PHYSICS REPORTS (Review Section of Physics Letters) 134, Nos. 2 & 3 (1986) 87—193. North-Holland, Amsterdam

THE CASIMIR EFFECT

Günter PLUNIEN, Berndt MULLER and Walter GREINER Institutfür Theoretische Physik der I. W. Goethe Universitüt Frankfurt; 60(X) Frankfurt/Main, Fed. Rep. Germany

Received 8 August 1985

Contents:

1. Introduction 89 4.3. Quarks and gluons in a bag 139 2. Energy of the vacuum state 91 5. The Dirac vacuum in external electromagnetic fields 147 2.1. Zero-point energies in field quantization and the 5.1. Vacuum energy and vacuum polarization 147 definition of the physical vacuum energy 91 5.2. The supercritical vacuum 159 2.2. Implications of the vacuum energy 102 5.3. The vacuum energy in nuclear scattering 167 3. Mathematical formulation and evaluation methods for 6. Casimir energy at finite temperature 172 vacuum energies 106 6.1. Partition functions and free energy 172 3.1. Method of mode Summation 106 6.2. Finite temperature propagators 174 3.2. The local formulation and Green function methods 110 6.3. Casimir energy 178 33. The multiple-scattering expansion for Green functions 7. Applications 180 and the Casimir energy 121 7.1. Casimir energy in dielectric media and the relation to 3.4. Phase-shift representation 130 the bag model of hadronic particles 180 4. Casimir effect 132 7.2. Boundary problems and Casimir energy in gauge 4.1. The Casimir effect between perfectly conducting plates 132 theory 186 4.2. The Casimir energy of a massive scalar field in a finite 8. Concluding remarks 189 cavity 137 References 190

Abstract: This report gives an introduction to the Casimir effect in quantum field theory and its applications. The interaction between the vacuum of a quantized field and an external boundary or a classical external field is investigated laying particular emphasis on Casimir’s concept of measurable change in the vacuum energy. The various methods for evaluation and regularization of the Casiinu- energy are discussed in detailfor specificfield configurations. Recent applications of the Casimir effect in supercritical fields, QCD bag models and electromagnetic media are reviewed.

Singleorders for this issue

PHYSICS REPORTS (Review Section of Physics Letters) 134, Nos. 2 & 3 (1986) 87—193

Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check.

Single issue price Dfi. 72.00, postage included.

0 370-1573/86/$37.45 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) THE CASIMIR EFFECT

Günter PLUNIEN, Berndt MULLER and Walter GRE1NER

Institut für Theoretische Physik der J. W. Goethe Universitât Frankfurt, 6000 Frankfurt/Main, Fed. Rep. Germany

NORTH-HOLLAND - AMSTERDAM G. Plunien et a!., The Casimir effect 89

1. Introduction

The occurrence of divergent zero-point energies is an inherent feature of quantum field theory. It appears as a direct consequence of canonical field quantization, which permits to establish the correspondence between classical (observable) quantities and quantum operators. Zero-point energies arise because the canonical quantization scheme does not fix the ordering of non-commuting operators in the field Hamiltonian. Thus, quantum field theory based on the operator concept, in principle, needs additional prescriptions in order to become a well-defined theory. Normally, this ambiguity is removed by requiring a certain ordering of operator products, Wick’s normal ordering. For the canonical field Hamiltonian this prescription implies a formal subtraction of the infinite zero-point energy and, per definition, the vacuum expectation value of the normal-ordered field Hamiltonian is then equal to zero. Accordingly, this formally introduced (mathematical) vacuum state has peculiar properties, namely, it carries neither energy nor momentum nor angular momentum. The standard arguments for such a subtraction lay particular emphasis on the fact that, in practice, it is not possible to measure the absolute value of the energy, but only differences in energy, allowing for an arbitrary choice of the origin on the energy scale. This statement is certainly valid for the microscopic Systems. However, the question remains whether the zero-point fluctuations of quantum fields must be taken seriously or not. For systems with a finite number of degrees of freedom zero-point energies have never been envisaged as problematic, because they are finite and measurable (e.g., the zero-point oscillations of crystals at zero temperature are observable) and, thus, they represent a real property of the ground state. That the situation should be different in the case of quantized fields is not a priori evident. The absolute value of the vacuum energy is, in principle, a measurable quantity, because it gravitates. The energy density of the vacuum must therefore be expected to contribute to some extent to the total energy density of the universe. However that may be, one would like to know how to deal with infinite zero-point energies in a meaningful way within a field theory based on the canonical field Hamiltonian and, in close connection with that, whether one can conclusively show possible con- sequences which indicate the presence of vacuum fluctuations. Such questions immediately lead to the Casimir effect. In 1948 Casimir showed [11that neutral perfectly conducting parallel plates placed in the vacuum attract each other. The attractive force can be considered as arising due to the change in the zero-point energy of the electromagnetic field when the plates are brought into position. The experimental verification [2—8]of this attraction has put the discussion about zero-point energies in field theory on firm ground. The general importance of the treatment of zero-point energy in the derivation of the original Casimir effect lies in the fact that it implies that the energy of the vacuum state of quantized fields cannot be correctly defined by normal ordering. The basic idea of what will later on be called “Casimir’s concept of vacuum energy” can be described like the following: A meaningful definition of the physical vacuum energy must take into account that in a real situation quantum fields always exist in the presence of external constraints, i.e. in interaction with matter or other external fields. An idealized description of such circumstances is obtained by forcing the field to satisfy certain boundary conditions. The zero-point modes are affected by the presence of these external constraints, and therefore, the zero-point energy is modified. Consequently, one is led to calculate the physical vacuum energy or Casimir energy of a quantized field with respect to its interaction with external constraints. It is generally defined as the difference between the zero-point energy referring to the distorted vacuum configuration and the free vacuum configuration, respectively. This formal definition only makes sense, if it is combined with appropriate regularization methods which guarantee a finite expression for this energy difference. Any variation of 90 G. Plunien et aL, The Casimir effect

the external boundary configuration induces a change in the vacuum energy, which is now considered as a measurable quantity. The main consequence of this concept is that the physical vacuum of a quantum field no longer represents a state with trivial global properties. Instead, the Casimir vacuum must be considered as a physical object reacting against distortions, i.e., possible vacuum effects can be interpreted as the “response” of the vacuum due to the presence of external constraints. This line of reasoning is spelled out in detail in the first part of section 2 of the present review article, where it is shown how zero-point energies arise when quantizing the scalar field, the electromagnetic field and the Dirac field. This introductory section continues with a more specific definition of the vacuum energy. Depending on the particular kind of external constraints distorting the free vacuum configuration, the vacuum energy may be understood as a contribution to the self-energy or to the interaction among the given boundary itself. Possibly it may cause boundary effects due to surface energies contributing, e.g., to surface tension or stresses inside the boundary. Basic investigations have been made of the Casimir energy of the electromagnetic field under constraints, which demonstrate the physical im- plication of the zero-point energy. As already shown by Casimir [301the zero-point energy of the electromagnetic field can be usefully applied to explain van der Waals attraction. However, the existence of repulsive Casimir forces [37] of electromagnetic origin seems to contradict this inter- pretation. This example already reveals that the physical content of the concept of Casimir energy embraces more than its use as an alternative way to understand certain known effects that can also be derived by conventional approaches. Presently, Casimir energies of quantized fields are studied in connection with a variety of problems, ranging from applications in particle physics, e.g. in QCD bag models, to gravitational physics, where its possible influence on the structure of space-time is studied. Examples for physical implications of Casimir energies are reviewed in the second part of section 2. In practice, the evaluation of vacuum energies remains a problematic exercise, because the available methods, in most cases, only allow an approximate calculation. Mainly two methods can be dis- tinguished: The mode-summation method is based on the direct evaluation of infinite sums over energy eigenvalues of the zero-point field modes. Within the local formulation on the other hand, one examines the constrained propagation of virtual field quanta and considers the vacuum stress tensor, which can be expressed in terms of propagators. Both treatments can be shown to be formally equivalent. Still, ambiguities between the results obtained by these methods cannot be excluded a priori since they imply inherently different regularization schemes, but correct results for the Casimir energy should be independent of the applied methods. Specific difficulties have to be solved in a successful application of the mode summation as well as the local Green function method. In the mode-summation method, which requires the knowledge of the total energy spectrum of the free and the constrained field modes particular cutoff procedures must be introduced in order to deal with divergencies. Whether a finite result for the vacuum energy is obtained after performing the involved infinite summations and cutoff removal crucially depends on the considered boundary problem. In the case of highly symmetric boundaries this method has been successfully carried out. The local Green function method represents a formally elegant way in order to derive the vacuum energy or the vacuum pressure. The main difficulty within this method is the determination of exact Green functions describing propagation in the presence of external boundaries. This can be solved by means of image source constructions [35] for plane geometries or, perturbatively by the multiple- scattering expansion [41, 42] in the case of general constraints. In section 3 we present the evaluation methods as they have been investigated in connection with the Casimir energy of the electromagnetic vacuum. G. Plunien el aL, The Casimir effect 91

Section 4 deals with selected cases of Casimir energies of quantum fields in the presence of simple geometric boundaries, in particular, the original Casimir effect, a massive particle constrained to a cavity, and the Casimir energy in the bag models of elementary particles. The extended class of Casimir problems involves all kinds of situations where a given quantum field interacts with an external field, which may be created by an arbitrary configuration of external sources. Correspondingly, the vacuum energy is defined as the difference between the zero-point energy in the presence of the external field and that of the free vacuum. This definition of the vacuum energy of the fermion field in an external electromagnetic field goes back to the work of Weiskopf, Schwinger and others (97, 100, 101]. As typical examples for such problems, the vacuum energy of a scalar field in the presence of external electromagnetic fields [78] or the vacuum energy of Higgs field configurations [62] have recently attracted interest. In section 5 we consider the vacuum energy of the electron—positron field in the presence of arbitrarily strong external electromagnetic fields [115].It is common for both types of Casimir problems to derive interactions between boundary configurations due to the zero point oscillations of the constrained field. The formalism as it has been developed to evaluate the Casimir energy of the vacuum state is directly generalizable to constrained quantum fields in thermodynamic equilibrium at finite temperature. The mode-summation method may be directly applied for calculating the Casimir free energy [81]. The corresponding local treatment is based on thermal Green functions, combined with the multiple- scattering expansion [41]. Temperature corrections may be of considerable interest in experimental measurements [81].The Casimir effect at finite temperature will be discussed in section 6. At the present time Casimir energies are discussed with increasing interest in connection with various physical problems. This makes it impossible to give a complete survey of applications, and we have, instead, concentrated on providing a lucid introduction into the field. Accordingly this review article on the Casimir effect represents a particular selection of problems which to the authors seemed mostly convenient in order to show the basic methods and underlying ideas. We only give a few representative examples of applications. The investigation of the Casimir effect in dielectric media placed in the electromagnetic field [39, 60, 61] has recently found renewed interest, because it has a direct application in QCD bag models, where the “true” vacuum is regarded as a perfect colour-magnetic conductor [128]. The Casimir energy density on external boundaries may be also useful in order to decide whether certain gauge breaking terms in a given field Lagrangian [134] have to be considered. The analogy between the electromagnetic theory of Casimir energy in material media and QCD, as well as the influence of a non-vanishing photon mass will be discussed in section 7. The review ends with a summary and speculative remarks.

Throughout the review, natural units (h = c = 1) and the signature (+, —‘ —, —) for the Minkowski metric are used.

2. Energy of the vacuum state

2.1. Zero-point energies in field quantization and the definition of the physical vacuum energy

In order to approach the problem of vacuum (ground-state) energies in quantum field theory let us start with a short recapitulation of the canonical field quantization scheme. If ço~°(x)denotes the dynamical fields, from the classical Lagrangian 92 G. Plunien eta!., The Casimir effect

3x ~~[(p~°(x,t), ~o~°(x, t)] , (2.1) L(t) = J d one derives the conjugate fields

~L(t) H~(x,t) = ~(a~~°(x, t))~ (2.2)

For boson fields, quantization proceeds by replacing the c-number fields by operators that satisfy canonical equal-time commutation rules:

[~~(x, t), .~(k)(x1t)] = [I7~’~(xt), #~~(x’t)] = 0 , (2.3a)

[ç1~(x,t), fl’(k)(xP t)] = i~Ik45(X— x’) . (2.3b)

Accordingly, anticommutation rules are required for fermion fields. In terms of ~ and 11 the field Hamiltonian H, formally identical with a classical Hamilton function, is obtained as

i-Ti = J cPx (~Hwc9~~t)— ~ t9~(z)) (2.4)

At this point one has to keep in mind that to ensure the hermiticity of i-Ti in general one has to (anti-) symmetrize with respect to the adjoint fields 1i~and ~. Only the fundamental postulate for the quantization of wave fields, i.e., the replacement of canonical field variables by operators together with the requirement of appropriate commutation relations, has been used so far. Unfortunately, this postulate alone is not sufficient to determine the field Hamiltonian in a satisfactory way. For instance, the order of non-commuting factors in the Hamiltonian is a priori undefined. The difficulty is made worse by the singularities arising from taking field operators at equal space-time points which leads to divergent vacuum expectation values of operators like the total energy. To ensure their elimination one requires renormalizability as a fundamental property of every physical field theory. In order to make the first point mentioned above more transparent and to see how zero-point energies appear in different theories, let us now consider the free scalar field, the electromagnetic field and the electron—positron field, in sequence. We then go on to define their physical vacuum energies.

2.1.1. The zero-point energy of the massive scalar field We start with the neutral (real) scalar field characterized by the Lagrangian

= — m242, (2.5) which leads to the field equation

(L1+m2)~=0. (2.6) G. Plunien eta!., The Casimir effect 93

For the Heisenberg operators 4 and i~= d4 the following equal-time commutation rules are valid:

[~(x, t), ~(x’, t)] = [fl(x, t), ñ(x’, t)] = 0, (2.7a)

[t~(x,t), JTi(x’, t)] = iô(x — x’). (2.7b)

To discuss the vacuum energy it is more convenient to adopt the momentum representation. Inside a finite cubic box of volume 11, at any given time, the operators q5(x, t) and H(x’, t) can be expanded in terms of the Fourier series:

~(x, t) = ~ exp(ik . x)4a(t), (2.8)

J~(x,t) = exp(ik X)fi_k(t). (2.9)

The neutral fields ~ and iTi are Hermitian, i.e. t~ = ~, so consequently we have

4~(t)= 4_~(t), ~ö~(t)= J.3k(t). (2.10)

Inserting the above expansions into eq. (2.7a) and (2.7b) we derive the commutation rules for the operators ~lk and j3~.So, for example, from

[t~(x,t), 1Ti(x’, t)] = ~ -~ exp(ik . x) exp(ik’ . x’)[4a(t), P_k’(t)]

= exp(ik’ . x) exp(—ik’ x’)[c~k(t),13k.(t)]

=ic5(x—x’) one can identify

[4~(t),p~’(t)]= ~ (2.11) since

exp[ik. (x — x’)} = 8(x — x’). (2.12)

Correspondingly the other commutators are

[4k(t), 4k.(t)] = [j3~(t),pk’(t)I = 0. (2.13) 94 G. Plunien eta!., The Casimireffect

Now we consider the field Hamiltonian. According to eq. (2.4) it reads

iTi=~Jd3x{I~+(V~)2+m2~2}, (2.14) or after partial integration of the gradient term one obtains i-Ti as a quadratic form of the fields ~ and H:

3x {fl2 + ~~(— V2 + m~)t~}, (2.15) i-Ti = ~J d

which can be expressed in terms of the operators /3k and 4~: - -

i-Ti = ~ J d3x exp[i(k + k’) x}{ I-al-a’ + (k’2 + m2)4~4~.}

~ (uk=Vk+m. (2.1-6)

This is exactly the Hamiltonian of a system of uncoupled one-dimensi9nal harmonic oscillators, which is not surprising because we have just decomposed the field qS into nOrmal modes 4~. Note that j3~and 4~ correspond to the canonical variables of the classical oscillator problem. To obtain the more compact energy representation of H, one usually transforms to a new basis ~ a~}of the Fock-space:

Ilk = V~w~(4a +-~--p~), (2.17a)

ak=V~cvk(qk__pk). (2.17b)

These creation and annihilation operators satisfy the commutation rule

[ak, Ilk’] = ‘5kk’, (2.18) while all other commutators vanish. Expressed in terms of the ~k and á~the Hamiltonian takes the symmetric form

(2.19)

where iI~= akak is the number operator that has eigenvalues ii,. = 0, 1, 2 The last step shows explicitly how the zero-point energy of the free massive scalar field appears. The vacuum 0) is defined by the equation â~I0)= 0. Obviously the vacuum expectation value E 0 = ~0IH~0)diverges since each oscillator contributes with ~Wk and one has an unbounded sum over all momenta. G. Plunien et al., The Casimireffect 95

2.1.2. The zero-point energy of the electromagnetic field We turn now to the discussion of the zero-point energy of the electromagnetic field. From the free field Lagrangian

2= ~ F,~,.= — ~ (2.20) one derives the conjugate fields as

= 92/9(90A~). (2.21)

In particular, the time-like component H°vanishes, whereas the spatial components coincide with the electric field H°=0, (2.22a)

= 3°A” — 3kAO = Ek. (2.22b)

Difficulties in quantization arising from the relation (2.22a) can be removed by the Gupta—Bleuler method [9, 10] in a Lorentz-covariant manner. Sufficing for our purpose here, the manifest Lorentz covariance can be abandoned by choosing the transverse (Coulomb) gauge for quantization. In this gauge the free electromagnetic field satisfies the conditions

A°=0, V•A=0, (2.23) and Maxwell’s equations reduce to

LIIA=0. (2.24)

Now quantization is achieved by the following commutation rules:

[A~(x,t), A1(x’, t)] = [1TI~(t),1, 1Ti1(x’, t)] = 0, (2.25a)

[A1(x,t), JZ(x’, t)] = iô(s.r). (x — x’), (2.25b) where in the last relation the usual 5-function is replaced by the divergence free transverse 6-function

ô(tr) (x — x’) (ôq — z1~9,ô~)6(x— x’) (2.26) to reconcile eq. [2.25b]with the gauge condition V~A = 0 and Gauss’ law V~H = V~E = 0. Relation (2.25b) guarantees the absence of photons with longitudinal polarization. To represent A and H in terms of Fourier series, one conveniently introduces for given k a particular orthogonal basis of polarization vectors {e~(A= 1, 2); k/!kP}, which satisfy the conditions

4~•k=0, E~~”ôAA’, A=1,2. (2.27) 96 G. Plunien et aL, The Casimir effect

The completeness relation for the basis requires

~ E~E~+= 6~. (2.28)

The Fourier representation of the fields A and 1? can then be written as

A(x, t) = ~ ~~exp(ik ‘x)e~4~(t) (2.29) and

fl(x, t) = ~ ~ç~exP(ik . x)r~4~(t). (2.30)

As in the case of the neutral scalar field the hermiticity of A and H yields the relations

(A) ‘(A) — (A) “(A)± (A) -‘(A)±— (A) -‘(A) — ~ q, , ~-aPa — ~a P—a, which allow one to derive the commutation rules for the operators 4~ and j3~

F4~A)(t)j3~’~(t)]= [j~~(t), j3~’~(t)]= 0 , (2.32a)

[4~(t), j3~’~(t)]= i6AA’6 kk’~ (2.32b)

We can now consider the Hamiltonian of the electromagnetic field, which is derived according to eq.

(2.4) in its canonical form

i4=~Jd3x{ft2+A.(_v2A)}. (2.33)

In the derivation use has been made of the identity f~d3x (V x A)2 = f 3x A (— V2A), which can be proved by applying the gauge condition (2.23) and integrating by parts. 0Expressedd in terms of j3~and 4~ the Hamiltoriian becomes again an infinite sum of uncoupled harmonic oscillators:

i-Ti = ~ {j5~j3~+ ~ (2.34)

The transformation to new creation and annihilation operators ~ and ê~defined by

ê(A) = VT~ (4~ + 1 j3(A)+) (2.35) G. Plunien et aL, The Casimir effect 97 satisfies the commutation rule

[-‘(A) ‘(A’)±l— , c~. — ~-‘AA’~’kk’

leads to the representation in terms of photon number operators ñj~=

~ Wk(fl~+~). (2.37)

The vacuum of the quantized electromagnetic field is defined by ê~I0)= 0. Consequently, the quantized free electromagnetic field also carries an infinite zero-point energy, E 0 —(OIHIO). = ~l~A Wk. In this context we note that in the framework of the Gupta—Bleuler quantization method one would obtain a Hamiltonian of similar form, but with additional contributions from the unphysical longitudinal or scalar photons [11].However, it can be shown [12]that the zero-point energy of the unphysical photons is exactly cancelled by the introduction of Fadeev—Popov ghost fields [13, 14], which carry a negative zero-point energy [15—17].In this sense, the covariant treatment of zero-point energy is equivalent to that in the transverse gauge.

2.1.3. Zero-point energy of fermion fields After having treated two boson fields which exhibit divergent zero-point energies because they correspond to an infinite collection of harmonic oscillators, let us now turn to the zero-point energy of fermion fields taking as an example the electron—positron field. This particular case will be of further interest in later sections. Before taking charge conjugation invariance into account, we discuss the quantization of the free Dirac field based on the Lagrangian

(2.38) which gives the field equation

(iJ—m)~P=0. (2.39)

The conjugate fields follow as

H~=i~P~. (2.40)

For fermion fields quantization is achieved by anticommutation relations

{#,,(x, t), #~(x’,t)} = {1?a(x, t), .ñ~~(xl,t)} = 0, (2.41a)

{1P,,(x, t), 1Ti,3(x’, t)} = ~5ajjS(X — x’). (2.41b)

According to eqs. (2.4) and (2.39) the field Hamiltonian reads

H= Jd3x i~i~ôo!P=Jd3x t~(—iy°y”~k+ y°m)~P. (2.42) 98 G. Plunien et a!., The Casimir effect

In order to derive the more convenient energy representation, the field operators are expanded in terms of a complete set of single-particle states

~P(x,t) = ~ i/i~(x)exp(—i~~t)l~+ ~ ~1i~(x)exp(—iE~t)l~. (2.43)

The first sum contains electron states referring to a positive energy e, > 0, whereas the negative energy states with ~. <0 are included in the second sum. With respect to eqs. (2.41a) and (2.41b) the corresponding anticommutation relations for the creation- and annihilation operators b, b~and b~,b~ are obtained

{l~,,1;}= {l~,l~,}= {l~,,l~,}= 0, (2.44a)

{6.,,, l~.}= ~ {l~,/~.}= ~ (2.44b)

The expansion (2.43) allows the expression of the Hamiltonian in the form

H = ~ l~l~+ ~ ~ (2.45)

In order to ensure astable ground state one requires that all electron states referring to anegative energy ~ (the negative energy continuum) are occupied. Accordingly, the Dirac vacuum is defined in terms of the operators b~and b~as

= 0, ~ 0, - (2.46a)

l~j0)=0, ~,,<0. (2.46b)

Making use of the relations (2.44a) and (2.44b) the Hamiltonian can be rewritten as

i-Ti = ~ — ~ 1J~+ Eo. (2.47)

The last term of H represents the infinite zero-point energy of the Dirac vacuum E 0 = (OIHIO) ~. ~, which arises from the occupied negative energy continuum. Instead of dealing with electron states of negative energy, Dirac’s hole picture implies that holes in the negative energy spectrum b~~0)n) are reinterpreted as positron (antiparticle) states with a positive energy E~= (—e,~)and positive charge. Formally the concept of antiparticles is introduced by the change to positron creation operators d~= b~. Then, in terms of the operators b2 and d~,the vacuum is defined by the equations

l~0)=0, ~>0, (2.48a)

d~I0)=0, s11<0. (2.48b)

The complete set of single-particle states can be divided into a subset of states describing particles G. Plunien eta!., The Casimireffect 99

(e,> E~) and one describing antiparticles (e~< CF). Therefore the boundary between particle and antiparticle states has the quality of a generalized “Fermi surface” (indicated by the symbol F). For the distinction between electron- @article) and positron- (antiparticle) states one introduces the Fermi energy CF. In the case of the non-interacting Dirac field the conventional choice for the Fermi energy is CF = 0. As we shall discuss in more details in section 5, the most general and consistent choice of the Fermi energy, also in the case when the Dirac field interacts with an external electromagnetic field of arbitrary strength and therefore bound states are present, turns out to be at a Fermi energy —meF~m[18]. We can rewrite the expansion of the field operator ~Pin terms of single-particle and antiparticle states

~P(x,t) = ~ t/Jk(X) exp(—iEkt)1 5 + ~ t/i~(x)exp(iEkt)d~, (2.49) k>F k

~=~Jd3x{[~i~, (_iyoykak+ yom)~fI]+[~fr+(_iyoykak + y°m),~I~]}. (2.50)

This leads to the final expression in terms of particle and antiparticle operators:

H= ~ Ekb~bk+~ Ekc2~dk+EO. (2.51) k>F k

Accordingly one obtains the result that the symmetrized non-interacting Dirac vacuum carries the infinite zero-point energy

E0=_~(~Ek+ ~ Ek). (2.52) k>F k

2.1.4. Zero-point energy and normal ordering So far we have shown that the occurrence of infinite zero-point energies is inherent within the canonical field quantization scheme because it does not fix the ordering of quantum operators in the field Hamiltonian. To remove this ambiguity one has several options. In the most accepted procedure one arbitrarily imposes a certain order of the non-commuting creation and annihilation operators in the Hamiltonian. It is based on the observation that zero-point energies always appear in connection with a rearrangement of terms containing a product of annihilation and creation operators. One circumvents this by introducing the so-called “normal ordering” of operator products and replacing all relevant expressions by their normal-ordered counterparts. Normal ordering of a product (indicated by embrac- ing the expression between colons) is defined as rearrangement of its factors in such a way that all annihilation operators stand to the right. For fermions the product must be multiplied by the sign of the required permutation. For instance, for boson operators

~:(â~ák + akak): = a,~ak, - (2.53) 100 G. Plunien eta!., The Casi,nir effect and for fermion operators

(!~l~— bkbk): = b~bk. (2.54)

Applying this procedure to the field Hamiltonians considered above, one obtains

:i-Ti: = i-Ti — (Oh-Tb) an i-Ti’. (2.55)

One observes that replacing the original Hamiltonian i-Ti by the normal-ordered Hamiltonian i-Ti’ is equivalent to a formal subtraction of zero-point energies. This has the effect that the energy of the corresponding free vacuum state becomes zero. It must be stressed that the normal ordering cannot be justified by referring to the correspondence principle which can be envisaged as the underlying guide in canonical field quantization. One has to accept normal ordering in this context as an additional postulate required to render the energy of the vacuum state finite. It becomes acceptable by the fact that it brings about consistency with the heuristic notion of a non-interacting vacuum as a state of zero energy. However, the general argument that the subtraction of the zero-point energy can be performed arbitrarily since it corresponds to a redefinition of the energy scale, and since only energy differences are in reality measurable quantities, must be regarded with caution. In relativistic quantum field theory the Hamiltonian, which is one of the generators of the Poincaré group, is determined in a unique manner for a given set of field operators [19]. Takahashi and Shimodaira [19, 20] have in fact shown that its expectation value must vanish in order that the generators satisfy the correct commutation rules. This provides a convincing argument for the usual notion of the vacuum state, but not a unique one for introducing H’. It does not exclude alternative approaches towards a satisfactory definition of the vacuum state. This problem is particularly important in the context of gravitational physics where the ad hoc subtraction of zero-point energies seems dubious on formal and physical grounds, because the expectation value of the energy—momentum tensor determines the geometry of space-time. In principle, quantum fluctuations may influence the local space-time structure in microscopic domains which are of the order of the Planck length (hG/c3)1”2 = 1.6 x 1O~cm. 2.1.5. The Casimir energy Searching for a general concept to define the properties of the vacuum state of quantized fields, attention is drawn to Casimir’s work on the zero-point energy of the electromagnetic field [1]. The underlying idea can be generalized to the case of other fields, leading to various versions of the Casimir effect. It is essential for the understanding of this effect to realize that quantized free fields, as they have been understood above, are abstract mathematical constructions. In reality fields exist and can be measured only in the presence of sources or as fields interacting with each other. A special case of particular interest is a field that is confined to a finite spatial volume. Generally speaking a physical field may be forced to satisfy certain boundary conditions, an important fact which has to be considered in the mathematical description. The presence of boundaries for the field induces a change in the energy spectrum and therefore modifies also the zero-point energy. Consequently, it is reasonable to define the physical vacuum energy of quantized fields as a difference in zero-point energy. Let aT’ stand for an arbitrary boundary required for the considered field. If E 0[aF] denotes the zero-point energy in the presence of boundaries, and E0[0] that without such boundaries, then the vacuum energy (Casimir energy) is formally defined as G. Plunjen et aL, The Casimir effect 101

Evac[t91’] = E 0[aF] — E0[O]. (2.56)

In order to make this formal definition more transparent, a few remarks are necessary to specify what possible situations or systems can be envisaged and what kind of “boundary conditions” must be dealt with. In a typical situation a field is given in the presence of macroscopic objects or it is confined to a finite cavity. Typical examples for this case are conductors or dielectrics constraining the electromagnetic field or the QCD vacuum confining quarks and gluons in a hadron bag. The field then has to satisfy certain geometrical boundary conditions, e.g., to vanish on the surface or in the outside of a cavity. In such cases it is convenient to treat the vacuum energy as a function of a suitable set of parameters, here abbreviated by A, which characterize the given geometrical configuration (fig. 2.1). This means that we write

Evac[A] = E0[A] — E0[A0], (2.57) where A0 denotes the parameter set corresponding to the situation without boundaries. Any variation in these parameters may induce a change in vacuum energy. Boundary conditions can be considered as idealizations of the real conditions where configurations of matter or external forces act on a given field. We are thus led to a generalized concept of the Casimir energy by considering the change in the energy of the vacuum of, for instance, the electron—positron field in the presence of an external electromagnetic field A,~.The external field now plays the role of the parameters A, which represent the boundary condition. The analogous definition of the vacuum energy then is

Evac[A] = E0[A] — E0[A = 0]. (2.58)

Alternatively, the vacuum energy may be treated as a function of parameters which determine the geometry of the given external source configuration. More detailed considerations will be necessary when the vacuum energy is explicitly evaluated (see section 5). To summarize, we note that (1) the ambiguity of divergent zero-point energies may be removed by retaining the original field Hamiltonian but introducing the definition, eq. (2.56), for the physical vacuum energy; (2) the heuristic notion of a non-interacting vacuum is not affected, since Eyac[0] is, per definition, equal to zero; (3) changes in zero-point energy are reintroduced in quantum field theory as in principle measurable corrections.

Fig. 2.1. The parameter A may describe the relative position or the geometry of certain objects in an external field. 102 G. Plunien eta!., The Casimir effect

This point of view implies that properties of quantized fields may be determined from the “response” of the vacuum state to external constraints or fields. Since it can be experimentally tested (at least in principle) whether such changes in the vacuum energy as a function of certain parameters A do exist or not, it is a question of physics (and not only of formalism) whether the zero-point energy must be included in the field Hamiltonian or not.

2.2. implications of the vacuum energy

After introducing the vacuum energy, and having made its formal definition plausible, we will now outline some implications of the vacuum energy. To do so we have to examine the meaning of zero-point energies in the context of various physical effects. The forces acting between neutral but polarizable objects, often generally classified as van der Waals forces, historically gave a stimulus to Casimir’s work on the zero-point energy of the electromagnetic field, which was motivated by the search for an explanation of the simplicity of the results for van der Waals attraction obtained in perturbation theory [28]. It is principally possible to distinguish three classes of van der Waals forces [29],viz., orientation, induction and dispersion forces. The latter class of forces indicates that two polarizable particles will attract each other because quantum charge fluctua- tions in one particle will produce a dipole moment, which will in turn induce a dipole moment in the second particle. Because of the relation of these forces to the phenomenon of optical dispersion they are termed “dispersion forces”. The concept of electromagnetic zero-point energy allows to calculate a particular subclass of the van der Waals forces, the so-called “long-range retarded dispersion forces”. For example, calculating the attraction of a polarizable particle to a perfect conducting wall (fig. 2.2), one can use the vacuum energy properly defined for this situation [30, 31]. Let E 0 = ~,, w~denote the zero-point energy of the electromagnetic field in a large cavity with conducting walls. The normal frequencies ~ are determined by requiring periodic boundary conditions. The presence of a particle inside the cavity, with dielectric polarizability a, separated by a distance R from the wall, will modify the frequency of each mode and consequently change the zero-point energy by the R-dependent amount

~w~(R)~~ [w~(R)—w~], which depends on R. For the combined system “particle plus conducting wall” the distance R

______/ Fig. 2.2. Boundary configuration assumed in the derivation of the van der Waals attraction between a polarizable particle and a perfectly conducting wall. G. Plunien et aL, The Casimir effect 103 represents the significant parameter. One obtains the corresponding vacuum energy as the energy difference between a configuration where the particle is placed at the distance R and that obtained in the limit R —~ ~ when ~ —~0 by definition. For the considered system the vacuum energy represents a contribution to the potential energy describing the attraction between the particle and the wall

Evac(R) = ~ (w~°~+&o~(R))—~~ (w~+~wn(co)). (2.59)

Explicit evaluation of the expression Evac(R) leads to the result

3a 1 Evac(R) = — ~, (2.60) which agrees with that first obtained by Casimir and Polder [28] from perturbation theory. The same method which leads to the above result can also be employed to derive the long-range retarded potential between two neutral polarizable particles separated by a distance R [30]:

7. (2.61) Evac(R) = —23a1a2/41TR Again, the result is identical with that found by Casimir and Polder [28]from perturbation theory. The successful use of the zero-point energy in this context is further supported by Boyer’s [32, 33] recalculation of long-range van der Waals potentials, where he also included the magnetic polarizability of the particle. The full expressions corresponding to eqs. (2.60) and (2.61) coincide with those obtained by Feinberg and Sucher [34] from dispersion-theory techniques. The original effect [1]considered by Casimir was the attraction of two uncharged parallel conducting plates at zero temperature. His work concerning this simple configuration shows most impressively the existence of forces which may be ascribed entirely to the change in the zero-point energy of the electromagnetic vacuum brought about by the presence of boundaries. It was not until 30 years later, however, that the local form of Casimir’s result, i.e., the formulation in terms of vacuum energy density and vacuum pressure, was exploited by Brown and Maclay [351.They showed that the vacuum expectation value of the regularized energy—momentum tensor is spatially constant, and derived the expression

= —(n-2/180a4)(~g~”+ z’~z~), (2.62) where z~= (0, 0, 0, 1) denotes a space-like four-vector orthogonal to the plates separated by a distance a. The form of this result becomes plausible by the following general arguments. First of all, the only two symmetric tensors of rank 2 that can be formed are g~”and z”z~because of the symmetry of the configuration. @~‘ is invariant with respect to arbitrary Lorentz boosts parallel to the plates, but the perpendicular direction is favoured. Since the electromagnetic field is massless and conformally invariant, the energy—momentum tensor, eq. (2.62), must be traceless and divergenceless. These facts require t9’”’ to be independent of position and only allow the combination (~g~’+ z~’z’~).Finally, on dimensional grounds it follows that Ø’~’must be proportional to the inverse fourth power of the separation. 104 G. Plunien et aL, The Casimir effect

The numerical factor (~i~2/18O),including the sign, can only be obtained by explicit calculation. This is an important point in that it shows that the result (2.62) is unlikely to be representative for the general situation. Whether the zero-point pressure on surface configurations tends to contract or to expand them, whether the vacuum energy is negative or positive, depends crucially on the particular geometry. Unfortunately the geometry enters in a non-trivial way. Thus, for example, the hope that the parallel-plate result may serve to stabilize the classical model of the electron as a charged shell [36] turned out to be a fallacy. Boyer’s result for the vacuum energy on perfectly conducting spherical shells [37, 38] is positive and consequently tends to expand this configuration:

E 0.093/2a. (2.63)

Therefore it cannot play the role of the Poincaré stress in electrodynamics. Nevertheless, over the years this idea instigated further efforts to evaluate the Casimir effect [39,40] for solid balls with certain dielectricity and permeability. Here in fact attractive energy contributions are obtained, but they are quantitatively too small to explain the observed charge quantization. In this context one has to prove whether the repulsive Coulomb energy of a charged sphere may be balanced by an appropriate attractive Casimir energy. If this would be the case, then one would obtain an eigenvalue for the fine-structure constant. From an approximate calculation it had also been suggested that the Casimir energy for perfectly conducting cylindrical shells may be zero [42].This would be intuitively reasonable from a geometrical point of view, classifying the cylindrical configuration somewhere in between the parallel-plate and the spherical-shell configuration. Explicit calculations for this geometry, however, have yielded a negative vacuum energy [43]. The obvious complexity of the relation between the magnitude of the electromagnetic Casimir energy and the shape of the surface has provided a stimulus for studying vacuum energy in the presence of arbitrarily shaped objects in the framework of the local representation. Balian and Duplantier [42] studied the Casimir effect (including temperature) concerning the electromagnetic field in regions bounded by thin perfect conductors with arbitrary smooth shapes. In this context a further aspect of the vacuum energy became obvious, namely that it can represent a contribution to the surface tension. Conceptually, this is easy to understand when one starts from an infinite plane conductor placed in the electromagnetic vacuum as a situation of reference. When a smooth deformation of the conductor is introduced, this induces a change in the zero-point energy. Suitable parameters for measuring the local deviation from a planar geometry are the principal curvatures of the conductor surface. Treated as a function of these parameters, the vacuum energy contributing to surface tension may be rigorously defined as the difference between the zero-point energy at a certain deformation and that which corresponds to the planar geometry. This interpretation of the Casimir effect can also be applied to more realistic objects in the electromagnetic field characterized by a dielectric constant and magnetic permeability. An essential property of such surface contributions of the vacuum energy is that they generally remain cutoff- dependent [44, 45]. This fact, however, makes it possible to obtain reasonable values for measurable quantities as, for example, the surface tension of liquid helium [44] and many metals [46, 47] by a suitable choice of the high-frequency cutoff. Having the means of deriving such geometrical expansions for the vacuum energy in the presence of arbitrarily shaped objects, one is close to the capability of considering vacuum energy in curved space-time [49, 50]. Accordingly, as a parallel development, remarkable progress has been made in revealing further aspects of the Casimir energy in gravitational physics [21—27].In the framework of such studies one can try to investigate cosmological consequences G. P!unien et aL, The Casirnir effect 105 of changes in vacuum energy due to deviations from Minkowskian geometry, to mention only one application. Reviewing at first only the various work that has been done concerning the electromagnetic vacuum, one is led to the following judgement of the zero-point energy problem: Casimir’s work stimulated studies about the zero-point energy problem in field theory. Such studies are generally important in order to derive an accurate field quantization scheme as a fully satisfactory “first principle theory”. Questions about the physical reality of the zero-point energy can only be settled by demonstrating observable effects that reveal the influence of the vacuum. The electromagnetic field represents a convenient example to examine the physical meaning of the zero-point energy, because its fundamental interaction is well known. In this way it has been shown that the concept of Casimir energy can be successfully applied to alternative calculations of the van der Waals attraction [32, 33]. But in contrast to that, the Casimir energy can also cause repulsive forces as in the case of the conducting spherical shell [37, 48]. This already indicates two possible implications of the vacuum energy: On the one hand the concept of a Casimir energy can lead to a different point of view or allows an alternative interpretation of known effects. On the other hand, however, this concept brings to light certain effects which may not be necessarily understandable in the framework of conventional field theory which neglects the zero-point energy. Directly connected with that is the clarification of the question of how to determine the “true” field Hamiltonian. Having realized that the concept of Casimir energy together with its evaluation methods can be applied successfully in the case of the electromagnetic field, it seems legitimate to look for the possible role played by zero-point energies considering various other constrained quantum fields. Let us now mention some fields of application for the use of zero-point energy concerning boson and fermion fields. First we have to note that Wentzel [51,52], even before Casimir, used the zero-point energy of the Klein—Gordon field to calculate the forces in the meson-pair theory of fixed sources. Wentzel’s pair model recently reemerged in the context of the self-energy calculations for massless scalar fields confined to a spherical cavity [53]. The concept of the Casimir effect has recently become a heuristic guide in particle physics [54—72]. The idea that hadrons can be described as confined fermion and (vector) boson fields, the fields of quarks and gluons, has become generally accepted, although the precise nature of the mechanism responsible for confinement remains unknown. The confinement itself gives rise to new effects connected with the Casimir energy due to quantum fluctuations in the fields. In the framework of the established phenomenological bag model [73—75],which represents a widely used approximation to the confinement mechanism existing in quantum chromodynamics (QCD), one has exactly the situation where fields are forced to satisfy certain boundary conditions on the bag surface in order to guarantee colour and quark confinement. Much work has been done in approaching and understanding the role of the zero-point energy in the internal structure of hadrons in the bag model [54—60].Originally it was suggested [76, 69] that the Casimir energies of quark and gluon fields could explain a part of the potential (—zia, where a is the bag radius and z is a pure positive number) necessary for the phenomenology. Good fits were obtained for the value z = 1.84. In order to find an explanation for this empirical value one has to examine all the energy contributions which are of the form 1/a. Beside the Casimir energy of confined quarks and gluons (their 1/a behaviour already follows on dimensional grounds) the center-of-mass correction has to be taken into account. The latter turns out to be the dominant 1/a contribution [139]to the static bag energy and leads to the value ZCM 1.95. The use of the Casimir energy in the framework of the bag model to explain the empirical value of z fails quantitatively. Both the zero-point energies of confined quarks and gluons give rise to a repulsive 106 G. Plunien et aL, The Casimireffect

Casimir energy [58, 61]. Taking all these effects together one is left with a value which is approximately about z 1.5. The evaluation of the Casimir energies treating quark and gluon fields as free and perfectly confined within the bag is plagued with divergencies (cutoff-dependent terms) when restricting to field modes in the interior of the bag [55,57]. These cutoff terms cancel when considering also the exterior field modes [58, 61]. The situation becomes more difficult considering massive quarks. The result for the fermion stress contains new ultraviolet-divergent terms in addition to those occurring in the massless case [59]. Concerning the Casimir energy in the hadron model, only further studies will allow a deeper understanding and a satisfactory explanation of their role. It remains to be said that Casimir energies give rise to interesting considerations in connection with the structure of the QCD vacuum [66—72]as well as the properties of the vacuum of scalar fields [77]. The investigation of the response of the vacuum to the presence of external fields represents a further field of application of Casimir energies. A problem of this type considers the zero-point energy of the electron—positron field in the presence of classical electromagnetic fields, created by an arbitrary configuration of external sources. The vacuum energy Evac[A] = E 0[A] — EO[A = 0] may be treated as a function of parameters which characterize the geometrical shape or the relative positions of the external sources for a given configuration. Which properties of the vacuum energy can be expected under such conditions? One could at first argue that the induction of polarization currents represents the main reaction of the Dirac vacuum to the presence of external electromagnetic fields. Any variation of the parameters which determine the external charge configuration will change the external field and, correspondingly, cause a displacement of the polarization charge density. Accordingly, such “dis- tortions” of the vacuum will lead to a vacuum pressure. Consequently one would expect that the vacuum energy represents a contribution to the interaction potential. In addition, the corresponding vacuum energy may contain self-energy contributions. Whether these expectations prove to be true has to be shown by explicit calculations. An investigation along this line has been carried out by Ambjorn and Wolfram [78] who discussed the vacuum energy of a charged scalar field in an external electric field. Another extension of the concept of Casimir energies is obtained by considering the combined system, field plus boundaries, at finite temperature, assuming thermodynamical equilibrium. Applying the standard formalism, the quantity which must be calculated under such conditions is the Helmholz free energy

F=—-~--ln2, (2.64) where ~ is the partition function of the system. The temperature-dependent Casimir energy for conducting plates in the electromagnetic vacuum has been calculated by several authors [44, 35, 79—81]. Corrections arising from the fact that real metals are not perfect conductors have also been considered [82]. These investigations are important in order to derive a better approximation to the measurable effects in real experiments.

3. Mathematical formulation and evaluation methods for vacuum energies

3.1. Method of mode summation

The most direct method to evaluate vacuum energies is that of mode summation. Starting from its definition as a difference in zero-point energies this method involves the performance of infinite G. P!unien eta!., The Casimir effect 107 summations over eigenenergies of corresponding field normal modes in the presence of external constraints and for the unperturbed vacuum configuration. In order to make this kind of approach in principle applicable, one has first of all to introduce a large but finite quantization volume which is bounded by a surface .~ (space cutoff) of suitable geometry adapted to the constraints present in the considered field. This suggests, for example, in case of the parallel-plate configuration to use a rectangular box as quantization volume. Quantizing the field under the usual assumption of periodic boundary conditions determines the free eigenmode frequencies, and the zero-point energy inside this volume becomes a discrete sum. Let us illustrate the situation for boson fields (scalar or vector fields), where the corresponding zero-point energy takes the form E 0 = ~k Wk. The free field zero-point energy inside the volume depends only on the space cutoff Z

E0[.~,0] = ~ Wk[I, 0], (3.1) where the symbol 0 indicates that the volume is otherwise empty. The vacuum inside the volume becomes disturbed by the presence of certain objects. As an idealization those objects can be treated as additional arbitrary surfaces S, on which the physical fields are constrained to fulfill certain boundary conditions. Many possible configurations can be studied in this way. The surface may be disconnected or simply connected confining a finite subspace or a combination of the two. Also different types of boundary conditions can be required, depending upon the physical situation. The simplest ones are Dirichlet conditions, where the field (or some of its components) must vanish on S:

Dirichlet: ço(x)!s = 0. (3.2)

Such a condition is applicable when perfect conductors are placed in the electromagnetic vacuum. The Neumann boundary condition, which states that the normal derivative of the field has to vanish on 5, is another possibility:

Neumann: n~Vç(x)js = 0. (3.3)

This condition is analogous to the linear bag condition applied to quarks inside a hadron, where the normal component of the quark current must vanish on the bag surface. In certain cases one may also assume an arbitrary linear combination of these, the Robin boundary condition. The eigenmode energies are obtained by solving the field equation with respect to these constraints. Accordingly, for a given surface configuration the zero-point energy depends also on S. In deriving the vacuum energy, it seems convenient to consider the energy difference

~ S] = wk [1’5] — Wk [.~,0]. (3.4)

This expression is still indefinite, which indicates that in fact such “perfect conductor” boundary conditions are too rough an approximation to the real situation. Instead of dealing with those “sharp” boundary conditions it seems to be more appropriate to “smooth” them out by taking into account certain microscopic details or physical surface effects (for instance, the finite penetration depth of electromagnetic waves in real materials). On the other hand one can expect that a more realistic 108 G. P!unien eta!., The Casimireffect

treatment becomes rather cumbersome. It is possible to circumvent these problems by introducing a suitable high-frequency cutoff x(w), assuming that it simulates the ignored physical details. Here an inherent problem of the mode-summation method becomes obvious, i.e., the embarrassing fact that it gives an indefinite expression (3.4), and requires the further introduction of a suitable regularization scheme in order to yield a finite result. This regularization should be, however, legitimized on physical grounds. For example, the conductivity of physical materials actually decreases at high frequencies, and the surfaces become transparent to electromagnetic waves of very short wavelength. Thus one can expect that the high-frequency modes are not influenced by any boundary S and, consequently, they should not contribute to ~ S]. Transparency of the surface can be simulated, for example, by choosing a cutoff x(w) with the behaviour x(w) 1 for w ~ w~and x(w) 0

for w ~‘ w~,where the cutoff frequency w~is determined by the properties of the material. Now one has ~deaLwiththeregularizedquantity ~

~Ereg[.~, 5, x] = E (wk[I, S] — wk[.~:, O])X(wk), (3.5)

from which the Casimir energy is obtained by removing the space cutoff (moving I to infinity). At least for a conducting surface S the Casimir energy should be the limit of 1~Ereg[I, S, x] when both the space cutoff I and the high-frequency cutoff x are removed. Such a limiting process has been worked out in several studies of the Casimir effect. However, only particular shapes for I and x have been introduced in connection with the considered shapes for S, for instance the rectangular box [1, 83], and the sphere [37, 48] for I, and an exponential decrease for x. After detailed analysis it turns out in these cases that the results are cutoff-independent. In order to make the above representation more specific let us consider their application to the one-dimensional version of the conducting parallel-plate configuration and to the conducting spherical shell configuration [37]. In the one-dimensional Casimir configuration (see fig. 3.1) the system con- sidered consists of a large one-dimensional box of size L bounded by perfectly conducting “walls” which is divided into two regions of length a and (L— a) by a perfectly conducting “plate”. This configuration encloses the electromagnetic field which is forced to satisfy periodic boundary conditions on the surfaces. The Casimir energy is obtained as the difference in zero-point energy corresponding to

the configurations where the “plate” is placed at a given distance a ‘~ L and that where the distance a is large, say L/2. The zero-point energy of the electromagnetic field inside the region (I) is given by E 2) ~a is formally divergent.1(a) = ~ Thereforemr/a. Obviouslyone has toalsointroducethe infinitesimala regularizationchangescheme.~E1(a)The= ~simplest(nir/aone employs a space cutoff (box L) and an exponential frequency cutoff exp(—Anir/a). The Casimir energy follows according to the subtraction of zero-point energies (see fig. 3.1):

Ec(a) = lim lim {(E 1(a, A) + EIJ(L — a, A)) — 2E511(L/2, A)}. (3.6) L-=~A—.O

E11 — ~ Em Epj

Q [-a Liq L-Lh1

Fig. 3.1. Subtraction of zero-point energies in the case of the one-dimensional version of the Casimir effect. G. P!unien eta!., The Casimir effect 109

The zero-point energies of the field inside regions (II) and (III) are obtained by a simple replacement of the length parameter a in E1(a, A). The regularized zero-point energy inside the region (I) is given by

E~(a,A) = ±.-~-~exp(—A~)= —-f- [i_ exp(_-A ~)]

= +—exp(—A—~)[1—exp(—A—~l . (3.7a) a \ alL \ a/i

In the limit A —~ 0 this energy contribution behaves like

2 — ii-/12a + C(A2). (3.7b) E1(a, A) = a/rrA

Performing the cutoff removal A —+0 and L —~~, the Casimir energy of the one-dimensional version of the parallel-plate configuration turns out to be

E~(a) —ir/12a. (3.8)

The same techniques can be applied in order to calculate the Casimir energy of the conducting spherical shell configuration. In this case the system consists of a large conducting sphere of radius R enclosing the quantization volume, which concentrically surrounds a small sphere of variable radius a ~ R (see fig. 3.2). The corresponding energy difference, eq. (3.4), is the change in the zero-point energy of the total system when the radius of the inner sphere grows from radius a to another radius R/~(with a fixed value ~ > 1). The subtraction of zero-point energies referring to the different regions will be performed as indicated in fig. 3.2:

z~E(a,R)= (E 1(a)+E11(a, R))— (E111(R/,~)+E1~(Rhj,R)). (3.9)

Since every single contribution in eq. (3.9) diverges, one is forced to introduce a suitable cutoff function which vanishes for large arguments, e.g., as before x(w) = exp(—Aw). The Casimir energy is obtained after performing the limit

E(a) = lim lim i~E(a,R, A), (3.10)

A-.O R-+oo and turns out to be cutoff-independent (eq. (2.63)).

Fig. 3.2. Subtraction of zero-point energies for the conducting spherical shell configuration. 110 G. P!unien eta!., The Casimir effect

Evaluating Casimir energies in similar cases according to this method — for instance the parallel-plate

configuration for a massive scalar field in a finite rectangular box [77]— one also has to introduce a high-frequency cutoff or some other regularization scheme. Difficulties arise in understanding the role of the cutoff frequency when it would appear in the final result, because in this case the regularization method is not well-motivated on physical grounds and must be considered as a formal technique designed to produce a finite quantity. Fortunately, the cutoff removal can be performed and the resulting Casimir energy is cutoff-independent also in this case. We add a few comments concerning the applicability of the mode summation method. It has already been mentioned that the eigenmode energies Wk are obtained by solving the field equation with respect to the required boundary conditions. The cases where analytical expressions for 10k can be obtained, and the summation and regularization can be performed, are severely limited when more complex geometries are considered. Apart from the enormous numerical effort associated with the application of this evaluation method in general, further technical difficulties occur in connection with the required boundary conditions. Such problems exist inherently in the case of confined spinor fields, where the boundary condition has to be satisfied simultaneously for both the upper and lower component, or for the electromagnetic field where one has to distinguish between transverse electric (TE) and transverse magnetic (TM) modes. For the highly symmetrical spherical shell configuration it is possible to divide the total zero-point energy into the contributions from TE and TM modes, respectively, because particular steps of the whole calculation can be performed analytically. Thereby one recognizes that the

Casimir energy becomes finite (in the limit R —~~ and A —~ 0) because of a delicate balance between the contributions of the TE and TM modes. The spherical shell calculation already indicates that the application of the mode summation method in the case of more general configurations can be expected to be very cumbersome and even unsatisfactory, when guidance from analytical calculations is absent. Numerical convergence is not a priori guaranteed. Consequently one may argue that the success of the mode summation method is limited to problems, where the evaluation is accomplished either analytic- ally or, at least, semi-analytically as in the case of the spherical shell. On the other hand, it should be emphasized that this method always may be used, in principle, to define the Casimir energy. Nevertheless, due to the severe problems encountered in solving non-trivial examples using the direct summation method, it is necessary to look for alternative formulations.

3.2. The local formulation and Green function methods

Another field theoretical approach for studying the properties of the vacuum starts from an analysis of the behaviour of local field quantities. For our purpose, the energy—momentum tensor T’~”represents the appropriate quantity: the integral overT00 represents the total energy, the components T°~’are related to the flow of energy and momentum, and the stress components T~are useful to deduce the mechanical properties of the vacuum. A local formulation implies the introduction of the energy— momentum tensor of the vacuum f9~according to eq. (2.56) in the form:

~ (0hi’~’t0)ar(0hI’~’b0)o, - (3.11) i.e., the measurable energy density of the vacuum is defined as the difference between that in the constrained field configuration and the one corresponding to the unconstrained field. Since space-time integrals over particular components are related to direct observable quantities, in addition to eq. (3.11) one has to require certain invariance properties under fundamental symmetry transformation of the G. P!unien eta!., The Casi,nir effect 111 system. For instance, the vacuum energy of a relativistic charged field should be invariant under charge conjugation. @~should also be symmetric in its indices because it acts, in principle, as a gravitational source. Equation (3.11) has to be regarded as a formal definition, i.e., ~ is not necessarily well-determined by this equation, as we shall see. The advantage of the local definition is that it permits a different point of view and a deeper understanding of the nature of vacuum energy and vacuum stress. It turns out that 9~is expressable in terms of field propagators. The occurrence of quantum field fluctuations and the associated observable vacuum effects can thus be understood from the modifications in the propagation of virtual field quanta under external constraints. In order to elucidate this connection we shall now derive the relation between the vacuum expectation value of the energy—momentum tensor and the propagator for two conformally invariant field theories, viz., the electromagnetic field and the massless scalar field. Beside the dynamical fields themselves, the energy—momentum tensor also contains first-order derivatives of the field. Such terms may be constructed by means of a suitable differential operator acting on the time-ordered product of the field operators T(~°(x)~’~(x’)).The operator for the energy—momentum tensor then formally follows after performing the Lorentz-covariant limit x’ —~x. By taking the vacuum expectation value the relationship to the propagator, which is generally defined as

iG~°~’~(x,x’) = (01 T~’~(x)~k)(xl))I0) (3.12) becomes clear. For a specific example, let us consider the energy—momentum tensor of the free electromagnetic field with the Lagrangian 2= ~ which leads to the field equations: (g”Ll— a~a~)A~= 0. We want to relate its symmetrical energy momentum tensor

T~~A= —F’~F~+ ~g~AFsF~uJ (3.13) to the propagator. The first step is to derive an equation between the time-ordered product of the electromagnetic field tensors

t~AP(X x’) = T(P’~(x)F~’(x’)) (3.14) and that of the vector fields as

= TX~,AP;~$T(A~(X)AP(X~))_(n~5(x~—x~)[A’~(x),ft~’(x’)] — n”6(x~—x~)[A~’(x),P~’(x’)]), (3.15a) where n” = (1, 0, 0, 0) denotes a time-like vector and the differential operator explicitly reads

3 a’~~) (3.15b) = (g”~a~’— ~ 9~)(g1~9~A— gAl The second term in eq. (3.15a) arises from the differentiation of the function ~9(x 0— x~)due to the presence of time ordering. In the derivation of eq. (3.15a) terms of the form g°’~6(x0— x~,)[A,~(x),Al3(x’)] have been omitted since for equal times

t5(xo— X~)[Aa(X), Al3(x’)] = 6(x0— Xt)[Aa(X, x0), Al3(x’, x0)] = 0. (3.16) 112 G. P!unien eta!., The Casimir effect

According to eqs. (3.15a) and (3.15b) and replacing all commutators which appear together with a temporal delta function by the corresponding equal-time commutators, the energy—momentum tensor follows as

j~~A(x)= —{~1~ T(A,,(x)A~(x’))+ 2i(n”n” — ~g~A)~5(x — x’)}I~~, (3.17a) with the abbreviation

A~~A;~~l3_p~A ;a$l ~A ,‘p; — T~~’~. 4g ~

The delta-function term in eq. (3.17a) arises from explicit evaluation of equal-time commutators. After taking the vacuum expectation value the desired relation between the unperturbed energy—momentum tensor (Of T~~A(x)I0)o and the free photon propagator

iDas(x — x’) = (01T(Aa(X)A~(X’))f0) (3.18) is found to be

(Of I’~(x)fO)o= —i{~i~ — x’) + 2(n~~nA— ~ig~A)6(x — x’)}I~.~. (3.19)

On the right-hand side of eq. (3.19) D,,~(x— x’) can be expressed in terms of the scalar Green function

G0(x — x’), which in the Feynman-gauge is simply related to the photon propagator according to

D~$(x—x’)=g~l3GO(x—x’)=——-’-~ g~ . , (3.20) 4ir (x—x) +137 leading to the result

(Of t~(x)f0)0 = —i{r~Go(x— x’) + 2(n~n”— ~g”jô(x — x’)}f ~ (3.21a)

= ~ = 2(3~’a —~g~”8”a.~)- (3.21b) 4 as x’ approaches x. In Ø~ the delta-function term will be cancelledThe first bytermthedivergescorrespondinglike 1/(xcounterpart— x’) of (O!T~”(x)f0)~.That the space integral over (Of T”~’(x)f0)~ actually yields the zero-point energy E 0[O] becomes clear by inserting an appropriate eigenfunction representation of Go(x — x’) and integrating over space. Introducing boundaries, the propagation is modified due to the surface interactions, and con- sequently this perturbs the homogeneity of the corresponding propagator. In addition, the propagator G(x, x’) must now satisfy the appropriate boundary conditions, for instance Dirichlet or Neumann conditions as in the case of perfect conductors (G(x, x’) = 0 or n~V G(x, x’) = 0 on the surface S). Under these conditions the differential operator r~, which relates (OIT’~IO)sto G(x, x’), is the same. Thus the energy—momentum tensor of the vacuum for the constrained field configuration can be written as [49] G. Plunien eta!., The Casirnireffect 113

~9~(x)= —i{r~.(G(x,x’)— G 0(x — x’))}j~’=~. (3.22)

The above expression looks formally quite simple. The vacuum subtraction appears now as the difference between the constrained and the free propagators. For explicit evaluations all one has to do is to construct G(x, x’) for the considered configuration, which is, however, not a trivial exercise. This is not surprising in view of the mathematical difficulties already encountered in the context of the mode summation method. One cannot expect to circumvent these problems simply by choosing a different calculational method. Except for some special cases with simple geometry the exact evaluation of ~ for an arbitrary configuration is of comparable intricacy. On the other hand, the local method has the advantage that allows one to apply systematic approximation schemes for the constrained propagator. For instance, the asymptotic behaviour of ~ in the vicinity of a smooth boundary can be obtained explicitly by this method [49]. Brown and Maclay [35] were the first to use the Green function method in calculating the vacuum stress tensor for the Casimir configuration. By virtue of the highly symmetrical parallel-plate configura- tion. The exact expression for the constrained Green function G(x, x’) can be constructed in terms of an infinite sum of image source functions (see fig. 3.3). If there are no boundaries in the electromagnetic field, only “direct” propagation (emission of a photon at a space-time point x’ and absorption at x) occurs. Introducing the parallel plates separated by a distance a, several additional contributions occur corresponding to certain reflections on the plates (mirrors). These contributions can be treated as if they arise from an infinite sequence of alternating sources placed along the direction z’~= (0, 0, 0, 1) normal to the plates, propagating2freely, —x’3).fromAccordingly,the imagethepointsexactx’ Green+ 2lz andfunctioni’ + 2alzDap(X,(1 =x’)0, ±1,..can be.,expressed±c~)towardas ax, sumwhereofi”free= (x’°,Greenx”,functions.x’ After the explicit calculatibn the energy—momentum tensor of the vacuum appears as

~9~(x)= —i28~’8” ~‘ G~(x— x’ — 2alz)I~=~, where the prime indicates that the I = 0 term (“direct” vacuum contribution) is excluded. Performing the summation and the differentiations leads to the result, eq. (2.62), already mentioned in the previous section. It is interesting to note that only contributions corresponding to an even number of reflections enter in ~ Lukosz [84, 85] shows in detail for Casimir’s original parallel-plate configuration that this cancellation with an odd number of reflections arises from the particular symmetry of the Green functions corresponding to the electric and the magnetic field modes, respectively.

0 • I I ~

Fig. 3.3. Illustration of the image-source construction of the constrained Green function for the parallel-plate configuration. 114 G. Plunien et aL, The Casimir effect

3.2.1. Casimir stress between perfectly conductingparallel plates In order to make the previous considerations more specific and to give an example for a successful application of the Green function method, let us now present the calculation of the Casimir effect between infinite plates in some detail. It has already been mentioned that the energy—momentum tensor can be derived from the Green function

~ x’) = (O~F~’(x,x’)!O) = (OJTP~”’(x)E~(x’)JO). (3.23)

This Green function itself can be constructed by means of a suitable differential operator which acts on the photon propagator jDa~~(x, x’) = (O~T(Aa (x)A~(x’))jO). In Feynman gauge, where the photon pro- pagator is simply related to the scalar Green function, one derives the relations

G~~AP(X,x’)~.5~APG(X x’), (3.24a)

1~9~~a’~. (3.24b) ~5~~AP = g~’P~91~31A— g#~P3~9lA + g~’8~9”— g

The notation = in eq. (3.24a) indicates that the equal-time commutators are omitted (see eq. (3.15a)).

The energy—momentum tensor is then given by

(Of k(x)IO)s~(i){G~,(x,~ ~ , (3.25) omitting gauge-dependent delta-function terms, which arise from the evaluation of commutators, and which cancel in ~ The exact Green function can be constructed in terms of an infinite sum over free Green functions corresponding to image sources of alternating sign. As illustrated in fig. 3.3, one has to deal with two types of images. Those which carry a positive sign are placed at positions x’ + 2alz. They correspond to those signals propagating from space-time point x’ to x, where an even number of reflections between the plates has taken place. The Green function referring to an even number of reflections (omitting delta-function terms) is given by

G~(x, x’) ~ ~ G(x - x’- 2alz). (3.26)

In order to derive the contribution from an odd number of reflections ~ which refers to negative sources placed at i’ + 2alz, one has to consider that in each single propagation function Da~(X— X’ — 2alz) the normal component of the potential changes sign, i.e.,

+ 2alz) = (g~+ 2z0~‘~z)A 3A 13(f + 2alz) ~‘ 13(f + 2alz), (3.27a) with

(3.27b)

It was convenient to introduce the modified tensor ~ which generates the transformation from normal variables to the reflected ones: x’~—* ~ and A’~—~ N’. In view of the property ~ = â’~the field A” remains unchanged for an even number of reflections and has to be replaced by the transformed field G. Plunien eta!., The Casimir effect 115

(eq. (3.27a)) if an odd number of reflections has taken place. The corresponding part of the total Green function G”” Ap takes the form

G~°(x,x’) — r~’~’~~ ~ (i)

= — ô~AP G(x -1’— 2alz), (3.28)

where the tilde indicates that instead of the normal Minkowski metric the modified tensor ~“~‘ now enters in the differential operator (3.24b). The total Green function describing photon propagation between parallel plates follows after subtraction of the free vacuum contribution, given by the term with /=0:

W”~’AP(x, x’) ~ 6~”~G(x— x’ — 2alz) — ~ ~ G(x - x’ - 2alz). (3.29)

The boundary conditions for the electromagnetic field require that the tangential components of the electric11= J~field12= F~and2= the0 onnormalthe plates.componentOne canof provethe magneticthat thefieldcorrespondingmust be zerocomponentsat the conductorof w””~surface;AP(x, x’) J~ vani~hi~iaccordanc~WiThTheS Orid tions. The eflfgy—momentum eIi~ö11~flOWObfãi1iëdãS - -- -

,n~”i \—~ns” I \.~a~”

‘-‘ vac~,Xj— ‘—‘ vac; event,X) ‘—‘ vac; odd X

= (_i){r~ ~ G(x - x’ — 2alz)} — (~ {~“~‘~ G(x — i’ - 2alz)} . (3.30)

Explicit evaluation shows that each term contributing to i9~,odd vanishes:

{r~”~~G(x— i’— 2alz)}f~...~,= {[~g””s9~c9’+g~Afza~aF

+ 2z”z”3~8,—2Z~Z~0V8IA— 2z”z~a”a,]G(x— i’ — 2alz)}f,,~

= — ~ (2x3 — 2al)4(—32z”z”z~z,,.z”z~

+ ~ + 16z’~z~z”z,,~)= 0. (3.31)

Accordingly, the energy—momentum tensor of the constrained electromagnetic vacuum only contains the contribution from an even number of reflections. One obtains explicitly,

@~(x)=—-~ j ~ ~ _4(x— xl_2a1zY(x_x~_2a1z)A}~ ~=_~ (x — x — 2alz) (x — x — 2alz)

= — 2 2ira 4{~g~A + z”z~}I~1l~ -~. (3.32) 116 G. P!unien et aL, The Casimir effect

Performing -the-infinite-summation, one is ledto the finairesult, eq~(262) -Thus forthe paralle[-pIate configuration one is able to derive the energy—momentum tensor of -the- constrained- vacuum—in- analytical form. This is feasible because of the simplicity of the boundary geometry, which allows one to construct the exact Green function in terms of free image functions, since no retardation effects appear.

3.2.2. Relationship between the local formulation and the method of mode summation In the previous section we have shown that the definition of the vacuum energy in terms of a difference between infinite summations of eigenmode energies referring to the constrained and the free vacuumconfiguration,respectively, ina incaLversion flndsJts~counterpart,in terms of a difference between the energy momentum tensors for corresponding field configurations. For the electromagnetic field we show now explicitly that both formal definitions of the vacuum energy are equivalent. We assume that the boundary conditions do not depend explicitly on time. Such a case is realized, e.g., for the electromagnetic field inside a static cavity of volume V enclosed by a perfectly conducting surface S. Accordingly, the photon propagator is homogeneous in time, i.e., D 4~13(x,x’) = DaJ~(X,x’, i-) where 0k=[S] of— 4.theTheconstrainedsame is alsoelectromagneticvalid for thefield,scalartheGreenboundaryfunction.valueInproblemorder towithobtainrespectthe eigenfrequenciesto the boundary conditions~ n HI 5 = 0 and ii X El5 =0 has tube ~ol-sred. Atcordirrgtu D1 ,i1iwch~nd13or~iiis[86, 871 this - problem reduces to two scalar boundary value problems for electric [E) and magnetic (H) modes, respectively, which can be treated separately:

~~E.H)(x x0) = 0. (3.33)

With respect to the boundary conditions (see eqs. (3.35b) and (3.35c)) one determines the normal modes

~ ~. H)(~ x0) = ~ ~,E.H)(x) exp(—iw~’H)[S]Xo), (3.34)

where the p ~‘ H)(x) are solutions of the scalar Helmholtz equation

(zl + o~’~~’~(x) = 0, (3.35a) and satisfy the boundary conditions:

= 0 (Dirichlet), (3.35b)

pj. V~t’’~(x)f~= 0 (von Neumann). (3.35c)

They also form a complete set of orthogonal eigenfunctions:

~ Q~ (x)p~E~~I)(x)~ 6(x— x’), (3.36a) k

J dsX 1~(~)E.1~(~)= 6kk’. (3.36b) G. Plunien eta!., The Casimireffect 117

Obviously one can define two scalar Green functions which satisfy the equation

L~G~’H)(x x’, r) = 5(x — x’). (3.37)

For brevity the indices (E), (H) and S indicating the mode type and the boundary dependence are omitted from now on. The scalar Green function can be expressed in terms of eigenfunctions, eq. (3.34), obeying the Feynman condition

* , exp(—iw~frf) G(x, x, T) = I ~, cc’k(X)cvk(X) . (3.38) k ZWk

Turning back to our purpose, we start from the equation for the energy—momentum tensor

(Of ~I”‘~(x)f0) = —2i{(~9”3’~’— kg” 9”~9,~)G(x,x’, r) + (n”n~’— ~g”~)ô(x— x’)}I~~. (3.39)

Owing to the homogeneity of the Green function and eq. (3.37) the vacuum energy density can be written as

19~)G(x,x’, - (3.40) (Of i’°°(x)JO) = —i{(~ô’9 ~ 1— ~a

Inserting the eigenfunction expansion (3.38) and performing the limit x’ —s x one obtains the expression

(Of i00(x)IO) = ~ w~(pk(x)(p~(x) — ~ —~— 8 [(ô,ço,,(x)q~~(x)]. (3.41)

The energy follows after spatial integration:

E = J d~x(OIT0°(x)IO)= — — J dI18~ok(x)co~(x). (3.42)

One sees that the first term of eq. (3.42) is already of the form which we have expected. The surface term, in fact, vanishes in view of the required boundary conditions eqs. (3.35b) and (3.35c). So what one obtains is in fact the mode sum for the zero-point energy of the electromagnetic field, separately for (E)- and (H)-type modes. For several technical questions connected with the proper identification of these modes we refer to references [84, 87]. Thus we have explicitly shown that in the case of the electromagnetic field the definition of vacuum energy introduced by the mode summation method is exactly reproduced by the local definition based on ~ i.e., one can conclude:

EvacES] = J d3x (Of t°°(x)JO) 3x (Of i’°°(x)f0)0 —5f d (3.43) = ~Wk[S]~ Wk[O]. 118 G. P!unien et aL, The Casimireffect

The fact that the definitions are equivalent must be taken with caution, because both formulations require inherently different regularization schemes. Thus, one has in general to prove whether the final results for Evac obtained within these formulations are in fact identical and independent from the particular regularization procedure applied. The fact that ~ is finite for the Casimir configuration in the electromagnetic field does not represent its general behaviour. This becomes clear when one derives the vacuum stress on arbitrary smooth boundaries or its contribution to the surface tension. In the vicinity of a plane perfect conductor, the energy—momentum tensor of the vacuum diverges with the inverse fourth power of the distance e from the boundary, i.e., 7~-(1/e4)”~4~+finiteterms. Near a slightly curved, smooth boundary the asymptotic expansion for ~ is found to be of the form [49] ø~-~ ~i (1/e’)~”’~°.The coefficients 9”~ depend upon the shape of the boundary, and they are purely local functions of its geometry. Accordingly, the part of 9~contributing to the surface tension is also in general divergent near the boundary. This divergence is not of the type which can be removed by usual subtractions as in the parallel-plate configuration. It is real in the sense that it originates in the unphysical idealization of perfect conductor boundary conditions. This gives reason for a careful analysis of how the physical vacuum energy should be defined and how it is related to both the mode sum energy E~”~(eq. (3.4)) and to the result of the local formulation E~V’~~= .f d3x OC. The mode-sum energy can be expressed as an integral over mode-number densities 37(w):

~ = 1 J dww( 375(w) — flo(w)). (3.44)

First of all both E~’~and E~~d)are in general infinite and therefore cannot directly be identified with the physical vacuum energy. The divergence of E~’~arises from the high frequency modes. According to what we have already mentioned in section 3.1, when one takes into account the transparency of real conductors, this implies the introduction of a high-frequency cutoff x(w) that makes the expression (3.44) finite. Under these circumstances E~”~should represent the physical vacuum energy and thus also the quantity, eq. (3.44), should in some sense represent the limit of this physical vacuum energy when the cutoff frequency tends to infinity. Particularly in the case of the Casimir configuration it turns out that the result for ~ is cutoff-independent and identical with

Eo0~~var This fact alone does not justify that both are said to be equal in general. Deutsch and Candelas [49] have pointed out the fact that “renormalization”, as necessary in the case of deriving ~ in the presence of boundaries, does not generally correspond to the removal of purely local divergencies in the energy—momentum tensor. This implies that performing the volume integral over 0~C to obtain the total energy does not generally lead to the same result as one would obtain by a “renormalization”, indicated in the following relation:

3x (Of T°°fO)~—J d3x (Of T001o)o] ~ = J d3x ~ (3.45) ~ = [Jd ren The proof of the equivalence of both energies as well as the decision whether ~ or E~::~~best corresponds to the physical vacuum energy even at high conductivity can only be given after explicit calculation. This implies that one has to be careful with the identification of quantities calculated by different methods. G. Plun!en et aL, The Casim!r effect 119

However, for particular configurations of perfect conductors (parallel plates, spherical shell) identical resultsfor the vacuum energyhave beenobtainedby different methods. This gives confidence in thesetech- niques and in the physical reality of the quantities obtained by them. In spite of the inherent uncertainty in identifying E~~O(as it stands) with the physical vacuum energy, it seems reasonable to take eq. (3.11) as a guide for studying vacuum energies of other quantum fields under constraints. For the electron— positron field interacting with a classical electric field, we will derive a relation analogous to eq. (3.22) that leads to an expression for the corresponding vacuum energy which has a simple physical interpretation. We now turn to explain further Green function techniques employed for calculating vacuum energies and the relation between them.

3.2.3. The local formulation in terms of the electric and magnetic Green functions At the beginning of this subsection the two-point function r””~AP(x x’) was introduced in eq. (3.14) and used to derive the relation between the energy—momentum tensor and the time-ordered product of the electromagnetic field operators. In this way an expression for f9~of the constrained electromag- netic field was obtained in terms of the difference between two-photon propagators. The energy— momentum tensor of the vacuum, as it is defined by eq. (3.22), does not represent the only possible local formulation, as we will show now. For this purpose we combine eqs. (3.14) and (3.15a),

5”(x’))f0) iG””~°(x,x’) = (Of T(F”~(x)F = ~~i’;AP. aP(Of T(Aa(X)A 13 (x’))JO) — n”ô(x0 — x/)(Of[A”(x), P~”~(x’)]f0)

+ n”~(xo— x~)(Oj[A”(x),PAP(xl)]IO) (3.46) and we write explicitly the equation for the photon propagator

(ô~E — 8”o~)(OfT(A”(x)A~’(x’))fO)= ig”~(x— x’). (3.47)

We further make use of the fact that every commutator which appears together with a temporal delta function reduces to the corresponding equal-time commutator. The (dyadic) Green functions for the physical fields, i.e. the electric and the magnetic field, can be introduced and ~ may then be expressed in terms of these Green functions. We define the electric and the magnetic Green functions according to t(x, x’) = (Of T(E”(x)E’(x’))f0), (3.48) iF” iJ~”(x,x’) = (OjT($k(x)fr(xF))fO). (3.49)

Considering the quantities G°”°”(x,x’) one recognizes that only the spatial components are non-zero, and they coincide with the electric Green function

GOk;OI(X, x’) = (Of T(PTh(x)ft01(xl))fO)

= i[~1 (x, x’). (3.50) By looking at eq. (3.46) one recognizes that .1” can be expressed as 120 G. P!unien et al., The Casimir effect 13D~ iF”(x, x’) = ir°~7’~ 13x’)—(x,i6~ô(x— x’). (3.51)

It can be proved that the Green function 1M is divergence-free

l9kFkl(x, x’) = 0, (3.52) which is a direct consequence of Gauss’ law V~E = 0 for the source-free electromagnetic field. In order

to derive the magnetic Green function we consider the purely spatial components of G”~’Ap which read

i G~’“(x, x’) = 8~0I~ (Of T(A’(x)A”(x’))f0) — 813,~~~(Of T(Ai (x)A~’(x’))fO)

+ a’a”(of T(Ai (x)A (x’))f0) — 8’a”(Of T(A1(x)Am(x’))IO). (3.53)

Contracting with the totally antisymmetrical Levi—Civita tensor we obtain the identity

~ ‘\_i ij;mn( I’t’ ~X, X ~ — 4lEijkEmnl ~X, X

= EijkEmnIl9l9(OIT(A’(X)A(X))IO). (3.54)

The magnetic Green function also has zero divergence

3kt~I3”(x,x’) = 0, (3.55) due to the Maxwell equation V~B = 0. We now have explicitly obtained the electric and magnetic contributions of the two-point Green function and, equivalently, their relations to the normal photon propagator. By comparison with the expression for the energy—momentum tensor, eq. (3.19), we find the following equations for the energy density and the stress components

(OJi’°°(x)JO)= —~i(F~(x,x)+ P~(x,x)), (3.56a)

(Of i~1k(x)fO) = ~i[F~(x, x) + ~~J’Y”(x, x) — ~8°’(~(x,x) + f’~(x,x))] - (3.56b)

1 and ~‘. The normal componentsIn the presenceof oft~~”’conductingand the tangentialsurfaces 5,componentsboundary conditionsof I” mustare requiredvanish onforS.F”In order to derive the desired energy—momentum tensor of the vacuum ø~, one has to change to the Green functions F~ and ‘~C~ where the corresponding free vacuum counter parts are subtracted, i.e., F~C= F~’— TO”’. Inserting this into eqs. (3.56a) and (3.56b) yields the components of ~ It should be noted that it is sufficient to know F”, since the quantities ‘ii”’ are related to the electric Green function according to

~o mF’~(x,x’)+ (ö”’3’3 + 8”8’)~(x— x’), (3.57) 8lO.pkl(~ x’) = Eij~Emnil9~49’ which can be directly verified. A similar relation holds for the electric Green function:

8°8’°F”(x, x’) = ~ijkEmnll9ö (P~”(x,x’)+ (~“d~d~+ 0kaI),~(x — x’). (3.58)

The structure of these equations is derived from the Maxwell equations V x B = t9°Eand V X E = — 8°B. G. Plunien eta!., The Casimir effect 121

To be complete, we note that the Green functions F” and P” satisfy the same second-order differential equations:

LJFkt(x x’) = ( 3k13i3. + ~9”9’)5(x— x’), (3.59a)

L1~”(x,x’) = — (3”8~0,+ 8”i9’)3(x — x’). (3.59b)

If one is only interested in the Casimir force, it is convenient to consider the infinitesimal change in the vacuum energy induced by a change in the parameters which determine the conductor configuration (energy-variation method): 3x3(Of T°°(x)f0), ~Evac = J d

= -~iJ d3x (~T~k(x,x) + ~~k(X, x)). (3.60)

The Casimir force per unit area can also be obtained from the spatial components 1~J~.~Cwhich describe the mechanical stress of the electromagnetic vacuum on the conducting surface (stress-tensor method). This version of the local formulation has been applied by several authors [44, 45, 88]. Although they differ slightly in their definition of the electric and magnetic Green functions, their results for the Casimir force are identical. Since F1” and ~j.~kl satisfy differential equations of the same structure as Maxwell’s equations for the electric and the magnetic field, the generalization of the Casimir concept to cases where a homo- geneously polarizable medium is placed in the vacuum is also possible. The vacuum energy can be formulated in terms of generalized Green functions iF” = (Of T(D”(x)E’(x’))JO) and i’P”(x, x’) = (Of T(H”(x)B’(x’))f0). The dielectric constant s~and the magnetic permeability ~t are determined by the geometry or the relative position of polarizable macroscopic objects. In the case of homogeneous media the infinitesimal change in the vacuum energy then is given by

~Evac = —~iJ d~x(~st~+ 6i~~), (3.61) where continuity of the normal components of ~Ji”and for the tangential components of F” is required. This follows from the behaviour of the corresponding electric and magnetic field components at the boundary between the medium and the vacuum.

3.3. The multiple-scattering expansion for Green functions and the Casimir energy

We turn now to explain the underlying idea of the multiple-scattering expansion which has been developed by Balian and Duplantier [41, 42] (see also [89—92])in order to study the behaviour of electromagnetic waves near perfect conductors. It has already been mentioned that an exact evaluation of the vacuum energy fails in general configurations, since the constrained Green functions F” and I~’ are not analytically known. The multiple-scattering expansion allows for the generation of approximate expressions for the constrained magnetic and electric Green functions by iteration of integral equations 122 G. P!unien eta!., The Casimir effect

for these Green functions [41].In principle, one could use the expressions involving F”’ and 1” derived above in the context of the stress-tensor or the energy-variation method in order to calculate the Casimir force. Balian and Duplantier were inspired by the mode-summation method and obtain the Casimir energy by deriving the eigenmode density in the presence of boundaries. Via dispersion relations the eigenmode density is related to a so-called “mode generating function” which can be expressed in terms of the electric and magnetic Green functions. Let us first discuss the concept of the multiple-scattering expansion for the Green functions. We consider the electromagnetic field in a region that is bounded by a perfectly conducting wall of arbitrary shape. In the presence of the conducting wall, free propagation of electromagnetic waves is disturbed by surface interactions. The exact Green function which describes the modified propagation could be constructed from terms each of which corresponds to a certain number of localized surface interactions. In other words, a single term in this expansion represents a process for which the wave scatters several times off the boundary and propagates freely in between (see fig. 3.4). The multiple-scattering expansion can be applied for the magnetic as well as the electric Green function. When the magnetic Green function is used, the multiple-scattering expansion has a direct physical interpretation in terms of induced currents flowing on the boundary. The exact Green function can be divided into a part describing free propagation, and the surface Green function which contains the interaction with these surface currents. The construction of the surface contributions therefore reduces to the determination of the induced currents. For these currents an integral equation can be derived which may be solved by iteration. Each term in this expansion corresponds to a certain number of scattering processes, which can be interpreted in the following way: The free magnetic Green function ~‘ represents the magnetic field produced by an oscillating dipole of unit strength Jo. In the presence of a boundary this source induces a primary surface current j~,which acts as the source of the scattered wave (single-scattering process). This in turn induces a secondary surface current which is the source of the next scattered wave (double-scattering process), and so on. A formally analogous expansion for the electric Green function is also possible. The full geometry of conducting boundaries enters in the Green function. It is suggestive to compare the multiple-scattering expansion with the semiclassical approximation of ray optics, in which the ray propagates freely between any number of reflections on the surface. These two expansions can be distinguished in the way they describe a touching of the surface. As an

çxiçX2

Fig. 3.4. Illustration of the multiple-scattering expansion: (a) single-scattering and (b) double-scattering processes. Free propagation takes place between the source point x’, the scattering points x 1 and x2, and the point x. G. P!unien et aL, The Casimir effect 123

idealization, the interactions with the boundary are described as pure mirror reflections in the framework of ray optics. The multiple-scattering expansion, in principle, represents an exact treatment of the full wave problem. The surface interaction is treated in such a way that each surface point acts as a source of an elementary wave (in analogy to Huygens’ principle) and ray optics is recovered as the high-frequency limit. In many cases the mirror reflection represents the dominant contribution in the multiple-scattering expansion, but additionally the ray may be scattered off any point on the surface in an arbitrary direction. It should be emphasized that the multiple-scattering expansion developed by Balian, Bloch and Duplantier [89—92,41, 42] is similarly applicable for deriving propagators of other constrained quantum fields. Hansson and Jaffe [93, 941 have applied this method to QCD in a cavity where they used these techniques to derive a systematic expansion for cavity propagators in order to calculate, e.g., the self-energy of quarks confined in a spherical cavity.

3.3.1. Multiple-scattering expansion of the electric and magnetic Green function We turn now to the derivation of the integral equation for magnetic and electric Green functions ~J” and F”’ given by Balian and Duplantier [41].As already mentioned above (eqs. (3.59a) and (3.59b)) the defining equations for the Fourier-transformed cti”(x, x’, w) are

2)~”= ~ (3.62a) (LI + w

= 0. (3.62b)

For reasons of brevity and transparency in the following we suppress the tensor indices and make use of the dyadic notation, e.g. Cu” ~P.The required boundary conditions for the normal and transverse components on the surface 5, are (Cu) 0 =0 and (Vx P~j,= 0 for x on S. Balian and Duplantier have identified j~’(x,x’) = ~mnri9t5(X — x’)6~’an Jo as the current density distribution at the position x of the unit magnetic dipole source with the direction indicated by the index r which is located at x’ and oriented along the direction indicated by the index 1. The magnetic Green function describes the magnetic field (the index k refers to its direction) produced by the dipole. The constrained magnetic Green function can be divided into the free and the surface part, i.e. 1i = Cu~+ Cu~,which satisfy the equations 2)~orr_Vxj 2)Cu.~O. (LI+w 0, (LI+-w The current j 0(x, x’) induces a surface current Js(xi, x’) which depends upon the surface point Xi and the position x’ (fixed parameter) of the source. The total vector potential A follows as the solution of the defining equation 2)A—(J (LI+w 0+ Js), (3.63) which is formally given by

3x” G A(x, x’) = J d 0(x, x”)j0(x”, x’)+ J do-1 G0(x, Xi) js(Xi, x’), (3.64) 124 G. P!unien eta!., The Casimir effect where G 0(x, x’) denotes the scalar Green function of eq. (3.63). The magnetic Green function t~is given by

i~P(x,x’)=VXA(x,x’)

= Cu)0(x, x’)+J do-1 Vx (G0(x, x1)j5(x1, x’)). (3.65)

The surface current can be deduced from the discontinuity of ‘I across the surface and its value on the surface [41]

j5(x1, x’) = 2n1 x cP(x1, x’). (3.66) n1 denotes the inwardly oriented ,~nitnormal vector on S at the surface point x1, and x’ denotes a point in the interior of S. Expressing P(x1, x’) according to eq. (3.65) leads to the integral equation for the surface current

Is(x1, x’) = j1(x1, x’) + 2J do-2 n1 x (V1 x (G0(x,, x2)j5(x2, x’))), (3.67)

with J ~ x’) = 2n1 x cb0(x1, x’). By iteration one generates the current up to an arbitrary order, i.e.,

J~(x1x’), = j1(x1, x’)+ 2 J do-2 n, X (V1 X (G0(x1, x2)j1(x2, x’)))

+ 4J do-2 do-3 n1 X (V1 + (G0(x1, x2)n2 x (V2 x (G0(x2, X3)75(X3, x’))))) ~ (3.68)

As discussed above, each term of this expansion has a natural interpretation. When we insert eq. (3.66) into eq. (3.65) we arrive at an integral equation for the exact Green function Cu~:

P(x, x’)= Cu(x, x’)+ 2J do-1 Vx (G0(x, x1)n1 X P(x1, x’)), (3.69) which may be iterated according to the same scheme as the current j~.In order to simplify the notation it is convenient to introduce the following integral kernels:

Kan2n1X(V1G0(x1x2)X) and M= VG0(x,x,)x.

The expression

MJ5_=J do-, VG0(x, x,)X j5(x,) involves the performance of surface integrations. In this short-hand notation the expansions for the G. Plunien eta!., The Casimir effect 125 surface current and the magnetic Green function can be written as 2j j=j1+Kj1+K 1+”~ (3.70a) and 2j u= ‘P+Mj1+MKj1+MK 1+~~~ (3.7Ob)

Note that terms which carry an odd power of K correspond to a process with an even number of scatterings. Equations (3.70a) and (3.7Ob) are valid when both points, x and x’, are located inside of S. Due to the symmetry of the magnetic Green function Cu(x, x’) =~,Cu’(x’,x) and the structure of the discontinuity across the surface, analogous expressions for Cu~ and J follow immediately for the case when x’ is outside S. Only the right-hand side of eq. (3.66) changes sign and, consequently, successive terms of the resulting expansions (3.7Oa, b) are of alternating sign. For the electricGreen function similar equations can be obtained by an analogous treatment. This is possible because F(x, x’, cv) satisfies the same differential equations (see also eq. (3.59a)): 2)F=—VX Jo (3.71a) (ii+w V~~=O. (3.71b)

The boundary conditions are now (F) 5 = 0 and (Vx F),, = 0 for x on the surface S. The resulting equations, which are valid for x and x’ both inside the bounded region, are

F(x, x’) = P0(x, x’) + J do~V x (G0(x, x1) gs(xi, x’)) (3.72) with the current

gs(Xi, x’) = — j1(x1, x’)— 2J do-2 n1 x (V1 x (G0(x1, x2) gs(x2, x’))). (3.73)

The expansion for g obtained by iterating eq. (3.73) differs from that for the current J (eq. (3.70a)) only in sign. Thus the corresponding expansion of the electric Green function is also of opposite sign, i.e., odd number scattering processes carry a negative sign: 2J i=~0-MJ1+MKJ1-MK 1+~~•. (3.74)

3.3.2. Relation between the Green function, the eigenmode density and the mode generatingfunction We now come to the relation between the Green functions and the eigenmode density of the constrained electromagnetic field. After this has been worked out, we shall turn directly to the calculation of the Casimir energy. The intermediate quantity between the Green functions and the eigenmode density is the so-called 126 G. P!unien et a!., The Casimir effect mode generating function. The meaning of this function and its construction become obvious by considering the spectral representation of the magnetic and the electric Green function. Equivalent to the knowledge of the magnetic Green function cP is that of the eigenmodes of the magnetic field inside the cavity. The eigenmodes B,,(x, t) = B~(x)exp(—iw,,t) with real amplitude B,,(x) satisfy the equations

(LI + w~)B,,= 0, (3.75a)

V~B,,=O. (3.75b)

The B~should be normalized according to

J d3xBfl(x)Bk(x)=~,,k. (3.76)

In order to derive an expression for Cu(x, x’, cv) one can make the ansatz

‘~b(x,x’, cv) = ~ B~(x)®Z~(x’,cv) (3.77) and determine the unknown vectors Z,, by insertion in the defining equation (3.62a).

~ (cvi — w~)B~(x)®Z~(x’,cv) = — V x j 0(x, x’)

= — V x (V x ô(x — x’) 1). (3.78a)

Projection with Bk(x) leads to

(w~_w2)Z1’(x1,w)Jd3xB1’(x).Vx(Vx~(x—x’)1)

3x{(B = J d 1’(x)~V)Vô(x—x’)—B1’(x)z.1t5(x—x’)}

= cv~B1’(x’)= —LI’B1’(x’), (3.78b) and one can identify the spectral representation of the magnetic Green function (Feynman boundary conditions imply a small imaginary part ie in the denominator) as

2B,,(x)ØB,,(x’). (3.79) P(x, x’, cv) = ~ cv~—w Let us now define the mode generating function !P(cv) as the trace of the magnetic Green function taken at coincident spatial points

= Tr[Cu)(x x, cv)] an J d3x cP~(x,x, cv). (3.80) G. P!unien eta!., The Casimir effect 127

Inserting eq. (3.79) and making use of the orthogonality relation (3.76) one obtains the dispersion relation

Tr[~)(cv)] = ~ cv~-w~= J dcv’ (~3(w’- cv~))w~-cv2

= J dcv’ p(w’) ~ (3.81)

Here p(cv) is the eigenmode density inside a finite cavity which is given by a sum over delta functions at

the eigenmode energies cv,,:

~ ô(cv—cv,,). (3.82)

Using the fact that p(cv) is an analytic function in the upper half-plane and shifting the pole by an amount of is, eq. (3.81) can be solved by contour integration. One finds

p(cv) = —~-- Im{Tr[Cu(cv + iE)]} = —~-- Im{~P(w+ is)}. (3.83)

This equation relates the magnetic mode density to the imaginary part of the trace of the magnetic Green function. Now the role of çli(w) as the mode generating function becomes obvious. In order to obtain the total mode density of the electromagnetic field, the electric field modes must also be taken into account. It turns out that the electric Green function F has a spectral representation of similar form as eq. (3.79) since it satisfies the same differential equations. An analogous calculation leads to the total mode generating function

~P(cv)= ~(Tr[Cu(cv)]+ Tr[cP(cv)]), (3.84) or if one inserts the multiple-scattering expansions (3.7Ob) and (3.74) one obtains

2’~’J = Tr[Cu~o]+ Tr[MK 1]. (3.85)

This explicit expression for !P reveals the quite remarkable result that, based on the particular symmetry behaviour between the electric and the magnetic Green functions in the multiple-scattering expansion, terms referring to an odd number of scatterings cancel. Only even numbers of scatterings contribute to the distribution of eigenmodes. In this context it should already be noted that this property of the multiple-scattering expansion for ~J’ has its counterpart in the perturbation expansion of the exact Feynman propagator of the electron—positron field interacting with an external electromagnetic field. There one obtains the result that, according to Furry’s theorem, only terms of even power in the electromagnetic field A~,contribute to the expansion. This corresponds exactly to the situation where a fermion scatters at an even number of space-time points (see also section 5). 128 G. P!unien eta!., The Casimir effect

One further remark to the above relations: The equality (3.84) between the mode generating function and the trace of the Green functions has to be understood in the sense that irrelevant divergencies arising from the limiting process x’ -~ x rnust be subtracted. Such divergencies are contained in the first two terms, Tr[p 0] and Tr[MK j~]of the expansion eq. (3.85) which are 2/Ix — x’f) and to 1/Ix — x’j, respectively [41]. Beside that, p also contains contributions behavingproportionalas wto2 and(cv 1 (volume and curvature contribution). The relation between t/i(w) and p(cv) via a dispersion relation assumes that the mode density satisfies the condition p(cv)—~Ofor cv—~x. To guarantee this behaviour when necessary, dispersion relation must be subtracted. This means that the equation

1 r cv’2 dcv’p(cv’) ~2 2 (3.86) ITW cv —cv 0 defines !P only up to a real additive constant which must be determined from the boundary condition at the limit cv -~ 0.

3.3.3. The Casimir energy Now we are in possession of all quantities and relations required for the discussion of Casimir energies within the multiple-scattering expansion method [42]. We limit the following representation only to the conceptional points necessary to understand the evaluation of Casimir energies. Balian and Duplantier consider a perfectly conducting shell S in the electromagnetic field enclosed in a large box ~, the walls of which are assumed to be perfectly conducting as well. Both the box and the shell can be of arbitrary smooth shape. The introduction of the space cutoff ~ is convenient in order to obtain discrete sums for the zero-point energies. The interior of the box I without the conductor S is taken as a “reference vacuum”. The Casimir energy for the considered system is formally defined as the difference in zero-point energies referring to internal, external and reference vacuum modes:

Evac = cv~[int.] + ~ cvm[ext.] — ~ cv,,[ref.]}. (3.87)

Accordingly, the first term depends only upon S while the vacuum term depends only on I and the second term contains both. Figure 3.5 illustrates the situation, e.g., for a spherical shell in a box of identical shape. The expression (3.87) can be rewritten in terms of the densities p(cv) = ~,. ~(cv— cv,,) as the following integral

Evac~Jdcvcv~(cv), (3.88) where the measure j5 describes the change of the eigenmode density in the box I when the conducting shell S is introduced

j5(cv) = p(cv; int.)+ p(cv; ext.)— p(cv; ref.). (3.89) G. P!unien eta!., The Casimir effect 129

b ext.

it-it.

—~ Q e z

Fig. 3.5. Various types of paths contributing to the Casimir energy for the conducting spherical shell S and with a space cutoff £.

Since these formal expressions are still divergent, one has to introduce a high-frequency cutoff x(w) in order to deal with a well-defined integral:

5(cv)~(cv). (3.90) E[S, I, x] = ~J dcvcvt

For calculating E[S, I, x] use can now be made of the connection between mode density and the mode generating function

~5(cv)= -~-- Im{ ~!‘(cv)} (3.91) and its multiple-scattering representation. Accordingly, the mode generating function is given by

l~(~) ~P(cv;int.)+ !P’(cv;ext.)— V’(w;ref.). (3.92)

Expressing #(cv) in terms of the multiple-scattering expansion of the magnetic and electric Green functions (eq. (3.85)) one can analyze the scattering processes which may contribute to ~ The multiple-scattering expansion for ~Pinv~lvesclosed paths with scatterings either on S or on I. From the particular symmetry relating of ~Pand F one knows that the contributions to the Green functions which are represented by paths with an odd number of scatterings cancel. Only paths of even order contribute. Figure 3.4 illustrates the various types of paths corresponding to double-scattering processes. Obviously the contribution described by the first term of eq. (3.85), Tr[Cu 0], cancels in ~P.In order to get an impression of the various higher-order scattering processes which can take place in the constrained configuration, and to see which of them may contribute to ~P,let us consider the case of double- scattering processes. 130 G. Plunien el aL, The Casimir effect

Two catagories of paths contributing to ~P(cv,ref.) can be distinguished according to the position x of the source, which may be inside S or between S and I (path (a) and (b) of fig. 3.4). These paths involve scatterings on I only. The term ~P(cv;mt.) contains paths referring to the source point x inside S and which involve both scatterings on S and I such as paths (a, c, d). Finally in the exterior region processes of type (b, e, f) can take place. Accordingly, the contributions corresponding to 1I’(cv; ref.), e.g. paths (a), (b) cancel completely with contributions to 1P(cv; int) and ~P(cv;ext.). Consequently, the possible double-scattering processes contributing to V’(cv) are represented by the paths (c, d, e, f). Higher-order scattering processes which contribute to ~P(cv)could be classified in an analogous way. We thus see that !P(cv) is represented by a multiple-scattering expansion with an even number of scatterings where at least one occurs on the interior surface S. In order to obtain the Casimir energy from eqs. ~3.90)and~3.91),Balian andDuplantier have removed~ both the high-frequency cutoff and the space cutoff. They have made rather general assumptions about the cutoff function x’ which are reasonable in the case of perfectly conducting shells. The cv integration can be performed after a rotation in the complex cv-plane and a suitable choice of integration contours. The final result for the Casimir energy, when I is pushed to infinity (under these conditions the only surviving contributions come from paths to type (c, e) in fig. 3.4.), is found to be

Evac[S] = dk [V’(ik)— ~P(i01)]. (3.93)

The second term of the integrand has its origin in the high-frequency cutoff. Balian and Duplantier [42] have also applied the multiple-scattering expansion to the finite- temperature case where they derived a general expression for the Casimir free energy. By this method they recalculated the parallel-plate and the spherical-shell result. The expansion for ~Pup to second order (double-scattering processes) turns out to be a rather good approximation, which already allows the investigation of the Casimir energy of the electromagnetic field in the presence of perfect conductors with arbitrary smooth shapes.

3.4. Phase-shift representation

Let us consider once more the vacuum energy of a quantum field ~ inside a large but finite spatial region enclosed by a surface I. For simplicity the field is chosen to satisfy periodic boundary conditions on the surface. This field configuration represents the free reference vacuum, which is distorted by the presence of additional boundaries inside I or by the interaction with localized fields like, for instance, a potential V. The corresponding vacuum energy can be formally expressed in terms of the phase shifts of the field eigenmodes. The phase-shift representation which goes back to Friedel [150] and Schwinger [97], has recently become a useful technique in the context of various physical problems [151—153].It has also been successfully applied for the evaluations of vacuum energies in the presence of external potentials [154, 155, 62]. In order to explain the main points of this method we consider the following one-dimensional problem: Inside a large but finite “box” of size ~L fX1f a quantum field ~ is interacting with a static, localized external potential V of a finite range which is assumed to be small compared with the size of the “box”. Depending on whether the external potential5t, < is~minattractiveand continuuma generalstatesspectrumwith eigenenergiesconsists of a set of discrete bound states which have eigenenergies G. Plunien eta!., The Casimir effect 131

> ~min. In the case of a massive relativistic particle field one has fEmj,,J = m. The contribution of the vacuum energy due to the change of the continuum when the external potential is introduced can be derived from the phase shifts of each of these states. For this purpose one analyzes the asymptotic behaviour of the field eigenmodes, i.e. ço~(x 1 —* ±cx).For the free vacuum configuration one has

~cos(kx1);for even ~k, (Pk(X1)~lsin(kx. 1); for odd ~‘k, (3.94a) 2) = 0, i.e., where the wave numbers k are determined by the boundary condition wk(L/ kL—2rrn. (3.94b)

Each of the field modes (3.94a) is modified by the presence of the external potential. For distances x 1 much larger than the range of the potential the asymptotic form is given by

icos(kxi + z1~[V]/2); for even çok, ~1’(x1)~ tsin(kx1+LI~[V]/2; for odd ~k, (3.95a) where the bounday condition implies for even and odd field modes respectively:

kL+ii1’[V]=2irn. (3.95b)

The phase shifts LI~and LI~are functionals of the potential V. Since the system is restricted to a large box L, which induces the conditions (3.94b) and (3.95b) where L tends to infinity the level densities are obtained as

(3.96a)

In accordance with (3.94b) the first term

= L/2rr (3.96b) is nothing but the level density for vanishing external potential. Since the corresponding eigenmode energies are related to the wave numbers, i.e. s = s(k), one can define the zero-point energy density according to

w1’[V] = s(k)371’[V]. (3.97)

With respect to the definition of the vacuum energy one is led to the spectral representation (the part of the positive energy continuum)

Evac[V] = J dks(k)(37~[V]— ?7~[O]), (3.98) 132 G. Plunien et aL, The Cajimireffect which is of a similar form as eq. (3.44) and formally consistent with the expression (3.93) given in terms of the mode generating function introduced within the treatment of Balian and Duplantier [42] concerning the electromagnetic Casimir energy. Inserting eqs. (3.96a, b) the vacuum energy takes the form

Evac[V1 J 2~e~(LI:[V]+z1~[V]),1T de (3.99) Emin which is the defining equation for the phase-shift representation of the vacuum energy. Let us finish this section with several remarks. The fact that the high-energy field modes do not contribute to the vacuum energy is ensured by the property [d(zi~[V])Ids]j~...= 0. For application of the phase-shift representation one has to calculate the phase shifts explicitly. Analytically expressions may be derived only in specific cases. In order to evaluate Evac according to this method appropriate regularization procedures must be applied, e.g., in the caseof an external Coulomb field. In thiscontext we note that the vacuum energy of the Dirac field interacting with aCoulomb potential of arbitrary strength, whichweshall derive by means of the local Green function method (section 5), has been calculated earlier [154] by the phase-shift method.

4. Casimir effect

4.1. The Casimir effect between perfectly conducting plates

As an explicit example for calculating Casimir energies let us consider the parallel conducting plate configuration in the electromagnetic field. In this case the mode summation method can be applied successfully for an exact evaluation of the Casimir energy. Attention will be drawn to the involved subtractions and the regularization of infinite quantities which are necessary in order to derive a finite result for the Casimir energy. For this purpose it is convenient to consider a large cubic cavity of volume L3 bounded by perfectly conducting walls as quantization box, in which a perfectly conducting square plate of length L is placed at an adjustable distance parallel to the x, y face. To find the Casimir energy one has to consider the difference between the zero-point energy corresponding to the situation in which the plate is placed at a small distance a from the wall and the one when the distance is large, say LI 37 ~ a with ~ > 1. The two configurations to be compared are illustrated in fig. 4.1. Formally the Casimir energy is defined as

Ec(a) = lim{(E,(a)+ E11(L— a))— (E111(L/’q)+ E1~(L—L/’q))}. (4.1)

Each single term of eq. (4.1) represents the zero-point energy as the sum of eigenmodes inside a rectangular cavity. The corresponding fIeld eigeiiuiudes are -determined-~y--rcquiring4he-~Dirichlet . boundary condition for the electric and the magnetic field, i.e. n - B = 0 and n X E = 0, on the walls. It is sufficient to derive the energy E,(a), since it is2atheareonlycharacterizedcontributionbywhichthe eigenfrequenciessurvives in the limit L—~cc. The eigenmodes inside the cavity of volume L cv = Vk~+k~ k~=(nina)2 k~ (n~in/L)2+(n~ir/L)2. (4.2) G. P!unien et aL, The Casimir effect 133 /~ / 44 ~ E~ - E~ ~

a (L-a) L/~ (L-L/~)

Fig. 4.1. Subtraction of zero-point energies in the case of the conducting parallel-plate configuration.

Since the condition a ‘~ L is satisfied, the components of k11 can be treated as continuous variables, whereas n still takes discrete values. In this limit the eigenfrequencies referring to TE and TM modes are identical. To each component k, correspond two standing waves (polarizations) and only a single one, if one2a (fig.component4.1) can isbeequalwrittento aszero. The zero-point energy of the electromagnetic field inside the cavity L

Et(a) = (L/2ir)2 J d2k 1~{~fk~f+ ~ Vk~+ (niT/a)2}. (4.3)

As it stands this expression as well as eq. (4.1) are ill-defined and require regularization. Various procedures of how to derive a finite result for the Casimir energy will be discussed in the following. In the case of conductors present in the electromagnetic field regularization can be legitimized on physical grounds. As mentioned earlier, real conductors become transparent for electromagnetic waves with sufficiently high frequency. Accordingly, the high-frequency modes will remain unaffected when the plates are present, and eigenmodes with frequencies larger than a cutoff frequency cv~(in principle determined by the properties of the real material) should cancel in eq. (4.1). For the regularization of expression (4.3) this implies the introduction of a suitable (i.e. infinitely often continuously differenti- able) cutoff function f(k/k~)which satisfies f(k/k~) 1 for k ~ k~and tends to zero sufficiently rapidly for values k ~ k~.Here, k~denotes the cutoff wave number which is assumed to be of the order of the inverse atomic size andmaybe defined according to the conditionf(1) = ~. The simplest choice is given by an exponential cutoff f(k) = exp(—k/k~). In order to calculate the Casimir energy one has to perform the energy subtraction, for instance, as indicated in fig. 4.1. Doing this it has to be taken into account that, since the distances (L — a), LI37 and

(L — LI37) are assumed to be large compared with a, the remaining discrete summation occurring in E1(a) (eq. (4.3)) also may be replaced by an integration. This leads to the expression

L 2 2 1/2 2 1/2 E(a, k~)=(i—) fd2k11{~Jki1ff(fki1f/k~)+~[k~+ (~)] f([k~+ (~)] /i~)

+ ~— ((L - a) - Li37 - (L - LI37)) f dk~kf(k/k~)}, (4.4) 134 G. P!unien et a!., The Casimir effect

2in the third term. Obviously certain where the cutoff is already introduced and k = [k~+ k~]” contributions corresponding to the larger cavities cancel. The above equation reveals explicitly the fact that one has certain freedom dividing up the quantization box and in performing the energy subtrac- tions. For example, the subtraction according to E(a) = {E 5(a) + E55((L — a)I2 — E111(L)} is completely equivalent. After performing the angular integration eq. (4.4) takes the form:

E(a, k~)= L2rr2 {~JdyVyf(—~-Vy)+ ~ J dyVyf(—~-Vy)- J dn J dyVyf(_~-Vy)~,

(4.5) 2k2Rr2 + n2. Defining the function where we introduced the dimensionless variable y = a

F(n) = J dy\/yf(__ Vy) (4.6) 2 ak~ the expression (4.5) can be written as

E(a, k~)= {~F(o)+ ~ F(n) - J dn F(n)}. (4.7) a n~1

For the further evaluation one can apply the Euler—MacLaurin formula. By making use of the assumed asymptotic behaviour of the cutoff function, eq. (4.7) takes the form

E(a, k~)= — L2ir2 ~ B t2”1~(0), (4.8) 4a ~. 2kF 1(2k)! where B2~denotes the Bernoulli numbers. Since the first two derivatives of F(n) vanish at n = 0, only higher-derivative terms contribute to the sum. For m 3 the derivatives of the function F(n) explicitly read

3)(innIak~)+ 2(m — 1)nf(m~2)(innIak~)+ n2f(innIak~)]. (4.9) F(m)(n) = —2[(m — 1)(m — 2)f~m_ Obviously only the first term does not vanish for n = 0 and the first derivative contributing in the sum (4.8) is F°’(O)= —4. Insertion of all non-vanishing terms into eq. (4.8) leads to the total expression for the Casimir energy

E(a, k~)= L2ir2 ~_ 2~4C 2(2k)! B2k(2k — 2)(2k —3) (—~-) 2~_4. (4.10)

The coefficients C2~_4are determined as the values f(2k_4)(0), particularly one has C0 = 1. G. Plunien et aL, The Casimir effect 135

The above expression for the Casimir energy reveals two interesting facts: In the cutoff-dependent result (4.10), which has been derived under rather general and reasonable assumptions for the cutoff function, the leading term (— 1/ad) turns out to be cutoff-independent. All the other terms contain powers of (rrIak~)and vanish when the cutoff is removed (k~—*cc). In this limit one obtains the well-known result for the Casimir energy 27r21 Ec(a)~,L (4.11) 720 a which gives rise to an attractive force per unit area (vacuum pressure)

21 (4.12) 240 a

A somewhat different method for calculating the Casimir effect between conducting plates consists in using the Poisson’s sum formula together with a specific choice of an exponential cutoff function. We take the exponential cutoff yielding

F(n) J dyVy ~ (4.13)

for the function F defined in (4.6) with the abbreviation ~ = (rrIak~).In order to evaluate eq. (4.7) with this specific case one can use Poisson’s sum formula, which reads

c(a) = ~- J dn e~’F(n), (4.14a)

~ F(n) = 2~ ~ c(2rrn). (4.14b)

In the present case the function F(n) is symmetric in n. Thus, these equations simplify:

c(a) = J dn cos(an)F(n), (4.14c)

~ F(n) + ~F(O)= irc(O) + 2ir ~ c(2lTn). (4.14d)

Inserting eq. (4.14d) into eq. (4.7) we observe that irc(0) is cancelled by the term —f~°dnF(n),which 136 G. P!unien et aL, The Casimir effect leads to the following expression for the Casimir energy:

Ec(a) = —i-- ~ c(2irn). (4.15) 2a ~

According to eq. (4.14c) we obtain the cutoff-dependent function

c(a, ji) = -~- J dn cos(an) J dy \/y e~”~

2 e”~= - ~ ~ (4.16) —~- ~[ = ira J dn sin(an)n 0

At this point it is permissible to take the limit /L —~ 0 which gives c(a) = (—4Iira4). Nowonecan perform the infinite summation (4.15) with the result

2 2 L22

E~(a)= — 8ir2a3 n4 = — 8ir2a3 = — 720a3~

The Poisson sum formula can also be applied in order to evaluate the temperature correction to the Casimir effect, as we shall see in section 6. Letus nowrepresent a third regularization method in orderto evaluate the Casimir energy. It is based on the analytic continuation in the number of dimensions d [77, 95]. Using the relation

2ir’~2

J d’~kf(k)= F(d/2) J dk k”1f(k), (4.18) the Casimir energy inside a d-dimensional cavity with one direction of finite length a reads

(4.19)

For evaluating the integral one can use the representation of the beta function [148]:

F(1+x)F(—y—x—i) j dttx(1+t)Y=B(1+x,_y_x_1)= F(—y) (4.20)

which formally yields

~~““2 F(—dI2) Ec(a, d) = (LI2)”~1 a’~ F(—112) ~(—d). (4.21) 6. Plunien eta!., The Casimireffect 137

The gamma function and the zeta function can be defined for values d >0 by analytic continuation. This is achieved by the reflection formulas [148]: [‘((1x)I2)in~’~2~(1x), (4.22a) F(xI2)7r_z/2~(x)= — —

F(1 — x)F(x) = inlsin(irx). (4.22b)

With these relations expression (4.21) can be rewritten and one obtains the Casimir energy also as a function of the number of dimensions d:

E~(a,d) = — ~- (4ir)~~~~2E((l+ d)I2)~(1+ d). (4.23)

Obviously, the result (4.11) is recovered for d = 3. With this we end our discussion of the various methods used to evaluate the formal definition of the Casimir energy. In particular the method of dimensional regularization, which is widely used in the field theoretic renormalization program, provides a very efficient tool forthe calculation of Casimirenergy in all cases where meaningful definition can be given to the formal expression (2.57).

4.2. The Casimir energy of a massive scalar field in a finite cavity

In this section we describe in some detail the calculation of the Casimir energy in the case of a non-interacting scalar field p of mass m inside a cavity of volume L2a assuming L ~‘ a. The field inside satisfies the free Klein—Gordon equation with Dirichlet boundary conditions on the walls. The situation considered here is analogous to the electromagnetic Casimir configuration discussed above. The corresponding eigenfrequencies of the field are given by

cvk- [k~+(~)2+m2]U2. (4.24)

Since ~ is a spin-0 field each of these eigenmodes contribute only with an amount ~cvkto the total zero-point energy. The Casimir energy of the scalar field will be calculated as a function of the length a:

E(a, m) = (LI2rn)2 J d2k 2 + m2]~2— J dn [k~+ (nrnla)2 + m2]1/2}. (4.25) 11 ~ [k~+ (nirla)

Again a certain regularization scheme must be applied. Ambjørn and Wolfram [77],who discussed this configuration within a more general framework of quantum fields in finite cavities, performed the regularization by analytical continuation in the number of space dimensions. In the following derivation we simply use an exponential cutoff and apply Poisson’s sum formula. In order to evaluate the Casimir energy we consider the cutoff-dependent expression

~ (4.26) 138 6. Plunien eta!., The Casimir effect

with the abbreviations

b(n, ~, A)= J dyVye~’~, ~ = amlir, A = irIak~. . (4~27). 2±n2 ~‘ Using Poisson’s sum formula (eqs. (4.14c) and (4.14d)) one derives the cutoff-dependent expression

E(a, ~, A) = L2ir2 2ir E c(2irn, ~, A), (4.28) where the tilde indicates that the term (L2ir2Il6a3)b(0, /L, A) has been omitted, since it is independent from the separation a. We now have to evaluate the cutoff-dependent function

it- f c(a,~,A)=—I dncos(an) I dyVye~’~ in J 0

2 d2 (4.29) ira dA

The remaining integral in the above equation can be analytically calculated [147]:

I(a,~,A)= J dn sin(an)n(/.L2+ n2)112exp[—A(j.c2+ n2)~2]

= a/L(a2 + A2)112K 2+A2)~2]. (4.30) 1[~(a

Inserting this into eq. (4.29) and performing the limit A —~0 one obtains

2/L2 c(a, /1) = — —j K 2(~aa). (4.31) ira

According to eq. (4.28) we obtain the cutoff-independent Casimir energy of the massive scalar field:

Ec(a,m)~~8ir a,,~.~-~K2(2amn). (4.32) 1n

This result coincides with that obtained by Ambjørn and Wolfram [77]using dimensional regularization. The remaining summation cannot be performed analytically. Only in the limits of small and large mass one can derive approximate expressions of the Casimir energy. In the limit m ~ a~the modified Bessel function behaves like [148]

2 — + 0(m2). (4.33) K2(2man) = 2(man) 6. P!unien et aL, The Casimir effect 139

Performing the n-summation the Casimir energy up to the order m2 turns out to be L221 L2m21

Ec(a~m)j~~+ 96 (4.34) The first term represents the Casimir energy of a massless scalar field. Its value is half of the result found in the electromagnetic case, due to the fact that the scalar field has only one polarization state. In the opposite case, when the condition ma ~ 1 is satisfied, the Casimir energy is found to be exponentially small

L2 m2 i in \112 Ec(a, m) = — —~j — e~2”~, (4.35) 16in a ma reflecting the fact that the Casimir energy vanishes in the classical limit of particles with large mass.

4.3. Quarks and gluons in a bag

Until now we have discussed the vacuum energy of quantized fields in rectangular cavities. For this particular geometry of variable external constraints it has been clearly shown that the zero-point energy must be envisaged as a contribution to the potential energy of quantum fields which represents as least in principle a measurable quantity. Particularly the investigation of Casimir energies of quantized fields confined in a spherical cavity has recently become of interest in the context of phenomenological bag models in particle physics [54—72]. The bag model is based on the belief that hadrons are built out of quarks (and gluons), and that quantum chromodynamics (QCD) provides the appropriate description of hadronic matter. QCD is given by the following Lagrangian, which is invariant under the colour group SU(3) and — in the absence of mass terms — under the hadronic flavour symmetry group SU(N~):

-~OCD = ~ [~T’k(iy~a~— mk)~P’k — ~g~PkA,,y~PkA~,] — ~ (4.36a) where Aa denote the Gell—Mann matrices, g the coupling constant and Nf the number of flavours. The index k will be suppressed later on. The non-Abelian field strength tensor is given by

F~’= — a~A~+ gf~CA~A~. (4.36b)

Although the theory is not yet completely understood, one expects that QCD contains quark and colour confinement which would be in accordance with the empirical finding that free quarks do not exist and all known particles are colour singlets. In a simplified, approximate model of hadronic structure based on QCD, confinement must be introduced by hand assuming that the fields are localized inside finite regions and requiring appropriate boundary conditions. Such a treatment represents the common aspect in several phenomenological bag models [75]. Now, even in an “empty” bag, i.e., which contains no real quarks, there will be non-zero fields due to quantum fluctuations and these give rise to a Casimir energy. The Casimir energies corresponding to the quark gluon fields may be evaluated separately, considering the two simplified 140 G. P!unien et aL, The Casimir effect problems: Quarks confined in a finite (static) bag of volume V are described by the free Lagrangian

= ~(iy~9,.— m)’I’, (4.37a) together with the linear boundary condition

(iy~’n~.— 1)~PI~= 0, (4.37b) where n~’= (0, n) is the outward normal vector to the static bag surface S. The corresponding Casimir energy is the difference between the zero-point energy due to the cavity field modes of the sea quarks and the one of the unconstrained field modes. Concerning the colour gauge fields one considers the free Lagrangian (neglecting the non-linear term 4~4c’~in the field-strength tensor): fabc~

= ~ (4.38a)

Within such an approach colour confinement is achieved by treating the exterior physical vacuum as a perfect colour magnetic conductor, i.e., the vacuum inside the bag is characterized by the colour magnetic permeability p~= 1, while ~ is infinite in the exterior. Through the vacuum relation ep~= 1, this implies that the dielectric constant e~,= 0 (and ~ ci~)outside the bag (see section 7.1). The colour electric and magnetic fields must then satisfy the following boundary conditions on the surface:

n.E’~I~=O, flXBajs=o. (4.38b)

The evaluation of the gluonic Casimir energy is completely analogous to the procedure followed for the electromagnetic Casimir effect. Note however, that the role of the colour-electric and colour-magnetic fields is just the opposite to that in normal electrodynamics (see section 7.1). The following discussion of Casimir energies in a spherical bag is intended to give an outline of the applied evaluation method without going too much into technical details. We mention right-away that present results are not completely satisfactory, since the final expressions for the Casimir energies contain cutoff-dependent parts. The remaining finite parts of the Casimir energy of quarks andgluons - are both found to be proportional to the inverse of the bag radius a, as it must be since a~ is the quantity setting the energy scale.

4.3.1. Casimir energy in a spherical bag In order to derive general expressions for Casimir energies of quantum fields inside a spherical bag, it is convenient to apply the local formulation. The steps necessary to perform the derivation become more transparent if one first considers the case of a confined massless scalar field. All method which are applied there can be generalized to the other cases of confined fermion or vector fields. Only the technical details are more involved. The energy—momentum tensor of the vacuum of the massless scalar field can be derived in an analogous way as we have shown in details for the electromagnetic field. Since one considers the scalar field in the presence of static boundaries the exact Green function is homogeneous in time. The vacuum stress tensor thus reads:

= {i(9~’3”~ — ~g 8~8”)(G(x,x’, x 0 — x~)— G0(x — x’))}l~~..~. (4.39) G. P!unien eta!., The Casimir effect 141

Both Green functions satisfy the equation

LIG(x, x’, x 0 — x~)= — 3(x — x’), LIG0(x — x’) = —5(x — x’), (4.40) where the constrained (or cavity) Green function may either fulfill Dirichlet or Neumann boundary conditions on the surface, i.e., GD(x, x~,x0 — x~)j5= 0 or n~VGN(x, x’, x0 — x~)js= 0. Analogous conditions must be required when x’ is on the boundary. It is now of advantage to separate the exact

Green function into the inhomogeneous free vacuum part G0(x — x’) and a remainder:

G(x, x’, x0 — x~)= G0(x — x’) + f~s(x,x’, x0 — x~). (4.41)

The remaining part Fs(x, x’, x0 — x~),which describes the modified propagation due to the presence of boundaries, consequently satisfies the homogeneous equation

L1Fs(x, x’, x0 — x~)= 0. (4.42)

Thus, one obtains the energy—momentum tensor (4.39) in terms of the boundary part:

E~jx)= j{~j~I~s(xx’, x0— x~)}I~’~. (4.43)

Formally the Casimir energy follows after spatial integration of the energy density over the bag volume:

3x ~~(x)=lirn i -~ J d3xP(x, x, r), (4.44) E~= J d where the limit T = — x~—~ 0 must be performed at the end. For further evaluation one now needs an appropriate eigenfunction expansion for the Green functions where the boundary conditions are easy to take into account. Concerning the spherical symmetric boundary problem it is convenient to choose the angular momentum representation of the Green function. Introducing the Fourier-transformed Green function G(x, x, w) satisfying (A + w2)G(x, x’cv) = 6(x — x’), the partial wave expansion implies:

G(x, x’, cv) = g 1(r, r’) m~i Yim(12)Yi*m(flh). (4.45)

The radial part of the Green function is determined by the radial equation 2 1(1+1) 1 (_~_~r_1 d ,.~ +cv2) g,(r, r’)=~6(r—r’), (4.46a) subject to Dirichiet or Neumann conditions (a denotes the bag radius)

g~(a,r’) = 0, -~--g’~(r,r’)Pr.~a= 0. (4.46b) 9r

The general solutions of eq. (4.46a) which satisfies the boundary conditions (4.46b) are well known [93, 142 G. Plunien et aL, The Casimir effect

55] as certain combination of spherical Bessel functions. The Dirichlet and Neumann scalar Green

function in the interior of the bag, i.e., r, r’ < a and which are regular at the origin are given by (r’ < r): GD(x, x’, w)= —ik ~ j,(kr’)h~1~(kr)~h~(ka)j(kr~)j(kr)]~ Ytm(fl)Ytm(Q’), (4.47a) 1.0 [ j 1(ka) m

1~(kr)—(i- h~1)(y)/~_ji(y))y=ka 1i(kr’)Ji(kr)] GN(x, x’, cv) = —ik ~ [j~(kr’)h~

X~ Yim(Q)Y~m(f2’), (4.4Th)

k = Icv~.The first terms, in eqs. (4.47a, b) correspond to the free particalwave propagator, while the second terms belong to the surface part .P(x, x’, cv). With these informations one is able to derive closed expressions for the Casimir energy of confined scalar fields satisfying either Dirichlet or Neumann boundary conditions. In the case of Dirichlet conditions one obtains in accordance with eq. (4.44)

cc a

Ec(a, T)= ~ (21+ 1)-~-~J ~e~’~k h~(ka)j drr2j~(kr), (4.48) 1=0 9r 2ir j 1(ka) —~ 0

where the sum formula for spherical harmonics,

2 = 2~1, (4.49) m~1 Yim(Q)1 has been used. The radial integral over- the interior of the- sphere - is -related- to the --sec-on4-Lommel- -- integral [149]:

J drrJ~(kr)= ~- [(-~-J(y))~ + (i - -A) J~(ka)] 2 dy y=ka (ak) 0

= ~ [~~a) — J~i(ka)J~+i(ka)], p> -1. (4.50)

Inserting this into eq. (4.48) leads to

Ec(a, r) = - ~(21+1) J ~ e’~i(ak)3h ~‘~(ak)[j2(ak) — j, 1(ak)j,+1(ak)]. (4.51) 2ir j1(ak)

Performing now a Wick rotation [88], G. Plunien et aL, The Casimir effect 143

cv—*icv, r—*ir, k—*iIwj, (4.52) and substituting

x=Iw~a, t~=rIa, (4.53) the Casimir energy takes the form

2 cos(ôx) K E~(a)= lim ~ (2/ + 1)1 dx x ~ { 1÷112(x)[12() — I,1,2(x)I t=o 11+112(x) 0

= lim ~ (4.54) ~ ira

The cutoff-dependent constant c(ö) abbreviates the remaining integral including the infinite summation over 1. This formula has been obtained by Bender and Hays [54]. As could have been guessed on grounds of dimensions the Casimir energy of a massless scalar field confined in a spherical bag is proportional to lIa. The same behaviour is also found in the case of confined massless quarks and gluons [54—56].The Casimir energy contributions referring to these fields differ from (4.54) only in the coefficients. While the radius dependence of the Casimir energies is well determined, the calculation of the exact coefficients is beset with difficulties. As one sees, even in the case of the scalar field the coefficient c(5) cannot be obtained analytically. In addition, it turns out [54]that c(5) still contains at least one term which diverges as 1I~in the limit ô —*0. With the same method as applied above, one can derive the Casimir energies for confined quark and gluon fields in a similar way. Concerning the gluon field the evaluation reduces to solving two scalar Green function problems [55].The scalar Green function corresponding to transverse colour-magnetic field modes (TM) is related to the Green function problem (4.47a) by: GTht(x, x’, cv) = — G’~(x,x’, cv). Concerning transverse colour-electric field modes (TE) the Green function is determined as solution of

2)GTh(x, x’, cv) = ô(x — x’), (4.55a) (A + cv with respect to the boundary condition

~~~(rGTh(x,x’, cv))J~~~= a =0. (4.55b) 8r

The angular momentum representation is given by [55] (r, r’ < a)

GTE(x, x’, cv) = ik ~ (Ji(kr’)h~’(kr)_[~_ (yh~(y))/~_(yj,(y~,)] y=ka)

Xm~:~~iYim([2)Y~m(fl’). (4.56) 144 G. Plunien et aL, The Casimir effect

Due to the intrinsic spin one of the gluon field the term with / = 0 is excluded in both Green functions. The Casimir energy again follows from the homogeneous part

3x (rFE(x x, r) + ~M(x, X, T)), (4.57) Ec(a, T) = ~ J d and becomes explicitly

Ec(a, r)=~ (21+ i)J ~e~(ak)~ ~ [~(yh 0(y))/~(yje(y))j~yka)

x (j~(ka)— j, 1(ka)j,±1(ka)). (4.58)

This formula was derived by Milton [55].After performing the Wick rotation (eqs. (4.52) and (4.53)) a more convenient expression is obtained:

Ec(a, ~) = - ~-~-— ~ (21+1)1 dx cos(5x)x {~-Sl(X)/s1(X) + ~2s,(x)/~ s1(x)

2 S~(X))}, (4.59) —2 (~-e~(x)~- s~(x)— e,(x) th where the functions s, and e, are defined as

s,(x) = (inxI2)U2I,±i, 2K,÷ 2(x), e,(x) = (2xIir)” 112(x). (4.60)

In order to obtain an approximate expression for (4.59) one can make use of uniform asymptotic expansions for the Bessel functions. Milton [55, 58] has derived the following result for the Casimir energy of confined gluons:

Ec(a, ~ = I (_4/3~2+ ~). (4.61)

Apart from a quadratically divergent term (the leading divergence [57]) the cutoff-independent contribution turns out as to be positive and thus gives rise to a repulsive vacuum pressure. Let us now continue with a short consideration on the Casimir energy of the quark field inside a spherical bag. In order to calculate it from local quantities, we make use of the relation between the symmetrical fermionic energy—momentum tensor of the vacuum and the Feynman propagator (see eq. (5.10)). The derivation of this relation will be discussed in more details in section 5 when we consider the vacuum energy of the Dirac field interacting with external electromagnetic fields. 6. P!unien et aL, The Casimir effect 145

The energy—momentum tensor of the vacuum of confined fermions is defined as

t(x,x’, r)] + (/2’~-t.~)}I~~ , (4.62) EP’~’(x)= ~{tr[y’~(ô”— a’~)F where F’ denotes the difference between the cavity propagator S(x, x’, r) and the free propagator S 0(x — x’). The cavity propagator is defined as the solution of the equation

(iy~9~— m)S(x, x’, i-) = 6(x — x’), (4.63a) with respect to the linear boundary condition:

(1 + in~y)S(x, x’, r)I~= 0. (4.63b)

If one again separates out the inhomogeneous part S0(x — x’) from the exact cavity propagator (4.63a), the boundary part F~satisfies

(iy’~9~.— m)F’(x, x’, r) = 0. (4.64)

The Casimir energy is then obtained as the spatial integral of the energy density t9°°over the bag volume: 3x tr(y°f’~(x,x, r)), (4.65) E~(r)= -~- J d taking T —*0 at the end. For massless quarks the evaluation has been carried out by several authors [54, 56] using the angular momentum representation of the fermion propagator, which can be similarly constructed as in the case of confined scalar or vector fields, although the technical details are more involved. The spherical representation of the propagator is of the form

S(x, x’, cv) = ~ S 111(r, r’, W)~jkm(I1)44l’m(fl’), (4.66) j1l’m where the ~jtm are two-component spinor spherical harmonics. For determining the radial part S11t-(r, r’, cv) one uses the relation between the fermion and the scalar propagator: iy’~9~G(x,x’, T) = S(x, x’, T). Accordingly the components of S~,t-are spherical Bessel functions [56, 93]. Explicit expres- sions for the (cutoff-dependent) Casimir energy have been derived by Bender and Hays [54] and by Milton [56]:

Ec(a, r)= ~ (2j+ 1) J ~e~(ka)32{[(~jj÷i,2(y))~ j1112(ka) j=1/2 —cc 2ir dy yka

~‘ d . . 1 1h~i,2(ka)11÷ii2(ka)—h~2112(ka)j1112(ka)~~ — I — 11—1/2(Y) 1 j1±112(ka)I X 2 2 I - (4.67) \dy / y=ka -~ L [j1+112(ka)]— [j1112(ka)] ii 146 6. P!unien et aL, The Casimir effect

Performing again a Wick rotation and introducing the notation (4.60) the expression for the Casimir energy takes the form [561

E~(a,~)= -~-- ~ (I + 1) dxx cos(ôx) ln(s~± J [-~- 1(x)+ s~(x)) ira =~ dx t=0

—2 (~-s,±1(x)e,(x)+ ~- s,(x)ei±i(x))]. (4.68)

Using Debye expansion for the Bessel functions Milton has derived the approximate result

Efa,8)=-~-(~=-th)-~ - - - - which still contains a quadratic-ally divergent term. The finite part carries the opposite sign as the one which has been found for gluons and thus leads to an attractive pressure. This short review on the Casimir energy of confined massless, non-interacting quarks and gluons clearly reveals the similarities of their evaluation and the results. Both are proportional to lIa and contain a leading quadratic divergence. For the gluons E~does also contain a logarithmic divergence [57]. Recently Milton [58]has shown that the quadratic divergence in (4.69) appears as a contact term which cancels when exterior field modes are included. The corresponding calculation conceptually is very similar to Boyer’s treatment of the spherical shell configuration [37]. In the context of confined quarks such an approach might have some physical relevance in the picture of K. Johnson and collaborators where the QCD vacuum is regarded as a foam of densely packed bubbles of perturbative vacuum. The inclusion of the exterior modes proceeds similarly calculations and eq. (4.68) changes to [58]

Ec(a, ~)= ~ (/+1) dxx cos(&) ln[(s~±1(x)+ s~(x))(e~+i(x)+ e~(x))]. (4.70) J dx 0

In the asymptotic expansion the quadratic divergence cancels and the finite part of the Casimir energy is now found to be positive

Ec(a) = 0.02041a, (4.71) which again gives rise to a repulsive pressure. In view of the results (4.61) and (4.71) it seems unlikely that the Casimir energy of quarks and gluons can explain the “zero-point” contribution of the (static) MIT bag energy E(a) = —zla (with z = 1.84) which occurs in phenomenological bag model fits to the spectrumof hadronic particles. Calculating Casimir energies Bender andHays [54]have also included finite quark masses which were recently more completely treated by Baacke and Igarashi [59].They obtained mass-dependent contributions which contain further divergencies. These terms require detailed renor- malization prescriptions which have not yet been worked out. G. P!unien eta!., The Casimireffect 147

In the context of evaluating Casimir energies of confined quantum fields in the framework of the local Green function methods it should be noted that progress has been made in the derivation of exact cavity propagators. Hanson and Jaffe [93,94] have investigated the multiple-reflection expansion for the scalar and fermionic propagator which is formally similar to techniques used in the multiple-scattering expansion for the photon propagator by Balian and Duplantier [41,42]. Applying such techniques may be useful in order to calculate higher-order surface and curvature corrections to the Casimir energy inside arbitrarily shaped bags. However, the fact that divergences inherently occur in the formalism makes the theoretical treatment of zero-point energies in finite cavities with sharp boundaries prob- lematic.

5. The Dirac vacuum in external electromagnetic fields

5.1. Vacuum energy and vacuum polarization

In the previous sections we have seen that the concept of Casimir energy allows a satisfactory physical interpretation of the zero-point energy of quantum fields. It permits to regard physical vacuum energies as the response of the vacuum to external constraints. As an idealization of the real situation, fields under certain constraints (e.g., macroscopic bodies) must satisfy appropriate boundary conditions. We have already mentioned that within a generalized concept of Casimir energy systems can also be considered where external fields created by an arbitrary source configuration play the role of external constraints. As an example for such a situation we will discuss the vacuum energy of the electron—positron field in the presence of a classical external electromagnetic field. For this purpose one can take the following physical picture as a guide: The zero-point energy of the Dirac field appears as an infinite sum over eigenenergies of the properly symmetrized Dirac Hamiltonian (eq. (2.48)). It is convenient to introduce a large quantization box together with periodic boundary conditions, say, for the upper spinor components, in order to obtain a discrete energy spectrum. The energy spectrum of the free Dirac field is visualized in fig. 5.1. It consists only of the possible free electron states and free positron states. This free vacuum configuration is modified by external electromagnetic fields. For instance, in the presence of a Coulomb field created by a positive external charge distribution, particularly that of a nucleus, additional bound states appear in the energy spectrum (fig. 5.1). Any change in the external field (e.g. displacement of the sources) produces a change of the energy levels and thus in the zero-point energy. Correspondingly, the physical vacuum energy of the electron—positron field has to be defined in this case as the difference in the zero-point energy taking the free vacuum configuration as the point of reference (eq. (2.58)). The vacuum energy can be treated as a function of suitable parameters, here abbreviated by A, which characterize the relative positions of the external sources or their geometry. For instance, in the case of several charges the relative distances between each other can be taken as the variables A (e.g., the two-center distance R for two colliding nuclei). In the case of a connected charge distribution the vacuum energy may be analyzed as a function of some conveniently chosen parameters describing the shape or spatial extension (e.g., the radius a, when considering a charged sphere). Accordingly the vacuum energy is then obtained as the difference between the zero-point energy corresponding to a source-configuration characterized by certain fixed values for the parameters A and that for such values A0, which correspond to the situation of a vanishing external field. Particularly in the cases mentioned above the free vacuum configuration is realized when either the two-center 148 G. P!unien et at, The Casimir effect

m m

0 0- -

-m ______-m ______

a) b)

Fig. 5.1. (a) Energy spectrum of the free Dirac equation-and (b> the spectrumin-the presence-of an -external CouIomb~fieId.- - - distance between the nuclei or the radius of the charged sphere tends to infinity. The Casimir energy defined in this context has to be interpreted as the response of the Dirac vacuum to “distortions” caused by the presence of an external source configuration. One could expect that the vacuum energy envisaged here may be related to corrections of the potential energy carried by an external source configuration due to vacuum polarization effects. For instance, when considering the Dirac vacuum in the presence of the static electromagnetic field A0(x, a) of a charged sphere, a static vacuum polarization p~~(x,a) will be induced. Infinitesimal variations Fia of the radius cause a change ~A0(x,a) and thus also affect the vacuum energy. It seems obvious to interpret bEvac(a) as the amount of energy necessary to extend or to contract a sphere by ba against the action of the polarizable vacuum. The total vacuum energy carried by such a configuration is obtained as the integral

I ,o Evac(a) = — da Evac(a ), (5.1) .‘ i3a a

where the derivative F(a)= —aEyac(a)It9a represents the generalized Casimir force. A further mechanical quantity, the vacuum pressure, then follows according to

Pvac(a) = — Evac(a). (5.2) 4ira 3a

In this example the vacuum energy defined above would represent the self-energy correction of a sphere due to the vacuum polarization. A further question that can be raised is whether it is possible to identify Evac as a contribution to the interaction potential between the external sources, as would be expected 6. P!unien eta!., The Casimir effect 149 when the separation between the sources is taken as the variable. Whether these physical implications of the vacuum energy can be proven to be true has to be shown by an explicit calculation of the vacuum energy. Consequently one is led to the question of which evaluation method for the vacuum energy would be most convenient concerning the problem raised here. First of all the option of applying the mode summation method will be discussed. The ingredients necessary for this method are the eigenmode energies of the constrained Dirac field (assuming static external fields) and those of the free field configuration. In general, the eigenmode energies can only be obtained by numerical solution of the Dirac equation with respect to the boundary conditions required for the spinor wavefunction on the surface enclosing the configuration. In this context one has to consider that for instance periodic boundary conditions cannot be fulfilled by the large and the small spinor components simultaneously. Nevertheless, having obtained the space-cutoff dependent eigenmode energies and thus the zero-point energies E 0[A, .~] and E0[0, .~], the next step towards the evaluation of the vacuum energy would be to apply an appropriate regularization scheme (a high-energy cutoff, for instance) in order to make the formally divergent difference of the zero-point energies finite. If this involved procedure would be carried out; the last step would be to attempt the cutoff removal. According to this method the whole calculation is predominantly numerical. Apart from very simple external field configurations the whole procedure seems to be rather cumbersome and not very practical. For this reason and stimulated by the Green function methods successfully employed to evaluate the Casimir energy in the case of the electromagnetic field, we turn to investigate the local formulation of the QED vacuum energy in terms of the energy—momentum tensor. Accordingly, for this purpose it is necessary to derive an expression for the vacuum expectationvalue of the energy—momentum tensor in the free Dirac vacuum, (01 T~j0)0,and in the Dirac vacuum interacting with the electromagnetic field AM of an external source configuration, i.e., (01 TM~cI0)A.Based on the analogous definition (eq. (3.11)), the energy—momentum tensor of the QED vacuum is formally defined by subtraction,

= (Oj T~I0)A — (01 iI0)~. (5.3)

ø~ depends on the structure of the QED vacuum, since the exact Feynman propagator appears in as we shall see. Particularly in the case of static external fields, a general expression for the vacuum energy which reveals the dominant role of vacuum polarization effects can be derived.

5.1.1. The energy—momentum tensor of the interacting QED vacuum In order to derive the energy—momentum tensor of the interacting Dirac vacuum, one first needs the relationship between (01 TM~~0)~and. the free propagator S0(x — x’) and also that between (01 i’M~dI0)Aand the propagator in the external field SA(x, x’). t9~must be of a form which ensures consistency with the requirement that quantities characterizing the vacuum must be invariant under charge conjugation (including the external source) and that ø~ must be symmetrical in ~aand v. This means that the vacuum energy of the electron—positron field interacting with an external electromagnetic field remains the same even when the role of electrons and positrons is interchanged with a simultaneous change in sign of the external sources. The properly symmetrized energy—momentum tensor of the free Dirac vacuum is given by

Ma~’P]+ [~ yMô~t1])+(term ~-*v)}. (5.4) = ~{([~ y 150 6. P!unien et at, The Casimir effect

This bilinear form contains the dynamical fields and their first derivatives. Such a combination can be obtained by means of a suitable differential operator acting on the time-ordered product of the fermion field operators

T(~P(x)!P(x’))= O(x~—x,)~i’(x)~i’(x’)—O(x~—xo)~i~(x’)~I’(x). (5.5)

The appropriate differential operator for this purpose reads (suppressing the matrix indices)

D~ = ~ + d~ç) (5.6a) with the abbreviation

d~ = ~iyM(8v — o’~). (5.6b)

When evaluating the quantities {tr D~[T(1I1(x)1~i(x’))]}one has to be careful with expressions arising from derivatives of the step functions in the time-ordered product. Accordingly, terms containing a temporal delta function appear, such as

~in”S(x 0—x,) tr[y~’{#(x), 1I’(x’)}] = 2ig°MnP6(x— x’), (5.7) where n’~denotes the time-like vector n” = (1, 0, 0, 0). The delta function on the right-hand side of eq. (5.7) has been generated by replacing the anticommutator by the corresponding equal-time commutator. Performing the Lorentz covariant limit x’ —~x, we obtain the following expression for the energy— momentum tensor of the free Dirac field: -

T~(x)= {2i(g°Mn~’+ g°vnM)~(x— x’)— 2tr[D~çT(~t’(x)~1/(x’))]}Ix~cx. (5.8)

Substituting in eq. (5.8) (Of T1I1(x)V~~(x’)l0)= iSo(x — x’), which satisfies the Dirac equation

(iYMoM — m)S0(x — x’) = 3(x — x’), (5.9) leads to the desired equation linking the vacuum expectation value of T~ and the Feynman propagator:

(01 fM~dI0)o= {2i(gOMn~’+ g°PnM)5(x— x’) + ~(tr[yM(9~’— 9’~)S0(x— x’)] + (term ~a ~‘))}IX’cX. (5.10)

By making use of eq. (5.9) the energy density can be rewritten in the form

(01 ~I’~(x)0)~= (—i) tr[y° h0(x)S0(x — x’)]I~’~, (5.lla) where

ho(x)= ~,O ~k jt~ + y°in (5.llb) denotes the free Dirac Hamiltonian. That the vacuum energy density (5.llb) is related to the G. P!unien et at, The Casimir effect 151 zero-point..,energy of the free Dirac field is simply shown by expressing again the propagator in the form

(0jT~[’(x)V’(x’)I0) and integrating over all space: E 3x (0Ii’~(x)l)o= if d~x((0J[4’~,h 0[0] = f d 0lfr]l0)+ (0j[4’~h0, 4110)). (5.12)

We now turn to derive an expression similar to eq. (5.10) for the energy—momentum tensor describing the Dirac vacuum in the presence of an electromagnetic field. Let us first consider the total system, the Dirac field coupled with an electromagnetic field, which is characterized by the Lagrangian

4-~~P]— ~e[~PyM’I’]AM — ~FM,,FM~—J~xtAM, (5.13) 2?= ~i{[4~y~9~!P]+ [V~yM,9M~fl}_~m[ yielding the coupled field equations:

(iyM9M — eyMAM — m)!1’ = 0, (5.14a)

tJt(jyM~ MA,. + in) = 0, (5.14b) 9 + e7 3VFILZ, = ~ (5.14c)

The total electromagnetic field is a superposition of fields created by the external current j~and that of the Dirac particles j~= ~e[~P,y~P],i.e., ATM = ~ + Af~.The Lagrangian (5.13) is invariant under charge conjugation tqgether with the change from j~,to (—j~J.The properly symmetrized energy— momentum tensor of the Dirac—Maxwell field reads explicitly:

T~..M)= ~i{([~ yTM3~~1I/]+ [~1’y’~’41)+ (term ~ v)} — ~(j~A’~+ j~ATM—) (j~~ TM) 1A”+ j~~1A — FILaFV + gTM~~(~J~F~+ j~xtAa). (5.15)

In the following we are only interested in describing the response of the Dirac vacuum to the electromagnetic field of the external source j~.For this reason it is sufficient to consider only the first part of the total energy—momentum tensor (5.15), i.e.,

T~)= ~i{([~ yTM~9~~~P]+ [1J~,yMt 9~’11i})+(term 1a~-*v)}. (5.16)

The index (A) indicates that the spinors 11’ and ~[‘ are determined by the Dirac equations (5.14a, b), and thus T~) depends implicitly on the electromagnetic field. Such a division of the total energy— momentum tensor is based on the following restrictions: Firstly, only a localized external current j~.,is assumed to be the source of the electromagnetic field. Secondly, if there is no charged fermion current, then consequently the total field As,. reduces to the external field A~’”(A~’is shorthand noted by ATM in the following).TM~’=In T~)this+caseTTM~the(EM;energy—momentumext), where the firsttensorterm(5.15)containsconsiststhe spinorof the parttwo anddominantthe secondcon- onetributions:dependsT exclusively on the external electromagnetic sources. A situation where this approximation holds is when a single electron moves in the localized Coulomb field of a nucleus. Thus describing the distortion of the Dirac vacuum due to the presence of electromagnetic sources, this external field 152 6. P!unien eta!., The Casimir effect

approximation is legitimate and the part T~)is the appropriate quantity for continuing with our considerations. In the next step we must derive the relation between vacuum expectation value (01 TTM~~l0)Aof the quantized expression (5.16) and the exact Feynman propagator satisfying the equation

TM3,. — eyTMA,. — m)SA(x, x’) = ô(x — x’). (5.17) (iy This is done in a similar way to the free field case, because T~)can be obtained by means of the same differentialoperatoracting upon the time-orderedproduct of the field operators. Accordingly weobtain the expression

(0ITI’~~(x)l0)A= {2i(go~n’~+ go~n~)ô(x— x’) + ~(tr[y~(3”— t9’~)SA(x,x’)] + (term jz~-* (5.18)

Apart from the occurrence of the exact propagator, (Oj T~”l0)Ahas obviously the same structure as eq. (5.10). The energy density then simply reads

(0li~0(x)I0)A= {4iö(x—_x~)+~tr[y°(c9°—1~’°)SA(x,x’)]}j~_ (5.19a

or by making use of eq. (5.17), .~ - ______

(0ji’°O(x)l0)A = (—i)tr[y°hA(x)SA(x, x’)]I~~, (5.19b)

where

hA(x)= yoyk(i0+eA)+eA+ y°m. (5.19c)

Performing the spatial integration the zero-point energy of the Dirac field in the presence of an external electromagnetic field is recovered:

E 3x ((OlE~kF,hA4110) + (OlE 4’hA, 4110). (5.20) 0[A] = f d~x(O~i’°olO)A = — ~J d We now have derived all quantities and relations in order to define the energy—momentum tensor of the Dirac vacuum. The equations (5.12) and (5.20) show that the correct local definition of the vacuum energy is achieved by means of the subtracted energy—momentum tensor

9~(x)= ~{tr[y~(r9~— ô’v)S(x — x’)] + (term 1a~-~v)}I~’~, (5.21)

where .~(x,x’) denotes the difference between the exact and the free Feynman propagator. Thus we are led to an expression for ø~ which has a similar structure as the one derived for the electromagnetic field (eq. (3.22)). The expression (5.21) reveals that all physical effects described by ø~ originate from the modified propagation of electrons and positrons in the vacuum under the action of external fields.

As it stands, ~ requires regularization, since S(x — x’) contains divergent contributions when x’ approaches x. For explicit evaluations it is convenient to deal with a differential form of eq. (5.21). The

) 6. Plunien et at, The Casimir effect 153 corresponding expression is obtained by considering infinitesimal changes of the external field A,.. Remembering that we are treating the electromagnetic field as a function of suitable parameters A characterizing the external source configuration, then, of course, any variation -in these parameters causes a change &~9~of the energy—momentum tensor. We assume that A,.(x, A) varies continuously with A. Although any change of the external source configuration modifies the propagation, we shall see that the corresponding variation ~SA(x,x’; A) is not necessarily a continuous function of A. Formally we write the infinitesimal change of the energy—momentum tensor as

~9~:~(x,A) = ~{tr[y~(3”— 0’~)&SA(x,x’; A)] + (term j~-~~ (5.22) and reobtain eq. (5.21) when integrating 69~ over A (A0 corresponds to the free field configuration) according to

@~(x,A) = J dA’ (01 ~TMv(x,A)JO)A. (5.23)

Thus we have formulated the local form of the QED vacuum energy in the context of a generalized concept of Casimir energies. It is satisfactory to note that the mathematical formulation applied successfully in the case of the electromagnetic and scalar field also carries over to spinor fields.

5.1.2. The vacuum energy of the electron—positron field in the presence of static electromagnetic fields We now turn to discuss the energy of the QED vacuum. Having defined the energy—momentum tensor (5.21), Evac[A] can now be evaluated by integration of the energy density ~C. We shall restrict ourselvesto static external electromagnetic fields created by a localized source configuration, for which an analytical expression can be derived. In the case of static external fields A,.(x) the exact Feynman propagator is homogeneous in time [96], i.e., SA(x, x’) = SA(X, x’, x0 — xi). Let us consider the infinitesimal change of the energy density (sup- pressing the A-dependence for a moment), which under such conditions reads

= tr[y°3°~SA(x,x’, x0— x~)]I~’=~. (5.24)

The right-hand side of this equation can be rewritten when considering eq. (5.17) from which the following differential equation for ~SA(x,x’) can be derived by considering infinitesimal changes in A,.:

TM 3,. — eyTM A,.(x)— m)~SA(x,x’)= eyTM6A,.(x)SA(x, x’). (5.25) (iy Making use of this equation one is led to the expression

~9gvac(x) = —i tr[y°hA(x)~SA(x,x’, x 0— x~)]l~~~— e~iA,.(x)itr[y~SA(x, x’, x0— x~)]I~’~,(5.26) where hA(x) is given by eq. (5.19c). In the second term of eq. (5.26) the vacuum polarization current, generally defined as [96] TM SA(x, ~ = ~e(OI[~ y~~]IO), (5.27) j~’~(x)= ei tr[y occurs explicitly. Together with the shorthand notation 154 6. Plunien et at, The Casirnir effect

~WN(X) = —i tr[y°hA(x)~SA(x,x’, x 0— x~)]l~~ (5.28) for the first contribution in eq. (5.26), we obtain the infinitesimal change of the vacuum energy of the electron—positron field in the presence of static external electromagnetic fields: 3x 6WN(X, A)_J d3x ~A,.(x, A)j~~(x,A) ~Evac[A1= f d = ~EN[A]+ ~EVP[A]. (5.29)

This is the basic equation for our further discussion. Stimulated by the concept of Casimir energy we have analogously derived the Dirac-vacuum energy based on the same techniques applied for evaluating vacuum energies of other quantum fields under constraints. It is very satisfying that we have obtained a result which allows a direct physical interpretation. The second term in eq. (5.29) reveals that one contribution to the vacuum energy refers to the effect of vacuum polarization. Conversely we could argue: Vacuum-polarization effects in QED can be understood as a manifestation of a change in the zero-point energy induced by the presence of external electromagnetic sources. The first term of eq. (5.29) is also of physical meaning. Its important role will become clear when we consider the electron—positron field in strong external electromagnetic fields. As we shall discuss in detail in section 5.2 this term is closely related to the effect of the phase transition from the neutral vacuum into a charged vacuum as the new stable ground state of the electron—positron field in strong electromagnetic fields. Let us now show that the first term of eq. (5.29), i.e. ~EN = f d3x &WN’ vanishes in the case of weak external fields A,.. For this purpose we consider the usual eigenfunction representation of the exact propagator:

iSA(x, x’, x 0 — x~)= O(x0 — x~)~ tfrk(x)IIJk(x ) exp[—irk(xo — x~)] k>F

— O(x~—Xo) ~ ~1’k(x’)~’k(x)exp[—iEk(xO— xi)], (5.30) k

i~SA(x,x, x 0— x~).=O(k0— x~)~ ~(Ifrk(x)4/k(x’))expHek(xO—x~)] k>F

— O(x~— X~)~ ~i(t//k(x’)t/Jk(x)) exp[—irk(xo — x~)] k

+ (terms proportional to i~ek(xo— x~)). (5.31) 6. P!unien et at, The Casimir effect 155

According to eqs. (5.28) and (5.29) only the first two terms will occur in SEN, while terms which are proportional to ~Ek(x0—x~) vanish in the limit x’—*x. Inserting the expansion (5.31) into eq. (5.28), the only remaining terms of ~WN are found to be of the form

5(ç1Jk(x)t/4(x))— ~ eko(t/4(x)t/Jk(x))}. (5.32) 6WN(X) = ~ e~t —~{k>F k

Integration over all space gives the result ~EN= 0, because each single term of eq. (5.32) is a variation of the norm integral and vanishes, i.e.,

J d3x ~(c~k(x)c&~(x))= ~J d3x t/4(x) c~k(x) 0.

Consequently the change of the vacuum energy of the electron—positron field in the presence of subcritical, static, external fields reduces to the part arising from vacuum polarization, i.e.,

~Evac(A)= ~E~~(A)= — f d3x ~A,.(x, A)j~~(x,A). (5.33)

This result coincides with that derived by J. Schwinger [97],where the vacuum energy is introduced in the same manner as here by the concept of Casimir energy. Before we turn to investigate the vacuum energy in supercritical external fields it is instructive to give a more specific explanation of the result (eq. (5.33)). The evaluation of the vacuum polarization is now reduced to the calculation of the vacuum polarization current j~,induced by the external electromagnetic field. In the case of subcritical fields this can be done by perturbation theory, which is based on the iterative solution of the integral equation for the exact propagator:

SA(x, x’)= S 4yS 0(x- x’)+ e f d 0(x— y)ft4(y)SA(y, x). (5.34)

According to the definition of the vacuum polarization current (5.27) j~can be calculated, in principle, up to arbitrary order. In this way one obtains the perturbation expansion for j~,where each contributing term carries an odd power of the external field A,.. This follows as a direct consequence from Furry’s theorem [98] which states that the series expansion for the vacuum energy contains only terms of even powers in A,.. This fact guarantees that ~Evac does not depend on the sign of the external sources in consistence with the requirement of charge conjugation invariance. First-order vacuum polarization (one-loop correction) represents the dominant contribution to the vacuum energy, and it is the only term which3x &A,.(x,needs A)j~regularization1~(x,A) in the[99].particularFor latercasepurposesof a staticwe shortlycharge distributiondiscuss the first-orderPext(X, A) astermthe~E~~(A)external=source—f d configuration. In general, the induced vacuum polarization current is related to the external current by

j~1~(x,A) = f d4yH~(x- y)j~(y), (5.35) 156 6. Plunien eta!., The Casimir effect where H~,,denotes the renormalized first-order (Uehling) polarization function. It has the Fourier representation [96]

2)= - ~ f dz z(1 - z) ln [i - p2z(1 - z)] (5.36) H~(p IT m—1~ 0 and its imaginary part reads [96]:

Im{H~(p2)}= —~a(1+ 2m2!p2)(1 — 4m2/p2)6(4m2/p2 — 1). (5.37)

For a static external charge distribution, Pex t(’, A), the change of the electrostatic potential ~iA0(x,A) is simply related to the change of the charge distribution according to 3x’ ~pext(X’,A)Ix—xl (5.38) ~A0(x,A) = J d The first-order term of ~Eyacthen takes the form

6E~(A)= — f d3x’ d3y ~pext(X’, A)pex 3x H~(x~ (5.39) t(y, A) J d — x The integral over x can be evaluated by inserting the Fourier representation of the polarization function:

J d3x = f -~-~jexp[ip . (x’ — y)]H~(—p2)J d3z exp(ip . z) (21T) zl x(lx’—yI) = , = W(lx —yI), (5.40) x where the function x(lx’ — ~I) is defined by (p = ~I)

x(Ix’ - ~I) = J dp sin(plx’ - yl)I1~(-p2). (5.41)

Inserting the expression (5.40) into eq. (5.39) one observes that the integrand is symmetrical in the variables x’ and y. This property allows one to rewrite eq. (5.39) in the following form:

~E~~(A)= — J d3x’ d3y ~pext(”, A )pext(y, A) W(jx’ — ~l]

= —~ f d3x’ d3ypext(X’, A)pex t(y, A)w(jx’ — yl)

= ~E~(A). (5.42) 6. P!unien et at, The Casimir effect 157

The function W(jx’ — ~l)represents the effective Coulomb interaction between the external charges due to the one-loop vacuum polarization correction. The corresponding Feynman diagram is shown in fig. 5.2. The above equation can now be formally integrated over the parameter A. In accordance with the definition of the vacuum energy as being the difference between zero-point energies, i.e., Evac(A) = E 0(A) — E0(A0) one directly obtains E~~(A). The physical meaning of the first-order vacuum energy depends on the role of the variable parameter A chosen to characterize the charge configuration. In the particular case that A describes the relative distance between two external charge distributions, one easily verifies that ~ is identical with the Uehling correction to the normal Coulomb interaction. Let us verify this point explicitly for the particular configuration of two point charges of opposite sign and taking the separation R = RI as the parameter:

pext(X, R) = e~(x— R). (5.43)

The vacuum energy we are interested to calculate is given by the difference E~~(R)=

E~(R)— E~”(R—~ cc). The configuration, where the charges are infinitely separated corresponds to the situation of vanishing external field, i.e., Ao(x, R —~cc) = 0. Inserting the charge distribution (5.43) into eq. (5.42) one obtains the expression

2 J d3x’ d3y ~(x’)3(y)W(lx’ — ~I) + e2 W(R). (5.44) E~(R)= —e The first divergent and R-independent term corresponds to the classical self-energy correction of the point charges due to the vacuum polarization, which cancel in the vacuum energy. As expected, the second term represents a potential energy contribution. Using the fact that the function ~(R) (eq. (5.41) can be expected in terms of the imaginary part of the polarization function (5.37) according to

2~ e’~ ~(R) = — dq Im{H~(q2)}, (5.45) IT j q 0

and due to the fact that W(R —~ cc) = 0, the vacuum energy becomes

e2 2a E~j~(R)= — — — j d~ (1 + 1/2~2)(1— 1/~2)h/2. (5.46) R3IT

This result is identical with the Uehling correction [102, 103] to the attractive potential between two point charges of opposite sign. The fact that one obtains the first radiative correction to the Coulomb

Fig. 5.2. One-loop vacuum-polarization correction contributing to the vacuum energy. 158 G. Plunien et at, The Casimir effect

interaction is not in itself surprising, but its derivation is the interesting point. In this example we have explicitly used the zero-point energy of the Dirac field in the presence of the static, external Coulomb field of two charges in order to derive the interaction correction between the sources as function of the separation. This derivation follows procedures successfully applied in derivations of van der Waals attraction between polarizable particles based on the zero-point energy of the electromagnetic field. The fact that in the case discussed above the concept of vacuum energy also leads to reasonable and physical results, gives confidence in the underlying ideas and methods, and it further supports the viewpoint that zero-point energies of quantum fields may be of general interest in the explanation of the origin of interactions. To give also an example for a vacuum energy treated as a function of a parameter which characterizes the geometry of an external source configuration, let us consider a homogeneously charged sphere of radius a. In order to evaluate the corresponding vacuum energy it. is useful to start from its derivative

3x’ d3y pext(X’, a)) pext(y, a) W(Ix’ — yI), (5.47) E~(a)= — f d (~ where the external charge distribution is given by

3Ze 1 pext(X’, a) = —~ O(a — jx’I). (5.48) 417 a

Inserting eq. (5.40) and performing the spatial integrations leads to the remaining integral:

E~(a)= - 18(Ze)2 j dPPH~(_P2)11(aP)12(ap) (5.49) IT ap ap 0

This expression can be evaluated analytically in the limits a —*0 and a —~ cc~ The first case may be applicable for calculating self-energy corrections of microscopic particles, when they are assumed to be homogeneously charged. Substituting -~ = ap and inserting the approximate expression of the -polariza- tion function in the case for am 4 1, one obtains

E~(a)= 6a~ d~ ln ~2 2)) J1(~)12(~) (5.50) ôa ~a J (~-((am) 0

Within this approximation it follows:

E~(a)= ~ a2 + 2 ln(am))c~+ 2c2], (5.51) where the constants c 1 and c2 are determined by the following integrals: 6. P!unien et a!., The Casimir effect 159

= J d~~Ji(~)J2(~) = ~, (5.52a)

C2= J d ~ ~ (5.52b)

with the Euler constant y = 0.577 ... The vacuum energy of a charged sphere is then obtained according to eq. (5.1):

2 ‘Z~ 1 E~(a)= — ~‘ “ (In (—‘~ — y + i’). (5.53) 5 ira \ ~2ami /

This vacuum energy is positive and supports the normal Coulomb repulsion which tries to expand the sphere. We can summarize so far: Within the local formulation one finds that the polarization of the Dirac vacuum is the important physical content of the vacuum energy. Evac(A) can be understood either as contribution to the interaction between electric charges or as a correction to the Coulomb energy of a given charge distribution.

5.2. The supercritical vacuum

In the previous section we have shown that the concept of Casimir energy is applicable in QED and allows for an interpretation of the zero-point energy of the Dirac vacuum in the field Hamiltonian. Defining the vacuum energy in the presence of external electromagnetic fields in terms of a difference between zero-point energies no additional formal difficulties occur. We now turn to discuss the vacuum energy of the electron—positron field in the presence of arbitrary strong, static electric fields. Let us first explain what is meant by “strong” fields. Consider a positive external charge distribution ~ A) as the source of the static field A0(x, A). Correspondingly, the energy spectrum of the stationary Dirac equation contains electron bound states at energies v~(A) between the gap, which are also functions of a certain parameter A. Assuming such an external charge configuration that the bound state energies are monotonically decreasing functions, i.e., e~(A1)>e0(A2) for

A1 < A2, we classify the external field5F =in—m,the i.e.,followingparticularlyway: (a)forAthefieldlowestis calledbound“subcritical”state r if all5F~bound(b) A valuestates lieA~aboveat whichthe Fermithe energyenergyof the lowest bound state reaches the lower energy continuum,0(A)> i.e., SF, is called “critical”. (c) When the bound state has joined the positron continuum, i.e. eo(A) < 5F, the external field is classified as “supercritical”. Figure 5.3 illustrates the energy spectrum of the Dirac equation in an attractive static potential as a function of A. In order to discretize the continuum the considered external charge configuration may be enclosed in a large but finite box. In the subcritical region the eigenenergies are obtained from solutions of the Dirac equation for localized states. Modified techniques are required in the case of supercritical fields, because the supercritical state appears only as a resonance in the positron continuum. The energy around which the resonance is localized, r~(A),is identified as the continuation of the energy eigenvalue so(A) into the region of supercritical external fields. In general, the distinction between subcritical and supercritical fields is 160 G. P!unien eta!., The Casimir effect

- --

C~(X) “

Fig. 5.3. Schematic dependence of the energy of lowest bound states on the strength parameter. closely connected with the fact, whether the deepest bound state appears in the gap between + m and —m or whether it is admixed to the lower continuum has an immediate consequence that is most easily understood in the framework of Dirac’s hole theory. As mentioned before, the distinction between particle and antiparticle states implies the definition of the Fermi energy, 5F = —m in potentials attractive to electrons. If, in supercritical external fields an unoccupied electron bound state crosses the Fermi energy, a hole is introduced into the Dirac sea. This hole can be filled without additional supply of energy by a sea electron that leaves a hole in the continuum. In terms of the hole picture this situation corresponds to the process of spontaneous electron—positron pair production. In the attractive potential the electron becomes strongly bound while the positron escapes to infinity. After this process has terminated, the source configuration will be surrounded by a strongly bound electron. The former neutral vacuum (only the bare sources and no other localized real charge) turns into the charged vacuum (with the surrounding electron cloud present) as a new stable ground state. The existence of the decay of the neutral vacuum in QED of strong external fields was predicted in the years 1970—73 by two groups at Frankfurt [104—109]and at Moscow [110—114].It has been experiment- ally studied in connection with heavy-ion collisions, where supercritical fields can be realized. We shall come back to this topic in the next section when we discuss the vacuum energy in the particular case of nuclear collisions. After these few introductory comments we now enter a discussion of the vacuum energy in supercritical (static) external fields. Let us consider static external fields produced by a charge distribution which is characterized by a parameter A. We will not specify A, but we assume that the field is attractive for electrons and that bound-state energies should vary with A as illustrated in fig. 5.3. We again start from the infinitesimal change of the vacuum energy:

~Evac(A) = ~EN(A)+ ~E~~(A), (5.54a)

6EN(A) = J d3x (—i) tr[y°hA(x)~SA(x,x’, x — x~~ (5.54b) 6. P!unien eta!., The Casimir effect 161

3x bA ~EVP(A) = — J d 0(x, A)p~~(x,A), (5.54c)

and calculate ~EN and ~ separately. We show now that ~EN(A) contributes to the vacuum energy in supercritical fields and that it is directly connected with the charge contained in the vacuum. For this purpose it is useful to expand the exact propagator in terms of eigenfunctions tfrk(X IA) [97, 115]

iSA(x, x’, x0— x0 A) = O(x0—x~)~ O(s~— ~F)t/’k(’I A)t/Ik(x’IA)exp[—iEk(xO—x/~)]

— O(x~—xo)~O(EF— rk)t//k(xIA)t/Fk(xlA)exp[—isk(xO—xo)]. (5.55)

Both summations run over the total energy spectrum of the Dirac Hamiltonian. The functions O(~k — ~F) and O(EF— ~k) guarantee the 5correctF = —m.distinctionThe representationbetween ofparticlethe exactandpropagatorantiparticle(5.55)statesis identicalwhich areto thaiseparatedof eq.by(5.30),the Fermibut it energyhas the advantage that the crossing of the Fermi level by a bound state can be considered in a simple way. This becomes obvious when we consider the infinitesimal change of the propagator and calculate the variation of the energy density ~WN, which reads:

~WN(X, A)= ~ {O(ek(A)— CF)ek(A)~pk(x,A)— O(EF— Ek(A))5k(A)~Pk(X,A)}

— ~ ~(r~(A)— E)65k(A)pk(x, A)ek(A), (5.56) where we have used the shorthand notation pk(x, A) = l//k(X I A)ç14(x I A) for the normalized single particle densities. One recognizes that the first term in (5.56) is the one already shown to vanish for subcritical fields (eq. (5.32)), whereas the second term contributes whenever a bound state reaches the Fermi level at a critical value A~,counting the number of states which dive into the Dirac sea. After spatial integration of eq. (5.56) we find simply:

— ~ 5(Ek(A)— sF)rk(A)~ek(A). (5.57)

In order to obtain the energy EN(A) contributing to Evac(A), the integration over the parameter A must be performed according to

E~(A)= — J dA’ ~ ~(ek(A’)— SF)Ek(A’)ÔA’Ek(A’)

= 5Ff dA’3A’ (~O(CF—

= ~F ~ (O(SF— sk(A))— 9(EF— ek(Ao))). (5.58) 162 6. P!unien et at, The Casimir effect

We must now recall that the k-summation (including spin degeneracy) only runs over bound states that become critical at values A~ when the parameters vary continuously between A 5F~ Consequently0 and A. Thewe findsecondthe resultstep-function term in (5.58) vanishes with respect to the condition sk(Ao)>

EN(A) = ~F ~ O(EF— Ek(A)) = EFN(A). (5.59)

This vacuum energy contribution is negative and it has the form of a step function (see fig. 5.4). Each diving bound state reduces the vacuum energy by an amount equal to the electron rest mass m. The sudden lowering of the vacuum energy reflects the spontaneous creation of an eIectron—positro~npair in a supercritical field. The vacuum energy inside a large box enclosing the system is reduced by the energy corresponding to the rest mass of the spontaneously emitted positrons, while the vacuum becomes charged due to the electron cloud surrounding the origináib~à á1ëEài~ëebffgUfäIiOñT This may be expressed in a different way by saying that a supercritical field must be treated as an open system, since it exchanges energy with its surroundings by particle emission. The change in particle number causes a sudden drop in the vacuum energy. The connection between particle exchange and vacuum charge can be made more explicit. Considering the change in the vacuum polarization charge density

~ A) = ie tr[y°aSA(x, x’, x 0— x~A)IIX.X (5.60) and performing the same manipulations that led to eq. (5.59) one finds the following expression for the change of the vacuum charge:

SQvac(A) = — e ~ 3(sk(A) — EF)~sk(A), (5.61) and after integration over A:

Qvac(i~t) = eN(A). (5.62)

The negative vacuum charge is carried by the electron cloud which is formed around the external

EN C

Xcr Acr I I A -2m-

-4m-

Fig. 5.4. Sudden decrease of the vacuum energy when electron bound states become supercritical. G. P!unien et a!., The Casimir effect 163

charges, when the electric field becomes overcritical. For reasons of charge conservation this vacuum charge is equal to the number of emitted positrons. Relation (5.62) allows one to rewrite eq. (5.59) in a form which makes the connection between vacuum charge and energy manifest:

EN(A) = ~ Qvac(A). (5.63)

Thus we have obtained a general expression for the vacuum energy contribution EN(A). We have done this without explicitly specifying the external charge distribution to underline that the phase transition to the supercritical vacuum has to be envisaged as a basic QED effect. When we now turn to evaluate the part E~~(A)arising from vacuum polarization, we will sometimes make reference to the results of theoretical investigations performed in the context of strong fields arising in heavy-ion collision. However, the basic methods which were adopted in this context should also be useful in more general cases. In the following evaluation of EVP(A) we restrict ourself to the simplest situation where only the lowest bound state dives into the positron continuum. Due to spin degeneracy then the vacuum becomes charged twice, i.e.,

3~pvp(X,A) = —2e0(A — A~). (5.64) Qvac(A) = J d Under such conditions it is legitimate to neglect the screening of the original field of the external charge configuration by the vacuum charge. One now needs an appropriate expression for the vacuum polarization charge distribution. This point has been studied intensely [106, 107] and only the main results necessary for our purpose will be reported here. Let us first consider the Feynman propagator in the external field SA(x, x’, x 0 — x~A) which can be expressed as a contour integral in the complex energy plane:

SA(X, x’, x0 — x~A) = I ~ exp[ie(xo — x~)]G(x,x’, s; A). (5.65)

The Green function G(x, x’, s; A) fulfils the equation

(hA(x, A)— s(A))G(x, x’, e; A)= 6(x—x’), (5.66a)

where hA(x, A) is again defined in eq. (5.19c). Its solution can be written in terms of a sum over the

spectrum of hA satisfying (hA — ek)’frk = 0, namely, t/fk(XIA)tI/k(xIA) (5.66b) k 55k(A)

Each singularity of the Green function (5.66b) corresponds to an energy eigenvalue of the stationary Dirac equation. The Feynman propagator is determined by the choice of the integration contour illustrated in fig. 5.5, in the case of a subcritical configuration. 164 6. P!unien eta!., The Casimir effect

ImE r ~ -mI X X m Re~

Fig. 5.5.The conventional choice of the contour detennining the Feynman propagator. The contour C crosses the real axis at the Fermi energy

= — m.

The two cuts beginning at e = ±mas well as the poles associated with the bound states between = —m and s = m are shown. The choice of the contour C plays the same role as the choice of the Fermi energy. The contour crosses the real axis at the Fermi energy 5F = —m and, thus, achieves the distinction between particle and antiparticle states. In accordance with the definition of the vacuum polarization density one is led to the representation in terms of contour integrals after Wichmann and Kroll [116]:

p~~(x,A)= £:~Ids tr[y°G(x, x, e; A)]

= —f- ds tr[y°G(x, x, e; A)] + ds tr[y°G(x, x, e; A)]}. (5.67) 4iri {J J C—

In fig. 5.6 the contours, i.e. C÷and C_, are illustrated for a subcritical situation. The contours C±and C_ enclosing the particle and antiparticle states, respectively, cross the real s-axis at the Fermi energy 5F= —m. The vacuum charge is equal to zero in the subcritical case, i.e., the vacuum polarization density (5.67) describes only local charge density fluctuations induced by the external electric field. For this reason it is called virtual vacuum polarization.

Im

-m m )( )C XXXI (1) (2) ReE

Fig. 5.6. The contour chosen by Wichmann and Kroll to define vacuum polarization in the subcritical case. 6. Plunien eta!., The Casimir effect 165

We now turn to the supercritical vacuum, where the total vacuum polarization can be divided into a virtual part and the real part, which is responsible for the non-vanishing vacuum charge of the supercritical vacuum. It is best to consider what happens in the complex s-plane when the external field becomes supercritical. When the field approaches the critical strength, the poles associated with certain bound states approach the Fermi energy. When the field exceeds the critical value, the pole (1) associated with the lowest bound state moves off the real axis into the upper half plane on the second Riemann sheet [117] (see fig. 5.7). The imaginary part of the pole energy, ~t = ~r + iF/2, reflects the fact that the former bound state has become an unstable resonance in the lower continuum. Maintaining the neutral vacuum as the reference state would imply choosing a contour that surrounds the poles in the same way as in a subcritical field. Consequently, the contour C in fig. 5.6 must be deformed to a contour C’ as illustrated in fig. 5.7. It can be shown that a choice of such a contour leads to an unstable vacuum state [106,107]. In order to define a new stable vacuum state in a supercritical field, one has to keep the contour D unchanged (see fig. 5.8). Correspondingly the Green function G(x, x, s; A) is defined such as to include only the poles remaining on the real axis. The vacuum polarization charge density is then calculated according to

p~~(x~A) = —f- J ds tr[y°G(x, x, s; A)]. (5.68) 4171 D This expression can be divided into the contributions of virtual and real vacuum polarization: Integration along contour C’ (see fig. 5.7) would represent the analytical continuation of the virtual vacuum polarization. In order to obtain the result (5.68), one has to add the contribution of the real vacuum polarization, which may be calculated by integrating along contour R (fig. 5.7). In terms of real and virtual vacuum polarization (5.68) can be written as [107]

p~~(x,A) = p~(x, A)+ p~(x,A)

= —~- de tr[y°G(x, x, s; A)] +—~- de tr[ y°G(x,x, s; A)]. (5.69) 4iri J 2iri J C~ R

Im C C’ --.~ ~

(~) ~ ~

Fig. 5.7. The contours determining virtual (C+; C_) and real (R) vacuum polarization in the supercritical case.

- 166 G. Plunien eta!., The Casimir effect

ImE

x (1) )( )C ~ (2) Re~

Fig. 5.8. The correct integration contour for the Feynman propagator in the case of supercritical external fields.

The two charge densities satisfy the conditions (including spin degeneracy)

f d3xp~(x,A)= 0, Jd3xp~(x~A)= —2e, (5.70) which explicitly states that the discontinuous behaviour in the vacuum charge originates from the real vacuum polarization which is present only in the supercritical vacuum. For the evaluation of the vacuum energy contribution E~~(A)in supercritical fields the representation (5.69) for p~is not very practical. In order to obtain a more convenient expression for it, one can make

-use of-the fact--that-the -exact resonance state approximately -behaves like a-bound--state4ll8]-.--Therefore - - it is reasonable to construct a quasi-bound state 1/Jr as an eigenstate of a slightly modified Dirac Hamiltonian (indicated by a tilde) which has an eigenvalue ~r equal to the real part of the resonance energy s~:

hA(x, A)çlir(x, A) = 5r(A)tfrr(X, A). (5.71)

Diagonalizing the positron states with respect to this Hamiltonian, one achieves that 1/er is orthogonal to all of these states. This procedure is carried out by a projection method [118] that was successfully applied in the case of the supercritical vacuum in the strong Coulomb field of superheavy nuclei. The real vacuum polarization charge density is then approximately given by

p~(x,A) = —2eI4’Jr(x, A)j2. (5.72)

It is important to note that in supercritical field the virtual vacuum polarization does not differ significantly from that in subcritical fields. It has also been shown conclusively [119—121]that no anomalous behaviour is found in higher-order contributions, and that the dominant contribution to the virtual vacuum polarization is given by the first-order Uehling term even in the supercritical case. Considering these facts, we calculate the part E~~(A)of the vacuum energy according to the relation

3AEV~(A)= — J d3x 3AAO(x, A)p~(x,A) + 20(A — A~)Jd3x t/4(x, A)e3AAO(x, A)1/Ir(X, A)

= 3AE~(A)+ 3AE~(A). (5.73) 6. Plunien eta!., The Casimir effect 167

With the help of the Hellmann—Feynman theorem [122] the part arising from the real vacuum polarization can be rewritten:

aAEVP(A) = 20(A — A~)8Ae~(A). (5.74)

After integration over A one obtains the result

E~~(A)E~(A)+ 20(A — A~)(s~(A)— EF), (5.75)

5F has been used. One observes that the real vacuum polarization lowers where the relation e/(A~)= the vacuum energy by an amount equal to the diving depth of the supercritical state. It is also obvious that E~~(A),is a continuous function of the parameter A. Adding both contributions (eqs. (5.59) and (5.75)) one obtains the total energy Evac(A) of the supercritical vacuum. If only the lowest bound state dives into the positron continuum it explicitly reads

Evac(A) = E~(A)+ 20(A — A~)s~(A). (5.76)

Thus, one is led to the following conclusion: The vacuum energy of the electron—positron field in the presence of a static external Coulomb field created by a given charge configuration, in general, consists of two parts which corresponds to the effect of vacuum polarization and the effect of the phase transition of the neutral vacuum into the charged vacuum as the new stable state in supercritical external fields. In the subcritical case the vacuum energy coincides with the interaction energy of the virtual vacuum polarization with the external field. In the supercritical case the vacuum energy is abruptly lowered by the resonance energies of supercritical states. Expression (5.76) can be easily generalized to the case when several bound states may become supercritical. Then, of course, the screening effect due to the real vacuum charge surrounding the original charge configuration must be taken into account when E~(A)and the eigenvalues s~(A)of supercritical states are calculated. Summarizing this section, we have shown that the zero-point energy of the electron—positron field in the presence of external electric fields may be utilized to calculate vacuum polarization and the effect of the phase transition from the neutral to the charged vacuum. This represents one more example for a successful application of the concept of a vacuum energy.

5.3. The vacuum energy in nuclear scattering

In order to make our general considerations concerning the vacuum energy of the electron—positron field more specific, we now turn to discuss the role of the vacuum energy in nuclear scattering. This problem is of considerable interest, because in heavy-nuclear collisions strong electromagnetic fields can be created which, in fact, permit experimental tests of the predicted phase transition of the supercritical vacuum to be carried out. The evaluation of the vacuum energy in the presence of the electromagnetic field of scattering nuclei, treated as a time-dependent problem, is no simple problem. In a semi-classical approach, where the nuclear motion is treated classically, one would parametrize the external field by the trajectories of the scattering nuclei. Instead of a full dynamical treatment we shall represent the nuclei by static charge distributions and take their distance R as the variable parameter. The geometry of the external charge distribution is illustrated in fig. 5.9. The two nuclei are represented by two homogeneously charged 168 6. P!unien et at. The Casimir effect

Fig. 5.9. Determination of the model charge distribution for two scattering nuclei. spheres with nuclear charges Z, and with fixed radii a. The latter are determined by the mass numbers 3.We take the charge distribution A, of the nuclei according to the empirical formula a, = 1.2 A~’

Pexs(X, R) = ~ —~-~ O(a, — Ix — R 4ira- 1I). (5.77) ~=~ Treating the vacuum energy as a function of the nuclear distance R = IRI = IR 1 — R2I, it is defined as the difference between the zero-point energies: Evac(R) = E0(R) — E0(R —* cc). The case of infinitely separated nuclei corresponds to the free vacuum configuration, since A0(x, R —* cc) = 0 in a finite box located at the center- between the two nuclei. For scattering nuclei the two-center distance can take values R > a1 + a2. In the previous section it has been shown that the vacuum energy (eq. (5.76)) consists of the contribution arising from the virtual vacuum polarization, which is suddenly lowered by the energy of the supercritical state when the external field becomes supercritical. Both effects can be calculated separately. The dominant contribution of E~j(R)is given by the first-order vacuum polarization, and one expects that it represents the Uehling potential for extended nuclei. This is easily verified by a simple calculation starting from eq. (5.42). Inserting the charge distribution (5.77) one obtains after obvious substitutions of the integration variables:

3z’ d3z p E~(R)= -~ J d 1(Iz’I)p,(IzI) W(Iz’ - zI) 3z’ d3zp - J d 1(z’)p2(Jz — RI) W(Iz’ — zI). (5.78)

The first two terms in this equation correspond to the Coulomb energy correction of the individual nuclei due to the vacuum polarization. This part cancels by the subtraction, since it is independent of the separation R. The interaction term can be expressed as a Fourier integral over momenta and vanishes in the limit R —* cc~Consequently one obtains the Uehling correction 6. Plunien eta!., The Casimir effect 169

2 J dp sin(PR) [J(l)(~2)J1(”1P) j EUh(R) ~!Z1Z2e 1(a2p) (5.79) ir R p a1p a2p 0

The spherical Bessel-functions j~arise from the Fourier transformed nuclear charge densities. With similar arguments as used in the derivation of the Uehling potential in the case of point charges Euh(R) can be also expressed in terms of the imaginary part of the polarization function

2 ~ J d~e (1+ 1/2~2)(1— 1/~2)h/2i EUh(R) = Z1Ze 1(2ma1~)i1(2ma2~) (5.80) R ir (2ma1~) (2ma2~) where i1 are modified spherical Bessel functions. The integral is well defined for values R > a1 + a2. Thus we have derived the (repulsive) Uehling potential between extended nuclei characterized by a homogeneous charge distribution. The result concerning point-like nuclei is regained performing the limit a1, a2-~0. In figs. 5.10 and 5.11 the Uehling potential is shown for two combinations of scattering nuclei, namely, Pb + Cm and U + U. The behaviour of these curves is dominated by the factor

E [MeV] EUH [MeV] ((H u+u \ Pb+ Cm (extended nuclei) (extended reictei) 2.0- 2.0-

1.5— 1.5-

1.0— 1.0-

0.5— 0.5-

REfm] R[fm] I I i 0 I I I i a 20 30 40 50 20 30 40 50 Fig. 5.10. The Uehling potential for the scattering extended nuclei Fig. 5.11. The same as in fig. 5.10 for U+U. Pb + Cm as a function of the separation. 170 6. P!unien eta!., The Casimir effect

(Z 2/R). The first-order radiation correction to the interaction potential between scattering nuclei is small1Z2comparede with the bare Coulomb interaction and becomes only of the order of 1 MeV in the case of heavy systems like U + U. The external Coulomb potential created by two such nuclei becomes supercritical for nuclear distances R smaller than a certain critical value R~ 1.Then, of course, the change in the charge of the vacuum must be taken into account. In order to determine for which combination of scattering nuclei the Coulomb field becomes supercritical, one considers the following problem [106]:Assume a bare “super nucleus” associated with a homogeneously charged sphere with total charge Z. = Z1 + Z2 and with a radius a = 1.2(A1 + A2)Us. Solving the Dirac equation with the corresponding Coulomb potential it turns out that the energy of the is state decreases monotonically as a function of Z. and joins the positron continuum at a total critical charge Zcr 173. According to this value we classify the subcritical (Z < Zcr) and the supercritical systems (Z> Zcr) of scattering nuclei. Concerning the scattering problem it only depends now on the two-center distance R whether the Coulomb potential remains subcritical or whether it becomes supercritical. If only the lowest molecular bound state is joins the positron continuum, the vacuum energy in accordance with eq. (5.76) reads

Evac(R) Euh(R) + 20(Rcr — R)s1~(R). (5.81)

In order to calculate the energy of the molecular is state as a function of R, one has to solve the stationary two-center Dirac equation. The energy of the supercritical is state is determined by the resonance energy of the supercritical state [106].Quantitative results for the vacuum energy are shown in figs. 5.12 and 5.13 for the supercritical systems Pb + Cm (Z11 = 178) and U + U(Z~= 184), respectively. In this adiabatic treatment the electric potential becomes critical at certain distances Rcr, which are found to be: Rcr = 23.3 fm for the system Pb + Cm and Rcr = 32.8 fm for the system U + U. The Uehling potential becomes suddenly lowered by the amount of twice the energyof the supercritical state (spin degeneracy)for separations R

E [MeV] ~ 1MeV] 10 — u+u Pb~Cm . (extendedrsiclei)

____ I I

R[fml 20\ N.~30~ 40 50

Q I Iii I I I I ~ -0.2 ~ I 20 30 40 - 50 Fig. 5.12. The vacuum energy for the scattering extended nuclei Fig. 5.13. The same as in fig. 5.12 for U + U. Pb + Cm as a function of the separation. G. Plunien et at, The Casimir effect 171 vacuum at critical distances. If the sudden drop in Evac(R) has been derived from an adiabatic model of a scattering process this jump would never appear in such a pronounced way, because the phase transition of the supercritical vacuum is not an instantaneous process. The time-scale in the process of spontaneous electron—positron pair production (the observable effect of the phase transition) is of the order -=i019s, while in Rutherford scattering the supercritical field may be present for only 1021s [118].Nevertheless, this adiabatic picture of nuclear scattering allows a simple interpretation of the vacuum energy. As already mentioned Evac(R) plays the role of a correction to the static Coulomb interaction potential. Two possible situations can be distinguished: In the case of subcritical systems, i.e., if no phase transition from the neutral to the charged vacuum can take place during the nuclear scattering process, the vacuum energy essentially represents the Uehling correction to the static interaction potential V(R) between the nuclei. Figure 5.14a illustrates the scattering potential for such a situation as a function of the nuclear distance R. The two nuclei feel the same interaction V(R) in both directions, i.e., when they approach each other and when they separate again. The situation is different, if there is a change to the charged vacuum while the nuclei come close together. Then the vacuum energy and, consequently, the interaction potential is abruptly lowered. When the distance R again increases, the two nuclei feel a potential which is lowered due to the binding effects of the two electrons in the supercritical is state (see fig. 5.i4b). In a plot of V versus R the system then follows a different potential curve. Such a change in the interaction potential, caused by the vacuum energy, may be envisaged as a kind of hysteresis effect in nuclear scattering. Let us shortly summarize: The aim of this section was to show that the concept of Casimir energy allows one to define a physical vacuum energy of the electron—positron field as a difference between zero-point energies. Applying arguments used in connection with the Casimir energy of the elec- tromagnetic field we showed that the vacuum energy of the electron—positron field is equivalent to the interaction with the vacuum polarization, and that the phase transition of the QED vacuum is reflected in a decrease of the vacuum energy in the presence of supercritical external electromagnetic fields.

E,,~ I Evac

\\\\\

R~RCrR (a) (b)

Fig. 5.14. Schematic diagram for the vacuum energy versus the separation of scattering nuclei: (a) for a subcritical field and (b) for a supercntical field. 172 6. Plunien eta!., The Casimir effect

6. Casimir energy at finite temperature

6.1. Partitionfunctions and free energy

In the previous sections we have shown how one can define the energy of the vacuum state of quantum fields based on the corresponding field Hamiltonian obtained by canonical quantization and we have seen how to deal with infinite zero-point energies in a meaningful way. The evaluation methods for Casimir energies discussed above allow one to extract their physical implications and observable consequences. So far we have considered the response of the vacuum due to the presence of external constraints. In reality fields exist at finite temperature and contain real quanta, i.e., a large number of states are occupied. Such conditions require a quantum statistical description. In the following the assumption is made that the considered quantum fields are to be described as systems in thermodynamical equilibrium at finite temperature T = 1//3 (Boltzmann constant kB = 1), instead of being in the vacuum state. From statistical mechanics one knows that an ensemble is characterized by the statistical operator j5. Let us consider the canonical ensemble, where the functional equation 13(H1 + H2) = ~5(H1)~(H2is) valid as a consequence of weak coupling and statistical independence of subsystems 1 and 2. This equation has the well-known solution

1~= ~exp(—/3$l, (6.1) where the normalization factor ~ represents the canonical partition function of the system, which is defined as trace in Fock space:

= tr(exp(—/3I~)). (6.2)

All relevant physical quantities describing the system in equilibrium can be derived from the partition function, e.g. the free energy

F = — T ln(2~), (6.3a) and the pressure (V denotes the volume)

P = — 3F/S9V. (6.3b)

The free electromagnetic field inside a large resonant cavity can be treated as a canonical ensemble determined by the statistical operator (6.1) and the partition function (6.2). The same description is justified in the case of a Klein—Gordon field, when the particle number is not conserved, i.e., for vanishing chemical potential /2. The situation is different in the case of the electron—positron field in the presence of strong external electromagnetic fields. As we have seen above, the supercritical electron—positron vacuum has to be treated as an open system which can exchange energy with its surroundings by particle emission. Accordingly, for a correct description of the supercritical vacuum the grand partition function must be 6. P!unien eta!., The Casirnir effect 173 introduced, i.e.,

= tr(exp[—/3(I~— ,aN)]), (6.4)

5F~ The field Hamil- where the Fermi energy SF = —m plays the role of the chemical potential: /2 = tonian which enters in the above equations contains the zero-point energy of the_particular quantum field, i.e., H = :H: + E 0. Consequently the partition functions factorizes as = ~ where ~ contains the zero-point energy and ~ represents the part which refers to real occupied states forming the present ensemble. This means that quantum fluctuations (or zero-point oscillations) and the real state of the field are statistically independent in the external field approximation. In accordance with eq. (6.3a) the free energy becomes a sum of the zero-point energy and the free energy of the quantum system referring to real occupied states:

F=E0+F=E0—Tln(~t). (6.5)

This relation shows that a correct generalization of the concept of Casimir energy at finite temperature is achieved by considering the free energy of a quantum field and defining a so-called Casimir free energy as the difference between the free energy of the field in the presence of external constraints and the one of the free field, i.e.,

F~[aF] = FEaT’] — F[0]. (6.6)

As in the case of the previously defined vacuum energy (eq. (2.56)) such a definition is, of course, only meaningful together with an appropriate regularization scheme leading to a finite result. The Casimir free energy reduces to the vacuum energy Evac[3F] when the temperature tends to zero. In order to calculate the thermal correction to the Casimir energy in principle one can apply both mode-summation method and the local fonnulation using finite-temperature Green functions. The direct mode summation is possible because the free energy of a considered field can be expressed in terms of field eigenmode frequencies or energy eigenvalues.3 (periodicConsidering,boundaryfor instance,conditionstheon freethe walls)electromagneticexplicit expressionsfield insideforathelarge,partitioncubicfunctioncavity ofandvolumethe freeL energy are easily derived. The complete Fock space of normalized multiple photon states IX) with n~photons of each momentum k and polarization A can be constructed according to

IX) = [I (n~!Yu2(c~~+Y2~i0), (6.7a)

where the occupation numbers can take values n~= 0, 1, 2 These states satisfy the eigenvalue equation.

n~IX)=n~JX). (6.7b)

Since the Hamiltonian of the photon field (eq. (2.37)) is also diagonal in this representation, the partition function becomes 174 6. P!unien eta!., The Casimir effect

= fl (~exp[—f3w~(n + 1/2)]) = H exp(—/3wk/2)[1— exp(—/3wk)]~, (6.8) and the free energy is given by

F = ~ {wk12+ T ln[i — exp(—wk/T)]}. (6.9)

The evaluation of the Casimir free energy according to the mode summation method implies a subtraction of infinite sums, and we have already seen that the first term of eq. (6.9) requires regularization. Similar regularization procedures are not necessary for the temperature-dependent part, which can be proved directly. In the case of the electromagnetic field inside a large empty cube (this represents the free field configuration) the contribution FLU] of the free energy is finite. Replacing the summation in eq. (6.9) by an integration over all momenta k, one obtains the free energy contribution:

3k ln[i — exp(—wk/T)] = T4 Jdxx2ln(i - e~) P[0] = 2T (~)3 J d (~)

= —2~-~(4)T~=—~IT2T4L3, (6.iOa) which gives rise to the pressure

P[0]= —ôF[0I/8V=~IT2T4, (6.iob) representing the Stefan—Boltzmann law. The free thermal radiation changes when external boundary conditions (e.g. perfect conductors) are introduced. Thus, it seems obvious to identify the thermal correction to the Casimir energy in terms of a derivation from free thermal radiation due to external constraints. As we shall see, in the case of the conducting parallel-plate configuration it turns out that in the high-temperature limit the effect due to the zero-point oscillation of the electromagnetic field becomes completely compensated by the thermal radiation.

6.2. Finite temperature propagators

We have seen that the thermal correction to the Casimir energy can be calculated from the free energy F of the constrained quantum field by the direct mode-summation method, because F can be expressed as a sum over the corresponding field eigenmode energies. Beside the direct evaluation of the Casimir free energy by means of this method, one has in principle the option to generalize the local version of Casimir energies in terms of the energy—momentum tensor to the finite-temperature case. The basic change in the formalism consists in that instead of considering the vacuum expectation values of relevant operators one now has to deal with their thermodynamical expectation values. Let us now point out the conceptual line of this treatment. The thermodynamical expectation value of an operator A referring to the quantum field, which is G. P!unien et at, The Casimireffect 175 characterized by the density operator ,5, is defined as trace in Fock space:

(A) = tr( 15A). - (6.11)

In accordance with such an ensemble averaging one defines the energy—momentum tensor for a field at finite temperature in the presence of external boundaries as the difference

ø~(x;/3) = (t~”’(x))8~— (i’~(x))~. (6.12)

The energy—momentum tensor of a quantum field consists of bilinear forms in the field operators. Such forms may be derived from the time-ordered product of the field operators by means of a suitable differential operator, as we have shown in detail before. Considering fields where such a construction is feasible, one is thus led to the representation of &~“(x,/3) in terms of finite temperature propagators. In principle, it is of the form

9(x, /3) = i{~ tXi)(~(~)(J)(x,x’)— ~(~)(J)(X — x’))}j~..,~~, (6.13) where the latin indices count the dynamical field and stand, e.g., either for Lorentz or spinor indices. Let us now consider the thermal Green function of a free massive scalar field. We note that the Green function obtained in the limit m -+0 is directly related to the thermal photon propagator, which represents, of course, the basic quantity in order to evaluate the temperature correction to the electromagnetic Casimir energy. Assuming a canonical ensemble of bosons at ~ = 0, but keeping the mass arbitrary, the thermal Green function is defined as the average

iG(x - x’; /3) = ~tr[exp(-/3~)T(~(x)~(x’))] = (T(~(x)~(x’))). (6.14)

For calculating the explicit expression one may use the discrete plane wave expansion (eq. (2.8)) for the operators ç~(x)and ç~(x’).In terms of the creation and annihilation operators â~and ~k one obtains:

~(x) = ~ —~ ,,.iL.. [exp(—iwkxo)exp(ik . x)âk(O)+ exp(iwkxo) exp(—ik . x)âi~ik(O)], (6.15) k vQv2wk where the time evolution ~~(X0) = exp(—iwkxo)âk(O) has already been substituted. The operators 8â~,kk’, ~kit followssatisfy thedirectly:Bose commutation relations, and considering the ensemble average ([ak, a~~])=

(ak’ak) l5kk.n(wk), ~ = 8kk’(i + n(wk)). (6.16)

The function n(wk) represents the average occupation number which is determined for Bose statistics as

n(wk) = (exp(13wk) — i)~. (6.17)

In accordance with the commutation relations [âk, ak’] = [a, a~.]= 0 all other averages vanish, i.e.,

(~k~k.) = (a~á.)= 0. (6.18) 176 G. P!unien et at, The Casimir effect

With these information, for instance, the average (~(x)c~(x’))can be calculated:

(çb(x)~(x’))= ~ ~ {exp[ik . (x — x’)] exp[—iwk(xo — x~)](âk(0)â~(0))

+ exp[—ik(x— x’)] exp[iwk(xo— x~)}(ái’ik(0)âk(0))}. (6.19)

Substituting eq. (6.17) and performing the continuum limit, eq. (6.19) takes the form

rd3k i (~(x)c~(x’))= J —i — {exp[—iwk(xo — x~)]exp[ik . (x — x’)](i + fl(Wk)) (2IT) 2wk

+ exp[iwk(xO — xi)] exp[—ik . (x — X’)]fl(Wk)}. (6.20)

The thermal Feynman propagator then follows after incorporating the time ordering (k 0 = = co_k):

2 - rn2 + ~ + 21ri[exp(/3wk) - i]15(k2 — m2)}. (6.21) (T(ç~(x)ç~(x’)))= J ~ exp[—ik(x — x’)] { k Thermal Green functions for other fields can be derived in a similar way. For fermions the thermal- Feynman propagator at finite chemical potential /2 is found to be

d4k ( ~‘k +m) S(x - x’; ~3)= J~— 2 + ~ + 2ITi6(k2 - m2)(y~’k~.+ m) 3exp[—ik(x-~ — x’)] {k~ rn

< F (9(k 0) + ø(—k0) iT (6.22) Lexp[/3(wk — /2)] + 1 exp[/3(wk + /2)] + iii

In an alternative formulation the propagator (6.21) can be represented by the series [35, 123]

(T(ç~(x)ç&(x’))) i ~ G(x — — if3pn), (6.23) where n denotes the time-like unit vector n’~= (1, 0, 0, 0). In this representation the thermal propagator appears as an infinite sum over free propagators corresponding to image sources displaced in imaginary time at i$v. The term with p = 0 is the free vacuum propagator G0(x — x’). A representation of the Green function of the form (6.23) is also valid for the thermal photon propagator (m = 0) and is thus useful in order to derive the thermal correction to the Casimir effect between conducting parallel plates. The scalar photon propagator reads

2. (6.24) G(x - x’; /3) = (- -~-~)~ [x - x’ - i/3pn] G. P!unien et at, The Casimir effect 177

Based on -this- explicit expression Brown and Maclay [35] have derived the thermal energy—momentum tensor. For the parallel-plate configuration all that is left to do is to consider the image sources displaced in space. According to eq. (3.23) the energy—momentum tensor of the electromagnetic field between the conducting plates at finite temperature follows as

(i~(a))= - -~ ~ k~ ~ + (2a1)zz~-(k~in~n~} (6.25)

In order to obtain @~“‘(a,/3) the free vacuum contribution at zero temperature must be subtracted, i.e. the term referring to 1 = k = 0. (T~(a))does also contain the free black-body radiation which appears explicitly in (6.25) when considering the limit of very large separation. This contribution corresponds to the terms with I = 0 and k 0. It explicitly reads

1~1T2T4(g’~”’— 4n~’n~), (6.26) -+ ca)) = —4 and gives the well-known results [124] for the energy density and pressure of a photon gas inside an infinite volume: -

(i’°°(a-+ cc) = 12T4, (6.27a)

(TI’~k(a-+ca))=18ik(j~~00(a_~cc)) (6.27b)

Excluding these terms as well as the zero-temperature contribution of the vacuum stress, i.e., I 0 and k = 0, one obtains the thermal correction to the energy—momentum tensor at finite temperature and plate separation:

ø~(a,/3) = — U(a, /3)(g~+ 4z~z~)+ g(a, $)(z~z~+ n~n~)}, (6.28)

where the functions f and g are defined as

f(a, /3) = ~ [(2a1)2+ (/3k)2]2, g(a, /3) = ~ ~ ~. (6.29) I,k=1 a 3a

Approximate expressions can be derived in the low- and high-temperature limit. The result for the energy density and pressure coincides with that obtained by Mehra [81] who applied the mode summation method (see the next section). We conclude this section with the remark that an alternative derivation of temperature-dependent Green functions may be based on an analytical continuation of the time evolution of Heisenberg operators to imaginary times. This implies that the operator exp(—/3H) is interpreted as the evolution operator to imaginary times i/3 (restricting to the interval [0,i$]). For details concerning this for- mulation of thermal Green function we refer to the literature (e.g. [125, 126]). 178 G. Plunien eta!., The Casimir effect

6.3. Casimir energy

For the parallel, conducting plate configuration in the electromagnetic field explicit results for the Casimir free energy have been obtained by the mode summation [79—81]as well as the local Green function method [35, 42]. For the case of this simple geometry of the boundary the exact temperature- dependent Green function can be derived. Balian and Duplantier have investigated the Casimir free energy concerning the spherical shell and in some details its behaviour for more general geometries [41, 421. Calculations on the free energy of a massive scalar field inside a rectangular box, which is finite in one direction, have also been carried out [77]. In order to give an explicit example we now turn to calculate the thermal correction to the original Casimir effect. Applying the mode summation method, one has to perform the energy subtraction in the same way as in the zero-temperature case. In accordance with the energy subtraction eq. (6.6) already used previously, we only need to calculate the temperature-dependent part of the Casimir free energy:

~

+-~-[(L-a)_(L_L/n)-L/n]Jdk2 ln[i_exp(_~Vk~+k~)]}

= 2 (~~)2 TJ d2kIl{ ~‘ln [i — exp(—~~Jk~+ (?2~)2)]

- Jdn ln[1_exp(_~ ~k~+ (~))]}. (6.30)

The prime on the summation sign indicates that the n = 0 term is taken with a factor ~. As already mentioned F~requires no regularization. In terms of the function

b(a, T, n) = ~ T J dy ln[i — exp(— ‘rr\/y/aT)], (6.3i) eq. (6.30) reads

Ec(a, T)=~{~b(a,~0)+~b(a, T,n)-Jdnb(a, T,n)}

2~( = ~ {_~i~(aT) 3)+~ b(a, T, n)+ 2T (~—~)3~(4)}~ (6.32)

Further exact evaluation for arbitrary temperature fails since the remaining sum cannot be performed analytically. Approximate expressions for Fc(a, T) can be obtained in the low-temperature and in the G. Plunien eta!., The Casimir effect 179 high-temperature limit. For low temperature (aT ~1) a series expansion of the logarithm in eq. (6.31) is justified yielding

2 ~ e’~”~~(~i~+i) n=l~ b(a, T, n)=— ~j T(aT)P ITt’ n=1 aT r/aT\2 aT] = — T{(~—_)+—_Je”~T+O(e2~T). (6.33) IT IT

Inserting this expression into eq. (6.32) leads to the finite-temperature correction

Pc(a, T) = L2air2{ [— ~ (.?1) ~_ (.?i)~(~.~.)e’~~] + 2 (.?i)~~(4)}. (6.34)

As is already clear from its definition, eq. (6.30), one recognizes that the first term of eq. (6.34) refers to the thermal radiation of a constrained electromagnetic field inside the cavity of volume L2a with L ~ a. The last term is the free field contribution (Stefan—Boltzmann term eq. (6.lOa)) corresponding to the same volume which was subtracted. Together with the zero-temperature part the Casimir free energy becomes

L2ir2 aT3 aT3 ir Fc(a, T) = — 720a3 { 1 + 360~(3)(_—) — (2aT)4 + 720 (__—) (i + —~)e_~n1aT}. (6.35)

This energy gives rise to a temperature dependent force per unit area:

~c(a, T) = - 240a4 {i + ~(aT)4—240 (~~)e~T}. (6.36)

The thermal correction can also be obtained from the definition of thermodynamic pressure

p(a,T)=—-c(V,T)=—-~--~--Pc(a,T), where V=L2a. av Lôa

In order to calculate the correction to the Casimir effect at high temperatures, i.e. aT>> 1, the summation over n in eq. (6.32) must be performed. Again one can use Poisson’s sum formula which in the present case states that if

c(a, T, a) = -~- J dn cos(an)b(a, T, n) (6.37a) then

T, 0)+ ~ b(a, 1’, n) irc(a, T,0)+2ir c(a, T, 2irn). (6.37b) 180 6. P!unien eta!., The Casimir effect

Due to the relation ITc(a, T, 0) J~°dn b(a, T, n) this term cancels in accordance with the energy subtraction and the free energy contribution F~becomes

T’c(a, T) = (~.)2 ~ c(a, T, 2ITn). (6.38) The function c(a, T, a) follows as

T) c(a, T, a) = -~-Jdn sin(an)n ln(i —e’~

I(—’~---~——coth(aTa)l, (6.39a) 2rradaL\aT!a a j which, in the high-temperature expansion, up to terms of the order ~ leads to the expression

1 T iT 2aT2\ c(a, T, a)= —~——--~— ~ (6.39b) aa 2cr a a /

Evaluating the sum over n, the result up to terms of the order C~(e_4~~T)reads

1 ~(3)T T 1 4irT Fc(a, T)= L21T2{ 72003_ 8IT4a2~4Ir3 (_~+_—_)e4~wT}. (6.40)

It is interesting to note that this expression contains exactly the zero-temperature Casimir energy with opposite sign which, therefore, cancels in the total Casimir free energy. Thus, for high temperatures the Casimir force becomes

2]e4’~T. (6.41) 4ira 2ira ~c(a, T) = — ____- —~--~ [1- 4rraT + ~(2ITaT) It is possible to show that this cancellation between the zero-temperature Casimir energy and the thermal contributions in the high-temperature limit is not an artefact of the one-dimensional box problem, but holds generally [156].

7. Applications

7.1. Casimir energy in dielectric media and the relation to the bag model of hadronicparticles

A main subject of the previous sections has been the Casimir effect between perfectly conducting plates that are placed in the electromagnetic vacuum. We restricted our presentation to very specific configurations, where the basic ideas and calculational methods that are also applicable in treating G. Plunien et at, The Casimir effect 181 zero-point energies of other quantum fields, could be most transparently demonstrated. In a generalized theory of the Casimir effect one considers the electromagnetic Casimir energy in the presence of dielectric and permeable media [44,39, 40, 60, 61]. These studies have become of considerable interest in particle physics, because there are some striking formal similarities between electromagnetism of continuous media and the phenomenological theory of bag models. As a basic assumption in such models, the QCD vacuum is modelled as a medium with infinite permeability and perfect dielectricity concerning the colour gauge fields [127, 128]. In order to point out this connection, let us spend a few remarks on the electromagnetic Casimir energy in material media. Investigations on this subject were started by Schwinger, DeRaad and Milton [39, 40, 44] based on source theory [129]. A review of this method has been given recently by DeRaad [130]. Particularly, the Casimir effect in a solid sphere with dielectricity r and permeability ,u has attracted interest. A general class of configurations is based on the following model assumption: A spherical ball of fixed radius a with constant dielectricity r~and permeability /2i (non-dispersive medium) may be embedded in an also homogeneous medium characterized by certain values ~2 and /22 (see fig. 7.1). The constitutive relations between the electromagnetic fields in vacuum and the ones in the media are given by D = sE and B = /LH. Also well known from electrodynamics, the following continuity conditions for normal and tangential field components have to be imposed on the spherical surface S which divides the interior and exterior medium:

EEr, E~,B f 5,Brj = continuous. (7.1) /2 is

These boundary conditions directly carry over to the electric and magnetic Green functions, as already mentioned in section 3.2.2. As a particular case Milton [39] has calculated the Casimir stress on a dielectric solid ball placed in the vacuum, i.e., for ~2 = /22 = = 1 and ~i = s. The special case of a conducting ball then is obtained when e tends to infinity. With respect to the boundary conditions, which imply the continuity of the field components {SEr, E5, B = H} on the surface, the problem can be solved. However, the final result for the Casimir stress (force per unit area) is found to be still cutoff-dependent, even after the performance of energy subtractions. The finite part of the stress turns out to be attractive. Milton derives explicit expressions for two limiting cases: For weakly polarizable media, i.e., e — 1 ~ 1 the

- / .- /7~

1 mt.

Fig. 7.1. The sphere of radius a divides interior and exterior medium. 182 6. Plunien eta!., The Ca.simir effect

Casimir force per unit area is found as

(s_1)2 16 1

= — 256’ira” ~ ~ (7.2) where 8 —*0 is the cutoff, and the finite part of the pressure arises from the Casimi-r energy

2/256a. (7.3) Ec(a) = —(s — 1)

For a perfectly conducting ball, i.e. in the limit s —~cc, the finite part of the Casimir energy is approximately given by [39]

Ec(a) —1/8ira. (7.4)

Taking these results for the Casimir energy one may be tempted to revive Casimir’s semiclassical electron model [36].We mention this point because, historically, elementary particle physics began with the classical theory of the electron [131].Endowing the electron with an extended charge distribution removes on one hand, the classical self-energy divergence but leads, on the other hand, to instability of the electron due to the Coulomb repulsion. In order to stabilize the electron within such a model, one is forced to introduce additional attractive forces (Poincaré stress). It seems reasonable that an attractive Casimir force may play the role of the Poincaré stress, regarding the electron as a spherical ball with some sort of effective dielectricity s. Assuming that the charge on the electron is preferably localized on the surface the Casimir energy should balance the repulsive Coulomb energy of a charged spherical shell:

E(a) = e2/8ira. (7.5)

Since the Casimir energy is also proportional to 1/a, such an electron model would allow one to calculate the fine-structure constant a = (e2/4rr) from the energy balance. In the case of s 1 one has to take eq. (7.3) as stabilizing Casimir energy. Accordingly it follows

a(s—1)2/128, (7.6) where the value e = 0.97 would reproduce the observed value a = 1/137 for the fine-structure constant. If the electron is imagined to be a conducting ball, according to eq. (7.4) a parameter-free model result is found to be

a—’1/4rr, (7.7) which is about a factor 10 too large. Nevertheless, it must be said that one should not take such “explanations” of the fine-structure constant too seriously, because they are essentially model-depen- dent. For instance, Brevik’s [40] calculation of the Casimir force on a dielectric sphere including electrostrictive contributions, where the medium is also assumed as non-dispersive and moreover to satisfy the Claudius—Mossotti relation (r — 1)/(e + 2) = const. p, leads to the total Casimir pressure (finite part) . G. Plunien et at, The Casimireffect 183

= — 24ira 4(0.308 — o.45V~). (7.8) If this pressure is assigned to balance the Coulomb repulsion ~= a/(8ira~),one sees that for the required value of a this can not be possible for any real values for s. Apparently, there is no evident way of explaining the fine-structure constant in this manner. A further notable calculation of the Casimir effect in a solid ball was carried out by Brevik and Kolbenstvedt [60,61, 72]. They considered a particular class of non-dispersive material media, assuming that in both interior and exterior media the vacuum relationship

E/.L = 1 (7.9)

is fulfilled. The boundary conditions on the surface dividing the two media then take the form

ii 1 -~ ~ Er, E 5, Br, — = continuous. (7.10) 1/2 /2

An obvious consequence of (7.9) is that within such a model the electric and magnetic fields are treated in a symmetric way, i.e. one has: E = /2D and B = /LH. In the presence of condition (7.9) the explicit calculation shows that the same delicate cancellations between exterior and interior contributions take place, which are characteristic for the case of a conducting spherical shell placed in the vacuum. In particular the cutoff problem normally present in the theory of Casimir energy of dielectric media disappears. The Casimir energy is found to be cutoff-independent and positive [61]:

1)[1+0.311 /212 2]. (7.lla) Ec(a)=Eo(a)(/212/212+ 1 (/212+ 1) Here E 0(a) = 0.0923/(2a) 3/Ma (7.llb) represents the well-known result for the idealized case of a perfectly conducting spherical shell [37]. Correspondingly the Casimir force is repulsive and depends on the properties of the interior and exterior media only through the ratio of their permeabilities /212 = /Ll/ic2. One observes that the Casimir energy (7.lla) is also invariant when interchanging interior and exterior medium, i.e., the substitution iL12--* 1//212 leads to the same result. Let us mention some limiting cases of eq. (7.lla). For /212~*0 or /L12—*cc (i.e. /2i—*0 or ic2--*O) one obtains the conducting-shell result. The case /22 = 1 (and ~2 = 1) corresponds to the situation of a compact sphere placed in the vacuum, a case which may be associated with some kind of semi-classical electron model. If in addition ~ —*0 (and g~—* cc) the electron becomes a perfect electric conductor where the E and B fields vanish in the interior. Conversely, if 1a1 -+ cc (and —*0) the electron becomes a perfect magnetic conductor with vanishing D and H fields in the interior. In both limiting cases the Casimir energy is equal to E0(a). The electromagnetic theory considered above is formally directly applicable to bag models of QCD. In section 4.3. we have already made reference to the duality between the electromagnetic field and the colour gauge fields. The correspondence of colour theory with the Maxwell theory of polarizable media 184 G. P!unien eta!., The Casimir effect

is invoked, if one considers the so-called perturbative vacuum in the interior of the bag and the exterior, “true” vacuum as some kind of colour-dielectric media. The duality between such a gauge theory and the electromagnetic theory described above may be summarized as follows:

Electromagnetic field Colour field

fields: B = ~ = SCE’~ (7.12a)

D = rE ~ > B~= (7.12b)

dielectric property: = 1 < > = 1 (7.12c)

boundary conditions:

Er, E {~ 5, Br, ~ B5}~ = continuous ( > B~,B~,E~,-~— E~’}= continuous. (7.12d)

The constraint (7.12c) in colour theory ensures that the gluonic modes propagate in the vacuum always with the velocity of light. The continuity conditions on the surface formally remain the same. The QCD case is now obtained as the limit p~—* cc and s~—*0 for the exterior vacuum and icc = s~= 1 for the interior vacuum. The situation assumed in QCD, namely, that a spherical bag is embedded in the physical vacuum may directly correspond to that of a spherical superconducting ball placed in the electromagnetic vacuum. The analogy is illustrated in fig. (7.2). The concept of considering the physical (exterior) QCD vacuum as a kind of material medium, which is characterized by the property p~s,= 1 in order to ensure relativistic invariance, has been discussed in detail by Lee [128].The assumption that the physical vacuum has properties comparable with a perfect colour magnetic conductor guarantees confinement in the sense that the bag surface is impermeable against the gluon field, i.e.,

D’~-+O, H”—10 (r>a). (7.13)

Accordingly, the continuity conditions imply that the interior field components E~and B~vanish when approaching the bag surface:

E~(r—*a—)=0, B~(r—*a—)=0. (7.14)

These conditions have already been mentioned in section 4.3, where we have discussed various efforts to evaluate the Casimir energy in a spherical cavity. The first of the two conditions (7.14) implies no colour electric flux through the bag surface: dfn . E’~= 0, whereas the second one implies vanishing energy flux of gluons through the surface, since it follows: n (Ea x B’~)I~= 0. As one sees, the conditions (7.13) are formally in accordance with the ones valid in the electromag- netic case (interchanging interior and exterior medium) when /22 = 1/s2 tends to zero. Thus, Brevik and Kolbenstvedt [61]conclude that the electrom-agnetic- Casimir energy (7.lla) is identical withiheoneof a 6. Plunien et at, The Casimireffect 185

~21 Ec~O‘~ -V7~-2Z - Z 2 . -C

~2-~-’- ~zv2z~z

a) b)

Fig. 7.2. Duality between the situation of a perfect conducting sphere in the electromagnetic vacuum (a) and the QCD vacuum assumed in the theory of bag models (b).

confined gluon field after performing the limit /L12—*O. Since each component of the colour field contributes to the energy with the amount of eq. (7.llb), the total Casimir energy of a gluonic bag is obtained by multiplying this result by eight:

E~(a)= 0.73880/(2a) 3/8a. (7.15)

This energy is positive and gives rise to a repulsive Casimir pressure. The above “evaluation” of the gluonic Casimir energy reveals that the appearance of cutoff-dependent terms (divergencies), former calculations [39, 40] are plagued with, are in fact a consequence of restricting exclusively to the contribution of interior bag modes. As an alternative to such calculations, Brevik and Kolbenstvedt have shown that it is possible to apply to the QCD case a limiting result, which originates from the electromagnetic theory of Casimir energy, where it is legitimate to take into account both the contributions of interior and exterior field modes. In a recent paper Milton [58] also considers the contribution of exterior field modes assuming a certain vacuum structure (foam of densely packed bubbles) which may legitimize such a treatment also in the case of calculating the fermion stress. It should be emphasized that the basic assumption (7.12c) is of great advantage in the sense that it allows a meaningful evaluation of the global Casimir energy referring to interior and exterior field modes. The same delicate cancellation of cutoff terms takes place as in case of the conducting spherical shell, and thus leads to a finite result. The details of these cancellations can be clearly analyzed when considering the Casimir energy density Wc(T) in the interior and exterior region [71, 72, 68]. The feature of finite global Casimir energy does not imply the regularity of the energy density, but this local quantity turns out to diverge at the bag surface in a particular way (wc(r—* a)--* —cc from inside and wc(r—* a)—*cc from outside) which makes the total Casimir energy finite [72]. The investigation of local quantities like the energy density or the expectation value (0IF~F~0)may be of interest in order to elucidate fundamental properties and structures of the physical vacuum (e.g. gluon condensate [71] or the formation of flux tubes [66—68]). The evaluation of Casimir energies of arbitrarily shaped bags is plagued with similar difficulties, which also occur in the electromagnetic theory of Casimir energy in the presence of arbitrarily shaped smooth boundaries. An appropriate treatment of corresponding bag-boundary conditions remains 186 G. P!unien eta!., The Casimir effect

problematic. However, the Casimir energy of bags deviating from spherical geometry may be of considerable interest in connection with the theory of highly excited meson states (Regge trajectories [144—146],bag fission, etc.). The Casimir energy of a perfect conducting cylindrical shell [132]may have some application in connection with the formation of QCD strings. A further considerable problem which could be of relevance in QCD bag models is based on the following electromagnetic Casimir problem: Consider two perfectly conducting spherical shells separated by a distance R which are placed in the electromagnetic vacuum. An approximate result for the Casimir force for this configuration (assuming a scalar “photon”) has recently been derived by DeRaad [130]. (The large-distance behaviour for spin-one photon has already been discussed by Balian and Duplantier [42].)The corresponding Casimir energy gives rise to an attractive interaction between the spheres. One should also remember that Casimir already used the zero-point energy in order to derive the retarded van der Waals attraction between static polarizable particles. It may be interesting to generalize these calculations and treating two spherical solid balls placed in an exterior medium, assuming Brevik’s condition p.s = 1 for both media. One can expect that from the generalized expression for the Casimir force a similar result as given by DeRaad would be obtained in some limiting cases of the permeability of the spheres and the imbedding medium. Of course, one would then like to know whether the QCD limit as considered by Brevik for a single sphere leads to a non-vanishing Casimir energy, or not. A remaining Casimir energy as a function of separation would represent a contribution to the interaction potential of two bags due to fluctuating colour fields.

7.2. Boundary problems and Casimir energy in gauge theory

It should be clear by now that zero-point energies in quantum field theory can be given a well-defined meaning. We now want to give one more example of boundary problems in field theory which have recently been investigated in connection with the Casimir effect. Let us consider a massive vector field in the presence of an infinite conducting plane. This problem has attracted interest for two reasons [133, 134]: Firstly, one would like to know whether the Maxwell theory is smoothly recovered as expected in the zero-mass limit [135],and secondly, it is interesting to know whether a non-zero mass could cause observable boundary effects. In addition, one likes to examine the influence of possible non-minimal coupling terms in the field Lagrangian which break gauge invariance in the massless limit.

- The local treatment is based on the vacuum-stress tensor &~ (the indication “vac” will be suppressed later on). Without loss of generality, the conducting plane may be phcéd~afthe pOsiTi~fl- x3 = 0. We first note that the corresponding situation for the massless electromagnetic field leads to a vanishing energy momentum tensor (all components are equal to zero). This is easily proved by representing 9~”explicitly in terms of photon propagators as discussed in section 3.2. As a consequence of a finite photon mass, the energy—momentum tensor has a non-vanishing trace, i.e. ~ 0. The general structure of the vacuum-stress tensor can be determined on the basis of symmetry arguments and conservation laws. Since the geometry of the boundary prefers the x3 direction, 9~”’can only consist of the tensors g~”and z~’z”which are weighted by arbitrary scalar functions of the position x, i.e.:

~(x)=f(x)g~+g(x)z~z”, (7.16) with the normal surface vector z” = (0, 0, 0, 1). The absence of external sources requires the con- G. P!unien et at, The Casimir effect 187

9~”~= 0 determines the scalar functions to be servation of energy and momentum. The condition t9~ equal and to depend only on the x3 coordinate:

~9~(x3)= f(x3)(g~’+ z”z~)+ ag”. (7.1-7)

Since the vacuum is expected to remain unaffected infinitely far away from the plane, ~9~”must vanish in the limit x31—*cc. This condition is satisfied if f(x3—*cc)--*0 and a = 0. Expressed in terms of the trace E~(x3)= 3f(x3), the vacuum-stress tensor takes the general form

= ~.(x3)(g~’ + z’~z”). (7.18)

This result implies that the vacuum pressure 933 vanishes everywhere, but the vacuum energy density may be different from zero. Davies and Toms [134] have investigated the field theory which is determined by the following classical action:

I[A] = J d~x\/—g{—~F~F~+ ~m2A~ + 1RA~’A~+ R~”A~A~}. (7.19)

The massive vector field A” is considered in a curved background space-time with metric tensor g”. From this given metric the Ricci tensor Ri”’ and the scalar curvature R are constructed. ~ and ~2 denote dimensionless coupling constants which describe a non-minimal interaction (i.e. ~, ~2 ~ 0) between the vector field and the background geometry. Although the Lagrangian in eq. (7.19) reduces to that of the Proca theory [136] in the Minkowski space limit, the energy—momentum tensor still retains an imprint of the non-minimal curvature-dependent terms. This turns out in the following limiting result for the trace [134]:

2A’’A,~— (3~~+ T~= —m 2)LIJ(A’’A,~)— ~29.~3~A’’A”. (7.20)

Based on this fact Davies and Toms [134]have examined the massless limit of ~9’”~for the Proca theory, i.e. ~ = ~2 = 0 (minimal coupling) as well as the case when non-minimal couplings are present. The appropriate expression for the renormalized trace ~ can be derived for both cases. Again one expresses the vacuum stress tensor in terms of free propagators of the massive photon field:

2{G~(x,x’) — G = —im 0’t~(x— x’)}I~.~. (7.21)

Constructing the renormalized massive vector field propagator, one has to consider the physical boundary conditions for the Proca field on the conducting plane. The finite mass breaks gauge invariance and in contrast to the Maxwell theory, the massive vector field has three independent polarization states, two transverse and a longitudinal (scalar) mode. Requiring Dirichlet boundary conditions on the conducting surface is only meaningful with respect to the transverse field modes. For the third spin state, the longitudinal modes, it is unrealistic to assume reflection on the boundary, too, since in the massless limit these modes decouple from matter and will penetrate the boundary even at infinite conductivity [137]. As shown by Bass and Schrödinger [138]they will remain unabsorbed in conducting media to a depth of the order of lim. The appropriate boundary condition for the massive photon field have been recently examined by Barton and Dombey, who classified the modes for a plane 188 G. Plunien eta!., The Casimir effect geometry. They have also analyzed the effect of a small mass correction to the Casimir force between conducting parallel plates [137] and have shown explicitly that contributions from the penetrating longitudinal modes are negligible. The remaining transverse modes, which will be reflected between the plates, lead to twice the Casimir energy of a massive scalar field under similar conditions (see eq. (4.34)). Considering only the non-penetrating field modes, Davies and Toms have explicitly derived the renormalized propagator, and they obtain the following result for the trace ~ in the minimal-coupling case:

ø~(x3)= — K 3). (7.22) 4IT (x) 1(2mx

Accordingly, when the mass of the vector field tends to zero the vacuum-stress tensor behaves like

— ~.~(m/ITx3)2(g’”+ z’~z~), (7.23) i.e., for arbitrary small but non-vanishing mass the energy density diverges on the boundary. In the massless limit ~9’~is seen to vanish, a result, which is consistent with the Maxwell case. Including non-minimal coupling terms one derives a modified expression for the trace:

= ~j-~--~(~+ 6~)[3 (rn)2 K 3) + 2- Ki(2mx3)] 423 K 3). (7.24) 2(2mx 1(2mx

In this case the expansion of 9’~for small m reads

_____— (1+ ~ + 6~~)](g~+ z~z~). (7.25)

The relations (7.23) and (7.25) reveal that even for an arbitrarily small mass there is an infinite vacuum energy per unit area at the boundary. In addition, eq. (7.25) shows that the non-minimal gauge-breaking terms survive even in the limit of vanishing mass and give also rise to a divergent vacuum energy density on the plane. The question arises whether this surface energy may be experimentally detectable, which would be a test for the masslessness of the photon. The present upper experimental limit for the photon mass is about 10’~g. In the above derivation, a perfectly conducting medium with a sharp boundary has been assumed. This is, of course, an idealization since real material media will be of finite conductivity, and the surface will not act as a perfect mirror. One may have to consider an effective penetration depth 8 which will serve to damp the divergence of the vacuum energy on the plane. Davies and Toms [134] conclude carefully that the surface energy induced by a non-zero photon mass, and in the presence of gauge-breaking terms in the electromagnetic field Lagrangian, could perhaps be of measurable mag- nitude ~ 1020 eV cm2, assuming 6 -~- 10_8 cm). However, one has to consider that even in the massless, gauge-invariant Maxwell case boundaries with extrinsic curvature produce in addition a divergent surface-energy contribution. G. P!unien eta!., The Casimir effect 189

8. Concluding remarks

During the last twenty years notable progress has been made in particle physics toward a unified field theory. In close connection with these efforts, our understanding of the nature of the vacuum state has been deepened and is now resting on a much firmer basis. Generally the (true) vacuum state JO) is defined as the state of lowest energy. According to the observation that space-time, on the small and on the very large scale, is isotropic and homogeneous, the vacuum state of a free quantum field must be invariant against rotations and translations. These symmetry properties imply that the free vacuum carries no energy, momentum or angular momentum, i.e., (OJP’’jO) = 0 and (0~M~”J0)= 0. On the other hand, the canonical field quantization leads to quantum operators with ill-defined vacuum expectation values like, for instance, the canonical field Hamiltonian which gives rise to an infinite zero-point energy. Such ambiguities which arise due to the existence of quantum fluctuations are removed by formal subtraction arguing that they do not produce observable effects. The situation is different in the case of fields interacting withexternal sources or boundaries. Since long it is known that the presence of external fields breaks fundamental symmetries and thus, quantum fluctuationsbecome observable. Already in the earlydays of quantum electrodynamics it was demonstrated that fluctuations in the vacuum charge density of the electron—positron field induced by an external electromagnetic field — the vacuum polarization — give rise to a correction to the interaction between electromagnetic sources (the Uehling potential). More recently, it has been recognized that external boundary conditions (as an idealization for real sources interacting with a given field) play a similar role as external fields, since they can also break symmetries and induce observable effects due to quantum fluctuations. This has revived the discussion about the vacuum energy and the Casimir effect. The basic feature of Casimir’s concept of vacuum energy is that the physical vacuum state of quantized fields must be determined with respect to the condition that fields usually exist in the presence of external constraints. In particular, this facilitates a meaningful treatment of zero-point energies examining their measurable consequences as contribution to the self-energy or to the interaction between the external constraints. Detailed studies on the Casimir energy of the constrained electromagnetic fields have represen- tatively demonstrated the role of zero-point energies. All evaluation methods which have been investigated in this context, particularly the mode summation method [1, 37, 81] and various local Green function methods [35, 42, 140], are similarly applicable in order to calculate Casimir energies of other constrained quantum fields (e.g. quarks and gluons confined in a bag). Both evaluation methods require inherently different regularization procedures in order to yield finite results. In the case of mode summation this can be achieved, for instance, by introducing high-frequency cutoffs or by means of dimensional regularization. The local treatments deal with the boundary part of the exact Green function where the free vacuum Green function is already subtracted (local regularization). Calculating the vacuum energy in terms of Green functions divergences which arise from taking expressions at equal space-time points, in addition, requires appropriate renormalization. Series expansion methods like the multiple scattering expansion or the perturbation expansion show that divergencies arise only for the lowest-order terms, as is typical for renormalizable field theories. The vacuum energy is unambiguously determined in both formulations. For the original Casimir effect, they lead to identical results. It is important to note that the same result for the Casimir force between conducting parallel plates is also derivable within a totally different approach dealing with fluctuating classical electromag- netic fields [141—143].This, of course gives confidence in the calculational methods mentioned above. 190 G. P!unien eta!., The Casimir effect

It is worth recalling that a fully analytical evaluation of Casimir energies is only possible for plane boundaries (parallel plates, rectangular box). In the case of boundaries of generic shape the Green function techniques facilitate approximate evaluations. As these are usually quite involved, it may be of interest to develop alternative calculational methods. Considering that one is mainly interested in global quantities like the Casimir energy or Casimir pressure, it seems somewhat inadequate to base the evaluation on quantities, such as a complete set of eigenmodes or the exact Green function, which contain more information about the constrained field configuration than necessary. It is therefore of great practical relevance to develop new approximation schemes for the calculation of the Casimir energy, e.g. variational methods. For example, it would be interesting to formulate the Casimir energy as a functional of the vacuum polarization charge density, which is stationary at the correct value. The concept of Casimir energy is further supported by the results obtained for the zero-point energy of the electron—positron field interacting with external electromagnetic fields. Here we saw that the vacuum polarization current can be expressed as the functional derivative of the vacuum energy with respect to the external electromagnetic field. This demonstrates the role of the vacuum energy as a contribution to the interaction potential between the external sources. The transition from a neutral to a charged vacuum state in strong fields is reflected in a discontinuous lowering of the vacuum energy. In the final chapter we discussed several applications of the Casimir energy, in particular the vacuum contribution to hadronic masses in the context of the MIT bag model, and alternative derivations of interactions in polarizable media, such as van der Waals forces. In addition to these applications, the Casimir effect is an essential ingredient in recent developments in high-energy physics, namely, Kaluza—Klein theories of unified interactions. As Appelquist and Chodos [25] have pointed out, the Casimir energy in a five-dimensional Kaluza—Klein theory with one compact dimension is attractive in the sense that it tends to shrink the size of the compact dimension. While this does not prove that space-time in higher dimensions must be compact, it lends support to the concept that the motion in higher dimensions may be frozen in due to their tiny size, if these are indeed compact. The existing literature on this and other applications of the Casimir energy is too vast to be comprehensively reviewed here. However, we hope that the present review article will help the interested reader to explore these fascinating developments by himself.

References

[11H.B.G. Casimir, Proc. Kon. Ned. Akad. wet. Si (1948) 793. [2] By. Deriagin and 1.1. Abrikosova, Soc. Phys. JEPT 3 (1957) 819. - [31 M.J. Sparnaay, Physica 24 (1958) 751. [41w. Black, J.G.V. De Jongh, J.Th.G. Overbeek and M.J. Sparnaay, Trans. Faraday Soc. 56 (1960) 1597. [5] A. van Silfhout, Proc. Kon. Ned. Akad. wet. B 69 (1966) 501. [6] D. Tabor and R.H.S. Winterton, Nature 219 (1968) 1120. [71ES. Sabisky and C.H. Anderson, Phys. Rev. A 7 (1973) 790. [8] A.A. Grib, S.G. Mamaev and VS. Mostepanenko, Kvantovye effekty v intensivnykh vneshnikh polyakh (Quantum effects in intense external fields) (Nauka, Moscow, 1980); cit. in: M. Bordag, E. Vitsorek and D. Robashik, Soy. J. NucI. Phys. 39 (1984) 663. [9] S. Gupta, Proc. Phys. Soc. London A 63 (1950) 681. [10] K. Bleuler, Helv. Phys. Acta 23 (1950) 567. [11] G. Källén, Quantumelectrodynamics (Handbuch der Physik V/i, Springer, 1958). 112] J. Ambjom and R.J. Hughes, NucI. Phys. B 217 (1983) 336. [13] R.P. Feynman, Acta Phys. Pol. 24 (1965) 697. [141L.D. Fadeev and V.N. Popov, Phys. Lett. 25B (1967) 29. [151T. Kugo and I. Ojima, Suppl. Prog. Theor. Phys. 66 (1979). [161I. Ojima, Prog. Theor, Phys. 64 (1980) 625. G. Plunien et at, The Casimireffect 191

[17]C. Becehi, A. Rouet and R. Stora, Ann. Phys. 98 (1976) 287. [18] J. Rafelski, L.P. Fulcher and A. Klein, Phys. Rep. 38C (1978) 227. [19]Y. Takahashi and H. Shimodaira, Nuovo Cimento 62A (1969) 255. [20] Y. Takahashi, An Introduction to Field Quantization (Pergamon, New York, 1969). [21] B.S. De witt, Phys. Rep. 19C (1975) 295. [22]L.H. Ford, Phys. Rev. D 11(1975) 3370. [23]iS. Dowker and R. Banach, J. Phys. A: Math. Gen. 11(1978) 2255. [24]T. Appelquist and A. Chodos, Phys. Rev. Lett. 50 (1983) 141. [25]T. Appelquist and A. Chodos, Phys. Rev. D28 (1983) 772. [26]T. Elster, Class. Quantum Gray. 1(1984) 43. [27]H. Muratani and S. Wada, Phys. Rev. D 29 (1984) 637. [28] H.B.G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360. [29] D. Langbein, Van der Waal Attraction, Springer Tracts in Modern Physics 72 (1974). [30] H.B.G. Casimir, J. Chim. Phys. 46 (1949) 407. [31] H.B.G. Casimir, Philips Rev. Rept. 6 (1951) 162. [32] T.H. Boyer, Phys. Rev. 180 (1969) 19. [33] T.H. Boyer, Phys. Rev. A 9 (1974) 2078. [34] G. Feinberg and J. Sucher, J. Chem. Phys. 48 (1968) 3333. [35]L.S. Brown and G.J. Maclay, Phys. Rev. 184 (1969)1272. [36] H.B.G. Casimir, Physica 19 (1953) 846. [37]T.H. Boyer, Phys. Rev. 174 (1968)1764. [38] T.H. Boyer, Ann. Phys. 56 (1970) 474. [391K.A. Milton, Ann. Phys. 127 (1980) 49. [4011. Brevik, Ann. Phys. 138 (1982) 36. [41] R. Balian and R. Duplantier, Ann. Phys. 104 (1977) 300. [42JR. Balian and R. Duplantier, Ann. Phys. 112 (1978) 165. [43] L.L. DeRaad, Jr. and K.A. Milton, Ann. Phys. 136 (1981) 229. [44] J. Schwinger, L.L. DeRaad Jr. and K.A. Milton, Ann. Phys. 115 (1978) 1. [45] P. Candelas, Ann. Phys. 143 (1982) 241. [46] J. Schmit and A.A. Lucas, Solid State Commun. 11(1972)419. [47] J. Schmit and A.A. Lucas, Solid State Commun 11(1972) 475. [48] B. Davis, J. Math. Phys. 13 (1972)1324. [49] D. Deutsch and P. Candelas, Phys. Rev. D 20 (1979)) 3063. [50]S. Sen, J. Math. Phys. 25 (1984) 2000. [51]G. Wentzel, Z. Phys. 118 (1941) 277. [52]G. Wentzel, Helv. Phys. Acta 15 (1942) 111. [53]M. Park and PB. Shaw, Phys. Rev. D 30 (1984) 437. [54]CM. Bender and P. Hays, Phys. Rev. D 14 (1976) 2622. [55]K.A. Milton, Phys. Rev. D 22 (1980) 1441. [561K.A. Milton, Phys. Rev. D 22 (1980) 1444. [57]K.A. Milton, Phys. Rev. D 27 (1983) 439. [58] K.A. Milton, Ann. Phys. 150 (1983) 432. [59]J. Baacke and Y. Igarashi, Phys. Rev. D 27 (1983) 460. [60] I. Brevik and H. Kolbenstvedt, Phys. Rev. D 25 (1982) 1731. [61] 1. Brevik and H. Kolbenstvedt, Ann. Phys. 143 (1982) 179. [62] H. Aoyama, Phys. Rev. D 29 (1984) 1763. [63]J.F. Donoghue, Phys. Rev. D 29 (1984) 2559. [64] L. Vepstas, AD. Jackson and AS. Goldhaber, Phys. Lett. 140B (1984) 280. [65] K. Kikkawa and M. Yamasaki, Phys. Lett. 149B (1984) 257. [66] H.B. Nielson and P. Olesen, Nuci. Phys. B 160 (1979) 381. [67]J. Ambjorn and P. Olesen, NucI. Phys. B 170 (1980) 60. [68] K. Olaussen and F. Ravndal, NucI. Phys. B 192 (1981) 237. [69]J.F. Donoghue and K. Johnson, Phys. Rev. D 21(1980)1975. [70] K. Johnson, in: Particles and Fieids—1979 (APS/DPF Montreal) (AlP New York, 1980). [71] K.A. Milton, Phys. Lett. 104B (1981) 49. [72] I. Brevik and H. Kolbenstvedt, Ann. Phys. 149 (1983) 237. [73] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf Phys. Rev. D 9 (1974) 3471. [74] A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Phys. Rev. D 10 (1974) 2599. 192 6. Plunien eta!., The Casimir effect

[75] P. Hasenfratz and J. Kutti, Phys. Rep. 40C (1978) 75. [76] T. De Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D 12 (1976) 206. [77] J. Ambjorn and S. Wolfram, Ann. Phys. 147 (1983) 1. [78] J. Ambjom and S. Wolfram, Ann. Phys. 147 (1983) 33. [79] M. Fierz, Helv. Phys. Acta 33 (1960) 855. [80] F. Sauer (Dissertation, Uniyersität Gottingen, 1962). [81]J. Mehra, Physica 37 (1967) 145. [82] CM. Hargreaves, Proc. Kon. Ned. Akad. Wet. B 68 (1965) 231. [83] W. Lukosz, Physica 56 (1971) 109. [841W. Lokosz, Z. Phys. 258 (1973) 99. [85]W. Lukosz, Z. Phys. 262 (1973) 327. [86]IS. Bromwich, Phil. Mag. 38 (1919) 143. [87] F. Borgnis, Ann. Phys. (Berlin) 35 (1939) 359. [88] K.A. Milton, L.L. DeRaad Jr. and J. Schwinger, Ann. Phys. 115 (1978) 388. [891R. Balian and C. Bloch, Ann. Phys. 60 (1970) 401. [90] R. Balian and C. Bloch, Ann. Phys. 64 (1971) 271. [91] R. Balian and C. Bloch, Ann. Phys. 69 (1972) 76. [92] R. Balian and C. Bloch, Ann. Phys. 85 (1974) 85. [93] T.H. Hansson and R.L. Jaffe, Phys. Rev. D 28 (1983) 882. [94] T.H. Hansson and R.L. Jaffe, Ann. Phys. 151 (1983) 204. [95] H. Verschelde, L. Wille and P. Phariseau, Phys. Lett. 149B (1984) 39. [96] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, Vol. 1 (McGraw-Hill, New York, 1964). [97] J. Schwinger, Phys. Rev. 82 (1951) 664; J. Schwinger, Phys. Rev. 94 (1954) 1362. [98] W.M. Furry, Phys. Rev. 51(1937)125. [99] C. ltzykson and J.B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980). [100]V. Weisskopf, Kgl. Danske Vid. Selskab. 14 (1936) 166. [101]w. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714. [102] E.A. Uehling, Phys. Rev. 48 (1935) 55. [103]R. Serber, Phys. Rev. 48 (1935) 41. [104]W. Pieper and W. Greiner, Z. Phys. 218 (1969) 327. [1051B. Muller, J. Rafelski and W. Greiner, Z. Phys. 257 (1972) 62; B. MOller, J. Rafeiski and W. Greiner, Z. Phys. 257 (1972) 183. [106]B. Muller, J. Rafelski and W. Greiner, Nuovo Cimento 18A (1973) 551. [107]J. Rafeiski, B. Muller and w. Greiner, Nuci. Phys. B 68 (1974)585. [108]L. Fulcher and A. Klein, Phys. Rev. D 8 (1973) 2455. [109] L. Fulcher and A. Klein, Ann. Phys. 84 (1974) 335. [110] S.S. Gershtein and Y.B. Zeldovich, Lettre Nuovo Cimento 1(1969) 835. [iii] S.S. Gershtein and YB. Zeldovich, Soy. Phys. JEPT 30 (1970) 358. [112]VS. Popov, Soy. J. Nucl. Phys. 12 (1971) 235. [113]VS. Popov, Soy. Phys. JEPT 32 (1971) 526. [114]YB. Zeldovich and VS. Popov, Soy. Phys. Uspekhi 14 (1972) 673. [115]6. Plunien, Diplomarbeit, Universität Frankfurt, 1984. [116]E.H. Wichmann and N.M. Kroll, Phys. Rev. 101 (1956) 843. [117]ML. Goldberger and KM. Watson, Collision Theory (Wiley, New York, 1964). [118]J. Reinhard, B. MOller and W. Greiner, Phys. Rev. A 24 (1981) 103. [119]M. Gyulassy, Phys. Rev. Lett. 32 (1974) 1393; M. Gyulassy, Phys. Rev. Lett. 33 (1974) 921. [120]M. Gyulassy, NucI. Phys. A 244 (1975) 497. [121]G.A. Rinker Jr. and L. Wilets, Phys. Rev. A 12 (1975) 748. [122]H. Hellmann, Einfuhrung in die Quantenphysik (Franz Deutike Verlag, 1937); ER. Davison, Physical Chemistry-An Advanced Treatise (Academic Press, New York, 1969). [123]ND. Birrel and P.C. Davies, Quantum Field Theory in Curved Space (Cambridge Univ. Press, London, 1982). [124]M. Plank, Verhandl. Deut. Physikalischen Ges. 2 (1900) 237. [125]L. Dolan and R. Jackiw, Phys. D 9 (1974) 3320. [126]G. Kennedy, R. Critchley and J. Dowker, Ann. Phys. 125 (1980) 346. [127]T.D. Lee,Phys. Rev. D 19 (1979) 1802. [128]T.D. Lee, Particle Physics and Introduction to Field Theory (Harwood, New York, 1981). G. P!unien eta!., The Casimir effect 193

[129]J. Schwinger, Particle, Sources and Fields, Vol. I (Addison-Wesley, Massachusetts, 1970). [1301L.L. DeRaad Jr., Forschntte Phys. 33 (1985) 117. [131]HA. Lorentz, The Theory of Electrons (Dover, New York, 1952). [132]K.A. Milton, Ann. Phys. 136 (1981) 229. [133]P.C.W. Davies and S.D. Unwin, Phys. Lett. 98B (1981) 274. [134]P.C.W. Davies and D.J. Toms, Phys. Rev. D 31(1985)1336. [135]E.C.G. Stueckelberg, Hely. Phys. Acts 14 (1941) 52; E.C.G. Stueckelberg, Helv. Phys. Acta 30 (1957) 209. [1361A. Proca, J. Phys. Radium Ser. VII, 8 (1936) 23. [137] 6. Barton and N. Dombey, Nature 311 (1984) 336. [138]L. Bass and E. Schrodinger, Proc. Roy. Soc. A 232 (1955) 1. [139]C.W. Wong, Phys. Rev. D 24 (1981) 1416. [140]J. Schwinger, Lett. Math. Phys. 1(1975) 43. [141] EM. Lifshitz, Soy. Phys. JEPT 2 (1956) 73. [142]T.H. Boyer, Phys. Rev. 174 (1968) 1631. [143]T.H. Boyer, Phys. Rev. D 11(1975)790. [144] K. Johnson and B. Thorn, Phys. Rev. D 13 (1976)1934. [145] RD. Viollier, S.A.Chin and AK. Kerman, Nucl. Phys. A 407 (1983) 269. [146] R. Shanker, D. Vasak, CS. Warke, W. Greiner and B. Muller, Z. Phys. C 18 (1983) 327. [147] IS. Gradshteyn and l.M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1980). [148] M. Abramowitz and l.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972). [149] G. Arfken, Mathematical Methods for Physicists (Academic Press, New York, 1970). [150]J. Friedel, Phil. Mag. 43 (1952) 153. [151]H. Yamagishi, Phys. Rev. D 27 (1983) 2383. [152]B. Grossman, Phys. Rev. Lett. 50 (1983) 464. [153]B. Grossman, Phys. Rev. Lett. 51(1983) 959. [154]B. MOller, Ann. Rev. NucI. Sci. 26 (1983) 351. [155]W. Greiner, B. Muller and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, Heidelberg, 1985). [156] G. Plunien, to be published.