Table Of ContentParity violating cylindrical shell in the framework of
QED
I V Fialkovsky†, V N Markov ¶ and Yu M Pis’mak†
8
0
† Department of Theoretical Physics, State University of Saint-Petersburg,Russia
0
¶ Department of Theoretical Physics, Petersburg Nuclear Physics Institute, Russia
2
n E-mail: †[email protected], [email protected],
a ¶[email protected],
J
5
Abstract. We present calculations of Casimir energy (CE) in a systemof quantized
] electromagnetic(EM)fieldinteractingwithaninfinite circularcylindricalshell(which
h
t wecall‘thedefect’). InteractionisdescribedintheonlyQFT-consistentwaybyChern-
-
p Simon action concentrated on the defect, with a single coupling constant a.
e For regularization of UV divergencies of the theory we use Pauli-Villars
h
regularization of the free EM action. The divergencies are extracted as a polynomial
[
in regularizationmass M, and they renormalize classical part of the surface action.
2
WerevealthedependenceofCEonthecouplingconstanta. CorrespondingCasimir
v
9 force is attractive for all values of a. For a → ∞ we reproduce the known results for
4 CE for perfectly conducting cylindrical shell first obtained by DeRaad and Milton.
0
As a future task for solving existing arguments on observational status of (rigid)
4
. self-pressure of a single object, we propose for investigation a system which we call
0
‘Casimir drum’.
1
7
0
:
v
i
X Submitted to: J. Phys. A: Math. Gen.
r
a
1. Introduction
Predicted in 1948 [1] Casimir effect has been for a long time a non-detectable theoretical
‘play of mind’. Prophesying an attractive force between two neutral parallel conducting
planes in vacuum, the Casimir effect is a pure quantum one — there is no such force
between the planes in classical electrodynamics.
Development of the experimental technique allowed first to observe Casimir effect
[2]andthentomeasureitwith5-10%accuracy[3]. Nowadays, duetoworksofMohideen
and his colleagues, as well as many other groups, the total experimental error for the
force between metal surfaces is reduced to 0.5% [4]-[6].
Original Casimir’s configuration of parallel (perfectly conducting) planes is well
studied both theoretically and experimentally. Still there are considerable lacunas in
our understanding of the effect for complicated geometries and non-perfect materials.
The particular interest for cylindrical geometry which we consider in this paper, is
P-odd cylindrical shell in the framework of QED 2
motivated by rapid development of the carbon nanotubes technology. For discussion
of possible role of the Casimir force in dynamics and stability of micro- and nano-
electromechanical devices (MEMS and NEMS) see [8, 9], and reference therein.
From the theoretical point of view, the Casimir force between distinct bodies is well
established, especially taking into account the main achievements (both analytical and
numerical) of the recent time [10]-[14]. On the other hand, calculations of the Casimir
effect for a single object (self-energy, self-stress, etc.) still provoke controversies [15, 16],
and a self-consistent description of systems with sharp material boundaries is yet to
be developed in the framework of the quantum field theory (QFT). In this paper we
address both of these issues.
The paper is organized as following. In Part 2 we discuss the construction of
the model for the system of quantum electrodynamics’ (QED) fields interacting with
the cylindrical shell. In Part 3 we sketch our approach to calculation of the modified
propagator of the system and the Casimir energy, and give their explicit form. In Part
4 we present the conclusions and discuss the perspectives of the work.
2. Statement of the problem
There are several ways to model the presence of matter (which we also call ‘spatial
defect’) in QFT. The simplest one is to fix the values of quantum fields and/or their
derivatives with boundary conditions (BC) on the surface of the defect. However,
imposing BC is physically unjustified as it constraints all modes of the fields. At the
sametime,inrealityfield’smodeswithhighenoughfrequencies propagatefreelythrough
any material boundary.
The most natural generalization of BC is to couple the quantum fields to classical
external field (background) supported spatially on the defect. The simplest case of
such background is a singular one with delta-function profile. Its introduction into
the classical action is equivalent to imposing matching conditions on the quantum fields
which model semitransparent boundaries. Delta-potential is an effective way to describe
a thin film present in the system, when its thickness is negligible compared to the
distances in the range of interest. In a certain limit (usually the strong coupling one)
delta-potentials reproduce simplest BC such as Dirichlet, Neumann or Robin ones in
the case of scalar fields [19].
In the context of QFT such interaction of quantum fields with delta-potentials
(introduced as a part of the action) must be constructed satisfying the basic principles
of the theory — locality, gauge and Lorenz invariance, renormalizability. For the first
time this issue was addressed and thoroughly studied (for the case of massless scalar
fields) bySymanzik [19]in1981. Sincethen, therewere anumber ofCasimir calculations
with delta-potentials, see for instance [20]-[23]. However, until very lately the issue of
renormalizability of a theory with delta-potential still invoked contradictions [15, 16],
and was apparently resolved in [17]. Still all of the existing papers deal only with
simplified scalar models, usually in lower dimensions. In a limited number of particular
P-odd cylindrical shell in the framework of QED 3
cases scalar fields can be combined as TM and TE modes [24] of EM field, to describe
some specific aspects of Casimir problems in QED. Until [25, 26] there were no attempts
to construct a self-consistent QED model with a delta-potential interaction satisfying
all QFT principles and allowing one to describe self-consistently all possible observable
consequences of the presence of a defect. In this paper we generalize the results of [25]
to the case of cylindrical geometry.
Followingtheapproachof[25],weconstructaQEDmodelwithphotonfieldcoupled
to the defect through a delta-potential supported on the surface of an infinite circular
cylinder. We neglect interaction of fermion fields with the defect since any observable
consequences of such interaction are exponentially suppressed at the distances larger
then inverse electron mass m−1 ≈ 10−10cm [26, 27]. Thus, massive fermion fields cannot
e
contribute to the Casimir force which has macroscopical (experimentally verified) values
at the scale of 10–100nm. Wecan neglect Diracfields and consider pure photodynamics.
For the photon field A and defect surface described by equation Φ(x) = 0 we
µ
construct the action as a sum
S = S +S (1)
0 def
ofusual Maxwell actionofelectromagnetic field(throughout the paper we set c = ~ = 1)
1
S = − d4xF (x)Fµν(x), F (x) = ∂ A −∂ A (2)
0 µν µν µ ν ν µ
4 Z
and defect action of Chern-Simon type [28]
a
S = d4xεµνρσ∂ Φ(x)δ(Φ(x))A (x)F (x). (3)
def µ ν ρσ
2 Z
Here ε is totaly antisymmetric tensor (ε = 1), a — dimensionless coupling
µνρσ 0123
constant. For a cylinder of radius R placed along the x axis we have
3
Φ(x) = x2 +x2 −R2. (4)
1 2
We must stress here that the form of the defect action (3) is completely determined by
above mentioned basic principles of QFT. In particular, introduction of any other local,
gauge and lorenz invariant terms (with higher derivatives, etc.) unavoidably brings
to the theory coupling constants of negative dimensions. Such theories have infinite
number of primitively divergent diagrams, being unrenormalizable in conventional sense
[19, 29]. Thus we are left with the Chern-Simon defect action that is space parity
violating and this unusual property quite naturally arises at the very beginning of our
consideration. We will show below that in the limit of a → ∞ the Casimir energy for
perfectly conducting cylindrical shell is reproduced.
3. Casimir energy and photon propagator
All properties of a QFT system can be described if its generating functional is known
G(J) = N DA exp{iS +JA}. (5)
Z
P-odd cylindrical shell in the framework of QED 4
To render the theory finite, instead of S (1) we set into G(J)
S = Sreg +S (6)
0 def
where we introduced Pauli-Villars [30] UV regularization of the theory
1
Sreg = − F Fµν −M−2∂ F ∂λFµν (7)
0 4 µν λ µν
here and below we omit the sign of integration, S is defined in (3). In the limit
def
M → ∞ we return to action S (1), but then the theory posses both standard UV
divergencies of QED and specific (geometry dependent) ones in the vacuum loops — in
Casimir energy in particular. These divergences can be canceled by counter-terms in
the framework of renormalization procedure. If M is finite there are no divergences in
the model with the action S.
For definition of the normalization constant N in (5) we use the following condition
G(0)| = 1 =⇒ N−1 = DA exp{iS| } (8)
a=0 Z a=0
which means that in pure photodynamics without a defect lnG(0) vanishes. This sets
the reference point for the values of the energy density (per unit length of the cylinder)
of the system as the later one is expressed through the value of G(0)
1
E = − lnG(0), T = dx , L = dx . (9)
0 3
iTL Z Z
For explicit calculations of G(J) we first choose the coordinate basis associated
with cylindrical geometry and transform the vectors accordingly
A = (A ,A ,A ,A ) → A¯ ≡ τA = (A ,A ,A ,A )
0 1 2 3 0 ⊥ k 3
with
1 0 0 0
0 cosφ sinφ 0
τ = , (10)
0 −sinφ cosφ 0
0 0 0 1
where φ is polar angle in x x plane. After such linear transformation of integration
1 2
variablesin(5)thedefectaction(3)doesnotcontainanymoreneithervectorcomponents
A nor derivatives ∂ transversal to the defect. This makes it possible to represent the
⊥ ⊥
defectactioncontributionasanintegraloverauxiliaryfieldsdefinedonthedefectsurface
only. Then functional integration becomes purely Gaussian and for G(J) one calculates
explicitly
1
G(J) = (DetQ)−1/2exp J D +τTH¯τ J . (11)
(cid:26)2 (cid:27)
(cid:0) (cid:1)
Here D ≡ D (x) is the standard (UV regularized) free photon propagator, H¯ defines
µν
its corrections due to the presence of the defect, and Q describes the dependence of the
P-odd cylindrical shell in the framework of QED 5
Casimir energy on the geometry of the defect. In Fourier-components respecting the
symmetries of the system the propagator in Feynman gauge can be written as
dp
D (x,y) = δ S eipS(xS−yS)D(p ;ρ ,ρ ) (12)
µν µν Z (2π)3 S x y
D(p ;ρ ,ρ ) = I (i|p |ρ )K (i|p |ρ )−I (q ρ )K (q ρ ), q = M2 −p2
S x y n S < n S > n S < n S > S S
q
where x = (x ,φ,x ), p = (p ,n,p ), dp = dp dp ∞ , I , K are modified
S 0 3 S 0 3 S 0 3 n=−∞ n n
Bessel functions, ρ is smaller (bigger)Rof ρ ,Rρ = xP2 +x2 — polar radius in the
<,> x,y x 1 2
plane x x . In the same Fourier representation for H¯(xp,y) and Q(x ,y ) we can write
1 2 S S
H¯ (p ;ρ ,ρ ) = aD(p ;ρ ,R)L Q−1(p )D(p ;R,ρ ) (13)
ab S x y S x ac cb S S y
Q (p ) = δ +aD(p ;R,R)L , (14)
ab S ab S ab
L = 2iRε pc, a,b,c = 1,2,3 (15)
ab abc S
Using the famous trln = lndet identity one can express the Casimir energy (9) as
1 1
E = dx tr(LnQ(x −y )) = dp trlnQ(p ) (16)
2iTL Z S S S xS=yS 8π2 Z S S
where trlnQ(p ) denotes the sum of diagonal elements of the 3 × 3-matrix lnQ(p ).
S S
Putting (14) into (16) one writes for the energy density
1 ∞ ∞
E = pdp ln 1+a2YM(p) (17)
4πR2 Z n
0 nX=−∞ (cid:16) (cid:17)
YM(p) = −4(I (p)K (p)−I′(q)K (q))×
n n n n n
n2(RM)2
× p2(I′(p)K′(p)−I′(q)K′(q))+ I (q)K (q)
(cid:18) n n n n p2 +(RM)2 n n (cid:19)
where q = p2 +(RM)2.
It is eapsy to check that (17) is finite for any fixed value of auxiliary mass M and
diverges in the limit M → ∞. With help of uniform (Debye) asymptotics of the Bessel
functions [31] we subtract the most divergent terms from the integrant to make it
finite when regularization is removed, and add them explicitly. Following then the
renormalization procedure we extract from the substraction exact (polynomial and/or
logarithmic) dependence on M, and construct the counter terms. This is done with help
of generalized Abel-Plana formula [32] and its modification [33].
As a result of the calculations we present the energy density (17) as a sum
E = E +∆ (18)
Cas
of finite Casimir energy
1 a2ln2π ∞
E = + pdp ǫfinite (19)
Cas 4πR2 (cid:18)4(1+a2) Z (cid:19)
0
1+a2Y (p) a2 p4
ǫfinite = ln 0 +
(cid:18) 1+a2 (cid:19) 1+a24(1+p2)3
∞ 1+a2Y (p) a2 p4
n
+2 ln +
(cid:18) (cid:18) 1+a2 (cid:19) 1+a24(n2 +p2)3(cid:19)
Xn=1
P-odd cylindrical shell in the framework of QED 6
here Y (p) = lim YM(p) = −4p2I (p)K (p)I′(p)K′(p), and the counter-terms
n M→∞ n n n n n
M
∆ = RM3A + A (20)
3 1
R
where
2a2 ∞ ∞ 1 ∞ ∞
A = pdp dn y(1)(p), A = pdp dnln 1+a2y(0)(p) ,
1 π(1+a2) Z Z n 3 2π Z Z n
0 0 0 0 (cid:0) (cid:1)
with
y(0) = 2−(n2 +p2 +1)−1 −2 n2 +p2/ n2 +p2 +1
n
p p
(1)
and y (p) is the first order term of uniform asymptotic of Y (p)| .
n n M=1
E as function of a is real for a ∈ ℜ and for a ∈ i(−1,1). The later region is out of
Cas
physical interest [21] as for this case the action (1), (6) acquires imaginary part. For all
physically sensible values of a, E is positive giving rise to attractive Casimir force. In
Cas
the limit a → ∞, one can easily derive that (19) coincides explicitly with known results
for the Casimir energy of a perfectly conducting cylinder [7].
For renormalization of the counter-terms (20) we must introduce into the action
(6) also the classical energy density
h
0
E = Rσ +
0
R
where bare parameters σ , h are the surface tension and inverse radius parameter
0 0
correspondingly. To renormalize the divergencies we make following redefinition
σ = σ −M3A , h = h−MA (21)
0 3 0 1
where σ and h must be taken as an ‘input’ parameters to the theory in the spirit of
[17]. They describe the properties of material of the defect. Just as electron mass
m and charge e in QED, values of σ and h cannot be predicted from the theory and
must be determined from appropriate experiments. Thus, in addition to standard QED
normalizationconditions, oneneeds threeadditionalindependent experiments toremove
all the ambiguities of our model — to determine σ and h, and to set the scale of the
coupling constant a.
4. Conclusions
We constructed the QED model which describes a photon field interacting with
semitransparent cylindrical shell (two-dimensional defect surface). The form of
interaction — Chern-Simon action — is completely determined by the basic principles
of QFT: locality, gauge and Lorenz invariance, renormalizability. The defect action
is parity odd and P-transformation is equivalent to the change of sign of the defect
coupling constant a. We calculated explicitly the modified photon propagator and the
Casimir energy. The later one appears to be P-invariant being even function of a and
tends with a → ∞ to its value for perfectly conducting shell. Parity violation manifests
P-odd cylindrical shell in the framework of QED 7
itself only when external field is applied [25]. Thus, we can say that such defect action
mimics the behavior of thin films of magnitoelectrics.
Consideration of two-dimensional defects within the scope of renormalizable QFT
is justified by existing experimental data. It unambiguously shows that Casimir force
for objects with sharp boundaries has 1/r3 (r — distance between the objects) behavior
and thus is governed by dimensionless parameters of the system. On the other hand,
from obvious dimensional reasoning it follows that the presence of any dimensional
constants like finite width of the film h, finite conductivity δ, or final UV cut-off in non-
renormalizable models, can only give corrections in the order h2/r5, δ2/r5 or higher.
Thus, to the next to leading order in inverse powers of r we can stay with surface
contributions only.
Effectively, with the defect action we model interaction of electromagnetic field
with a surface layer (thin film) of atoms constituting the media, which can naturally
be parity violating. This property of the media translates into the surface properties
described in our model by a single constant a. The parity in our model is preserved
in two cases: the trivial one a = 0, and a → ∞ that corresponds to perfect conductor
limit as we showed above. We predict that thin films which are parity even should have
universal amplitude of the Casimir force: either vanishing (with a = 0), or coinciding
with one of the perfect conductor (a = ∞). From theoretical point of view we can not
decide whether materials with finite defect coupling a exist or not.
Presented calculations of the Casimir energy (and subsequently the Casimir force)
indirectly presume that the energy change is measured between two adiabatic states of
the system which differ by the radius of the cylinder. In particular this means that if an
experiment is to be carried out, the cylinder must be deformed as a whole, uniformly
along all of its (infinite) length. Such experimental setup looks unrealistic but still
possible in principle. This rises arguments in the literature (see review [34]) that calcu-
lations of a renormalized (rigid) self-pressure has insignificant (if any) predictive power.
As a way to resolve contradictions, one have to calculate ‘soft’ self-pressure for
local deformations of the shape of the body and reveal its dependence on position along
the body’s surface. For example, consider a perfectly conducting cylinder shell of finite
length L and of radius R. It is clear that in the limit R/L → ∞ in a vicinity of the
cylinder axis the local self-pressure must reproduce the attractive Casimir force between
parallel plates — the two bases of the cylinder. On the other hand, when R/L ∼ 1 one
can consider the deformations of the cylinder as a whole, and should recover the rigid
stress. We call such system a Casimir drum. Explicit calculation of the Casimir energy,
rigid and soft self-pressures of Casimir drum is the next step for our research.
Acknowledgement
Authors would like to thank Prof R. Jaffe and Prof D. Vassilevich for fruitful discussions
on the subject of the paper. V.N. Markov and Yu.M. Pismak are grateful to Russian
Foundation of Basic Research for financial support (RFRB grant 07–01–00692).
P-odd cylindrical shell in the framework of QED 8
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