Electromagnetic Casimir densities for a wedge with a coaxial cylindrical
  shell
Electromagnetic Casimir densities for a wedge with a coaxial
cylindrical shell
A. A. Saharian∗
Department of Physics, Yerevan State University, 1 Alex Manoogian Street,
375025 Yerevan, Armenia
October 29, 2018
Abstract
Vacuum expectation values of the field square and the energy-momentum tensor for the
electromagnetic field are investigated for the geometry of a wedge with a coaxal cylindrical
boundary. All boundaries are assumed to be perfectly conducting and both regions inside
and outside the shell are considered. By using the generalized Abel-Plana formula, the
vacuum expectation values are presented in the form of the sum of two terms. The first one
corresponds to the geometry of the wedge without the cylindrical shell and the second term
is induced by the presence of the shell. The vacuum energy density induced by the shell
is negative for the interior region and is positive for the exterior region. The asymptotic
behavior of the vacuum expectation values are investigated in various limiting cases. It is
shown that the vacuum forces acting on the wedge sides due to the presence of the cylindrical
boundary are always attractive.
PACS numbers: 03.70.+k
1 Introduction
The Casimir effect is among the most interesting macroscopic manifestations of vacuum fluctu-
ations. It have important implications on all scales, from cosmological to subnuclear, and has
become in recent decades an increasingly popular topic in quantum field theory. In addition to
its fundamental interest the Casimir effect also plays an important role in the fabrication and
operation of nano- and micro-scale mechanical systems. The imposition of boundary conditions
on a quantum field leads to the modification of the spectrum for the zero-point fluctuations and
results in the shift in the vacuum expectation values for physical quantities such as the energy
density and stresses. In particular, the confinement of quantum fluctuations causes forces that
act on constraining boundaries. The particular features of the resulting vacuum forces depend
on the nature of the quantum field, the type of spacetime manifold, the boundary geometries
and the specific boundary conditions imposed on the field. Since the original work by Casimir [1]
many theoretical and experimental works have been done on this problem (see, e.g., [2, 3, 4, 5]
and references therein). Many different approaches have been used: mode summation method
E-mail: saharian@ictp.it
http://arxiv.org/abs/0704.1898v1
with combination of the zeta function regularization technique, Green function formalism, mul-
tiple scattering expansions, heat-kernel series, etc. Advanced field-theoretical methods have
been developed for Casimir calculations during the past years [6, 7, 8]. However, there are still
difficulties in both interpretation and renormalization of the Casimir effect. Straightforward
computations of geometry dependencies are conceptually complicated, since relevant informa-
tion is subtly encoded in the fluctuations spectrum [8]. Analytic solutions can usually be found
only for highly symmetric geometries including planar, spherically and cylindrically symmetric
boundaries. Recently the Casimir energy has been evaluated exactly for several less symmetric
configurations of experimental interest. These include a sphere in front of a plane and a cylinder
in front of a plane [9].
A great deal of attention received the investigations of quantum effects for cylindrical bound-
aries. In addition to traditional problems of quantum electrodynamics under the presence of
material boundaries, the Casimir effect for cylindrical geometries can also be important to the
flux tube models of confinement [10, 11] and for determining the structure of the vacuum state
in interacting field theories [12]. The calculation of the vacuum energy of electromagnetic field
with boundary conditions defined on a cylinder turned out to be technically a more involved
problem than the analogous one for a sphere. First the Casimir energy of an infinite perfectly
conducting cylindrical shell has been calculated in Ref. [13] by introducing ultraviolet cutoff
and later the corresponding result was derived by zeta function technique [14, 15, 16]. The local
characteristics of the corresponding electromagnetic vacuum such as energy density and vacuum
stresses are considered in [17] for the interior and exterior regions of a conducting cylindrical
shell, and in [18] for the region between two coaxial shells (see also [19]). The vacuum forces
acting on the boundaries in the geometry of two cylinders are also considered in Refs. [20].
The scalar Casimir densities for a single and two coaxial cylindrical shells with Robin boundary
conditions are investigated in Refs. [21, 22]. Less symmetric configuration of two eccentric per-
fectly conducting cylinders is considered in Ref. [23]. Vacuum energy for a perfectly conducting
cylinder of elliptical section is evaluated in Ref. [24] by the mode summation method, using the
ellipticity as a perturbation parameter. The Casimir forces acting on two parallel plates inside
a conducting cylindrical shell are investigated in Ref. [25].
Aside from their own theoretical and experimental interest, the exactly solvable problems
with this type of boundaries are useful for testing the validity of various approximations used
to deal with more complicated geometries. From this point of view the wedge with a coaxial
cylindrical boundary is an interesting system, since the geometry is nontrivial and it includes
two dynamical parameters, radius of the cylindrical shell and opening angle of the wedge. This
geometry is also interesting from the point of view of general analysis for surface divergences
in the expectation values of local physical observables for boundaries with discontinuities. The
nonsmoothness of the boundary generates additional contributions to the heat kernel coefficients
(see, for instance, the discussion in [26, 27, 28] and references therein). The present paper is
concerned with local analysis of the vacuum of the electromagnetic field constrained to satisfy
perfectly conducting boundary conditions on boundary surfaces of a wedge with a coaxial cylin-
drical boundary. Namely, we will study the vacuum expectation values of the field squares and
the energy-momentum tensor for the electromagnetic field for both regions inside and outside
the cylindrical shell. In addition to describing the physical structure of the quantum field at
a given point, the energy-momentum tensor acts as the source of gravity in the Einstein equa-
tions. It therefore plays an important role in modelling a self-consistent dynamics involving the
gravitational field. The vacuum expectation value of the square of the electric field determines
the electromagnetic force on a neutral polarizable particle. Some most relevant investigations to
the present paper are contained in Refs. [2, 29, 30, 31, 32, 33], where the geometry of a wedge
without a cylindrical boundary is considered for a conformally coupled scalar and electromag-
netic fields in a four dimensional spacetime. The total Casimir energy of a semi-circular infinite
cylindrical shell with perfectly conducting walls is considered in [34] by using the zeta function
technique. For a scalar field with an arbitrary curvature coupling parameter the Wightman
function, the vacuum expectation values of the field square and the energy-momentum tensor
in the geometry of a wedge with an arbitrary opening angle and with a cylindrical boundary
are evaluated in [35, 36]. Note that, unlike the case of conformally coupled fields, for a general
coupling the vacuum energy-momentum tensor is angle-dependent and diverges on the wedge
sides. Our method here employs the mode summation and is based on a variant of the gen-
eralized Abel-Plana formula [37] (see also Refs. [19, 38]). This enables us to extract from the
vacuum expectation values the parts due to a wedge without the cylindrical shell and to present
the parts induced by the shell in terms of strongly convergent integrals. Note that the closely
related problem of the vacuum densities induced by a cylindrical boundary in the geometry of
a cosmic string is investigated in Refs. [39, 40] for both scalar and electromagnetic fields.
We have organized the paper as follows. In the next section we describe the structure of
the modes for a wedge with a cylindrical shell in the region inside the shell. By applying
to the corresponding mode sums the generalized Abel-Plana formula, we evaluate the vacuum
expectation values of the electric and magnetic field square. Various limiting cases of the general
formulae are discussed. Section 3 is devoted to the investigation of the vacuum expectation
values for the energy-momentum tensor of the electromagnetic field in the region inside the
shell. The additional vacuum forces acting on the wedge sides due to the presence of the
cylindrical boundary are evaluated. In section 4 we consider the vacuum densities for a wedge
with the cylindrical shell in the exterior region with respect to the shell. Formulae for the
shell contributions are derived and the corresponding surface divergences are investigated. The
vacuum forces acting on the wedge sides are discussed. The main results are summarized and
discussed in section 5.
2 Vacuum expectation values of the field square inside a cylin-
drical shell
Consider a wedge with the opening angle φ0 and with a coaxial cylindrical boundary of radius
a (see figure 1) assuming that all boundaries are perfectly conducting. In accordance with the
problem symmetry, in the discussion below the cylindrical coordinates (r, φ, z) will be used. We
are interested in the vacuum expectation values (VEVs) of the field square and the energy-
momentum tensor for the electromagnetic field. Expanding the field operator in terms of the
creation and annihilation operators and using the commutation relations, the VEV for a quantity
F {Ai, Ak} bilinear in the field can be presented in the form of the mode-sum
〈0|F {Ai, Ak} |0〉 =
F {Aαi, A∗αk} , (1)
where {Aαi, A∗αk} is a complete set of solutions of the classical field equations satisfying the
boundary conditions on the bounding surfaces and specified by a set of quantum numbers α.
In accordance with formula (1), for the evaluation of the VEVs for the square of the electric
and magnetic fields and the energy-momentum tensor, the corresponding eigenfunctions are
needed. In this section we consider the region inside the cylindrical shell (region I in figure
1). For the geometry under consideration there are two different types of the eigenfunctions
corresponding to the transverse magnetic (TM) and transverse electric (TE) waves. In the
discussion below we will specify these modes by the index λ = 0 and λ = 1 for the TM and TE
waves respectively. In the Coulomb gauge, the vector potentials for the TM and TE modes are
Figure 1: Geometry of a wedge with a coaxial cylindrical boundary with radius a.
given by the formulae
Aα = βα
(1/iω)
γ2e3 + ik∇t
Jq|m|(γr) sin(qmφ) exp [i (kz − ωt)] , λ = 0
−e3 ×∇t
Jq|m|(γr) cos(qmφ) exp [i (kz − ωt)]
, λ = 1
, (2)
where e3 is the unit vector along the axis of the wedge, ∇t is the part of the nabla operator
transverse to this axis, Jν(x) is the Bessel function of the first kind, and
ω2 = γ2 + k2, q = π/φ0. (3)
In Eq. (2), m = 1, 2, . . . for λ = 0 and m = 0, 1, 2, . . . for λ = 1. The normalization coefficient
βα is found from the orthonormalization condition for the vector potential:
dV Aα ·A∗α′ =
δαα′ , (4)
where the integration goes over the region inside the shell. From this condition, by using the
standard integral involving the square of the Bessel function, one finds
β2α =
4qTqm(γa)
δm, δm =
1/2, m = 0
1, m 6= 0 , (5)
where we have introduced the notation
Tν(x) = x
ν (x) + (1− ν2/x2)J2ν (x)
. (6)
Eigenfunctions (2) satisfy the standard boundary conditions for the electric and magnetic
fields, n×E = 0 and n ·B = 0, on the wedge sides corresponding to φ = 0 and φ = φ0, with n
being the normal to the boundary. The eigenvalues for the quantum number γ are determined by
the boundary conditions on the cylindrical shell. From the latter it follows that these eigenvalues
are solutions of the equation
J (λ)qm (γa) = 0, λ = 0, 1, (7)
where we use the notations J
ν (x) = Jν(x) and J
ν (x) = J
ν(x). We will denote the corre-
sponding eigenmodes by γa = j
m,n, n = 1, 2, . . ., assuming that the zeros j
m,n are arranged in
ascending order. Consequently, the eigenfunctions are specified by the set of quantum numbers
α = (k,m, λ, n).
First we consider the VEVs of the squares of the electric and magnetic fields inside the shell.
Substituting the eigenfunctions (2) into the corresponding mode-sum formula, we find
〈0|F 2|0〉 = 4q
λ=0,1
m,n Tqm(j
m,n + k2a2
×g(ηFλ)[Φ(λ)qm(φ), Jqm(j(λ)m,nr/a)], (8)
where F = E,B with ηEλ = λ, ηBλ = 1−λ, and the prime in the summation over m means that
the term m = 0 should be halved. In formula (8), for a given function f(x), we have introduced
the notations
g(0)[Φ(φ), f(x)] = (k2r2/x2)
Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2
+Φ2(φ)f2(x), (9)
g(1)[Φ(φ), f(x)] = (1 + k2r2/x2)
Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2
, (10)
Φ(λ)ν (φ) =
sin(νφ), λ = 0
cos(νφ), λ = 1
. (11)
The expressions (8) corresponding to the electric and magnetic fields are divergent. They may
be regularized introducing a cutoff function ψµ(ω) with the cutting parameter µ which makes
the divergent expressions finite and satisfies the condition ψµ(ω) → 1 for µ → 0. After the
renormalization the cutoff function is removed by taking the limit µ → 0. An alternative way
is to consider the product of the fields at different spacetime points and to take the coincidence
limit after the subtraction of the corresponding Minkowskian part. Our approach here follows
the first method.
As we do not know the explicit expressions for the zeros j
m,n as functions on m and n, and
the summand in formula (8) is strongly oscillating function for large values of m and n, this
formula is not convenient for the further evaluation of the VEVs of the field square. In order to
obtain an alternative representation, we apply to the series over n the generalized Abel-Plana
summation formula [37] (see also [38])
Tqm(j
m,n)f(j
m,n) =
dx f(x) +
qm (z)
qm (z)
qm (x)
qm (x)
e−qmπif(eπi/2x) + eqmπif(e−πi/2x)
, (12)
where Yν(z) is the Neumann function and Iν(z), Kν(z) are the modified Bessel functions. As
it can be seen, for points away from the shell the contribution to the VEVs coming from the
second integral term on the right-hand side of (12) is finite in the limit µ → 0 and, hence, the
cutoff function in this term can be safely removed. As a result the VEVs can be written in the
〈0|F 2|0〉 = 〈0w|F 2|0w〉+
, (13)
where
〈0w|F 2|0w〉 =
γ3ψµ(ω)
γ2 + k2
J ′2qm(γr) +
J2qm(γr)
+ J2qm(γr)
−(−1)ηF1 cos(2qmφ)
J ′2qm(γr)−
J2qm(γr)
, (14)
〈F 2〉cyl =
λ=0,1
qm (xa)
qm (xa)
G(ηFλ)[k,Φ
qm(φ), Iqm(xr)]√
x2 − k2
. (15)
In formula Eq. (15) we have introduced the notations
G(0)[k,Φ(φ), f(x)] = (k2r2/x2)
Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2
+Φ2(φ)f2(x), (16)
G(1)[k,Φ(φ), f(x)] = (k2r2/x2 − 1)
Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2
. (17)
The second term on the right-hand side of Eq. (13) vanishes in the limit a → ∞ and the first
one does not depend on a. Thus, we can conclude that the term 〈0w|F 2|0w〉 corresponds to
the part in the VEVs when the cylindrical shell is absent with the corresponding vacuum state
|0w〉. Hence, the application of the generalized Abel-Plana formula enables us to extract from
the VEVs the parts induced by the cylindrical shell without specifying the cutoff function. In
addition, these parts are presented in terms of the exponentially convergent integrals.
First, let us concentrate on the part corresponding to the wedge without a cylindrical shell.
First of all we note that in Eq. (14) the part which does not depend on the angular coordinate φ
is the same as in the corresponding problem of the cosmic string geometry with the angle deficit
2π − φ0 (see Ref. [40]), which we will denote by 〈0s|F 2|0s〉. For this part we have
〈0s|F 2|0s〉 =
γ3ψµ(ω)
γ2 + k2
J ′2qm(γr) +
J2qm(γr)
+ J2qm(γr)
= 〈0M|F 2|0M〉 −
(q2 − 1)(q2 + 11)
180πr4
, (18)
where 〈0M|F 2|0M〉 is the part corresponding to the Minkowskian spacetime without boundaries
and in the last expression we have removed the cutoff. To evaluate the part in (14) which
depends on φ, we firstly consider the case when the parameter q is an integer. In this case the
summation over m can be done by using the formula [41, 42]
cos(2qmφ)J2qm(y) =
J0(2y sin(φ+ φ0l)). (19)
The formulae for the other series entering in Eq. (14) are obtained from (19) taking the deriva-
tives with respect to φ and y. In particular, for the combination appearing in the angle-dependent
part we obtain
cos(2qmφ)
J ′2qm(y)−
J2qm(y)
J ′1(2y sin(φ+ φ0l)). (20)
Substituting this in formula (14), the integrals remained are evaluated by introducing polar
coordinates in the (k, γ)-plane. In this way one finds
〈0w|F 2|0w〉 = 〈0s|F 2|0s〉 −
3(−1)ηF1
sin−4(φ+ lπ/q). (21)
The sum on the right hand-side of this formula is evaluated by the double differentiation of the
relation [41]
cos−2(x+ lπ/q) = q2 sin−2(qx+ qπ/2). (22)
Finally, for the renormalised VEVs of the field square in the geometry of a wedge without a
cylindrical boundary we find
〈F 2〉w,ren = −
(q2 − 1)(q2 + 11)
180πr4
(−1)ηF1q2
2πr4 sin2(qφ)
2 sin2(qφ)
+ 1− q2
, (23)
with ηE1 = 1 and ηB1 = 0. Though we have derived this formula for integer values of the
parameter q, by the analytic continuation it is valid for non-integer values of this parameter as
well. The expression on the right of formula (23) is invariant under the replacement φ→ φ0 −φ
and, as we could expect, the VEVs are symmetric with respect to the half-plane φ = φ0/2.
Formula (23) for F = E was derived in Ref. [32] within the framework of Schwinger’s source
theory. For q = 1 from formula (23) as a special case we obtain the renormalized VEVs of the
field square for a conducting plate. In this case x = r sinφ is the distance from the plate and
one has
〈F 2〉pl,ren = −
3(−1)ηF1
. (24)
Another special case q = 1/2 corresponds to the geometry of a half-plane. In (23) taking the
limit r → ∞, with x0 = rφ0 being fixed, we obtain the corresponding results in the region
between two parallel plates located at the points x = 0 and x = x0:
〈F 2〉2pl,ren = −
180x40
− (−1)
ηF1π3
2x40 sin
2(πx/x0)
2 sin2(πx/x0)
. (25)
Now, we turn to the investigation of the parts in the VEVs of the field square induced by
the cylindrical boundary and given by formula (15). By using the formula
dk km
xf(x)√
x2 − k2
dxxm+1f(x), (26)
these parts are presented in the form
〈F 2〉cyl =
λ=0,1
qm (xa)
qm (xa)
G(ηFλ)[Φ(λ)qm(φ), Iqm(xr)]. (27)
Here, for given functions f(x) and Φ(φ), we have introduced the notations
G(0)[Φ(φ), f(x)] = Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 + 2Φ2(φ)f2(x), (28)
G(1)[Φ(φ), f(x)] = −Φ2(φ)f ′2(x)− Φ′2(φ)f2(x)/x2. (29)
As we see the parts in the VEVs induced by the cylindrical shell are symmetric with respect to
the half-plane φ = φ0/2.
The expression in the right-hand side of (27) is finite for 0 < r < a including the points on the
wedge sides, and diverges on the shell. To find the leading term in the corresponding asymptotic
expansion, we note that near the shell the main contribution comes from large values of m. By
using the uniform asymptotic expansions of the modified Bessel functions (see, for instance, [43])
for large values of the order, up to the leading order, for the points a−r ≪ a| sinφ|, a| sin(φ0−φ)|
we find
〈F 2〉cyl ≈ −
3(−1)ηF1
4π(a− r)4
. (30)
For the points near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymp-
totic expansions are the same as for the geometry of a wedge with the opening angle φ0 = π/2.
The leading terms given by formula (30) are the same as for the geometry of a single plate (see
(24)). They do not depend on φ0 and have opposite signs for the electric and magnetic fields.
In particular, the leading terms are cancelled in the evaluation of the vacuum energy density.
Surface divergences originate in the unphysical nature of perfect conductor boundary conditions
and are well-known in quantum field theory with boundaries. In reality the expectation values
will attain a limiting value on the conductor surface, which will depend on the molecular de-
tails of the conductor. From the formulae given above it follows that the main contribution to
〈F 2〉cyl are due to the frequencies ω . (a − r)−1. Hence, we expect that formula (27) is valid
for real conductors up to distances r for which (a− r)−1 ≪ ω0, with ω0 being the characteristic
frequency, such that for ω > ω0 the conditions for perfect conductivity fail.
Near the edge r = 0, assuming that r/a≪ 1, the asymptotic behavior of the part induced in
the VEVs of the field square by the cylindrical shell depends on the parameter q. For q > 1+ηF1,
the dominant contribution comes from the lowest mode m = 0 and to the leading order one has
〈F 2〉cyl ≈ −(−1)ηF1
21−ηF1q
)2ηF1
K1(x)
I1(x)
. (31)
In this case the quantity 〈B2〉cyl takes a finite limiting value on the edge r = 0, whereas 〈E2〉cyl
vanishes as r2. For q < 1 + ηF1 the main contribution comes form the mode with m = 1 and
the shell-induced parts diverge on the edge r = 0. The leading terms are given by the formula
〈F 2〉cyl ≈ −
(−1)ηF1q(r/a)2(q−1)
22q−1πΓ2(q)a4
dxx2q+1
Kq(x)
Iq(x)
K ′q(x)
I ′q(x)
. (32)
As for the points near the shell, here the leading divergences in the VEVs of the electric and
magnetic fields are cancelled in the evaluation of the vacuum energy density. For q = 1+ηF1 the
main contribution comes from the modes m = 0, 1 and the corresponding asymptotic behavior
is obtained by summing the right-hand sides of Eqs. (31) and (32). In accordance with (23),
near the edge r = 0 the total VEV is dominated by the part coming from the wedge without
the cylindrical shell. Here we have considered the VEVs for the field square. The VEVs for the
bilinear products of the fields at different spacetime points may be evaluated in a similar way.
Now, we turn to the investigation of the behavior of the VEVs induced by the cylindrical
boundary in the limit q ≫ 1. In this limit the order of the modified Bessel functions is large for
m 6= 0. By using the corresponding asymptotic formulae it can be seen that the contribution of
these terms is suppressed by the factor exp[−2qm ln(a/r)]. As a result, the main contribution
comes from the lowest mode m = 0 and the VEVs induced by the cylindrical shell are propor-
tional to q. Note that in this limit the part corresponding to the wedge without the cylindrical
shell behaves as q4.
3 Vacuum energy-momentum tensor inside the cylindrical shell
Now let us consider the VEV of the energy-momentum tensor in the region inside the cylindrical
shell. Substituting the eigenfunctions (2) into the corresponding mode-sum formula, for the non-
zero components we obtain (no summation over i)
〈0|T ii |0〉 =
2π2a3
λ=0,1
m,n Tqm(j
m,n + k2a2
f (i)[Φ(λ)qm(φ), Jqm(j
m,nr/a)], (33)
〈0|T 12 |0〉 = −
m sin(2qmφ)
λ=0,1
(−1)λ
m,nTqm(j
m,n + k2a2
J2qm(j
m,nr/a), (34)
where we have introduced the notations
f (j)[Φ(φ), f(x)] = (−1)i
2k2/γ2 + 1
Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/y2
+Φ2(φ)f2(x), (35)
f (l)[Φ(φ), f(x)] = (−1)lΦ2(φ)f ′2(x)−
Φ2(φ) + (−1)lΦ′2(φ)/x2
f2(x), (36)
with j = 0, 3 and l = 1, 2. As in the case of the field square, in formulae (33) and (34) we
introduce a cutoff function and apply formula (12) for the summation over n. This enables us
to present the vacuum energy-momentum tensor in the form of the sum
〈0|T ki |0〉 = 〈0w|T ki |0w〉+ 〈T ki 〉cyl, (37)
where 〈0w|T ki |0w〉 is the part corresponding to the geometry of a wedge without a cylindrical
boundary and 〈T ki 〉cyl is induced by the cylindrical shell. By taking into account (26), the latter
may be written in the form (no summation over i)
〈T ii 〉cyl =
λ=0,1
qm (xa)
qm (xa)
F (i)[Φ(λ)qm(φ), Iqm(xr)], (38)
〈T 12 〉cyl =
m sin(2qmφ)
λ=0,1
(−1)λ
qm (xa)
qm (xa)
I2qm(xr), (39)
with the notations
F (i)[Φ(φ), f(y)] = Φ2(φ)f2(y), i = 0, 3, (40)
F (i)[Φ(φ), f(y)] = −(−1)iΦ2(φ)f ′2(y)−
Φ2(φ)− (−1)iΦ′2(φ)/y2
f2(y), i = 1, 2. (41)
The diagonal components are symmetric with respect to the half-plane φ = φ0/2, whereas the
off-diagonal component is an odd function under the replacement φ → φ0 − φ. As it can be
easily checked, the tensor 〈T ki 〉cyl is traceless and satisfies the covariant continuity equation
∇k〈T ki 〉cyl = 0. For the geometry under consideration the latter leads to the relations
r〈T 12 〉cyl
〈T 22 〉cyl = 0, (42)
r〈T 11 〉cyl
〈T 21 〉cyl = 〈T 22 〉cyl. (43)
As it is seen from formula (39), the off-diagonal component 〈T 12 〉cyl vanishes at the wedge sides
and for these points the VEV of the energy-momentum tensor is diagonal. By using the in-
equalities I ′ν(x) <
1 + ν2/x2Iν(x) and −K ′ν(x) >
1 + ν2/x2Kν(x) for the modified Bessel
functions, it can be seen that K ′ν(x)/I
ν(x) + Kν(x)/Iν(x) < 0. From this relation it follows
that the vacuum energy density induced by the cylindrical shell in the interior region is always
negative.
The renormalized VEV of the energy-momentum tensor for the geometry without the cylin-
drical shell is obtained by using the corresponding formulae for the field square. For the corre-
sponding energy density one finds
〈T 00 〉w,ren =
〈E2〉w,ren + 〈B2〉w,ren
= −(q
2 − 1)(q2 + 11)
720π2r4
. (44)
As we see the parts in the VEVs of the field square which diverge on the wedge sides cancel
out and the corresponding energy density is finite everywhere except the edge. Formula (44)
coincides with the corresponding result for the geometry of the cosmic string (see [44, 45]) with
the angle deficit 2π − φ0 and in the corresponding formula q = 2π/φ0. Other components are
found from the tracelessness condition and the continuity equation and one has [29, 30] (see also
〈T ki 〉w,ren = −
(q2 − 1)(q2 + 11)
720π2r4
diag(1, 1,−3, 1). (45)
As we could expect this VEV vanishes for the geometry of a single plate corresponding to q = 1.
In the limit r → ∞, for fixed values x0 = rφ0, from (45) the standard result for the geometry of
two parallel conducting plates is obtained.
The force acting on the wedge sides is determined by the component 〈T 22 〉ren of the vacuum
energy-momentum tensor evaluated for φ = 0 and φ = φ0. On the base of formula (37) for the
corresponding effective pressure one has
p2 = −〈T 22 〉ren|φ=0,φ0 = p2w + p2cyl, (46)
where
p2w = −
(q2 − 1)(q2 + 11)
240π2r4
, (47)
is the normal force acting per unit surface of the wedge for the case without a cylindrical
boundary and the additional term
p2cyl = −〈T 22 〉cyl|φ=0,φ0 = −
λ=0,1
qm (xa)
qm (xa)
F (λ)qm [Iqm(xr)], (48)
with the notation
F (λ)ν [f(y)] =
ν2f2(y)/y2, λ = 0
−f ′2(y)− f2(y), λ = 1 , (49)
is induced by the cylindrical shell. From formula (47) we see that the corresponding vacuum
forces are attractive for q > 1 and repulsive for q < 1. In particular, the equilibrium position
corresponding to the geometry of a single plate (q = 1) is unstable. As regards to the part
induced by the cylindrical shell, from (48) it follows that p2cyl < 0 and, hence, the corresponding
forces are always attractive.
Now, let us discuss the behavior of the boundary-induced part in the VEV of the energy-
momentum tensor in the asymptotic region of the parameters. Near the cylindrical shell the
main contribution comes from large values of m. Thus, using the uniform asymptotic expansions
for the modified Bessel functions for large values of the order, up to the leading order, for the
points a− r ≪ a| sinφ|, a| sin(φ0 − φ)| we find
〈T 00 〉cyl ≈ −
〈T 22 〉cyl ≈ −
(a− r)−3
60π2a
, 〈T 11 〉cyl ≈
(a− r)−2
60π2a2
. (50)
These leading terms are the same as those for a cylindrical shell when the wedge is absent. For
the points near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymptotic
expansions are the same as for the geometry of a wedge with the opening angle φ0 = π/2. The
latter are given by (45) with q = 2. Near the edge, r → 0, for the components (no summation
over i) 〈T ii 〉cyl, i = 0, 3, the main contribution comes from the mode m = 0 and we find
〈T ii 〉cyl ≈
4π2a4
K ′0(x)
I ′0(x)
= −0.0590
, i = 0, 3. (51)
For the components (no summation over i) 〈T ii 〉cyl, i = 1, 2, when q > 1 the main contribution
again comes form m = 0 term and one has 〈T ii 〉cyl ≈ −〈T 00 〉cyl, i = 1, 2. For q < 1 the main
contribution into the components 〈T ii 〉cyl, i = 1, 2, comes from the term m = 1 and we have (no
summation over i)
〈T ii 〉cyl ≈
(−1)iq cos(2qφ)
22q+1π2Γ2(q)a4
)2(q−1)
dxx2q+1
Kq(x)
Iq(x)
K ′q(x)
I ′q(x)
, i = 1, 2. (52)
In this case the radial and azimuthal stresses induced by the cylindrical shell diverge on the
edge r = 0. In the case q = 1 the sum of the contributions of the terms with m = 0 and m = 1
given by formulae (51) and (52) should be taken. For the off-diagonal component the main
contribution comes from the m = 1 mode with the leading term
〈T 12 〉cyl ≈
q sin(2qφ)
22q+1π2Γ2(q)a3
)2q−1
dxx2q+1
Kq(x)
Iq(x)
K ′q(x)
I ′q(x)
, (53)
and this component vanishes on the edge for q > 1/2.
In the limit q ≫ 1, the contribution of the modes with m > 1 is suppressed by the factor
exp[−2qm ln(a/r)] and the main contribution comes from the m = 0 mode. The leading terms
are given by the formulae (no summation over i)
〈T ii 〉cyl ≈
4π2a4
K ′0(x)
I ′0(x)
I20 (xr/a), i = 0, 3, (54)
〈T ii 〉cyl ≈ −
4π2a4
K ′0(x)
I ′0(x)
I20 (xr/a) + (−1)iI21 (xr/a)
, i = 1, 2. (55)
Though in this limit the vacuum densities are large, due to the factor 1/q in the spatial volume,
the corresponding global quantities tend to finite value. In particular, as it follows from Eq.
(55), in the limit under consideration one has 〈T ii 〉cyl > 0. Note that in the same limit the parts
corresponding to the wedge without the cylindrical shell behave as q4 and, hence, for points not
too close to the shell these parts dominate in the VEVs.
In figures 2-5 we have plotted the parts in the VEVs of the energy-momentum tensor induced
by the cylindrical shell, a4〈T ki 〉cyl, as functions of x = (r/a) cos φ and y = (r/a) sinφ, for a wedge
with the opening angle φ0 = π/2.
In figure 6 we have presented the dependence of the effective azimuthal pressure induced by
the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a for different values of the
parameter q.
Figure 2: The part in the VEV of the energy density, a4〈T 00 〉cyl, induced by the cylindrical
boundary as a function on x = (r/a) cos φ and y = (r/a) sin φ for a wedge with φ0 = π/2.
Figure 3: The part in the VEV of the radial stress, a4〈T 11 〉cyl, induced by the cylindrical boundary
as a function on x = (r/a) cos φ and y = (r/a) sinφ for a wedge with φ0 = π/2.
Figure 4: The part in the VEV of the azimuthal stress, a4〈T 22 〉cyl, induced by the cylindrical
boundary as a function on x = (r/a) cos φ and y = (r/a) sin φ for a wedge with φ0 = π/2.
Figure 5: The part in the VEV of the off-diagonal component, a4〈T 21 〉cyl, induced by the cylin-
drical boundary as a function on x = (r/a) cos φ and y = (r/a) sinφ for a wedge with φ0 = π/2.
0.1 0.2 0.3 0.4 0.5
Figure 6: The effective azimuthal pressure induced by the cylindrical shell on the wedge sides,
a4p2cyl, as a function of r/a. The numbers near the curves correspond to the values of the
parameter q.
There are several special cases of interest for the geometry of boundaries we have consid-
ered. The case φ0 = π corresponds to the semi-circular cylinder. The Casimir energy for the
corresponding interior region is evaluated in Ref. [34] by using the zeta function technique. The
case φ0 = 2π corresponds to the geometry of a cylindrical shell with a coaxial half-plane. And
finally, the limit φ0 → 0, r, a → ∞, assuming that a − r and aφ0 ≡ b are fixed, corresponds
to the geometry of two parallel plates separated by a distance b, perpendicularly intersected
by the third plate. In the latter case it is convenient to introduce rectangular coordinates
(x′1, x′2, x′3) = (x, y, z) with the relations x = a − r, y = rφ. The components of the tensors
in these coordinates we will denote by primes. The corresponding vacuum energy-momentum
tensor is presented in the form
〈0|T ′ik |0〉 = 〈T ′ik 〉(0) + 〈T ′ik 〉(1), (56)
where 〈T ′ik 〉(0) is the vacuum expectation value in the region between two parallel plates located
at y = 0 and y = a and 〈T ′ik 〉(1) is induced by the intersecting plate at x = 0. The latter is
related to the quantities investigated above by formulae
〈T ′ii 〉(1) = lim 〈T ii 〉cyl, 〈T ′12 〉(1) = − lim
〈T 12 〉cyl, (57)
with lim corresponding to the limit a → ∞, φ0 → 0 for fixed a − r and aφ0. Taking this limit
in the term with m = 0 of formula (38) we replace the modified Bessel functions by the leading
terms of the corresponding asymptotic formulae for large values of the argument and the integral
is taken elementary. For the terms with m 6= 0 in formulae (38), (39) we note that in the limit
under consideration one has q = π/φ0 → ∞, and the order of the Bessel modified functions tends
to infinity. Introducing a new integration variable x→ qmx, we can replace these functions by
their uniform asymptotic expansions for large values of the order. After these replacements the
integration and the further summation over m are done by using the formulae from [46].
4 Vacuum densities in the exterior region
In this section we consider the VEVs for the field square and the energy-momentum tensor in the
region outside the cylindrical boundary (region II in figure 1). The corresponding eigenfunctions
for the vector potential are obtained from formulae (2) by the replacement
Jqm(γr) → g(λ)qm(γa, γr) = Jqm(γr)Y (λ)qm (γa)− Yqm(γr)J (λ)qm (γa), (58)
where, as before, λ = 0, 1 correspond to the waves of the electric and magnetic types, respec-
tively. Now, the eigenvalues for γ are continuous and in the normalization condition (4) the
corresponding part on the right is presented by the delta function. As the normalization inte-
gral diverges for γ′ = γ, the main contribution into the integral comes from large values of r and
we can replace the cylindrical functions with the argument γr by their asymptotic expressions
for large values of the argument. By this way it can be seen that the normalization coefficient
in the exterior region is determined by the relation
β−2α =
J (λ)2qm (γa) + Y
qm (γa)
. (59)
Substituting the eigenfunctions into the corresponding mode-sum formula, for the VEV of the
field square one finds
〈0|F 2|0〉 =
λ=0,1
k2 + γ2
g(ηFλ)[Φ
qm(φ), g
qm(γa, γr)]
qm (γa) + Y
qm (γa)
, (60)
where the functions g(ηFλ)
qm(φ), g
qm(γa, γr)
are defined by relations (9), (10) with the
function f(x) = g
qm(γa, x). To extract from this VEV the part induced by the cylindrical shell,
we subtract from the right-hand side the corresponding expression for the wedge without the
cylindrical boundary. The latter is given by formula (14). The corresponding difference can be
further evaluated by using the identity
g(ηFλ)[Φ
qm(φ), g
qm(γa, γr)]
qm (γa) + Y
qm (γa)
= g(ηFλ)[Φ(λ)qm(φ), Jqm(γr)]
qm (γa)
(s)(λ)
qm (γa)
g(ηFλ)[Φ(λ)qm(φ),H
qm(γr)], (61)
where H
(1,2)
qm (z) are the Hankel functions. In order to transform the integral over γ with the
last term on the right of (61), in the complex plane γ we rotate the integration contour by the
angle π/2 for the term with s = 1 and by the angle −π/2 for the term with s = 2. Due to the
well-known properties of the Hankel functions the integrals over the corresponding parts of the
circles of large radius in the upper and lower half-planes vanish. After introducing the modified
Bessel functions and integrating over k with the help of formula (26), we can write the VEVs of
the field square in the form (13), where the part induced by the cylindrical shell is given by the
formula
〈F 2〉cyl =
λ=0,1
qm (xa)
qm(xa)
G(ηFλ)[Φ(λ)qm(φ),Kqm(xr)]. (62)
In this formula the functions G(ηFλ) [Φ(φ), f(x)] are defined by formulae (28), (29). Comparing
this result with formula (27), we see that the expressions for the shell-induced parts in the
interior and exterior regions are related by the interchange Iqm ⇄ Kqm. The VEV (62) diverges
on the cylindrical shell with the leading term being the same as that for the interior region. At
large distances from the cylindrical shell we introduce a new integration variable y = xr and
expand the integrand over a/r. For q > 1 the main contribution comes from the lowest mode
m = 0 and up to the leading order we have
〈E2〉cyl ≈
, 〈B2〉cyl ≈ −
15πr4
. (63)
For q < 1 the dominant contribution into the VEVs at large distances is due to the mode m = 1
with the leading term
〈F 2〉cyl ≈ −
4q2(q + 1)
cos(2qφ)
2q + 3
+ (−1)ηF1 q + 1
2q + 1
. (64)
For the case q = 1 the contributions of the modes m = 0 and m = 1 are of the same order
and the corresponding leading terms are obtained by summing these contributions. The latter
are given by the right-hand sides of formulae (63) and (64). As we see, at large distances the
part induced by the cylindrical shell is suppressed with respect to the part corresponding to the
wedge without the shell by the factor (a/r)2β with β = min(1, q).
Now we turn to the VEVs of the energy-momentum tensor in the exterior region. Substi-
tuting the eigenfunctions into the corresponding mode-sum formula, one finds (no summation
over i)
〈0|T ii |0〉 =
λ=0,1
k2 + γ2
f (i)[Φ
qm(φ), g
qm(γa, γr)]
qm (γa) + Y
qm (γa)
, (65)
〈0|T 12 |0〉 = −
m sin(2qmφ)
λ=0,1
(−1)λ
qm (γa, γr)
k2 + γ2
. (66)
Subtracting from these VEVs the corresponding expression for the wedge without the cylindrical
boundary, analogously to the case of the field square, it can be seen that the VEVs are presented
in the form (37), with the parts induced by the cylindrical shell given by the formulae (no
summation over i)
〈T ii 〉cyl =
λ=0,1
qm (xa)
qm (xa)
F (i)[Φ(λ)qm(φ),Kqm(xr)], (67)
〈T 12 〉cyl =
m sin(2qmφ)
λ=0,1
(−1)λ
qm (xa)
qm (xa)
K2qm(xr). (68)
Here the functions F (i) [Φ(φ), f(y)] are defined by formulae (40), (41). By using the inequality
given in the paragraph after formula (43), we can show that the vacuum energy density induced
by the cylindrical shell in the exterior region is positive.
In the way similar to that for the interior region, for the force acting on the wedge sides
is presented in the form of the sum (46), where the part corresponding to the wedge without
a cylindrical shell is determined by formula (47) and for the part due to the presence of the
cylindrical shell we have
p2cyl = −〈T 22 〉cyl|φ=0,φ0 = −
λ=0,1
qm (xa)
qm(xa)
F (λ)qm [Kqm(xr)]. (69)
In this formula, the function F
ν [f(y)] is defined by relations (49) and the corresponding forces
are always attractive.
The leading divergence in the boundary induced part (67) on the cylindrical surface is given
by the same formulae as for the interior region. For large distances from the shell and for q > 1
the main contribution into the VEVs of the diagonal components comes from the m = 0, λ = 1
term and one has (no summation over i)
〈T ii 〉cyl ≈ −
15π2r4
, c0 = c3 = 2, c1 = 1, c2 = −5. (70)
In the case q < 1 the main contribution into the VEVs of the diagonal components at large
distances from the cylindrical shell comes from the m = 1 mode. The leading terms in the
corresponding asymptotic expansions are given by the formulae
〈T ii 〉cyl ≈ −q2(q + 1)ci(q)
cos(2qφ)
, (71)
with the notations
c0(q) = c3(q) =
2q + 3
, c1(q) =
2q2 + q + 1
(2q + 1)(2q + 3)
, c2(q) = −
q + 1
2q + 1
. (72)
In the case q = 1 the asymptotic terms are determined by the sum of the contributions coming
from the modes m = 0 and m = 1. The latter are given by formulae (70), (71). For the off-
diagonal component, for all values q the main contribution at large distances comes from the
m = 1 mode with the leading term
〈T 12 〉cyl ≈ −
q3(q + 1)
2q + 1
sin(2qφ)
. (73)
For large values of q, q ≫ 1, the contribution of the terms with m > 0 is suppressed by the factor
exp[−2qm ln(r/a)] and the main contribution comes form the m = 0 term with the behavior
〈F 2〉cyl ∝ q and 〈T ki 〉cyl ∝ q. In figure 7 we have plotted the dependence of the effective azimuthal
pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a for
q = 1.
1.4 1.5 1.6 1.7 1.8 1.9 2
Figure 7: The effective azimuthal pressure induced by the cylindrical shell on the wedge sides,
a4p2cyl, as a function of r/a in the exterior region for q = 1. The curves for the values q = 0.5, 3
are close to the plotted one.
5 Conclusion
In this paper we have investigated the polarization of the electromagnetic vacuum by a wedge
with a coaxial cylindrical boundary, assuming that all boundaries are perfectly conducting. Both
regions inside and outside of the cylindrical shell (regions I and II in figure 1) are considered. In
section 2 we have evaluated the VEVs of the field square in the interior region. The corresponding
mode-sums contain series over the zeros of the Bessel function for TM modes and its derivative
for TE modes. For the summation of these series we used a variant of the generalized Abel-
Plana formula. The latter enables us to extract from the VEVs the parts corresponding to the
geometry of a wedge without a cylindrical shell and to present the parts induced by the shell
in terms of integrals which are exponentially convergent for points away from the boundaries.
For the wedge without the cylindrical shell the VEVs of the field square are presented in the
form (23). The first term on the right of this formula corresponds to the VEVs for the geometry
of a cosmic string with the angle deficit 2π − φ0. The angle-dependent parts in the VEVs of
the electric and magnetic fields have opposite signs and are cancelled in the evaluation of the
vacuum energy density. The parts induced by the cylindrical shell are presented in the form
(27). We have discussed this general formula in various asymptotic regions of the parameters
including the points near the edges and near the shell. In section 3 we consider the VEV of the
energy-momentum tensor in the region inside the shell. As for the field square, the application
of the Abel-Plana formula allows us to present this VEV in the form of the sum of purely wedge
and shell-induced parts, formula (37). For the geometry of a wedge without the cylindrical
boundary the vacuum energy-momentum tensor does not depend on the angle φ and is the same
as in the geometry of the cosmic string and is given by formula (45). The corresponding vacuum
forces acting on the wedge sides are attractive for φ0 < π and repulsive for φ0 > π. In particular,
the equilibrium position corresponding to the geometry of a single plate (φ0 = π) is unstable.
For the region inside the shell the part in the VEV of the energy-momentum tensor induced
by the presence of the cylindrical shell is non-diagonal and the corresponding components are
given by formulae (38), (39). The vacuum energy density induced by the cylindrical shell in the
interior region is negative. We have investigated the vacuum densities induced by the cylindrical
shell in various asymptotic regions of the parameters. For points near the cylindrical shell the
leading terms in the asymptotic expansions over the distance from the shell are given by formulae
(50). These terms are the same as those for a cylindrical shell when the wedge is absent. For
a wedge with φ0 < π the part in the vacuum energy-momentum tensor induced by the shell
is finite on the edge r = 0. For φ0 > π the shell-induced parts in the energy density and the
axial stress remain finite, whereas the radial and azimuthal stresses diverge as r2(π/φ0−1). The
corresponding off-diagonal component behaves like r2π/φ0−1 for all values φ0. For the points
near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymptotic expansions
are the same as for the geometry of a wedge with the opening angle φ0 = π/2. In the limit of
small opening angles, φ0 ≪ π, the shell-induced parts behave like 1/φ0. In the same limit the
parts corresponding to the wedge without the shell behave as 1/φ40, and for points not too close
to the shell these parts dominate in the VEV of the energy-momentum tensor. The presence
of the shell leads to additional forces acting on the wedge sides. The corresponding effective
azimuthal pressure is given by formula (48) and these forces are always attractive.
The VEVs of the field square and the energy-momentum tensor in the region outside the
cylindrical shell are investigated in section 4. As in the case of the interior region, these VEVs
are presented as sums of the parts corresponding to the wedge without the cylindrical shell and
the parts induced by the shell. The latter are given by formula (62) for the field square and by
formulae (67), (66) for the components of the energy-momentum tensor. In the exterior region
the vacuum energy density induced by the cylindrical shell is always positive. Additional forces
acting on the wedge sides due to the presence of the shell are given by formula (69). As in the
case of the interior region these forces are attractive. For large values of the parameter q, the
contribution into the parts induced by the cylindrical shell coming from the modes with m 6= 0
is exponentially suppressed, whereas the contribution of the lowest mode m = 0 is proportional
to q. Though in this limit the vacuum densities are large, due to the factor 1/q in the spatial
volume element, the corresponding global quantities tend to finite limiting values.
Acknowledgments
The work was supported by the Armenian Ministry of Education and Science Grant No. 0124.
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	Introduction
	Vacuum expectation values of the field square inside a cylindrical shell
	Vacuum energy-momentum tensor inside the cylindrical shell
	Vacuum densities in the exterior region
	Conclusion
