Casimir effect for thin films in QED

Abstract

We consider the problem of modeling of interaction of thin material films with fields of quantum electrodynamics. Taking into account the basic principles of quantum electrodynamics (locality, gauge invariance, renormalizability) we construct a single model for Casimir-like phenomena arising near the film boundary on distances much larger then Compton wavelength of the electron. In this region contribution of Dirac fields fluctuations are not essential and can be neglected. In the model the film is presented by a singular background field concentrated on a 2-dimensional surface and interacting with quantum electromagnetic field. All properties of the film material are described by one dimensionless parameter. For two parallel plane films the Casimir force appears to be non-universal and dependent on material property. It can be both attractive and repulsive. In the model we study scattering of electromagnetic wave on the plane film, an interaction of plane film with point charge, homogeneously charged plane and straight line current. Here, besides usual results of classical electrodynamics the model predicts appearance of anomalous electromagnetic phenomena. Comment: 8 pages,1 figure

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arXiv:hep-th/0606058v1 7 Jun 2006
Casimir effect for thin films in QED
V N Markov ¶ and Yu M Pis’mak †‡
† Department of Theoretical Physics, State University Saint-Petersburg, Russia
¶ Department of Theoretical Physics, Saint-Petersburg Nuclear Physics Institute,
Russia
‡ Institute for Theoretical Physics, University Heidelberg, Germany
Abstract.
with fields of quantum electrodynamics. Taking into account the basic principles of
quantum electrodynamics (locality, gauge invariance, renormalizability) we construct
a single model for Casimir-like phenomena arising near the film boundary on distances
much larger then Compton wavelength of the electron. In this region contribution of
Dirac fields fluctuations are not essential and can be neglected. In the model the film is
presented by a singular background field concentrated on a 2-dimensional surface and
interacting with quantum electromagnetic field. All properties of the film material are
described by one dimensionless parameter. For two parallel plane films the Casimir
force appears to be non-universal and dependent on material property. It can be both
attractive and repulsive. In the model we study scattering of electromagnetic wave on
the plane film, an interaction of plane film with point charge, homogeneously charged
plane and straight line current. Here, besides usual results of classical electrodynamics
the model predicts appearance of anomalous electromagnetic phenomena.
We consider the problem of modeling of interaction of thin material films
PACS numbers: 12.20.Ds

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Casimir effect for thin films in QED
2
1. Introduction
In 1948 it was shown by Casimir that vacuum fluctuations of quantum fields generate an
attraction between two parallel uncharged conducting planes [1]. This phenomena called
the Casimir effect (CE) has been well investigated with methods of modern experiments
[2, 3, 4].The CE is a manifestation of influence of fluctuations of quantum fields
on the level of classical interaction of material objects. Theoretical and experimental
investigation of phenomena such a kind became very important for development of
micro-mechanics and nano-technology.
Though there are many theoretical results on the CE [5], however the majority
of them are received in framework of several models based not on the quantum
electrodynamics (QED) directly. Usually, one assumes that the CE can be investigated
in the framework of free massless quantum scalar field theory with fixed boundary
conditions or δ-function potentials [6, 7] ignoring restrictions following from gauge
invariance, locality and renormalizability of QED. By means of such methods one
can investigate some of the CE properties, but there is no possibility of studying
other phenomena generated by interaction of the QED fields with considered classical
background within the same model.
An approach for construction of the single QED model for investigation of all
peculiar properties of the CE for thin material films was proposed in [8, 9]. We consider
its application for simple case of parallel plane films. We show that gauge invariance,
locality and renormalizability considered as basic principles make strong restrictions for
constructions of the CE models in QED, which make it possible to reveal new important
features of the CE-like phenomena.
2. Construction of models
We construct models for interaction of the material film with QED fields on the basis
of most general assumptions. We suppose that the film is presented by a singular
background (defect) concentrated on the 2-dimensional surface. Its interaction with
QED fields has most general form defined by the geometry of the defect and restrictions
following from the basic principles of QED (gauge invariance, locality, renormalizability).
The locality of interaction means that the action functional of the defect is represented by
an integral over defect surface of the Lagrangian density which is a polynomial function
of space-time point in respect to fields and derivatives of ones. The coefficients of this
polynomial are the parameters defining defect properties. For the quantum field theory
(QFT) with singular background the requirement of renormalizibility was analyzed by
Symanzik in [10]. He showed that in order to keep renormalizability of the model, one
needs to add a defect action to the usual bulk action of QFT model. The defect action
must contain all possible terms with nonnegative dimensions of parameters and not
include any parameters with negative dimensions. In case of QED the defect action
must be also gauge invariant.

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Casimir effect for thin films in QED
3
From these requirements it follows that for thin film (without charges and currents)
which shape is defined by equation Φ(x) = 0, x = (x0,x1,x2,x3), the action describing
its interaction with photon field Aµ(x) reads
SΦ(A) =a
2
where Fνρ(x) = ∂νAρ− ∂ρAν, ελµνρdenotes totally antisymmetric tensor (ε0123= 1), a
is a constant dimensionless parameter. The action (1) is a surface Chern-Simon action
[11, 12]. The fermion defect action can be written as
?
ελµνρ∂λΦ(x)Aµ(x)Fνρ(x)δ(Φ(x))dx(1)
SΦ(¯ψ,ψ) =
?
¯ψ(x)[λ + uµγµ+ γ5(τ + vµγµ) + ωµνσµν]ψ(x)δ(Φ(x))dx (2)
Here, γµ, µ = 0,1,2,3, are the Dirac matrices, γ5= iγ0γ1γ3γ3, σµν= i(γµγν− γνγµ)/2,
and λ, τ, uµ,vµ, ωµν= −ωνµ, µ,ν = 0,1,2,3 are 16 dimensionless parameters.
Expressions (1), (2) are the most general forms of gauge invariant actions
concentrated on the defect surface being invariant in respect to reparametrization of
one and not having any parameters with negative dimensions.
We consider in this paper CE-like phenomena arising on the distances from the
defect boundary much larger then Compton wavelength of the electron. In this case
one can neglect the Dirac fields in QED because of exponential damping of fluctuations
of those on much smaller distances (∼ m−1
e
for proton [8]). Thus, for constructing of model we can use the action of free quantum
electromagnetic field (photodynamic) with additional defect action (1).
For description of all physical phenomena it is enough to calculate generating
functional of Greens functions. For gauge condition φ(A) = 0 it reads
≈ 10−10cm for electron, ∼ m−1
p
≈ 10−13cm
G(J) = C
?
eiS(A,Φ)+iJAδ(φ(A))DA (3)
where
S(A,Φ) = −1
4FµνFµν+ SΦ(A),(4)
and the constant C is defined by normalization condition G(0)|a=0= 1. The first term
on the right hand side of (4) is the usual action of photon field. Along with defect action
it forms a quadratic in photon field full action of the system which can be written as
S(A,Φ) = 1/2 AµKµν
ΦAν. The integral (3) is gaussian and is calculated exactly:
G(J) = exp
?1
2Trln(DΦD−1) −1
2JDφJ
?
where DΦis the propagator DΦ= iK−1
and D is the propagator of photon field without defect in the same gauge. For the static
defect, function Φ(x) is time independent, and lnG(0) defines the Casimir energy.
In order to expose essential Feature of CE-like phenomena in constructed model,
we calculate the Casimir force (CF) for simple case of two parallel infinite plane films
and study a scattering of electromagnetic wave on the plane defect. We consider also an
interaction of the plane film with a parallel to it straight line current and an interaction
of film with a point charge and homogeneous charge distribution on parallel plane.
Φ of photodynamic with defect in gauge φ(A) = 0,

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Casimir effect for thin films in QED
4
3. Casimir force
We consider defect concentrated on two parallel planes x3= 0 and x3 = r. For this
model, it is convenient to use a notation like x = (x0,x1,x2,x3) = (? x,x3). Defect action
(1) has the form:
S2P=1
2
?
(a1δ(x3) + a2δ(x3− r))ε3µνρAµ(x)Fνρ(x)dx.
The defect action S2P was discussed in [13] in substantiation of Chern-Simon type
boundary conditions chosen for studies of the Casimir effect in photodynamics. This
based on boundary conditions approach is not related directly to the one we present.
The defect action (1) is the main point in our model formulation, and no any boundary
conditions are used.The action S2P is translationally invariant with respect to
coordinates xi, i = 0,1,2. The propagator DΦ(x,y) is written as:
D2P(x,y) =
1
(2π)3
?
D2P(?k,x3,y3)ei?k(? x−? y)d?k,
and D2P(?k,x3,y3) can be calculated exactly. Using latin indexes for the components of
4-tensors with numbers 0,1,2 and notations
Plm(?k) = glm− klkm/?k2, Llm(?k) = ǫlmn3kn/|?k|,?k2= k2
0− k2
1− k2
2, |?k| =
?
?k2
(g is metrics tensor), one can present the results for the Coulomb-like gauge ∂0A0+
∂1A1+ ∂2A2= 0 as follows [9]
D33
2P(?k,x3,y3) =−iδ(x3− y3)
|?k|2
, Dl3
2P(?k,x3,y3) = D3m
2P(?k,x3,y3) = 0,
Dlm
2P(?k,x3,y3) =Plm(?k)P1(?k,x3,y3) + Llm(?k)P2(?k,x3,y3)
2|?k|[(1 + a1a2(e2i|?k|r− 1))2+ (a1+ a2)2]
where
P1(?k,x3,y3) = [a1a2− a2
+[a2
−ei|?k||x3−y3|[(1 + a1a2(e2i|?k|r− 1))2+ (a1+ a2)2],
P2(?k,x3,y3) = a1[1 + a2(a2+ a1e2i|?k|r)]ei|?k|(|x3|+|y3|)+
+a2[1 + a1(a1+ a2e2i|?k|r)]ei|?k|(|x3−r|+|y3−r|)−
?
The energy density E2P of defect is defined as
1a2
2(1 − e2i|?k|r)][ei|?k|(|x3|+|y3−r|)+ ei|?k|(|x3−r|+|y3|)]ei|?k|r+
2(1 − e2i|?k|r)]ei|?k|(|x3|+|y3|)+ [a2
1+ a2
1a2
2+ a2
1a2
2(1 − e2i|?k|r)]ei|?k|(|x3− r|+|y3−r|)−
−a1a2(a1+ a2)ei|?k|(|x3|+|y3−r|)+ ei|?k|(|x3−r|+|y3|)?
ei|?k|r.
lnG(0) =1
2Trln(D2PD−1) = −iTSE2P
where T =
expressed in an explicit form in terms of polylogarithm function Li4(x) [9]. For identical
?dx0 is duration of defect, and S =
?dx1dx2, is the area of film. It is

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Casimir effect for thin films in QED
5
Figure 1. Function f(a) determining Casimir force between parallel planes
films with a1= a2= a it holds: E2P= 2Es+ ECas,Es=
?ln
a2
?
(1 + a2)
d?k
(2π)3,
ECas= −
1
16π2r3
?
Li4
?
a2
(a + i)2
?
+ Li4
?
(a − i)2
??
.
Here Esis an infinite constant, which can be interpreted as self-energy density on the
plane, and ECas is an energy density of their interaction. Function Li4(x) is defined
as Li4(x) =?∞
given by
F2P(r,a) = −∂ECas(r,a)
∂r
The force F2P is repulsive for |a| < a0and attractive for |a| > a0, a0≈ 1.03246 (see
Figure 1). For large |a| it is the same as the usual CF between perfectly conducting
planes. The model predicts that the maximal magnitude of the repulsive F2Pis expected
for |a| ≈ 0.6. For two infinitely thick parallel slabs the repulsive CF was predicted also
in [14].
Real film has a finite width, and the bulk contributions to the CF for nonperfectly
conducting slabs with widths h1, h2 are proportional to h1h2. Therefore it follows
directly from the dimensional analysis that the bulk correction Fbulkto the CF is of the
form Fbulk≈ cFCash1h2/r2where FCasis the CF for perfectly conducting planes and c
is a dimensionless constant. This estimation can be relevant for modern experiments
on the CE. For instance, in [4] there were results obtained for parallel metallic surfaces
where width of layer was about h ≈ 50 nm and typical distance r between surfaces was
0.5µm ≤ r ≤ 3µm. In that case 3 × 10−4≤ (h/r)2≤ 10−2. In [4] authors have fitted
the CF between chromium films with function CCas/r4. They claim that the value of
CCascoincides with known Casimir result within a 15% accuracy. It means that bulk
force can be neglected, and only surface effects are essential. In our model the values
a > 4.8 of defect coupling parameter a are in good agreement with results of [4].
Now we study the scattering of classical electromagnetic wave on plane defect and
effects generated by coupling of plane film with a given classical 4-current.
k=1
xk
k4 = −1
2
?∞
0k2ln(1 − xe−k)dk. The force F2P(r,a) between planes is
= −
π2
240r4f(a).