spacetime dimensions D > 2, the divergent
self-energy of a single plate can be related to the lack of conformal invariance of the
electromagnetic Lagrangian for dimensions D > 4.
Two parallel, metallic plates separated by the distance L in vacuum, will interact
due to the modifications of the quantum fluctuations of the electromagnetic field caused
by the boundary conditions at the plates. The resulting force was first calculated by
Casimir[1] who found it to be given by the attractive pressure P = −π2/240L4. Using
the conformal symmetry of the electromagnetic field in D = 4 spacetime dimensions,
Brown and Maclay[2] later obtained the vacuum expectation values of all the components
of the electromagnetic energy-momentum tensor
Tµν= FµαFα
ν− ηµνL (1)
where L = −(1/4)F2
constant between the plates, the corresponding fluctuations of the separate electric and
magnetic fields were found by L¨ utken and Ravndal to be in general non-constant and
actually divergent as one approaches one of the plates[3]. These divergences are caused
by imposing ideal boundary conditions valid for arbitrarily small wavelengths of the
field. A physical boundary would only affect fluctuations down to a finite wavelength
which is expected to result in an increasing, but finite value of the fluctuations near the
plates. The quantitative effects of such more realistic boundary conditions have been
investigated during the last few years but a complete and satisfactory description is still
lacking[4].
Casimir forces in spacetimes with dimensions D > 4 were first systematically
calculated by Ambjørn and Wolfram[5]. For the electromagnetic field between two
αβis the standard Lagrangian. While these expectation values were

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Resolution of an apparent inconsistency in the electromagnetic Casimir effect
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parallel hyperplanes with separation L, the attractive pressure was found to be
P = −(D − 1)(D − 2)Γ(D/2)ζR(D)
(4π)D/2LD
(2)
where the factor D−2 is the number of physical degrees of freedom in the field resulting
from gauge invariance. If the energy density between the plates is constant, it would
just be this pressure divided by the factor D − 1. This is the case when D = 4 and it
is of interest to see if it holds also in the more general case D > 4. For this purpose
we calculate in the following the separate fluctuations of the electric and magnetic
components of the field which then allows us to find all the vacuum expectation values
of the components of the energy-momentum tensor (1).
Today these quantum effects could be of relevance for stacks of parallel branes
where the electromagnetic field is replaced by one or more of the abelian Ramond-
Ramond fields. Any divergent energy density would then have serious implications for
the stability of such configurations due to the resulting large gravitational interactions.
The electromagnetic field tensor Fµν = ∂µAν− ∂νAµ in D = d + 1 spacetime
dimensions has d electric components Ei= F0iand d(d − 1)/2 magnetic components
Bij= Fij. For the geometry under consideration, the simplest and most natural choice
of gauge is the axial gauge nµAµ = 0 where the unit D-vector nµis normal to the
plates. Taking this along the z-axis, we thus have Az = 0. The component A0is no
longer a free variable, but depends on the others via the Maxwell equation ∂iFi0= 0.
It gives A0= −∆−1∂i˙Aiwhere the operator ∆ = ∂2
degrees of freedom described by the spatial field components Aiwhere i ?= z. The full
Lagrangian then follows as
L =1
2
after a few partial integrations and neglecting surface terms.
In order to quantize the system, we must solve the classical wave equation following
from the Lagrangian. For this purpose we impose the boundary condition nµFµν = 0
at the plates. This is the same as for the MIT quark bag where it had a physical
justification[6]. Here it is just taken for convenience. In the axial gauge it gives ∂zAi= 0
at the plates which is the Neumann boundary condition for each physical field component
Ai(x) = Ai(t;xT,z). We then have the general mode expansion
i. There are thus D −2 independent
?
ddx
?˙Ai
?
δij− ∂i∆−1∂j
?˙Aj− Ai
?
∂i∂j− δij∆
?
Aj
?
(3)
Ai(t;xT,z) =
?
2
L
∞
?
n=1
?dd−1kT
(2π)d−1Ain(t,kT)eikT·xTcos
?nπz
L
?
(4)
which satisfies the wave equation and the boundary conditions. The factor
normalization factor. In the mode sum we have dropped a term with n = 0 since it will
not contribute to any physical results after regularization.
Quantization can now be done in the standard way. We introduce orthonormal
polarization vectors eλ normal to the wavevector kT and a longitudinal polarization
vector eL along this direction. The coordinate components Ain of the field are then
?
2/L is a

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Resolution of an apparent inconsistency in the electromagnetic Casimir effect
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replaced by the polarization components (Aλn,ALn). After quantization at t = 0 the
transverse components can then be written on the standard form as
Aλn(kT) =
?
1
2ωn
?
aλn(kT) + a†
λn(−kT)
?
(5)
where ω2
the standard commutator
n= k2
T+k2
zwith kz= πn/L. The creation and annihilation operators now have
[aλn(kT),aλ′n′(k′
T)] = δλλ′δnn′(2π)d−1δ(kT− k′
T) (6)
However, the longitudinal component
ALn(kT) =
?
1
2ωn
?ωn
kz
??
aLn(kT) + a†
Ln(−kT)
?
(7)
contains an extra factor when the corresponding creation and annihilation operators
have the same canonical commutator (6). The full field operator (4) is then expressed
in terms of these new operators corresponding to definite polarization states.
The field fluctuations between the two plates can now easily be calculated. As a
simple example, consider Ez = −∂z∆−1∂j˙Aj. If we isolate the mode with quantum
numbers (n,kT), we find the operator
∆−1∂j˙Aj=
?
L2ωn
ω2
n
acting on the vacuum state. Here we have used that ∆ gives k2
space. We see that the transverse modes will not contribute here since they satisfy the
orthogonality condition eλ·kT= 0. However, for the longitudinal mode we have instead
eL· kT = kT and it will give a non-zero contribution. The derivative ∂zgives a factor
kzand cos(nπz/L) → sin(nπz/L). For this mode alone we thus get the fluctuation
1
2ωnk2
L
Including all the modes, we thus have for the full fluctuation of this electric field
component
?dd−1kT
(2π)d−1
when we write k2
2
?
1
(ikj)(−iωn)
?
aλneλj+
?ωn
kz
?
aLneLj
?
eikT·xTcos
?nπz
nin momentum
L
?
+ H.c. (8)
T+k2
z= ω2
?E2
z?|n,kT=2
L
Tsin2nπz
(9)
?E2
z? =1
L
∞
?
n=1
?
ωn−k2
z
ωn
?
sin2nπz
L
(10)
T= ω2
n− k2
z. For the other components we similarly find
?dd−1kT
(2π)d−1
?E2
i? =1
L
∞
?
n=1
?
ωn(d − 2) +k2
z
ωn
?
cos2nπz
L
(11)
where there is an implied sum over the transverse index i.
fluctuations can be obtained the same way and become
?dd−1kT
(2π)d−1
i<j? =1
L
n=1
(2π)d−1
The magnetic field
?B2
iz? =1
L
∞
?
∞
?
n=1
?
?
ωn+k2
z
ωn(d − 2)
?
sin2nπz
L
(12)
?B2
?dd−1kT
ωn(d − 2) −k2
z
ωn(d − 2)
?
cos2nπz
L
(13)

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Resolution of an apparent inconsistency in the electromagnetic Casimir effect
4
when we again sum over the indices i and j. We have also confirmed these results by
performing the same calculations in Coulomb gauge instead of axial gauge.
Using now a combination of dimensional and zeta-function regularization as
previously used when D = 4[7], we can write the result on the form
1
2L
∞
?
n=1
?dd−1kT
(2π)d−1
Γ(D/2)
(4π)D/2LD
?
ζR(D) ±1
1 ± cos2nπz
L
??
ωn
z/ωn
??
k2
?
= −
?
2fD(z/L)
1
D − 1
?
(14)
Here ζR(D) is the Riemann zeta-function while fD(z/L) depends on the distance z from
the plates. When the spacetime dimension D is even, it can be written on the compact
form
πD
Γ(D) dθ
fD(z/L) =
?
−d
?D−1
cotθ(D = even)(15)
where θ = πz/L. But when D is odd, no such closed expression is easily derived.
However, using a different regularization based on the corresponding point-split Green’s
functions, one finds in general[8]
fD(z/L) =
∞
?
j=−∞
1
|j + z/L|D= ζH(D,z/L) + ζH(D,1 − z/L) (16)
where ζH(D,z/L) is the Hurwitz zeta-function. When D is even, this can be shown to
agree with (15).
The regularized fluctuations of the electric field normal to the plates thus become
ζR(D) −1
?E2
z? =(D − 2)Γ(D/2)
(4π)D/2LD
?
2fD(z/L)
?
(17)
while for the transverse components we find
?E2
i? = −2(D − 2)Γ(D/2)
(4π)D/2LD
?
ζR(D) +1
2fD(z/L)
?
(18)
For the magnetic fluctuations we similarly have
?B2
iz? = −(D − 2)(D − 2)Γ(D/2)
(4π)D/2LD
?
ζR(D) −1
2fD(z/L)
?
(19)
and
?B2
i<j? = (D − 3)(D − 2)Γ(D/2)
(4π)D/2LD
?
ζR(D) +1
2fD(z/L)
?
(20)
Notice again that in these expressions we have summed over the transverse indices i
and j, each taking D − 2 different values. All these correlators are seen to diverge near
the plates z → 0 or z → L where the function fD(z/L) diverges. This is the same
phenomenon which has previously been seen in D = 4 dimensions[3].
The pressure betwen the plates due to these fluctuations is defined by P = ?Tzz?.
From (1) we have Tzz= B2
?L? = −1
(4π)D/2LD
iz− E2
2(D − 1)(D − 2)Γ(D/2)
z+ L where now
fD(z/L) (21)

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Resolution of an apparent inconsistency in the electromagnetic Casimir effect
5
Together with the values for ?E2
function fD(z/L) cancels out in the pressure and gives the expected value (2).
So far there are no inconsistencies in the obtained results.
calculate the energy density E = ?T00? between the plates, with T00= E2
obtain
E = −(D − 2)Γ(D/2)
(4π)D/2LD
z? and ?B2
iz? from above, the z-dependence from the
But when we now
i+E2
z−L, we
[ζR(D) − (D/2 − 2)fD(z/L)] (22)
The z-dependence in the last term is non-zero when D > 4 and makes the energy density
diverge like z−Dwith distance z from the plates. As a result, the total energy of the
system is infinite, a result which seems to be impossible to reconcile with the finite
Casimir force (2). In fact, (2) corresponds to having a constant energy density equal
to the first term in (22). This apparent inconsistency has been verified in a different
approach based on Green’s function methods[8].
It is tempting to explain this problem by the imposed boundary conditions. We
have used the MIT boundary condition which is equivalent to letting the electromagnetic
vector potential satisfy Neumann boundary conditions in the axial gauge.
we instead imposed metallic boundary conditions, equivalent to Dirichlet boundary
conditions for the vector potential in the axial gauge, the only change in the above
results would be the replacement of the mode functions cos(nπz/L) with sin(nπz/L) in
(4) so that fD→ −fDin the above results. Needless to say, the problem would remain.
Only for periodic boundary conditions, as for finite temperature, would the disturbing
term be absent[9]. But this is not necessarily satisfying from a physical point of view. A
more mathematical discussion of such divergences near confining boundaries has been
initiated by Fulling but here only scalar fields are considered[10].
A physical explanation of the above conumdrum becomes apparent when we take
the limit L → ∞ and thus consider the quantum fluctuations around a single plate.
From (22) we then find the energy density
Had
E1= (D − 2)(D/2 − 2)
Γ(D/2)
(4π)D/2|z|D
(23)
which is non-zero on both sides of the plate and diverges when we approach it. This
situation is analogous to the diverging energy density surrounding a pointlike electron.
It is intrinsic to a single plate and should not contribute to the interaction between the
plates induced by the same vacuum fluctuations. To see the connection with the Casimir
force, we should subtract the self-energy (23) for both plates from the full energy density
(22), taking into account both sides of each plate. We thus obtain the interaction energy
density
˜E = −(D−2)
Γ(D/2)
(4π)D/2LD×







(D/2 − 2)(L/(L − z))D
ζR(D) − (D/2 − 2)˜fD(z/L) for 0 < z < L,
(D/2 − 2)(L/z)D
for z < 0,
for z > L,
(24)
where now