Casimir energy due to inhomogeneous thin plates

We study the Casimir energy due to a quantum real scalar field coupled to two planar, infinite, zero-width, parallel mirrors with non-homogeneous properties. These properties are represented, in the model we use, by scalar functions defined on each mirror's plane. Using the Gelfand-Yaglom's theorem, we construct a Lifshitz-like formula for the Casimir energy of such a system. Then we use it to evaluate the energy perturbatively, for the case of almost constant scalar functions, and also implementing a Derivative Expansion, under the assumption that the spatial dependence of the properties is sufficiently smooth. We point out that, in some particular cases, the Casimir interaction energy for non-planar perfect mirrors can be reproduced by inhomogeneities on planar mirrors.


I. INTRODUCTION
The evaluation of observables in the static Casimir effect presents many interesting challenges, both from the point of view of designing experiments capable of measuring its features with finer resolutions, as well as regarding the calculations involved in the prediction of those features [1]. In theoretical studies, the two main aspects usually considered in a system exhibiting the Casimir effect are the geometry of the mirrors and the properties of the media composing them.
In some special cases, for example when the mirrors are assumed to satisfy 'perfect' boundary conditions 1 , the ensuing simplifications allow one to study geometrical aspects in a simpler way. The Derivative Expansion (DE), is an approach that may be applied to the case of two smoothly curved surfaces, such that at each point their local separation is smaller than the curvature radii [2,3]. When the mirrors impose simple boundary conditions, the DE predicts the form of the corrections to the Proximity Force Approximation (PFA), which is the leading term in the expansion. Indeed, the next to leading order (NTLO) term is completely fixed except for a single, global, dimensionless factor. That factor depends exclusively on the kind of boundary condition used, and it may be determined, once and for all, from a calculation which is perturbative in the departure from planar mirrors, to the second order in that departure. Note that the DE so obtained is not perturbative in the departure, yet it can be determined, for example, from the knowledge that perturbative expansion.
The opposite, complementary case, would amount to systems with the simplest possible, yet non trivial, geometry, like media having planar and parallel (inter)faces. Here, one can focus on the effects of the media inhomogeneities on the detailed properties of the Casimir force, in this case, its dependence on the distance between the mirrors. We are particularly interested in situations where the planar media are inhomogeneous at a macroscopic or mesoscopic level, that is, composed by regular patches or periodic arrays of homogeneous media. On general grounds, we expect the Casimir energy to exhibit a rich dependence on the inhomogeneities, similar to those that appear when considering perfect conductors with non trivial geometries, as trench arrays and other periodic structures [4]. Such rich dependence was measured in good agreement with calculations for sinusoidally corrugated 1 Namely, boundary conditions which do not involve any parameter or function; typically, Dirichlet or Neumann for the case of a real scalar field, and perfect conductor conditions for the electromagnetic field.
In this article, we consider the static Casimir effect for a special class of system, consisting of a quantum real scalar field in the presence of two zero-width, parallel and infinite plates.
The latter are not assumed to be homogeneous; rather, each plate will be characterized, regarding its properties, by a real scalar potential defined on the plane. This can be considered as a toy model for the interaction of the electromagnetic field with a thin mirror characterized by position-dependent electromagnetic properties, and generalizes previous works for scalar fields in the presence of homogeneous thin plates, named δ-potentials [6]. Under those assumptions about the system, and using the Gelfand-Yaglom's (GY) theorem [7], we shall obtain a Lifshitz-like formula [8], which yields the Casimir energy as a functional having as arguments the 'scalar potentials', i.e., the two functions which characterize the plates [9].
We then use this generalized Lifshitz-formula to obtain more explicit expressions, making further assumptions about the potentials. We firstly consider the case of scalar potentials describing small departures from homogeneity, implementing a perturbative approach Casimir.
We compare and relate the results to those corresponding to two Dirichlet mirrors with non planar geometries. On the other hand, if the space-dependence of the properties is smooth, a DE for the energy becomes a useful tool. We use the perturbative results to obtain the usual PFA and its NTLO for this particular system. This paper is organized as follows: in Sect. II, we derive the would-be Lifshitz formula for the system. Then, in Sect. III, we present the expansion up to the second order in the departures from homogeneity, and discuss some particular limiting cases. In Sect. IV we apply the previous result to the calculation of the DE to the second order in derivatives.
Section V contains our conclusions.

II. LIFSHITZ FORMULA FOR NON HOMOGENEOUS PLATES
We derive here an expression for the vacuum energy corresponding to a quantum real scalar field ϕ(x) in 3 + 1 dimensions, coupled to two non-homogeneous zero-width, infinite, parallel plates. It may be regarded as a Lifshitz formula but for zero-width inhomogeneous plates, or even as a special case of the TGTG formula [10]. The kind of model we deal with may be defined in terms of its Euclidean action S(ϕ), which is assumed to be of the form: with spacetime coordinates x = (x 0 , x 1 , x 2 , x 3 ), and spatial coordinates x = (x 1 , x 2 , x 3 ). The 'potential' V accounts for the presence of two plates, denoted by R and L, and located on the planes x 3 = l 2 and x 3 = − l 2 , respectively. Specifically, that function is assumed to be of the form: where x = (x 1 , x 2 ) are coordinates on the mirrors. We could also include a bulk potential V bulk (x 3 , x ) between mirrors. This is a very interesting problem that raises mathematical and physical questions about the proper derivation of finite Casimir forces [11]. We plan to address this issue in a forthcoming paper. It is worth to stress that the potentials V R,L on the plates mimic inhomogeneities of the mirrors, that could be formed by a juxtaposition of homogeneous materials. Other kind of situations, like frequency-dependent electromagnetic properties, would require, to be modelled, the presence of terms in the action which are nonlocal in time.
In the approach that we shall follow to derive the generalized Lifshitz formula, the vacuum energy will be obtained from the functional integral: Indeed, assuming the system to be defined within a 'time interval': − T 2 < τ < T 2 and considering the T → ∞ limit, since the potential has been assumed to be time-independent, the leading behaviour of the effective action Γ is Γ ∼ T × E, E being the vacuum energy.
Thus, we may relate E to a functional determinant, by using the formal result for Γ which, ignoring irrelevant factors, is given by: with ∂ 2 ≡ ∂ µ ∂ µ , µ = 0, 1, . . . , d. Therefore where and the determinant and trace are understood to be in the functional sense, for operators acting on functions of x.
Our strategy to derive the generalized Lifshitz formula consists of 'dimensionally reducing' the determinant appearing in (5). By this, we mean transforming it into one where the x 3 coordinate has been dealt with. Equivalently, at the level of the functional trace, it amounts to tracing out the x 3 coordinate, writing the result in terms of operators now acting on ). That can be done by a suitable application of GY theorem, as follows. We first assume the system to be defined within a large 'box' along the x 3 direction, namely: − L 2 < x 3 < L 2 where L >> l, and we shall take the L → ∞ limit at the end of the calculation. To proceed, we will need to assume some boundary conditions on x 3 = ± L 2 . We shall, for the sake of simplicity, use Dirichlet conditions 2 .
We first rewrite the operator K introduced above in a way which makes it simpler to perform the dimensional reduction: where: with The reason for this seemingly formal splitting is that x 3 is the variable about which the functional determinant of K will be reduced. To each value of x 3 , we may associate an operator kernel along the x coordinates; indeed: where The corresponding result for the determinant is usually presented in terms of its ratio with a 'reference' operator K 0 , which in our case we assume to correspond to a vanishing potential. Using this ratio instead of just K in (5) is indeed convenient, since it then produces the energy of the system taking as a reference the energy of the vacuum in the absence of the plates.
The ratio between determinants may be put in terms of two determinants of dimensionally reduced operators [12]: where ψ(x d ) and ψ 0 (x d ) are operatorial solutions to the homogeneous equations The matrix elements of these solutions may naturally be denoted by: (and analogously for ψ 0 ). The initial conditions on the solutions to the homogeneous equations are We need a more explicit solution to Eq. (13); that can be done by first converting it to a first order system, by introducing which renders the original second-order equation into with where I denotes the identity operator. Note that Eq. (17) is a Schroedinger-like equation in which the coordinate x 3 plays the role of time, and H the role of a 'Hamiltonian'. In terms of the evolution operator U, we then have the solution: with the path-ordering operator P, which acts in the same way as the time-ordering operator, but with x 3 playing the role of the time.
Equipped with the solution for Ψ just presented, we note that or, using indices A and B, which can assume the values 1 or 2, to distinguish the 4 (opera- we see that This is a more explicit form of the reduction, where we still need to take the L → ∞ limit. To do this we note that, since V vanishes for |x 3 | > l 2 , we may write: where the → 0 + limit has been introduced, since the potential is discontinuous at ± l 2 . The explicit form of U (0) (x 3 , x 3 ) may be found by exponentiation, because it corresponds to an x 3 -independent potential: Using this result, a rather lengthy but otherwise straightforward calculation shows that, when L → ∞, the ratio between determinants may be written in terms of µ AB , a shorthand notation for the matrix elements as follows: The vacuum energy, referred to the vacuum in the absence of the plates, is then given by In the previous expression, no use has been made yet of the precise form of the potential in the region occupied by the plates: this will determine the objects we have denoted by µ AB .
Dealing with the effect of the δ functions, one can derive the explicit form: which, when inserted in (28), yields for the energy a result with the structure: with E L (E R ) denoting the self-energy of the L (R) plate: where, we recall, H 0 = −∇ 2 + k 2 0 , and E I is the interaction energy, responsible for the Casimir force between mirrors. It is given by a generalized form of Lifshitz formula: with the operatorial 'reflection coefficients': Note that, except when the potentials are uniform, i.e., independent of x , these reflection coefficients involve non commuting objects in their definitions, since H 0 depends on the planar Laplacian operator. That makes it difficult to obtain more explicit expressions, unless extra assumptions are made.

CONSTANT POTENTIALS
Let us now expand Eq.(32), the general expression for the interaction part of the vacuum energy, up to the second order in the departure from constant functions: where v L and v R are constants. We assume they are chosen in such a way that the spatial average of each departure vanish. The expansion for the energy is then: where: where L 2 is the area of the plates, and we have introduced Note that the Dirichlet limit is obtained as σ L,R (ρ) → 1. Eq.(36) is easily obtained from Eq.(32) by setting V L,R = v L,R in the reflection coefficients of Eq.(33).
The first-order term, on the other hand, vanishes. Indeed, it has the form: We have adopted above some conventions that we shall continue to use: a shorthand notation for the integration over x , and indices from the beginning of the Greek alphabet to denote each one of the two mirrors, namely, α = L, R.
Now, the functional derivatives in (38) are evaluated for (simultaneously) vanishing η L and η R , and are thus x -independent. Therefore, The second-order term, E (2) , is a quadratic form in the departures, where the γ αβ , being second functional derivatives at vanishing departures, can only depend on the differences between the arguments. Moreover, one can also show that the depend just on the modulus of x − x , since for constant potentials the plates are homogeneous and isotropic.
It is convenient to use, in what follows, Fourier transforms. We use a tilde on the Fourier transformed version of an object, and analogous conventions for Fourier transformation along the three coordinates x 0 , x 1 , x 2 (≡ x ), on the space-time of the mirrors. As a technical remark: it is convenient, since the expressions become more symmetrical, to use space-time dependent departures, and to regard them as time-independent at the end of the calculation. Thus, where, having in mind the static limit mentioned above, To compute theγ αβ , we need to expand the reflection coefficients in Eq.(33) up to second order in η L,R , then insert the result in Eq.(32), and keep the quadratic terms. The expansion of the reflection coefficients, which has of course the same form for each plate, may be written as follows (we only write the one for the L plate, the expansion for the other plate is obtained by replacing L → R below): where: The form of the order n term, for any n ≥ 1, is in fact: Inserting the previous expansion for each one of the plates to the second order, and expanding the logarithm accordingly, we may write down the different components ofγ αβ more explicitly. The first ones we write down involve a departure in just one of the mirrors, to the second order: and then we have the ones where each departure appears to the first order, which are symmetrical:γ LR =γ RL : .
The term proportional toγ LL (RR) describes corrections to the Casimir energy produced by the inhomogeneities of the L(R) plate, while the term proportional toγ LR is the relevant one for the computation of lateral forces and torques induced by the interaction between the inhomogeneities on both mirrors.
The rather complex expressions can be simplified for "quasi-Dirichlet" mirrors for which x L,R 1. In this limit we have σ L,R (ρ) 1 − ρ/x L,R and therefore the form factorsγ αβ can be approximated by and where we omitted a k -independent term inγ LL andγ RR . We can see that in this limit the γ αβ are all of the same order of magnitude if x R x L .
It is interesting to compare these results with the one for curved Dirichlet mirrors. For simplicity we will compare the interaction energy between Dirichlet mirrors (a curved mirror in front of a planar one) with that of a Dirichlet mirror (R) in front of a quasi-Dirichlet planar mirror with inhomogeneities (L). The form factor for the former configuration, that we will denote here byγ G (k ) is similar toγ LL [13] : Note that the integrands ofγ LL andγ G differ in a term proportional to e −|p +2lk | , and therefore they have the same behavior in the limit |k |l 1. Indeed, one can easily show that both are proportional to |k |l in this limit. Therefore, one can mimic a perfect mirror with non-planar geometry with inhomogeneities on a planar mirror. The departure from the planar geometry of a Dirichlet mirror, η G (x ), should be proportional to the departure from the constant potential η L (x ). This is valid when the scale of variation of the geometry (and of the inhomogeneities) is much smaller that the distance between mirrors. The opposite limit will be studied in the next section.

IV. DERIVATIVE EXPANSION
Let us now consider the case in which the inhomogeneities are smooth, that is, its scale of variation is much larger than l. We may extract from the results of the previous Section the would be zero-order in the DE ('PFA') term. Indeed, from Eq.(36) we find with In Fig. 1 we plot the function f (x L , ∞), that describes the PFA to the interaction between a Dirichlet (R) and an inhomogeneous mirror (L). As expected, f vanishes in the limit of transparent mirrors (x L → 0), and tends to its Dirichlet value (−π 4 /45) for x L 1. It interpolates monotonically between these two extreme cases. linearly as x L → 0, and tends to −π 4 /45 as x L → ∞.
To obtain the second-order term in a DE, which we denote by E DE , we follow our previous works [2,3]. We need to extract the term which is quadratic in k from eachγ αβ . Namely, for small k :γ We have found: where: c LL = c RR | R↔L , and: (58) It is again instructive to analyze the case of quasi-Dirichlet mirrors. As before, we will consider the case in which the R mirror is perfect (x R → ∞), and the L mirror is quasiperfect. Expanding the leading order result for x L 1 we obtain where ... denotes the mean value over the surface. On the other hand, in this limit As a consequence, the NTLO correction reads E (2) DE 9.14 96π 2 l 3 d 2 x These results can be written in a simpler way by introducing a sort of "conductivity" ρ L = V −1 L (note that ρ L → 0 for perfect mirrors) E DE A = − π 2 1440 l 3 1 − 3 l ρ L + 9.14 96π 2 l 3 (∇ρ L ) 2 .
V. CONCLUSIONS In this paper we evaluated the vacuum energy for a scalar field in the presence of two inhomogeneous thin plates. The interaction between the field and the plates is modelled by potentials localized on the plates. The calculation generalizes previous works on δ-potentials and can be considered as a toy model for the interaction of the electromagnetic field with microstructured flat mirrors.
From a mathematical point of view, we used the GY theorem to compute the corresponding functional determinant, and arrived to a formal result for the vacuum energy, Eq.(32), that can be thought as a generalized Lifshitz formula, in which the reflection coefficients are replaced by nonlocal operators that become local for homogeneous plates.
We have obtained explicit integral expressions for the vacuum energy in two different situations that involve either small or smooth inhomogeneities. For small inhomogeneities, we compared the result with the corresponding one for Dirichlet plates with non trivial geometries, pointing out that when the scales of variation of the geometry and the inhomogeneity are much smaller than the distance between mirrors, geometry can traded off by inhomogeneity. For smooth inhomogeneities, we used the DE as a expansion to obtain the NTLO correction to the PFA approximation.
The formalism developed in this paper can be adapted to compute the vacuum energy in the electromagnetic case, including an inhomogeneous medium between plates. The calculation of Casimir-Polder forces on atoms near an inhomogeneous plate is also of interest.
We plan to address these issues in a forthcoming work.