Unifying Theory for Casimir Forces: Bulk and Surface Formulations

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Article Unifying Theory for Casimir Forces: Bulk and Surface Formulations

Giuseppe Bimonte 1,2,* and Thorsten Emig 3

1 Dipartimento di Fisica E. Pancini, Università di Napoli Federico II, Complesso Universitario di Monte S. An-gelo, Via Cintia, I-80126 Napoli, Italy 2 INFN Sezione di Napoli, I-80126 Napoli, Italy 3 Laboratoire de Physique Théorique et Modèles Statistiques, CNRS UMR 8626, Université Paris-Saclay, 91405 Orsay CEDEX, France; [email protected] * Correspondence: [email protected]

Abstract: The principles of the electromagnetic fluctuation-induced phenomena such as Casimir forces are well understood. However, recent experimental advances require universal and efficient methods to compute these forces. While several approaches have been proposed in the literature, their connection is often not entirely clear, and some of them have been introduced as purely numerical techniques. Here we present a unifying approach for the Casimir force and free energy that builds on both the Maxwell stress tensor and path integral quantization. The result is presented in terms of either bulk or surface operators that describe corresponding current fluctuations. Our surface approach yields a novel formula for the Casimir free energy. The path integral is presented both within a Lagrange and Hamiltonian formulation yielding different surface operators and expressions for the free energy that are equivalent. We compare our approaches to previously developed numerical methods and the scattering approach. The practical application of our methods is exemplified by the derivation of the Lifshitz formula. Keywords: casimir forces; fluctuation induced interactions; stress tensor; path integral Citation: Bimonte, G.; Emig, T. Unifying Theory for Casimir Forces: Bulk and Surface Formulations. Universe 2021, 7, 225. https://doi. 1. Introduction org/10.3390/universe7070225 The interaction induced by quantum and thermal fluctuation of the electromagnetic field is an everyday phenomenon that acts between all neutral objects, both on atomic Academic Editors: Alexei A. and macroscopic scales [1–5]. For the Casimir interaction between macroscopic bodies, Starobinsky and Gerald B. Cleaver the last two decades have witnessed unparalleled progress in experimental observations and the development of novel theoretical approaches [6,7]. In most of the recent theoreti- Received: 30 May 2021 cal approaches, the computation of Casimir forces between multiple objects of different Accepted: 30 June 2021 Published: 4 July 2021 shapes and material composition has been achieved by the use of scattering methods or the so-called TGTG formula [8–16]. These approaches have the advantage of relatively

Publisher’s Note: MDPI stays neutral low numerical effort; they are rapidly converging and can achieve in principle any desired with regard to jurisdictional claims in precision [17–22]. Another merit of these methods is the exclusion of UV divergencies by published maps and institutional affil- performing the subtraction analytically before any numerical computation. Other efficient iations. approaches that have been developed before the scattering approaches include path integral quantizations where the boundary conditions at the surfaces are implemented by delta functions [23]. These approaches are limited to scalar fields with Dirichlet or Neumann boundary conditions [24], or the electromagnetic field with perfectly conducting boundary conditions [25], with the exception of a similar approach for dielectric boundaries [26]. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. However, such analytical (and semi-analytical) methods have been restricted to symmetric This article is an open access article and simple shapes, like spheres, cylinders or ellipsoids [27–31]. Geometries where parts of distributed under the terms and the bodies interpenetrate, such as those shown in Figure1a, cannot be studied with scatter- conditions of the Creative Commons ing approaches. For general shapes and arbitrary geometries, new methods are needed. Attribution (CC BY) license (https:// Purely numerical methods based on surface current fluctuations have been developed [32], creativecommons.org/licenses/by/ but they rely on a full-scale numerical evaluation of matrices and their determinants, which 4.0/). complicates these approaches when high precision of the force is required.

Universe 2021, 7, 225. https://doi.org/10.3390/universe7070225 https://www.mdpi.com/journal/universe Universe 2021, 7, 225 2 of 34

Hence, there is a need to develop methods that predict Casimir interactions between objects of arbitrary geometries composed of materials with arbitrary frequency-dependent electromagnetic properties. The Casimir force can be viewed as arising from the interaction of fluctuating currents distributions. In fact, these effective fluctuating electric and magnetic currents can be considered to be localized either in the bulk of the bodies or just on their surfaces. The surface approach relies on the observation that the electromagnetic response of bodies can be represented entirely in terms of their surfaces, known as the “equivalence principle”, which is based on the observation that many source distributions outside a given region can produce the same field inside the region [33]. The surface approach has been introduced in the literature as a method for a purely numerical computation of Casimir interactions [32]. There are two different methods to implement the idea of computing Casimir forces from fluctuating currents. One can either integrate the Maxwell stress tensor over a closed surface enclosing the body, directly yielding the Casimir force, or integrate over all electromagnetic gauge field fluctuations in a path integral, yielding the Casimir free energy. We shall consider both approaches here. Compared to scattering theory-based approaches, the surface formulation has the advantage that it does not require the use of eigenfunctions of the vector wave equation that are specific to the shape of the bodies. Hence, our approach is applicable to general geometries and shapes, including interpenetrating structures. In fact, the power of the surface approach has been demonstrated by numerical implementations in Reference [32], where it was used to compute the Casimir force in complicated geometries. In this paper, we present both the Maxwell stress tensor and path integral-based approaches for the Casimir force and free energy in terms of bulk or surface operators. Our main advancements are • A new, compact and elegant derivation of the Casimir force from the Maxwell stress tensor within both a T-operator approach and a surface operator approach; • A new surface formula for the Casimir free energy expressed in terms of a surface operator; • A new path integral-based derivation of a Lagrange and Hamiltonian formulation for the Casimir free energy. We also compare the approaches presented here to methods existing in the literature. For the special case of bodies that can be separated by non-overlapping enclosing surfaces, along which one of the coordinates in which the wave equation is separable is constant, our approach is shown to be equivalent to the scattering approach. Our approaches also show the general equivalence of the use of the Maxwell stress tensor in combination with the fluctuation-dissipation theorem on one side and the path integral representation of the Casimir force on the other side. As the most simple application of our approaches, we re-derive the Lifshitz formula for the Casimir free energy of two dielectric slabs. Other analytical applications of our approach will be presented elsewhere. The geometries and shapes to which our approaches can be applied are shown in Figure1a. For comparison, in Figure1b, we display non-penetrating bodies to which scattering theory-based approaches are limited. In general, we assume a configuration composed of N bodies with dielectric functions er(ω) and magnetic permeabilities µr(ω), r = 1, ... , N. The bodies occupy the volumes Vr with surfaces Σr and outward pointing surface normal vectors nˆ r. The space with volume V0 in between the bodies is filled by matter with dielectric function e0(ω) and magnetic permeability µ0(ω). Universe 2021, 7, 225 3 of 34

(a) (b)

Figure 1. Conﬁguration of bodies: (a) general shapes and positions that can be studied with the approaches presented in this work, (b) non-penetrating conﬁgurations that can be studied within the scattering approach.

2. Stress-Tensor Approach 2.1. Bulk and Surface Expressions for the Force Consider a collection of N magneto-dielectric bodies in vacuum. In the stress-tensor (bare r) approach, the (bare) Casimir force Fi | on body r is obtained by integrating the expectation value T of the Maxwell stress tensor h iji 1 1 T (x) = E (x)E (x) + H (x)H (x) δ [ E (x)E (x) + H (x)H (x) ] (1) h ij i 4π h i j i h i j i − 2 ij h k k i h k k i over any closed surface Sr drawn in the vacuum, which surrounds that body (but excludes all other bodies): (bare r) 2 Fi | = d σ nˆ j(x) Tji(x) , (2) ISr h i where nˆ is the unit normal oriented outside Sr, and the angular brackets denote the expectation value taken with respect to quantum and thermal fluctuations. For a system in thermal equilibrium at temperature T, the (equal-time) expectation values of the products of field components (at points x and x0 in the vacuum region) are provided by the fluctuation- dissipation theorem [34,35]:

∞ ˆ ˆ (EE) Ei(x)Ej(x0) = 2kBT ∑ 0 ij (x, x0; i ξn) , h i n=0 G ∞ ˆ ˆ (HH) Hi(x)Hj(x0) = 2kBT ∑ 0 ij (x, x0; i ξn) , (3) h i n=0 G

where ξn = 2πnkBT/¯h are the Matsubara imaginary frequencies, and the prime in the summations mean that the n = 0 term is taken with a weight of one half. When the r.h.s of the above equations are plugged into Equation (1), one obtains for Tij(x) a (αβ) h i formally divergent expression, since the Green functions (x, x ; i ξ ) are singular in the Gij 0 n Universe 2021, 7, 225 4 of 34

coincidence limit x = x0. This divergence can however be easily disposed of by noticing that the Green’s functions admit the decomposition:

(αβ) (αβ;0) (αβ) (x, x0; i ξ ) = (x x0; i ξ ) + Γ (x, x0; i ξ ) , (4) Gij n Gij − n ij n

(αβ;0) (αβ) where (x x ; i ξ ) is the Green’s function of free space, while Γ (x, x ; i ξ ) de- Gij − 0 n ij 0 n scribes the effect of scattering of electromagnetic fields by the bodies. In the coincidence (αβ;0) (αβ) limit, only (x x ; i ξ ) diverges, while Γ (x, x ; i ξ ) attains a finite limit. When the Gij − 0 n ij 0 n decomposition in Equation (4) is used to evaluate the r.h.s. of Equation (1), one finds that the expectation value of the stress tensor is decomposed in a way analogous to Equation (4):

( ) T (x) = T 0 (x) + Θ (x) , (5) h ij i h ij i ij ( ) where T 0 (x) is the divergent expectation value of the stress tensor in empty space, and h ij i Θij(x) is the ﬁnite expression

∞ kBT (EE) (HH) Θ (x) = 0 Γ (x, x; i ξ ) + Γ (x, x; i ξ ) ij 2π ∑ ij n ij n n=0 h 1 (EE) (HH) δ Γ (x, x; i ξ ) + Γ (x, x; i ξ ) . (6) − 2 ij kk n kk n ( ) Since the divergent contribution T 0 (x) is independent of the presence of the bodies, h ij i one can just neglect it and then one obtains the following ﬁnite expression for the Casimir force on body r due to the presence of the other bodies,

(r) 2 Fi = d σ nˆ j(x) Θji(x) . (7) ISr The further development of the theory starts from the observation that the dyadic (αβ) Green’s functions Γij (x, x0) (for brevity, from now on we shall not display the dependence of the Green’s functions on the Matsubara frequencies ξn) can be expressed in two distinct possible forms. The ﬁrst representation is general, since it is valid for arbitrary constitutive equations of the magneto-dielectric materials constituting the bodies, which can possibly be non- (αβ) homogeneous, anisotropic, and non-local. For all points x and x0, it expresses Γij (x, x0) in the form of an integral of the T-operator Tˆ (for its deﬁnition, see Appendix B.1) over the volume V occupied by all bodies:

N (αβ) 3 3 (αρ;0) (ρσ) (σβ;0) Γij (x, x0) = ∑ d y d y0 ik (x y)Tkl (y, y0) lj (y0 x0) . (8) Vr V G − G − r,r0=1 Z Z r0 The above formula has a simple intuitive interpretation, if one recalls that according to its definition, the T-operator provides the polarization induced in the volume of the bodies when they are immersed in the electromagnetic field generated by a certain distribution of external sources. The second representation is less general than Equation (8) because it applies only to magneto-dielectric bodies that are (piecewise) homogeneous and isotropic 1. It expresses (αβ) ˆ 1 Γij (x, x0) in the form of an integral of the surface operator M− defined in Equation (A44), Universe 2021, 7, 225 5 of 34

over the union Σ = Σr of the surfaces Σr of the bodies. For two points x and x0 both lying 2 (αβ) in the vacuum regionS outside the bodies , the surface representation of Γij (x, x0) reads:

N (αβ) Γ (x, x ) = d3y d3y δ(F (y)) δ(F (y )) ij 0 ∑ 0 r r0 0 − Vr V r,r0=1 Z Z r0 (ρσ) (αρ;0) 1 (σβ;0) (x y)) M− (y, y0) (y0 x0) , (9) × Gik − kl Glj − where Fr(y) = 0 is the equation of the surface Σr. This representation also has a simple intuitive meaning, if one considers that Mˆ 1 (see Appendix B.2 for details) is defined as − − the operator that provides the fictitious surface polarizations that radiate outside the bodies the same scattered electromagnetic field as the one radiated by the physically induced volumic polarization, in response to an external field. The derivations of Equations (8) and (9) are presented in AppendixB. It is apparent that both representations have the same mathematical structure, consisting of a two-sided convolution of a certain kernel (αβ) (αβ;0) (x, x ) with the free-space Green’s functions (x x ): Kij 0 Gij − 0

(αβ) 3 3 (αρ;0) (ρσ) (σβ;0) Γij (x, x0) = d y d y0 ik (x y) kl (y, y0) lj (y0 x0) . (10) ZV ZV G − K G − The only difference between the two representations consists in the expression of , (ρσ) K which in the case of Equation (8) is the three-dimensional kernel Tkl (y, y0) supported in the volume V occupied by the bodies:

(ρσ) (ρσ) (y, y0) = T (y, y0) , (11) Kkl kl (ρσ) while in Equation (9) is the two-dimensional kernel M 1 (y, y ) supported on the K − − kl 0 union Σ of their surfaces: N (ρσ) (ρσ) 1 (y, y0) = δ(Fr(y)) δ(Fr (y0)) M− (y, y0) (12) Kkl − ∑ 0 kl r,r0=1 In both cases, the above equation can be concisely written using the operator notation described in AppendixA: Γˆ = ˆ(0) ˆ ˆ(0) . (13) G K G (αβ) In AppendixC, we prove that the structure of the representation of Γij (x, x0) given in Equation (10), allows to re-express the Casimir force Equation (7) in the following remarkably simple form:

∞ (r) 3 3 (αβ) ∂ (βα;0) Fi = 2kBT ∑0 d y d y0 lj (y0, y; i ξn) jl (y y0; i ξn) . (14) V V K ∂y G − n=0 Z r Z i A crucial role in the derivation of Equation (14) is played by the fact that the kernel K satisﬁes a set of reciprocity relations analogous to those satisﬁed by the Green’s functions:

(ρσ) s(ρ)+s(σ) (σρ) (y, y0) = ( 1) (y0, y) , (15) Kkl − Klk where s(E) = 0 and s(H) = 1. It is possible to verify that the reciprocity relations satisﬁed (αβ;0) (αβ) (self r) by (y y ) and (y, y ) ensure vanishing of the “self-force” F | : Gij − 0 Kij 0 i ∞ (self r) 3 3 (αβ) ∂ (βα;0) Fi | = 2kBT ∑0 d y d y0 ij (y0, y; i ξn) ji (y y0; i ξn) = 0 . (16) V V K ∂y G − n=0 Z r Z r i Universe 2021, 7, 225 6 of 34

This implies that in Equation (14) the y integral is in fact restricted to V V , in 0 − r accord with one’s intuition that the force on body r is due to the interaction with the other bodies. It is possible to present Equation (14) in a more compact and symmetric form, by deﬁning the derivative ∂/∂x of any kernel (y, y ), with respect to rigid translations of r A 0 the r-th body:

∂ ∂ ∂ (y, y0) ψr(y) (y, y0) + ψr(y0) (y, y0) , (17) ∂xr A ≡ ∂y A ∂y0 A

where ψ (y) is the characteristic function of V : ψ (y) = 1 if y V , ψ (y) = 0 if y / V . r r r ∈ r r ∈ r Using ∂/∂xr, we can rewrite Equation (14) as

∞ (r) 3 3 (αβ) ∂ (βα;0) F = kBT ∑0 d y d y0 ij (y0, y; i ξn) ji (y y0; i ξn) . (18) V V K ∂x G − n=0 Z Z r The expression on the r.h.s. of the above formula can be compactly expressed using the operator notation and the trace operation described in AppendixA:

∞ (r) ∂ (0) F = k T 0 Tr ˆ (i ξ ) ˆ (i ξ ) . (19) B ∑ K n ∂x G n n=0 r Depending on whether we use for the kernel Kˆ the T-operator of Equation (8) or rather the surface operator Mˆ of Equation (9), Equation (19) provides us with two distinct − but formally similar representations of the Casimir force, which is expressed either as an integral over the volume V occupied by the bodies or as an integral over their surfaces Σ. One feature of Equation (19) is worth stressing. Since the force is expressed as a trace, Equation (19) can be evaluated in an arbitrary basis, leaving one with complete freedom in the choice of the most convenient basis in a concrete situation. A representation of the Casimir force in the form of a volume integral equivalent to Equation (19) was derived in [11], while the surface-integral representation was obtained in [32]. Equation (19) can be computed numerically for any shapes and dispositions of the bodies, by using discrete meshes covering the bodies. An efﬁcient numerical scheme based on surface-elements methods is described in [32], where it was used to compute the Casimir force in complex geometries, not amenable to analytical techniques.

2.2. Casimir Free Energy In this section, we compute the Casimir free energy of the system of bodies, starting F from the force formula Equation (19). We shall see that the T-operator and the surface-operator approaches lead to two distinct but equivalent representations of the Casimir energy.

2.2.1. T-Operator Approach Plugging into Equation (19) the expression of the T-operator Equation (A36), we ﬁnd that the Casimir force can be expressed in the form:

∞ ∞ (r) ˆ ∂ ˆ(0) 1 ∂ ˆ(0) F = kBT ∑0 Tr T(i ξn) (i ξn) = kBT ∑0 Tr χˆ , (20) ∂xr G 1 χˆ ˆ(0) ∂xr G n=0 n=0 − G where in the last passage, we made use of the fact that the polarization operator χˆ deﬁned in Equation (A24) is invariant under a rigid displacement of the body. The r.h.s. of the above equation can be formally expressed as a gradient:

(r) ∂ F = bare , (21) − ∂xr F Universe 2021, 7, 225 7 of 34

where is the bare free energy: Fbare ∞ ˆ(0) bare = kBT ∑0 Tr log[1 χˆ ] (22) F n=0 − G

Unfortunately, bare is formally divergent. To obtain the ﬁnite Casimir free energy , F (r) F one has to subtract from the divergent self-energies of the individual bodies: Fbare Fself ∞ (r) ˆ(0) self = kBT ∑0 Tr log[1 χˆr ] , (23) F ( n=0 − G )

where χˆr = ψˆrχˆψˆr is the polarizability operator of body r in isolation. In the case of a system composed by two bodies, the renormalized Casimir free energy can be recast in the following TGTG form:

∞ (1) (2) ˆ ˆ(0) ˆ ˆ(0) = bare self self = kBT ∑0 Tr log[1 T1 T2 ] , (24) F F − F − F n=0 − G G

where 1 Tˆr = χˆr , (25) 1 χˆ ˆ(0) − r G is the T-operator of body r in isolation. To prove Equation (24), one notes that for each Matsubara mode the operator identity holds:

(1 ˆ(0)χˆ ) ˆ(0)Tˆ ˆ(0)Tˆ (1 ˆ(0)χˆ ) = ˆ(0)χˆ ˆ(0)χˆ . (26) − G 1 G 1G 2 − G 2 G 1G 2 The above identity in turn allows to prove the following chain of identities:

Tr log[1 ˆ(0)χˆ ] + Tr log[1 ˆ(0) Tˆ ˆ(0)Tˆ ] + Tr log[1 ˆ(0)χˆ ] − G 1 − G 1 G 2 − G 2 = Tr log[(1 ˆ(0)χˆ )(1 ˆ(0) Tˆ ˆ(0)Tˆ )(1 ˆ(0)χˆ )] − G 1 − G 1 G 2 − G 2 = Tr log[(1 ˆ(0)χˆ )(1 ˆ(0)χˆ ) (1 ˆ(0)χˆ ) ˆ(0) Tˆ ˆ(0)Tˆ (1 ˆ(0)χˆ )] − G 1 − G 2 − − G 1 G 1 G 2 − G 2 = Tr log[(1 ˆ(0)χˆ )(1 ˆ(0)χˆ ) ˆ(0)χˆ ˆ(0)χˆ ] = Tr log[(1 ˆ(0)(χˆ + χˆ )] − G 1 − G 2 − G 1G 2 − G 1 2 = Tr log[1 ˆ(0)χˆ] . (27) − G Equating the ﬁrst line with the last line, we obtain the identity:

Tr log[1 ˆ(0)χˆ] − G = Tr log[1 ˆ(0)χˆ ] + Tr log[1 ˆ(0) Tˆ ˆ(0)Tˆ ] + Tr log[1 ˆ(0)χˆ ] . (28) − G 1 − G 1 G 2 − G 2 Upon summing the above identity over all Matsubara modes (with weight one half for the n = 0 term), and then multiplying it by kBT, we ﬁnd:

( ) ( ) 1 + 2 + = , (29) Fself Fself F Fbare which is equivalent to Equation (24). The energy formula Equation (24) was derived in [13] using the path-integral method and in [11], using Rytov’s ﬂuctuational electrodynamics [36].

2.2.2. Surface Operator Approach Now we derive the surface-operator representation of the Casimir energy. To do that, we start from the surface-operator representation of the force, which is obtained by Universe 2021, 7, 225 8 of 34

replacing Kˆ in Equation (19) with minus the inverse of the surface operator Mˆ deﬁned in Equation (A44): ∞ (r) 1 ∂ (0) F = k T 0 Tr Mˆ − (i ξ ) ˆ (i ξ ) . (30) − B ∑ n ∂x G n n=0 r This can also be written as:

∞ (r) ˆ 1 ∂ ˆ ˆ(0) ˆ F = kBT ∑0 Tr M− (i ξn) Π (i ξn)Π , (31) − ∂xr G n=0 where Πˆ is the tangential projection operator deﬁned in Appendix B.2. Now, one notes the identity: ˆ ∂ (0) ∂M Πˆ ˆ (i ξn) Πˆ = , (32) ∂xr G ∂xr which is a direct consequence of Equation (A44) since

N ∂ ˆ ˆ(s) ˆ ∑ Πs Πs = 0 . (33) ∂xr s=1 G

Plugging Equation (32) into Equation (31), we obtain:

∞ (r) 1 ∂ F = k T 0 Tr Mˆ − (i ξ ) Mˆ (i ξ ) , (34) − B ∑ n ∂x n n=0 r The r.h.s. of the above equation can be formally expressed as a gradient:

(r) ∂ ˜ F = bare , (35) − ∂xr F

where ˜ is the bare free energy: Fbare ∞ ˜ ˆ bare = kBT ∑0 Tr log M(i ξn) . (36) F n=0

Similarly to what we found in the T-operator approach, the surface formula of the bare- ˜ energy bare is formally divergent. The finite Casimir free energy is obtained by subtracting F ( ) from ˜ the limit ˜ ∞ of the bare energy when the bodies are taken infinitely apart Fbare Fbare from each other. From Equation (A44), one sees that in the limit of infinite separations, the operator Mˆ approaches the limit Mˆ ∞

N ˆ ˆ M∞ = ∑ Mr , (37) r=1

where Mˆ = Πˆ ( ˆ(r) + ˆ(0)) Πˆ . (38) r r G G r Notice that the surface operator Mˆ r is localized onto the surface Σr of the r-th body. This implies that: Mˆ Mˆ = 0 , for r = s . (39) r s 6 ( ) Using Equation (37), we ﬁnd that ˜ ∞ is the formally divergent quantity: Fbare ∞ N ∞ N ˜ (∞) ˆ ˆ ˜ (r) bare = kBT ∑0 Tr log M∞(i ξn) = ∑ kBT ∑0 Tr log Mr(i ξn) ∑ self . (40) F n=0 r=1 n=0 ≡ r=1 F Universe 2021, 7, 225 9 of 34

( ) (r) The additive character of ˜ ∞ allows to interpret ˜ as representing the (inﬁnite) Fbare Fself the self-energy of the bodies in the surface approach. Upon subtracting Equation (40) from Equation (36), we arrive at the following formula for the Casimir energy:

∞ Mˆ (i ξ ) = k T log det n . (41) B ∑0 ˆ F n=0 M∞(i ξn)

An easy computation shows that

1 1 ( ) Mˆ = 1 + ˆ 0 . (42) ˆ ∑ ˆ rs M∞ r=s Mr G 6 where ( ) ˆ 0 = Πˆ ˆ(0) Πˆ . (43) Grs r G s Substitution of the above formula into Equation (41) results in the following surface formula for the Casimir energy:

∞ 1 ( ) = k T log det 1 + ˆ 0 . (44) B ∑ 0 ∑ ˆ rs F n=0 " r=s Mr G # 6 In the simple case of two bodies, the above formula reduces to:

∞ 1 (0) 1 (0) = kBT 0Tr log 1 ˆ ˆ . (45) F ∑ − Mˆ G12 Mˆ G21 n=0 1 2 The surface formulas for the Casimir energy given in Equations (44) and (45) were not known before and are presented here for the ﬁrst time. Comparison of Equation (45) with Equation (24) reveals the striking similarity of the T-operator and surface-approach representations of the Casimir energy. Indeed we see that both formulas can be written in the form: ∞ (0) (0) = k T 0Tr log 1 ˆ ˆ ˆ ˆ , (46) F B ∑ − K1 G K2G n=0 h i where ˆ is the kernel, which gives the scattering Green’s function of body r in isolation: Kr Γˆ = ˆ(0) ˆ ˆ(0) . (47) r G Kr G 3. Equivalence of the Surface-Formula with the Scattering Formula for the Casimir Energy In the previous sections, we have shown that, both in the T-operator and in the surface approaches, the Casimir energy of two bodies can be expressed by the general F Equation (46). This formula is valid for any shape and relative dispositions of the two bodies, and in particular for two interleaved bodies (see Figure1a). Now we show that when the two bodies can be enclosed within two non-overlapping spheres (see Figure1b), Equation (46) is the same as the well-known scattering formula [8–12]:

∞ (1) (12) (2) (21) = k T 0tr log 1 . (48) F B ∑ − T U T U n=0 h i where (r) is the scattering matrix of body r (see Equation (A76) for the deﬁnition of (r)), T T (rs) are the translation matrices deﬁned in Equation (A82) and tr denotes a trace over U multipole indices. To prove equivalence of Equation (46) with Equation (48), one starts from the observation that the trace operation in Equation (46) involves evaluating the Green functions (αβ;0) ↔ (y, y0) at points y and y0, one of which (call it y ) belongs to body 1, while the other G 1 Universe 2021, 7, 225 10 of 34

(call it y2) belongs to body 2. For two bodies that can be separated by non-overlapping spheres, it is warranted that y1 X1 < y2 X1 and y2 X2 < y1 X2 , where X1 | − | | − | | (1)− | (2)| − | and X2 are the positions of the centers of the spheres S and S , respectively, and d = X2 X1 is their distance. This condition satisﬁed by y1 and y2 ensures that it is | − | (αβ;0) legitimate to express ↔ (y , y ) (with r = s = 1, 2) by the partial-wave expansion (see r s 6 AppendixD): G

(αβ;0) s( ) (α reg) (β out) ↔ β | (yr, ys) = λ( 1) ∑ Φplm (yr Xr) Φpl | m (ys Xr) G − plm − ⊗ − − s( ) (α reg) (rs) (β reg) β | | = λ ( 1) ∑ ∑ Φplm (yr Xr) pl;p l (d) Φp l m(ys Xs) , (49) − − ⊗ U 0 0 0 0− − plm p0l0

(reg/out) where Φ (y X ) are a basis of regular and outgoing spherical waves with origin at plm r − r Xr. When the above expansion is substituted into Equation (46) and the trace is evaluated, one ﬁnds that can be recast in the form: F ∞ = kBT ∑ 0 tr log[1 ] . (50) F n=0 − N

where is the matrix of elements: N (21) (12) plm;p l m (d) (d) N 0 0 0 ≡ ∑ ∑ ∑ Upl,p00l00 Up000l000;p0000l0000 p00l00 p000l000m000 p0000l0000 (αµ) s(α) 3 3 (α reg) ↔ (µ reg) λ ( 1) d y1 d y0 Φ | (y1 X1) (y1, y0 ) Φ | (y0 X1) ∑ 1 p00l00 m 1 1 p000l000m000 1 × α,µ − ZV1 ZV1 − − · K · − (νβ) s(ν) 3 3 (ν reg) ↔ (β reg) λ ( 1) d y2 d y0 Φ | (y2 X2) (y2, y0 ) Φ | (y0 X2) . ∑ 2 p0000l0000 m000 2 2 p0l0m0 2 × β,ν − ZV2 ZV2 − − · K · −

Recalling the formula Equation (A79) for the scattering matrices (r) of the two bodies, T we see that is the matrix: N = (21)(d) (1) (12)(d) (2) . (51) N U T U T Upon substituting the above expression into the r.h.s. of Equation (50), and using cyclicity of the trace, we see that Equation (50) indeed coincides with the scattering formula Equation (48).

4. Path Integral Approach As in the previous sections, we consider again N dielectric bodies occupying the volumes Vr, r = 1, ... , N, bounded by surfaces Σr. Their electromagnetic properties are described by the dielectric functions e(r) and magnetic permeability µ(r). The bodies are embedded in a homogeneous medium occupying the outside volume of the bodies, V0, with dielectric function e(0) and magnetic permeability µ(0). In the Euclidean path integral quantization of the electromagnetic ﬁeld, the Casimir free energy at ﬁnite temperature T can be obtained as

∞ (κn) = k T 0 log Z , (52) F − B ∑ (κ ) n=0 Z∞ n

where the sum runs over the Matsubara momenta κn = 2πnkBT/¯hc, with a weight of 1/2 for n = 0. The partition function is given by a path integral that we shall derive now. Z The partition function describes the conﬁguration of inﬁnitely separated bodies and Z∞ subtracts the self-energies of the bodies from the bare free energy. In the following two Universe 2021, 7, 225 11 of 34

sections, we shall derive both a Lagrangian and a Hamiltonian path integral expression of the partition function. In both cases, we employ a ﬂuctuating surface current approach. A path integral approach that is based on bulk currents can be found, e.g., in Reference [10].

4.1. Lagrange Formulation

The action of the electromagnetic ﬁeld coupled to bound sources Pind in the absence of free sources is in general given by

1 1 S = d3x e E2 B2 + P E . (53) EM 2 x − µ ind · Z x In the following, we express the action in terms of the gauge field A choosing the transverse or temporal gauge with A0 = 0. The functional integral will then run over A only. The electric field is given by E = ikA κA and the magnetic field by B = A. → − ∇ × Then the action in terms of the induced sources at fixed frequency κ is given by

N ˆ 1 3 2 2 1 2 3 S[A] = d x A exκ + ( A) κ ∑ d x A Pr , (54) − 2 3 µ ∇ × − V · ZR x r=1 Z r

for fluctuations A of the gauge field, and induced bulk currents Pr inside the objects. The inverse of the kernel of the quadratic part of this action is given by the Green tensor G(x, x0), which is defined by

(AA) (AA) 1 ↔ 2 ↔ (x, x0) + exκ (x, x0) = 4π 1 δ(x x0) . (55) ∇ × µx ∇ × G G −

For spatially constant e and µ with body r, this yields the free Green’s tensor

(AA;r) √erµrκ x x0 ↔ 1 e− | − | (x, x0) = µr 1 2 , (56) G − erµrκ ∇ ⊗ ∇ x x | − 0| which is symmetric, reflecting reciprocity. From the relation between the gauge field A and (AA;r) (EE;r) the electric field E follows the relation κ2↔ (x, x ) = ↔ (x, x ), which allows to − 0 0 compare the results below to those of the stress-tensor-basedG G derivation. Next, we define the classical solutions Ar of the vector wave equation in each region Vr, obeying A + e µ κ2A = κµ P (57) ∇ × ∇ × r r r r − r r We use this definition together with the fact that A has no sources inside Vr, i.e., obeys above wave equation with vanishing right-hand side, to rewrite the source terms of Equation (54) as

3 3 1 2 κ d x A Pr = d x A Ar + erκ Ar (58) − V · V · µ ∇ × ∇ × Z r Z r r 1 = d3x [A ( A ) ( A) A ] µ r r r ZVr · ∇ × ∇ × − ∇ × ∇ × · 1 = d3x [ (( A ) A) (( A) A )] µ r r r ZVr ∇ · ∇ × × − ∇ · ∇ × × 1 = d3x [n (( A ) A) n (( A) A )] µ r r r r r ZΣr · ∇ × × − · ∇ × × 1 = d3x [A (n ( A )) + ( A) (n A )] . µ r r r r r ZΣr · × ∇ × ∇ × · × Now we have to consider the electric ﬁeld E = κA only on the surfaces Σ . However, − r the values of the electric ﬁeld E and its curl E are those when the surface is approached ∇ × Universe 2021, 7, 225 12 of 34

from the inside, denoted by E and ( E) . It is important to realize that in the above − ∇ × − surface integral, A and A multiply vectors that are tangential to the surface, and hence ∇ × only the tangential components of A and A contribute to the integral. Hence, we can ∇ × use the continuity conditions of the tangential components of E and H,

1 1 nr E = nr E+ , nr ( E) = nr ( E)+ , (59) × − × µr × ∇ × − µ0 × ∇ ×

to write the source terms as

3 1 3 κ d x A Pr = d x[A (nr ( Ar)) + ( A) (nr Ar)] µ − − − ZVr · r ZΣr · × ∇ × ∇ × · × 3 1 1 = d x A+ (n ( A )) + ( A)+ (n A ) , (60) µ · r × ∇ × r µ ∇ × · r × r ZΣr r 0 where the first form applies to A inside the objects and the second form to A outside the objects. There is another advantage of having expressed the latter integrals in terms of the values of A and A when the surfaces are approached from either the outside or the ∇ × inside of the objects. In the region V , the field A A is fully determined by its values on 0 ≡ 0 the surfaces Σr and the dielectric function e0 and permeability µ0, which are constant across V0. When integrating out A0, in fact, one computes the two-point correlation function of A+ and ( A)+ on the surfaces Σ , and hence the behavior of A inside the regions V ∇ × r 0 r with r > 0 is irrelevant. Following the same arguments for A A inside the objects, the ≡ α behavior of Ar outside of region Vr is irrelevant for computing the correlations of A and − ( A) on the surfaces Σr. Hence, we can replace in the action Sˆ[A, Ar ] the spatially ∇ × − { } dependent ex by e0 when the coupling of A0 to the surface fields Ar is represented by the second line of Equation (60), and similarly replace ex by er when the coupling of Ar to the surface fields Ar is represented by the first line of Equation (60). That this is justified can also be understood as follows. The field A0 in region V0 can be expanded in a basis of functions that obey the wave equation with e0. The same can be done for Ar in the interior of each object, i.e., Ar can be expanded in a basis of functions that obey the wave equation with er in Vr. For each given set of expansion coefficients in V0 there are corresponding coefficients within each region Vr that are determined by the continuity conditions at the surfaces Σr. The functional integral over A then corresponds to integrating over consistent sets of expansion coefficients that are related by the continuity conditions. The two-point correlations of A+ and ( A)+ on the surfaces Σ are then ∇ × r fully determined by the integral over the expansion coefficients of A0 in V0 only, and the interior expansion coefficients play no role. Equivalently, the two-point correlations of A and ( A) on the surfaces Σr are then fully determined by the integral over − ∇ × − the expansion coefficients of Ar in Vr only, and now the exterior expansion coefficients are irrelevant. Hence, in the functional integral, the integration of A can be replaced by N + 1 integrations over the fields Ar, r = 0, ... , N, where each Ar is allowed to extend over unbounded space with the action for a free field in a homogeneous space with er, µr. However, it is important that the correct of the two possible forms of the surface integral in Equation (60) is used. The multiple counting of degrees of freedom that results from N + 1 functional integrations poses no problem since the (formally infinite) factor in the partition function cancels when the Casimir energy is computed from Equation (52). With this representation, we can write the partition function as a functional integral over A, separately in each region Vr, and the surface fields Ar on body r, leading to the partition function

N N ˆ (κ) = ∏ Ar ∏ Ar exp βS[ Ar , Ar ] . (61) Z r=0 Z D r=1 Z D − { } { } Universe 2021, 7, 225 13 of 34

with the action

N ˆ 1 3 2 2 1 2 S[ Ar , Ar ] = d x Ar erκ + ( Ar) (62) { } { } − 2 ∑ 3 µ ∇ × r=0 ZR r N 1 1 + d3x A (n ( A )) + ( A )(n A ) ∑ µ 0 r × ∇ × r µ ∇ × 0 r × r r=1 ZΣr r 0 N 1 1 + d3x A (n ( A )) + ( A )(n A ) ∑ µ r r × ∇ × r µ ∇ × r r × r r=1 ZΣr r r Now, the ﬂuctuations A can be integrated out easily, noting that the two point correlation function Ar(x)A (x ) = 0 for all r, r = 0, ... , N with r = r , and for equal- h r0 0 i 0 6 0 region correlations

(AA;r) ↔ A (x)A (x0) = (x, x0) (63) h r,j r,k i G jk (AA;r) ↔ ( A) (x)A (x0) = (x, x0) (64) h ∇ × r,j r,k i ∇ × G jk (AA;r) (AA;r) ↔ ↔ A (x)( A) (x0) = (x, x0) = (x, x0) (65) h r,j ∇ × r,k i − × ∇ ∇ × G jk G jk (AA;r) ↔ ( A) (x)( A) (x0) = (x, x0) h ∇ × r,j ∇ × r,k i − ∇ × × ∇ G jk (AA;r) ↔ = (x, x0) (66) ∇ × ∇ × G jk

(AA;r) (AA;r) where always acts on the argument x of ↔ and the notation ↔ means ∇ (AAG;r) (AA;r) ∇ × G that acts column-wise on the tensor ↔ whereas ↔ means that acts ∇ (AA;r) G G × ∇ ∇ row-wise on the tensor ↔ . We obtain for the partition function G N N β 3 3 (κ) = A exp d x d x0A (x)L (x, x0)A (x0) ∏ r 2 ∑ r r r Z r=1 Z D "− r=1 ZΣr ZΣr N + d3x d3x A (x)M (x, x )A (x ) . (67) ∑ 0 r rr0 0 r0 0 Σr Σ r,r0=1 Z Z r0 !# Universe 2021, 7, 225 14 of 34

with the kernels

(AA;r) 1 ↔ Lr(x, x0) = (x, x0)(nr )(~ n0 ~ ) µ2 ∇ × ∇ × × · r × · r G (AA;r) ↔ + (x, x0)(nr ( ))(~ n0 ~ ) ∇ × G × ∇ × · r × · (AA;r) ↔ + (x, x0)(nr )(~ n0 ( 0 ~ )) ∇ × G × · r × ∇ × · (AA;r) ↔ + (x, x0)(n ( ))(~ n0 ( 0 ~ )) r × ∇ × · r × ∇ × · G (AA;0) 1 ↔ M (x, x0) = (x, x0)(nr )(~ n0 ~ ) rr0 2 r0 µ0 ∇ × ∇ × G × · × · (AA;0) 1 ↔ + (x, x0)(nr ( ))(~ nr0 ~ ) µ0µr ∇ × G × ∇ × · 0 × · (AA;0) 1 ↔ + (x, x0)(nr )(~ n0 ( 0 ~ )) µ µ r0 0 r0 ∇ × G × · × ∇ × · (AA;0) 1 ↔ + (x, x0)(nr ( ))(~ n0 ( 0 ~ )) (68) µ µ r0 r r0 G × ∇ × · × ∇ × ·

(AA;r) ↔ where (x, x0) is the free Green function of Equation (56), and the arrow over the placeholderG indicates to which side of the kernel M acts. This notation implies that the · derivatives are taken before the kernel is evaluated with x and x0 on the surfaces Σr.

4.2. Hamiltonian Formulation The representation of the partition function in the previous subsection sums over all configurations of the surface fields Ar, and the action depends both on Ar and the tangential part of its curl, which is functionally dependent on Ar. Hence, the situation is similar to classical mechanics where the Lagrangian depends on the trajectory q(t) and its velocity q˙(t). The Lagrangian path integral runs then over all of path q(t) with q˙(t) determined by the path automatically. To obtain a representation in terms of a space of functions that are defined strictly on the surfaces Σr only, it would be useful to be able to integrate over Ar and its derivatives independently. In classical mechanics, this is achieved by Lagrange multipliers that lead to a Legendre transformation of the action to its Hamiltonian form. Here the situation is similar. To see this, it is important to realize that the bilinear form described by Lr is degenerate on the space of functions over which the functional integral runs, i.e., d3x d3x A(x)L (x, x )A(x ) = 0 for all A(x) that are Σr Σr 0 r 0 0 A + 2A = regular solutions of the vector waveR equationR erµrκ 0 inside region Vr. (reg,r) ∇ × ∇ × With a basis A (x) for this functional space, the elements of L can be expressed as { ν } r 3 3 (reg,r) (reg,r) Lr(ν, ν0) = d x d x0 Aν (x)Lr(x, x0)A (x0) ν0 ZΣr ZΣr 1 ( r) ( r) = d3x ( A (x)) n A reg, (x) + A (x) n A reg, (x) 2 ν0 r ν ν0 r ν µr Σr ∇ × × × ∇ × Z h i 1 ( r) ( r) = d3x A (x) A reg, (x) A reg, (x)( A (x)) 2 ν0 ν ν ν0 µr Vr ∇ × ∇ × − ∇ × ∇ × Z h i = 0 (69) Universe 2021, 7, 225 15 of 34

where we used the relations of Equation (58), and deﬁned

(AA;r) 3 ↔ (reg,r) Aν (x) = d x0 (x, x0) nr0 A (x0) 0 ∇ × × ν0 ZΣr G (AA;r) ↔ (reg,r) + (x, x0) nr0 0 A (x0) , (70) G × ∇ × ν0 and made use of the fact that A (x) is also a solution of the vector wave equation inside ν0 Vr. This implies that the kernel Lr can be ignored in the above functional integral over regular waves Ar inside the objects. However, the appearance of the kernel Lr is important in what follows. Let us consider the part of the action Sˆ[ Ar , Ar ] which, after functional integration over Ar, generates { } { } the kernel Lr. It is given by

1 3 2 2 1 2 Sr = d x Ar erκ + ( Ar) − 2 3 µ ∇ × ZR r 1 + d3x [A (n ( A )) + ( A )(n A )] . (71) µ r r r r r r r ZΣr × ∇ × ∇ × × The exponential of this action can be written as a functional integral over two new vector ﬁelds Kr and Kr0 that are deﬁned on the surfaces Σr and are tangential to the surfaces,

exp( βS ) − r (AA;r) β 1 3 3 ↔ = r Kr K0 exp d x d x0 Kr(x) (x, x0) Kr(x0) Z D D r − 2 µ2 · ∇ × ∇ × · I r ZΣr ZΣr G (AA;r) (AA;r) ↔ ↔ + Kr(x) (x, x0) K0 (x0) + K0 (x) (x, x0) Kr(x0) · ∇ × G · r r · ∇ × G · (AA;r) ↔ + K0 (x) (x, x0) K0 (x0) r · · r G 1 3 + d x A (n ( A )) K0 + ( A ) (n A K ) (72) µ r · r × ∇ × r − r ∇ × r · r × r − r r ZΣr where is some normalization coefficient, and we have used K K to indicate that Zr D rD r0 the functional integral extends only over vector fields that are tangential to the surface Σ . H r This representation shows that Ar acts as a Lagrange multiplier. Integration over this field removes the imposed constraints between the dependent tangential fields n A , n r × r r × ( A ) by replacing them with the independent tangential fields K and K , respectively. ∇ × r r r0 Substituting Equation (72) for each object into the expression for the partition in Equation (61), we obtain with Kr = (Kr, Kr0 ) the partition function

N (κ) = A0 ∏ Kr (73) Z Z D r=1 I D N β 3 3 exp[ βS [A , K ]] exp d x d x0K (x)Lˆ (x, x0)K (x0) eff 0 r 2 ∑ r r r × − { } "− r=1 ZΣr ZΣr # with the kernel from Equation (72),

(AA;r) (AA;r) ↔ ↔ ˆ 1 (x, x0) (x, x0) Lr(x, x0) = 2 ∇ × ∇ ×(GAA;r) ∇ ×(GAA;r) , (74) µr  ↔ ↔  (x, x0) (x, x0)  ∇ × G G  Universe 2021, 7, 225 16 of 34

and with the effective action

1 3 2 2 1 2 Seff[A0, Kr ] = d x A0e0κ + ( A0) { } 2 3 µ ∇ × ZR 0 N 3 1 1 + d x A K0 + ( A )K , (75) ∑ µ 0 r µ ∇ × 0 r r=1 ZΣr r 0

where we have integrated out Ar for r = 1, ... , N, constraining the functional integral over A to be replaced by the substitutions n A K and n ( A ) K . Integrating r r × r → r r × ∇ × r → r0 out A0, ﬁnally yields

N N β 3 3 (κ) = K exp d x d x0 K (x)Lˆ (x, x0)K (x0) ∏ r 2 ∑ r r r Z r=1 I D "− r=1 ZΣr ZΣr N + d3x d3x K (x)Mˆ (x, x )K (x ) , (76) ∑ 0 r rr0 0 r0 0 Σr Σ r,r0=1 Z Z r0 !# with the additional kernel

(AA;0) (AA;0) 1 ↔ (x, x ) 1 ↔ (x, x ) µ2 0 µ0µ 0 Mˆ (x, x ) = 0 ∇ × ∇ × G r0 ∇ × G . (77) rr0 0  (AA;0) (AA;0)  1 ↔ (x, x ) 1 ↔ (x, x ) µ0µr 0 µrµ 0  ∇ × G r0 G    It should be noted again that the functional integral in Equation (76) runs over tan- ˆ ˆ gential vector ﬁelds Kr, Kr0 deﬁned on the surfaces Σr only. The kernels L and M can be combined into the joint kernel

Nˆ = Lˆ δ + Mˆ . (78) rr0 r rr0 rr0 Since Nˆ acts in the path integral only on tangential vectors, the projections of Nˆ on the tangent space of the surfaces Σr have to be taken. Let tr,1(x), tr,2(x) be two tangent vector ﬁelds that span the tangent space of Σ at x. The 4 4 matrix kernels then become r ×

L˜ r,mn(x, x0) = tr,m(x)Lˆ r(x, x0)tr,n(x0) 2 1 (tr,m(x). )(tr,n(x0). )gr tr,m(x).tr,n(x0) gr (tr,m(x) tr,n(x0)). gr = ∇ ∇ − ∇ 1 − × ∇ µ (tr,m(x) tr,n(x0)). gr 2 (tr,m(x). )(tr,n(x0). )gr + tr,m(x).tr,n(x0)gr r − × ∇ − erµrκ ∇ ∇ ! and

M˜ (x, x ) = t (x)Mˆ (x, x )t (x ) rr0,mn 0 r,m rr0 0 r0,n 0 2 µ0 (tr,m(x). )(tr,n(x0). )g0 tr,m(x).tr,n(x0) g0 (tr,m(x) tr,n(x0)). g0 1 µr = ∇ ∇ − ∇ − 0 × ∇2 µ0 µ0 µ0 µ0  (tr,m(x) tr,n(x0)). g0 2 (tr,m(x). )(tr,n(x0). )g0 + tr,m(x).tr,n(x0)g0 µr e0µrµ κ µrµr − × ∇ − r0 ∇ ∇ 0   which expresses all kernels in terms of tangential and normal derivatives of the scalar Green function g ( x x ) = e √erµrκ x x0 / x x . These expressions simplify when an r | − 0| − | − | | − 0| orthonormal basis t (x), t (x), n (x) = t (x) t (x) is used. The Casimir free energy r,1 r,2 r r,1 × r,2 is then given by ∞ 1 = k T 0 log det Nˆ (κ )Nˆ − (κ ) , (79) F − B ∑ n ∞ n n=0 h i where the determinant runs over all indices, i.e., x, x0 located on the surfaces Σr, and r, r0 = 1, ... , N. The kernel Nˆ ∞ is obtained from the kernel Nˆ by taking the distance between all bodies to inﬁnity, i.e, by setting Mˆ = 0 for all r = r . In the following we shall again rr0 6 0 denote the form of the partition function in Equation (67) as Lagrange representation, and Universe 2021, 7, 225 17 of 34

the one of Equation (76) as a Hamiltonian representation. By a simple computation, one can verify that the Hamiltonian representation of the Casimir free energy in Equation (79) is indeed equivalent to the surface formula Equation (44).

5. Application: Derivation of the Lifshitz Theory As a simple example to demonstrate the practical application of the surface formulations, we consider two dielectric half-spaces, one covering the region z z = 0, with the ≤ 1 surface Σ1 and dielectric function e1 and magnetic permeability µ1, and the other covering the region z z = H, with the surface Σ and dielectric function e magnetic permeability ≥ 2 2 2 µ2. We shall consider both the Lagrange and Hamiltonian representation in the following.

5.1. Lagrange Representation We compute the matrix elements of the kernels L and M of Equation (68) in the basis of transverse vector plane waves, given by

ik x +p1z ik x +p1z M1,k = e− k k zˆ = ( iky, ikx, 0)e− k k (80) k ∇ × − 1 ik x +p1z 1 2 ik x +p1z N1,k = e− k k zˆ = ( ikx p1, iky p1, k )e− k k (81) k κ ∇ × ∇ × κ − − k ik x p2(z H) ik x p2(z H) M2,k = e− k k− − zˆ = ( iky, ikx, 0)e− k k− − (82) k ∇ × − 1 ik x p2(z H) 1 2 ik x p2(z H) N2,k = e− k k− − zˆ = (ikx p2, iky p2, k )e− k k− − (83) k κ ∇ × ∇ × κ k 2 2 with pr = erµrκ + k and the sign of z is ﬁxed so that the waves are regular inside the k half-spaces.q Note that we include here a z dependence to be able to compute the curl on the surfaces. For the Green tensor, we use the representation

e µ κ2 + q2 q q q q (AA;r) 1/(e κ2) r r x x y x z ↔ iq(x x0) r q q e µ κ2 + q2 q q (x, x0) = e − 2 2 y x r r y y z G q erµrκ + q  2 2 Z qzqx qzqy erµrκ + qz µ G˜ (κ, q)   eiq(x x0) r r , (84) − 2 + 2 ≡ Zq erµrκ q which yields after the curl operations

(AA;r) 0 iqz iqy ↔ iq(x x ) µr − (x, x0) = e − 0 iqz 0 iqx ∇ × q e µ κ2 + q2  −  G Z r r iq iq 0 − y x µ G˜ (q)   eiq(x x0) r r0 (85) − 2 + 2 ≡ Zq erµrκ q 2 2 (AA;r) qy + qz qxqy qxqz ↔ iq(x x0) µr −2 2 − (x, x0) = e − 2 2 qxqy qx + qz qyqz ∇ × ∇ × G q erµrκ + q  − −2 2 Z qxqz qyqz qx + qy  − −  µ G˜ (q) eiq(x x0) r r00 . (86) − 2 + 2 ≡ Zq erµrκ q Universe 2021, 7, 225 18 of 34

We also need the following expressions for the operators that appear in the kernels, acting on the basis functions, which are tangential to the surfaces. On surface Σ1 we have

ik x ik x zˆ M1,k = ( ikx, iky, 0)e− k k um1e− k k (87) × k − − ≡ 1 ik x ik x zˆ N1,k = (iky p1, ikx p1, 0)e− k k un1e− k k (88) × k κ − ≡ ik x ik x zˆ M1,k = (iky p1, ikx p1, 0)e− k k vm1e− k k (89) × ∇ × k − ≡ 1 2 2 ik x ik x zˆ N1,k = (ikxe1µ1κ , ikye1µ1κ , 0)e− k k vn1e− k k (90) × ∇ × k κ ≡

and similarly on surface Σ2,

ik x ik x zˆ M2,k = ( ikx, iky, 0)e− k k um2e− k k (91) × k − − ≡ 1 ik x ik x zˆ N2,k = ( iky p2, ikx p2, 0)e− k k un2e− k k (92) × k κ − ≡ ik x ik x zˆ M2,k = ( iky p2, ikx p2, 0)e− k k vm2e− k k (93) × ∇ × k − ≡ 1 2 2 ik x ik x zˆ N2,k = (ikxe2µ2κ , ikye2µ2κ , 0)e− k k vn2e− k k . (94) × ∇ × k κ ≡

It is straightforward to show that the matrix elements of Lr in the above basis all vanish, as the basis functions are regular solutions of the vector wave equation. This observation is in agreement with the above ﬁnding that the kernel Lr is degenerate on the space of those solutions. We proceed with the computation of the elements of kernel M. We ﬁnd for the case r = r0,

3 3 M M Mrr(k , k0 ) = d x d x0 (x)Mrr(x, x0) (x0) (95) k k Σr Σr N N Z Z r,k r,k0 k k ∞ dqz 1 um um = δ(k + k0 ) G˜ 00(k , q ) 2π 2 u 0 z u k k Z ∞ " µ0 n r,k k n r, k − k − k 1 vm ˜ um 1 um ˜ vm + G0 (k , qz) + G0 (k , qz) µ0µr vn r,k k un r, k µ0µr un r,k k vn r, k k − k k − k 1 v v µ m ˜ m 0 iqz(z z0) + G0(κ, k , qz) e − z,z z µ2 vn k vn e µ κ2 + k2 + q2 | 0→ r r r,k r, k # 0 0 z k − k k 2 2 2 2 2 µ0 pr µr p0 µ0k − 2 0 = δ(k + k ) k (µ0µr) 0 e2 p2 e2 p2 k k 2p0  0 0 r − r 0  − e0µ0   δ(k + k0 )Mrr(k ) ≡ k k k and for the case r = r we get 6 0 Universe 2021, 7, 225 19 of 34

M M M (k , k ) = d3x d3x (x)M (x, x ) (x ) (96) rr0 0 0 rr0 0 0 k k Ωr Ω N N Z Z r0 r,k r0,k0 k k ∞ dqz 1 um ˜ um = δ(k + k0 ) 2 G000(k , qz) k k ∞ 2π µ un k un Z " 0 r,k r0, k − k − k 1 vm ˜ um 1 um ˜ vm + G0 (k , qz) + G0 (k , qz) µ0µr vn k un µ0µr un k vn r,k r0, k 0 r,k r0, k k − k k − k 1 v v µ r + m G˜ (κ, k , q ) m 0 eiqz( 1) H 0 z 2 2 2 − µrµr vn k vn e0µ0κ + k + q 0 r,k r0, k # z k − k k 2 k (µ0 p1 µ1 p0)(µ0 p2 µ2 p0) − − 0 p H = (k + k ) k µ1µ2 e 0 δ 0 µ0 − , k k 2µ0 p0 0 (e0 p1 e1 p0)(e0 p2 e2 p0) ! − − − e0 δ(k + k )M (k ) , 0 rr0 ≡ k k k iq ( 1)r H where the sign in e z − determines upon integration over qz the sign of the terms iq . The vanishing of the off-diagonal elements reﬂects the fact that the two polarizations ∼ z described by the basis functions M and N do not couple for planar surfaces. The total kernel M can be written as

M(k ) = k µ2 p2 µ2 p2 (µ p µ p )(µ p µ p ) 0 1− 1 0 0 1 1 0 0 2 2 0 p0 H 2 0 − µ µ µ − e− 0 µ0µ1 0 1 2 2 2 2 2 2  e0 p1 e1 p0 (e0 p1 e1 p0)(e0 p2 e2 p0) p H  k 0 − 0 − − e− 0 k − e0 − e0  (µ p µ p )(µ p µ p ) µ2 p2 µ2 p2  2p0  0 1 1 0 0 2 2 0 p0 H 0 2− 2 0  − µ µ µ − e− 0 2 0  0 1 2 (µ0µ2)   2 2 2 2   (e0 p1 e1 p0)(e0 p2 e2 p0) p H e0 p2 e2 p0   0 − − e− 0 0 −   − e0 − e0    The Casimir free energy is given by

∞ d2k k 1 = kBT ∑0 2 log det MM∞− (k ) (97) F (2π) k κ=κn n=0 Z h i in terms of the determinant of the matrix

µ µ p µ p 1 0 2 0 1− 1 0 e p0 H 0 µ1 µ0 p2+µ2 p0 − e p e p 0 1 0 0 1− 1 0 e p0 H 1  e0 p2+e2 p0 −  MM∞− (k ) = µ1 µ0 p2 µ2 p0 p H k − e− 0 0 1 0  µ2 µ0 p1+µ1 p0   e0 p2 e2 p0 p H   0 − e− 0 0 1   e0 p1+e1 p0    which has four dimensions due to two sets of basis functions (polarisations) M and N per surface. This yields the ﬁnal result

∞ dk (e0 p1 e1 p0)(e0 p2 e2 p0) 2p H = k T 0 k log 1 − − e− 0 F B ∑ (2π)2 − (e p + e p )(e p + e p ) n=0 Z 0 1 1 0 0 2 2 0 (µ0 p1 µ1 p0)(µ0 p2 µ2 p0) 2p H 1 − − e− 0 . (98) × − (µ p + µ p )(µ p + µ p ) 0 1 1 0 0 2 2 0 κ=κn This result is in agreement with the Lifshitz formula [37].

5.2. Hamiltonian Representation Now we derive the Lifshitz expression for the free energy of two dielectric half-spaces in the Hamiltonian representation. Since the kernels Lˆ and Mˆ of Equations (74) and (77) act Universe 2021, 7, 225 20 of 34

on vector ﬁelds that are tangential to the surfaces, we need to compute the matrix elements in a basis of tangential vectors tr,1(x) and tr,2(x) that span the tangent space of surface Σr at position x. For a planar surface, one can simply set tr,1(x) = xˆ1 and tr,2(x) = xˆ2. For a given pair of positions x, x0 on the surface and ﬁxed surface indices r, r0 we obtain the following 4 4 dimensional matrices × (AA;r) (AA;r) xˆ ↔ (x, x ) ↔ (x, x ) xˆ Lˆ (x, x ) = 1 1 0 0 1 r 0 µ2 ∇ × ∇ ×(GAA;r) ∇ ×(GAA;r) r xˆ2  ↔ ↔  xˆ2 (x, x0) (x, x0) (AA;r)∇ × G G(AA;r) (AA;r) (AA;r) ↔  ↔  ↔ ↔ xˆ1. (x, x0).xˆ1 xˆ1. (x, x0).xˆ2 xˆ1. (x, x0).xˆ1 xˆ1. (x, x0).xˆ2 ∇ × ∇ × G (AA;r) ∇ × ∇ × G (AA;r) ∇ × G (AA;r) ∇ × G (AA;r)  ↔ ↔ ↔ ↔  1 xˆ2. (x, x0).xˆ1 xˆ2. (x, x0).xˆ2 xˆ2. (x, x0).xˆ1 xˆ2. (x, x0).xˆ2 2 ∇ × ∇ ×(GAA;r) ∇ × ∇ ×(GAA;r) ∇ ×(GAA;r) ∇ ×(GAA;r) ≡ µr    xˆ . ↔ (x, x ).xˆ xˆ . ↔ (x, x ).xˆ xˆ .↔ (x, x ).xˆ xˆ .↔ (x, x ).xˆ   1 0 1 1 0 2 1 0 1 1 0 2   ∇ × G (AA;r) ∇ × G (AA;r) G (AA;r) G (AA;r)   ↔ ↔ ↔ ↔   xˆ2. (x, x0).xˆ1 xˆ2. (x, x0).xˆ2 xˆ2. (x, x0).xˆ1 xˆ2. (x, x0).xˆ2  ∇ × G 2 2 ∇ × G G G  qy + qz qxqy 0 iqz  −2 2 − iq (x x ) qxqy qx + qz iqz 0 − 0 2 1 e k k k  − q qxqy  = 2 2 x µr q p +q 0 iqz 1 + 2 2 r z erµrκ erµrκ  − 2  R  qxqy qy   iqz 0 2 1 + 2   erµrκ erµrκ  2 2  qy pr qxqy 0 pr  − −2 2 ∓ qxqy q p pr 0 iq (x x0 ) x r 1 e −  − − ± 2  = k k k qx qxqy , µr q 2pr 0 pr 1 + 2 2 k erµrκ erµrκ  ∓ 2  R  qxqy qy   pr 0 2 1 + 2   ± erµrκ erµrκ    where we set x = (x , 0) and x = (x , H) for surfaces 1 and 2, respectively. We determined the sign of the termsk iq from thekq -integration by the observation that z, z have to be ∼ z z 0 taken to the surface with z z staying inside the object. The upper (lower) sign of p − 0 r refers to r = 1 (r = 2). Analogously, for kernel M we get for the case r = r0

(AA;0) (AA;0) 1 ↔ (x, x ) 1 ↔ (x, x ) xˆ 2 0 µ µr 0 xˆ ˆ 1 µ0 ∇ × ∇ × G 0 ∇ × G 1 Mrr(x, x0) =  (AA;0) (AA;0)  xˆ2 1 ↔ 1 ↔ xˆ2 µ µ (x, x0) 2 (x, x0)  0 r ∇ × G µr G  q2+q2 q q  y z x y 0 iqz 2 2 µ µr µ0 − µ0 − 0 2 2  qxqy qx+qz iqz  iq (x x0 ) 0 − 2 2 µ µr µ0 e k k k − µ0 µ0 0 2 =  iq q qxqy  q p2 + q2  z 1 x 1  Z 0 z 0 µ µ 2 1 + 2 2 2  − 0 r µr e0µ0κ µr e0µ0κ   2   iqz 1 qxqy 1 qy   0 2 2 2 1 + 2   µ0µr µr e0µ0κ µr e0µ0κ    2 2  qy p qxqy p  − 0 0 ± 0 2 2 µ µr µ0 − µ0 0 2 2  qxqy qx p p0  iq (x x0 ) − 0 ∓ 0 − 2 2 µ µr µ0 e k k k − µ0 µ0 0 =  2  (99) p  p0 1 qx 1 qxqy  q 2 0 0 ± 2 1 + 2 2 2 Z k  µ0µr µr e0µ0κ µr e0µ0κ   2   p0 1 qxqy 1 qy   ∓ 0 2 2 2 1 + 2   µ0µr µr e0µ0κ µr e0µ0κ      and for r = r , 6 0 Universe 2021, 7, 225 21 of 34

(AA;0) (AA;0) 1 ↔ (x, x ) 1 ↔ (x, x ) xˆ1 µ2 0 µ0µ 0 xˆ1 Mˆ (x, x ) = 0 ∇ × ∇ × G r0 ∇ × G rr0 0 (AA;0) (AA;0) xˆ2   xˆ2 1 ↔ (x, x ) 1 ↔ (x, x ) µ0µr 0 µrµ 0  ∇ × G r0 G   q2+q2 q q  y z x y 0 iqz µ2 µ2 µ0µ 0 − 0 − r0 2 2  qxqy qx+qz iqz  iq (x x0 ) iqz H 0 µ e k k− ∓ µ2 µ2 µ0µ 0 k − 0 0 r0 =  iq q2 q q  q p2 + q2  z 1 x 1 x y  Z 0 z 0 1 + 2 2  − µ0µr µrµr e0µ0κ µrµr e0µ0κ   0 0 2   iqz 1 qxqy 1 qy   0 2 1 + 2   µ0µr µrµr e0µ0κ µrµr e0µ0κ   0 0  2 2 qy p qxqy p  − 0 0 ∓ 0 µ2 µ2 µ0µ 0 − 0 r0 2 2  qxqy qx p p0  iq (x x0 ) − 0 ± 0 µ e k k− µ2 µ2 µ0µ 0 k − 0 0 r0 p0 H =  2 e− (100) 2p  p0 1 qx 1 qxqy  q 0 0 ∓ 1 + 2 2 Z k  µ0µr µrµr e0µ0κ µrµr e0µ0κ   0 0 2   p0 1 qxqy 1 qy   ± 0 2 1 + 2   µ0µr µrµr e0µ0κ µrµr e0µ0κ   0 0    where again the upper (lower) sign everywhere refers to r = 1 (r = 2). For the kernel Mˆ we determined the sign of the terms iq from the q -integration by the observation that z, z ∼ z z 0 have to be taken to the surface with z z staying outside the object. When combining the − 0 kernels Lˆ and Mˆ into the joint kernel N = Lˆ δ + Mˆ , it is diagonal in q -space with rr0 r rr0 rr0 the blocks N(q ) on the diagonal given by the 8 8 matrix shown in Figure2k. × The Casimirk free energy is given by

∞ d2q k 1 = kBT ∑0 2 log det NN∞− (q ) (101) F (2π) k κ=κn n=0 Z h i in terms of the determinant of the above matrix where N is the matrix with H ∞, ∞ → i.e., the matrix N with the off-diagonal 4 4 blocks e p0 H vanishing. This yields the × ∼ − ﬁnal result

∞ 2 d q (e0 p1 e1 p0)(e0 p2 e2 p0) 2p H = k T 0 k log 1 − − e− 0 F B ∑ (2π)2 − (e p + e p )(e p + e p ) n=0 Z 0 1 1 0 0 2 2 0 (µ0 p1 µ1 p0)(µ0 p2 µ2 p0) 2p H 1 − − e− 0 , (102) × − (µ p + µ p )(µ p + µ p ) 0 1 1 0 0 2 2 0 which is again identical to the Lifshitz formula. Note that in the Hamiltonian approach there is no need to express the kernels in a basis for the space of functions that are regular solutions of the wave equation inside the objects. However, the number of ﬁelds per surface is now doubled compared to the Lagrangian approach, resembling the situation in quantum mechanics where the Hamiltonian path integrals run over the canonical coordinates q and p independently. Universe 2021, 7, 225 22 of 34

2 2 q p q q y 1 x y 0 1 2p1µ1 2p1µ1 2µ1 2 2 qxqy q p 1 0 x 1 0 1 2p1µ1 2p1µ1 2µ1 2 1 1 qx 1 qxqy 0 0 1+ 2 2 B 2µ1 2p1µ1 ✏1µ1 2p1µ1 ✏1µ1 C B 2 C B 1 ⇣1 qxqy ⌘ 1 qy C B 0 2 1+ 2 C 2µ1 2p1µ1 ✏1µ1 2p1µ1 ✏1µ1 N(q )=B 2 2 C (II.68) B q p q q C k B ⇣ ⌘ y 2 x y 0 1 C 2p2µ2 2p2µ2 2µ2 B 2 2 C B qxqy q p 1 C x 2 0 B 2p2µ2 2p2µ2 2µ2 C B 0 2 C B 1 1 qx 1 qxqy C 0 1+ 2 2 B 2µ2 2p2µ2 ✏2µ2 2p2µ2 ✏2µ2 C B q2 C B 1 ⇣1 qxqy ⌘ 1 y C B 0 2 1+ 2 C B 2µ2 2p2µ2 ✏2µ2 2p2µ2 ✏2µ2 C @ ⇣ ⌘ A

q2 p2 q2 p2 y 0 qxqy p0 y 0 p0H qxqy p0H p0 p0H 2 2 0 2 e 2 e 0 e µ0 µ0 µ0µ1 µ0 µ0 µ0µ2 2 2 2 2 qxqy qx p0 p0 qxqy p0H qx p0 p0H p0 p0H 0 2 2 0 2 e 2 e e 0 1 µ0 µ0 µ0µ1 µ0 µ0 µ0µ2 2 2 p0 1 qx 1 qxqy p0 p0H 1 qx p0H 1 qxqy p0H B 0 2 1+ 2 2 2 0 e 1+ 2 e 2 e C B µ0µ1 µ1 ✏0µ0 µ1 ✏0µ0 µ0µ1 µ1µ2 ✏0µ0 µ1µ2 ✏0µ0 C 2 2 B q q q q q q C B p0 ⇣1 x y ⌘ 1 y p0 p0H ⇣1 x y ⌘p0H 1 y p0H C 0 2 2 2 1+ 2 e 0 2 e 1+ 2 e µ0 B µ0µ1 µ1 ✏0µ0 µ1 ✏0µ0 µ0µ1 µ1µ2 ✏0µ0 µ1µ2 ✏0µ0 C + 2 2 2 2 B q p q q q p q q C 2p0 B y 0 p0H x y p0H ⇣p0 p0H ⌘ y 0 x y ⇣ p0 ⌘ C B µ2 e µ2 e 0 µ µ e µ2 µ2 0 µ µ C 0 0 0 1 0 0 0 2 B q q q2 p2 q q q2 p2 C B x y p0H x 0 p0H p0 p0H x y x 0 p0 C 2 e 2 e e 0 2 2 0 B µ0 µ0 µ0µ1 µ0 µ0 µ0µ2 C B 2 2 C p0 p0H 1 qx p0H 1 qxqy p0H p0 1 qx 1 qxqy B 0 e 1+ 2 e 2 e 0 2 1+ 2 2 2 C B µ0µ2 µ1µ2 ✏0µ0 µ1µ2 ✏0µ0 µ0µ2 µ2 ✏0µ0 µ2 ✏0µ0 C 2 2 B q q q q q q C B p0 p0H ⇣1 x y ⌘p0H 1 y p0H p0 ⇣1 x y ⌘ 1 y C e 0 2 e 1+ 2 e 0 2 2 2 1+ 2 B µ0µ2 µ1µ2 ✏0µ0 µ1µ2 ✏0µ0 µ0µ2 µ ✏0µ0 µ ✏0µ0 C B 2 2 C @ ⇣ ⌘ ⇣ ⌘ A

2 2 2 2 Hp0 2 2 Hp Hp q p q p q q q q e (qy p0) e 0 q q 0 y 0 + y 1 x y x y 00 x y 0 e µ0p0 µ1p1 µ1p1 µ0p0 µ0p0 µ0p0 µ2 Hp 2 2 2 2 2 2 Hp0 0 Hp qxqy qxqy q p q p e qxqy e (qx p0) e 0 0 x 0 + x 1 00 0 1 µ1p1 µ0p0 µ0p0 µ1p1 µ0p0 µ0p0 µ2 Hp 2 2 2 Hp e 0 q + µ ✏ Hp0 (✏0p0+✏1p1)qx µ0p1+µ1p0 (p0✏0+p1✏1)qxqy e 0 ( x 0 0) e qxqy B 002 2 + 2 2 2 0 2 2 C B  µ1p0p1✏0✏1 µ1p0p1  µ1p0p1✏0✏1 µ1  µ1µ2p0✏0  µ1µ2p0✏0 C 2 Hp 2 2 B (✏ p +✏ p )q Hp Hp0 e 0 q + µ ✏ C B (p0✏0+p1✏1)qxqy 0 0 1 1 y µ0p1+µ1p0 e 0 e qxqy ( y 0 0) C 1 00 2 2 2 2 + 2 0 2 2 B  µ1p0p1✏0✏1  µ1p0p1✏0✏1 µ1p0p1 µ1  µ1µ2p0✏0  µ1µ2p0✏0 C = B Hp0 2 2 Hp Hp 2 2 2 2 C . e (qy p0) e 0 q q 0 q p q p q q q q 2 B x y 0 e y 0 + y 2 x y x y 00C B µ0p0 µ0p0 µ1 µ0p0 µ2p2 µ2p2 µ0p0 C Hp 2 2 B Hp0 0 Hp 2 2 2 2 C e qxqy e (qx p0) e 0 qxqy qxqy q p q p B 0 x 0 + x 2 00C B µ0p0 µ0p0 µ1 µ2p2 µ0p0 µ0p0 µ2p2 C B Hp 2 2 C Hp e 0 q + µ ✏ Hp0 2 B e 0 ( x 0 0) e qxqy (✏0p0+✏2p2)qx µ0p2+µ2p0 (p0✏0+p2✏2)qxqy C 0 2 2 002 2 + 2 2 2 B µ2  µ1µ2p0✏0  µ1µ2p0✏0  µ2p0p2✏0✏2 µ2p0p2  µ2p0p2✏0✏2 C B Hp 2 2 2 C Hp Hp0 0 B e 0 e qxqy e (qy + µ0✏0) (p0✏0+p2✏2)qxqy (✏0p0+✏2p2)qy µ0p2+µ2p0 C B µ 0 2µ µ p ✏ 2µ µ p ✏ 00 2µ2p p ✏ ✏ 2µ2p p ✏ ✏ + µ2p p C B 2 1 2 0 0 1 2 0 0 2 0 2 0 2 2 0 2 0 2 2 0 2 C

@ A 19 Figure 2. Matrix N(q ) forming the diagonal blocks of the matrix N. k Universe 2021, 7, 225 23 of 34

6. Conclusions To date, analytical and purely numerical approaches to compute Casimir interactions have been developed independently, and it remained an open question if and how they are related. Analytical methods build on ideas from scattering theory and hence require an expansion of the Green function and bulk or surface operators in terms of special functions that are solutions of the wave equation. Hence, the very existence of such functions and the convergence of the expansion limit these approaches to sufﬁciently symmetric problems. Purely numerical approaches, such as that developed in [32], can be applied to basically arbitrary geometries but the numerical effort can be extremely high. Hence, it appeared useful to us to study the relation between these approaches in order to develop methods that can serve as semi-numerical approaches that combine the versatility of the purely numerical approaches with the smaller numerical effort of analytical methods. Hence, we have presented in this work a new compact derivation of formulas for the Casimir force, which is based both on bulk and surface operators that also enable analytical evaluations. This we have demonstrated for the simplest case of two dielectric slabs. Further semi- analytical implementations of our approaches are underway. Our Hamiltonian path integral representation is equivalent to the one derived by Johnson et al. as a purely numerical approach using Lagrange multipliers to enforce the boundary conditions in the path integral. Interestingly, the here-presented derivation of this representation from a Lagrangian path integral demonstrates the relation of this approach to the scattering approach when the T-matrix is deﬁned, as originally by Waterman, by surface integrals of regular solutions of the wave equation over the bodies’ surfaces [38]. This shows the close connection of these approaches, motivating further research in the direction of new semi-analytical methods to compute Casimir forces.

Author Contributions: Both authors contributed equally to the present work. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A. Dyadic Green’s Functions and Notations It is well known [34] that the expectation values of quantum fields in systems in thermal equilibrium can be expressed, via the Matsubara formalism, in terms of the analytic continuation to the positive imaginary axis of the appropriate Green’s functions. Along the imaginary axis, response functions have a simpler behavior than along the real frequency axis. For example, it can be shown that the electric and the magnetic permittivities of a magneto-dielectric medium are real-valued and positive definite for imaginary frequencies [39]. In the same manner, retarded Green’s functions are real-valued for positive imaginary frequencies. In this Appendix, we briefly review their definitions and main properties. To be specific, consider a system consisting of a collection of possibly spatially dis- persive magneto-dielectric bodies in vacuum, occupying the (non-overlapping) regions of space V1,..., VN. We let V0 be the vacuum region that separates the bodies. The electro- Universe 2021, 7, 225 24 of 34

magnetic response of the system is described by the following real-valued and positive deﬁnite frequency-dependent electric and magnetic permittivities

N (r) eij(x, x0; i ξ) = ∑ ψr(x)eij (x, x0; i ξ)ψr(x0) , r=0 N (r) µij(x, x0; i ξ) = ∑ ψr(x)µij (x, x0; i ξ)ψr(x0) , (A1) r=0

( ) ( ) where e 0 (x, x ; i ξ) = µ 0 (x, x ; i ξ) = δ δ(3)(x x ) is the permittivity of the vacuum. ij 0 ij 0 ij − 0 For local media, e (x, x ; i ξ) = e (x)δ(3)(x x ), and µ (x, x ; i ξ) = µ (x)δ(3)(x x ). ij 0 ij − 0 ij 0 ij − 0 The principle of microscopic reversibility [40] requires that the permittivities satisfy the symmetry relations:

eij(x, x0; i ξ) = eji(x0, x; i ξ) ,

µij(x, x0; i ξ) = µji(x0, x; i ξ) . (A2)

The imaginary-frequency Maxwell’s Equations are:

E = κ(B + 4π M ) , −∇ × ext H = κ(D + 4π P ) , (A3) ∇ × ext

where κ = ξ/c, Pext and Mext are the external polarization and magnetization, and

3 Di(x) = d x0 eij(x, x0; i ξ) Ej(x0) , 3 ZR 3 Bi(x) = d x0 µij(x, x0; i ξ) Hj(x0) . (A4) 3 ZR At the boundary separating media r and s, the tangential components of E and H are continuous: nˆ [E(r) E(s)] = nˆ [H(r) H(s)] = 0 , (A5) × − × − (αβ) where nˆ is the unit normal to the boundary. The dyadic Green’s functions (x x ; i ξ ) Gij − 0 n are deﬁned such that:

(EE) (EH) 3 ↔ ↔ E(x) = d x0 (x, x0; i ξ) Pext(x0) + (x, x0; i ξ) Mext(x0) , R3 · · Z G G (HE) (HH) 3 ↔ ↔ H(x) = d x0 (x, x0; i ξ) Pext(x0) + (x, x0; i ξ) Mext(x0) . (A6) R3 · · Z G G It can be shown [33] that the Green’s function satisﬁes the reciprocity relations:

(αβ) s(α)+s(β) (βα) (x, x0; i ξ) = ( 1) (x0, x; i ξ) , (A7) Gij − Gji where s(E) = 0 and s(H) = 1. In a local medium:

(3) e (x, x0; i ξ) = δ (x x0) e (x; i ξ) , ij − ij (3) µ (x, x0; i ξ) = δ (x x0) µ (x; i ξ) . (A8) ij − ij Universe 2021, 7, 225 25 of 34

(EH) ↔ In a local medium, it is possible to express the three Green’s function (x, x0; i ξ), (HE) (HH) (GEE) ↔ ↔ ↔ (x, x0; i ξ) and (x, x0; i ξ) in terms of the electric Green’s function (x, x0; i ξ), as: G G G (HE) 1 (EE) ↔ 1 ↔− ↔ (x, x0; i ξ) = µ (x, i ξ) → (x, x0; i ξ) , (A9) G − κ ∇ × G (EH) (EE) 1 ↔ 1 ↔ ← ↔− (x, x0; i ξ) = (x, x0; i ξ) ‘ µ (x0, i ξ) , (A10) G − κ G × ∇ (HH) 1 (EE) 1 ↔ 1 ↔− ↔ ← ↔− (x, x0; i ξ) = µ (x, i ξ) → (x, x0; i ξ) ‘ µ (x0, i ξ) G κ2 ∇ × G × ∇ 1 ↔− (3) 4π µ (x, i ξ) δ (x x0) , (A11) − − where ←‘ denotes derivation w.r.t. x , acting from right. The electric Green function solves ∇ 0 the differential Equation:

1 (EE) (EE) ↔− ↔ 2 ↔ 2 (3) ~ µ (x, i ξ)~ (x, x0; i ξ) + κ ↔e (x, i ξ) (x, x0; i ξ) = 4πκ ↔1 δ (x x0) . ∇ × ∇ × − − G G In the case of a homogeneous isotropic medium, this Equation can be solved explicitly:

2 (EE) 1 ∂ 2 ij (x x0; i ξ) = + µ κ δij g0(x x0) , (A12) G − − e ∂xi∂x0j ! −

From Equations (A9)–(A11), one then gets:

2 (HH) 1 ∂ 2 ij (x x0; i ξ) = + e κ δij g0(x x0) , (A13) G − − µ ∂xi∂x0j ! − (HE) ∂ ij (x x0; i ξ) = κ eijk g0(x x0) , (A14) G − − ∂xk − (EH) ∂ ij (x x0; i ξ) = κ eijk g0(x x0) , (A15) G − − ∂xk0 − where κ√eµ x x e− | − 0| g (x x0) = . (A16) 0 − x x | − 0| is the Green’s function of the scalar Helmoltz Equation in free space:

2 2 (3) e µ κ g (x x0) = 4π δ (x x0) , (A17) ∇ − 0 − − − It is convenient to introduce a compact notation for the ﬁelds and the Green’s functions. We collect the ﬁelds and the sources into six-rows column-vectors:

Φ(E)(x) E(x) Φ = , ≡ Φ(H)(x) H(x) ! (E)(x) D(x) D = , D ≡ (H)(x) B(x) D ! (E) Kext (x) Pext(x) Kext (H) = . (A18) ≡ Mext(x) Kext (x) ! Universe 2021, 7, 225 26 of 34

The deﬁning Equations of the Green’s functions Equation (A6) can then be interpreted as deﬁning the action of the linear operator ˆ onto the vector K : G ext Φ = ˆ K . (A19) G ext With this notation, the reciprocity relations Equation (A7) are expressed by the operator Equation: ˆT = Sˆ ˆ Sˆ , (A20) G G (αβ) where Sˆ is the 6 6 diagonal matrix of elements S = ( 1)s(α)δ δ . The electric and × ij − αβ ij the magnetic permittivities in Equation (A1) can be both collected into the permittivity operator eˆ. Equation (A1) becomes:

N (r) eˆ = ∑ ψˆr eˆ ψˆr , (A21) r=0

where the operator ψˆr is

ˆ (αβ) (3) ψr ψr ij (x, x0) = δαβ δij δ (x x0) ψr(x) . (A22) ≡ | − The constitutive Equations (A4) are expressed as:

= eˆ Φ . (A23) D For later use, it is convenient to deﬁne the polarization operator χˆ:

1 χˆ eˆ 1ˆ . (A24) ≡ 4π − Equation (A1) implies that the polarizability χ of a collection of bodies is:

N χˆ = ∑ χˆ(r) , (A25) r=1

(r) (r) where χˆ is the polarizability of body r. Clearly χˆ is supported in the volume Vr. This implies: χˆ(r)χˆ(s) = 0 , if r = s . (A26) 6 Finally, we deﬁne the trace operation Tr of any operator Aˆ:

ˆ 3 (αα) TrA = ∑ ∑ d yAii (y) . (A27) i α ZV

Appendix B. Derivation of Equation (10) In this Appendix, we derive the volumic and surface representations, Equation (8) (αβ) and Equation (9), respectively, for the scattering Green function Γij (x, x0).

Appendix B.1. Volumic Representation Let Φ = ˆ(0)K be the external field generated by a certain distribution of external ext G ext sources Kext. If one or more magneto-dielectric bodies are exposed to the field Φext, they get polarized, and we let Kind be the induced polarization field. The polarization field Kind depends linearly on the external field Φext:

Kind = Tˆ Φext . (A28) Universe 2021, 7, 225 27 of 34

The above equation defines the T-operator of the system. The definition of the T- operator implies that the scattered field Φsc produced by the bodies is:

Φ = ˆ(0)K = ˆ(0)Tˆ Φ = ˆ(0)Tˆ ˆ(0)K . (A29) sc G ind G ext G G ext However, by deﬁnition of Γˆ: Φsc = Γˆ Kext . (A30) A comparison of the above two equations gives:

Γˆ = ˆ(0)Tˆ ˆ(0) , (A31) G G which coincides with Equation (8). Now we show that the T-operator can be expressed in terms of the polarizability operator χˆ. Indeed, by deﬁnition of polarizability it holds:

Kind = χˆ Φtot , (A32)

where the total field Φtot = Φext + Φind is the sum of the external field and of the induced field Φ = ˆ K generated by K . Therefore: ind G0 ind ind K = χˆ(Φ + Φ ) = χˆ(Φ + ˆ(0)K ) . (A33) ind ext ind ext G ind

Using Equation (A28) to eliminate Kind from the previous equation, we ﬁnd:

Tˆ Φ = χˆ(1ˆ + ˆ(0)Tˆ )Φ . (A34) ext G ext

Since the above equation holds for an arbitrary distribution of sources Kext, i.e., for an arbitrary external ﬁeld Φext, it implies the operator identity:

Tˆ = χˆ(1ˆ + ˆ(0)Tˆ ) . (A35) G The above relation constitutes an equation of Lippmann–Schwinger form, which determines the T-operator. Its formal solution is:

1 1 Tˆ = χˆ = χˆ . (A36) 1 χˆ ˆ(0) 1 ˆ(0) χˆ − G − G Appendix B.2. Surface Representation In this section, we derive the surface formula Equation (9) of the scattering Green’s function . The existence of this representation is a direct consequence of the equivalence principle [33] of classical electromagnetism. This principle is applicable to bodies con- stituted by homogeneous and isotropic magneto-dielectric materials whose electric and magnetic permeabilities are of the form3:

N (3) (r) eij(x, x0; i ξ) = δij δ (x x0) ∑ ψr(x)e (i ξ)ψr(x0) , − r=0 N (3) (r) µij(x, x0; i ξ) = δij δ (x x0) ∑ ψr(x)µ (i ξ)ψr(x0) . (A37) − r=0

The equivalence principle expresses the scattered field in terms of fictitious equivalent currents in a homogeneous medium replacing the scatterer. Let Φ be the electromagnetic field that solves the Maxwell Equations (A3), with the constitutive Equation (A4) and the permittivities e and µ as in Equation (A37), subjected to the boundary conditions Equation (A5) on the surfaces of N bodies in a vacuum. According to the equivalence Universe 2021, 7, 225 28 of 34

principle, there exist tangential surface polarizations Ksurf concentrated on the union Σ of the surfaces Σr of the bodies: such that the ﬁeld Φ can be expressed as

N ˆ (0+) ˆ(0) ˆ (r+) ˆ(r) (r) Φ = ψ0(Φ + Ksurf) + ∑ ψr(Φ Ksurf) , (A38) G r=1 − G

(r) ˆ where Ksurf = ψrKsurf is the restriction of the surface current to the surface Σr of the r-th body, ˆ(r), r = 0, 1, ... N is the Green’s functions of an infinite homogeneous medium G having constant electric and magnetic permittivities equal to e(r)(i ξ) and µ(r)(i ξ), respectively, and (r) Φ(r+) = ˆ(r) K , (A39) G ext (r) is the external field generated by K . The Green’s functions ˆ(r) are obtained by replacing ext G e and µ by e(r)(i ξ) and µ(r)(i ξ), respectively, in Equations (A12)–(A15). The equivalence (r) principle shows [33] that the surface currents Ksurf coincide with the tangential values of the field Φ on the surface Σr:

(r E) ( ) K | (x) nˆ Hˆ (x) K r surf = (F ( )) r = N surf (r H) δ r x × ˆ , 1, . . . , (A40) ≡ K | (x) nˆ E(x) surf ! − × In concrete applications of the surface current method to scattering problems, the surface current Ksurf is unknown. However, it can be uniquely determined a posteriori by imposing the requirement that the ﬁeld Φ in Equation (A38) has continuous tangential components across the boundaries of the N bodies. This requirement leads to the following set of Equations:

(r) Πˆ (Φ(0+) + ˆ(0) K ) = Πˆ (Φ(r+) ˆ(r) K ) , (A41) r G surf r − G surf

where Πˆ r is the surface operator, which acts on the ﬁeld Φ(x) by projecting it onto the tangential plane to Σr at x:

↔ (3) ↔ Πr(x, x0) = δ (x x0) δ(Fr(x)) 1 nˆ (x) nˆ (x) , r = 1, . . . , N (A42) − − ⊗ h i Equation (A41) can be recast in the form:

N ˆ ˆ (r+) (0+) MKsurf = ∑ Πr(Φ Φ ) , (A43) r=1 −

where Mˆ is the surface operator

N ˆ ˆ ˆ(0) ˆ ˆ ˆ(r) ˆ M = Π Π + ∑ Πr Πr , (A44) G r=1 G

ˆ N ˆ ˆ where Π = ∑r=1 Πr . The operator M is invertible [41]. Equation (A43) then determines the unique surface current Ksurf that solves the boundary-value problem:

N ˆ 1 ˆ (r+) (0+) Ksurf = M− ∑ Πr(Φ Φ ) (A45) r=1 −

1 Note that according to its definition, Mˆ − acts on tangential fields defined on the union Σ of the surfaces Σr of the bodies and returns as a result a tangential polarization field Ksurf defined on Σ. Suppose now that the external sources Kext are localized in the vacuum region V0: (0) Kext = Kext . (A46) Universe 2021, 7, 225 29 of 34

Since Φ(r+) = 0, Equation (A45) now gives:

1 (0+) 1 (0) (0) K = Mˆ − Φ = Mˆ − ˆ K . (A47) surf − − G ext (0) ˆ According to Equation (A38), the scattered ﬁeld Φ(sc) = ψ0 Φ(sc) in the vacuum region V0 is: (0) (0) (0) 1 (0) (0) Φ = ˆ K = ˆ Mˆ − ˆ K . (A48) (sc) G surf −G G ext

The above equation shows that when x and x0 belong to V0, the scattering Green’s function is: (0) 1 (0) Γˆ = ˆ Mˆ − ˆ . (A49) −G G We thus see that in the vacuum outside the bodies Γˆ has precisely the form of Equation (13), with 1 ˆ = Mˆ − . (A50) K −

Appendix C. Proof of the Force Formula Equation (14) In this Appendix, we demonstrate the force formula Equation (14). The simple (ρσ) proof presented below holds for smooth kernels kl (y, y0). Unfortunately, the kernels (ρσ) K (y, y ) involved in both the volume and the surface representations of the scattering Kkl 0 Green’s functions lack the necessary smoothness. Indeed the T-operator in Equation (8) is in general discontinuous on the surfaces of the bodies. As to the surface operator Mˆ in Equation (9), it is a singular kernel supported on the surfaces of the bodies. Fortunately, (ρσ) it is possible to remedy this difﬁculty by representing the kernel (y, y ) as the limit Kkl 0 ˜ (ρσ) of a one-parameter family of smooth real kernels kl (y, y0; λ) that are supported on an K (ρσ) arbitrarily small open neighborhood of the domain of the original kernel (y, y ) : O Kkl 0 (ρσ) (ρσ) (y, y0) = lim ˜ (y, y0; λ) . (A51) Kkl λ 0 Kkl → (ρσ) The approximating kernels ˜ (y, y ; λ) can be so deﬁned as to satisfy the reciprocity Kkl 0 relations Equation (15)4:

(ρσ) s(ρ)+s(σ) (σρ) ˜ (y, y0; λ) = ( 1) ˜ (y0, y; λ) , (A52) Kkl − Klk where we recall that s(α) is deﬁned such that s(E) = 0 and s(H) = 1. It thus holds:

(αβ) (αβ) Γ (x, x0) = lim Γ˜ (x, x0; λ) , (A53) ij λ 0 ij → where we set

(αβ) 3 3 (αρ;0) (ρσ) (σβ;0) Γ˜ (x, x0; λ) d y d y0 (x y) ˜ (y, y0; λ) (y0 x0) . (A54) ij ≡ Gik − Kkl Glj − ZO ZO For brevity, below we shall omit writing the parameter λ, and we shall consider the λ 0 limit only at the end. In the ﬁrst step, we use reciprocity relations satisﬁed by the → free-space Green function to rewrite Equation (A54) as:

(αβ) 3 3 (αρ;0) (ρσ) s(σ)+s(β) (βσ;0) Γ˜ (x, x0) = d y d y0 (x y) ˜ (y, y0)( 1) (x0 y0) . (A55) ij Gik − Kkl − Gjl − ZO ZO Next, one notes that, by virtue of the reciprocity relations Equation (A52), the real (ρσ) kernel ( 1)s(σ) ˜ (y, y ) is symmetric, and therefore, it can be diagonalized: − Kkl 0 s(σ) ˜ (ρσ) (ρ) (σ) ( 1) kl (y, y0) = ∑ wmKm k(y) Km l (y0) , (A56) − K m | | Universe 2021, 7, 225 30 of 34

(ρ) where the eigenvalues wm are non-vanishing real numbers, and the eigenvectors Km k(y) are real smooth ﬁelds supported in . By plugging the above expansion into Equation| (A55), O ˜ (αβ) we see that Γij (x, x0) can be expressed as:

˜ (αβ) s(β) (α) (β) Γij (x, x0) = ( 1) ∑ wmΦm i(x) Φm j(x0) , (A57) − m | |

(α) where Φm i(x) are the real fields: | ( ) (αρ;0) (ρ) Φ α (x) = d3y (x y)K (y) . (A58) m i Gik − m k | ZO | (α) Now one notes that the fields Φm i(x) are well defined in all space, and by construction they satisfy the following euclidean-time| Maxwell Equations:

E = κ(H + 4π M ) , (A59) −∇ × m m m H = κ(E + 4π P ) , (A60) ∇ × m m m where we set Φ (E , H ), K (P , M ) and κ = ξ /c. When the expansion m ≡ m m m ≡ m m n Equation (A57) is substituted into Equation (6), we find for the dyad ↔Θ the following expression:

∞ (EE) (HH) ↔ ↔ ↔ Θ = 2k T 0 wm Θ Θ , (A61) B ∑ ∑ m − m n=0 m where

(EE) 2 ↔ 1 Em ↔ Θ = Em Em 1 , (A62) m 4π ⊗ − 2 (HH) 2 ↔ 1 Hm ↔ Θ = Hm Hm 1 . (A63) m 4π ⊗ − 2 (HH) ↔ It is worth noticing that the minus sign that multiplies Θm in Equation (A61) is a direct consequence of the factor ( 1)s(β) in the r.h.s. of Equation (A57). At this point, we − use the divergence theorem to convert the surface integral in Equation (7), giving the force F(r), into a volume integral. That gives:

∞ (EE) (HH) (r) 3 ↔ 3 ↔ ↔ F = d y Θ = 2k T 0 wm d y Θ Θ . (A64) ∇ · B ∑ ∑ ∇ · m − m ZOr n=0 m ZOr where is the portion of included within the surface S . By using standard identities Or O r of vector calculus, and the Maxwell Equations satisfied by the fields (Em, Hm) one finds:

(EE) (HH) ↔ ↔ Θ Θ = κEm Mm Em( Pm) + κHm Pm + Hm( Mm) ∇ · m − m × − ∇ · × ∇ · = ( H ) M E ( P ) ( E ) P + H ( M ) . (A65) ∇ × m × m − m ∇ · m − ∇ × m × m m ∇ · m Upon substituting the r.h.s. of the above equation into the r.h.s. of Equation (A64) we get ∞ (r) 3 Fi = 2kBT ∑ 0 ∑ wm d y Pm j∂iEm j Mm j∂i Hm j . (A66) m r | | − | | n=0 ZO h i If the ﬁelds Em and Hm are expressed in terms of the sources Pm and Mm, via Equation (A58) one can recast the r.h.s. of the above equation in the form:

∞ (r) 3 3 s(α) (α) ∂ (αβ;0) (β) Fi = 2kBT ∑ 0 ∑ wm d y d y0( 1) Km j(y) jk (y y0)Km k(y0) r − ∂yi G − n=0 m ZO ZO | | Universe 2021, 7, 225 31 of 34

∞ 3 3 ˜ (βα) ∂ (αβ;0) = 2kBT ∑ 0 d y d y0Kkj (y0, y; λ) jk (y y0) , (A67) r ∂yi G − n=0 ZO ZO where in the last passage, we made use of Equation (A56), and restored the dependence of K˜ on the parameter λ. By taking the limit λ 0 of the r.h.s., we see that Equation (A67) → reproduces the formula for force in Equation (14).

Appendix D. Partial Wave Expansion and the Scattering Matrix (αβ;0) ↔ The Green’s function (x, x0) admits an expansion in partial waves in any co- ordinate system in whichG the vector Helmoltz Equation is separable. When spherical coordinates are used, the partial-wave expansion takes the form of an inﬁnite series over spherical multipoles:

(E out/reg) 1 (out/reg) Φ | (x) = x φ (x, κ) , (A69) Mlm l(l + 1) ∇ × lm (E out/reg) i (E out/reg) Φ | (x) = p Φ | (x, κ) , (A70) Nlm κ ∇ × Mlm

(reg/out) where φlm (x, κ) are the following regular and the outgoing spherical waves:

(reg) φ (x, κ) = i (κ x )Y (xˆ) , (A71) lm l | | lm ( ) φ out (x, κ) = k (κ x )Y (xˆ) . (A72) lm l | | lm

Here, il(z) = √π/2zIl+1/2(z) is the modified spherical Bessel function of the first kind, and kl(z) = √π/2zKl+1/2(z) is the modified spherical Bessel function of the third kind. Notice the relation:

(E out/reg) i (E out/reg) Φ | (x) = Φ | (x) . (A73) Mlm κ ∇ × Nlm

(H reg/out) According to Maxwell Equations, the magnetic partial waves Φilm| (x) are obtained by taking a curl of the electric waves:

(H reg/out) 1 (E reg/out) Φ | (x) Φ | (x) . (A74) ilm ≡ − κ ∇ × ilm However, using the two relations Equations (A70) and (A73), one ﬁnds:

(H reg/out) (E reg/out) | | Φilm (x) = i Φτ(i)lm (x) . (A75)

where τ(M) = N, τ(N) = M. Next, we deﬁne the scattering matrix (r) of an isolated object r placed in vacuum. (αβ) T ↔ Let Γ r (x, x0) be the scattering part of the Green’s function of body r in isolation. The scattering matrix (r) is deﬁned such that at all points (x, x ) lying outside a sphere S(r) T 0 Universe 2021, 7, 225 32 of 34

(αβ) ↔ containing the body in its interior and centered at the point Xr, Γ r (x, x0) has the partial wave expansion:

(αβ) s(β) (α out) (β out) (r) ↔ | | Γ r (x, x0) = ( 1) λ ∑ ∑ Φplm (x Xr) Φp l m (x0 Xr) plm,p l m , (A76) − − ⊗ 0 0− 0 − T 0 0 0 plm p0l0m0

(αβ) ↔ The fact that all Green’s functions Γ r (x, x0) can be expressed in terms of a single scattering matrix (r) follows from Equation (A74), together with the following identi- T(αβ) ↔ ties satisﬁed by Γ r (x, x0) at all points in the vacuum outside the body, which in turn result from the identities of Equations (A9)–(A11) satisﬁed by the full Green’s functions (αβ) ↔ G (x, x0) in the presence of body r:

(HE) 1 (EE) ↔Γ (x, x0; i ξ) = → ↔Γ (x, x0; i ξ) , r − κ ∇ × r (EH) (EE) 1 ← ↔Γ (x, x0; i ξ) = ↔Γ (x, x0; i ξ) ‘, r − κ r × ∇ (HH) (EE) 1 ← ↔Γ (x, x0; i ξ) = → ↔Γ (x, x0; i ξ) ‘. (A77) r κ2 ∇ × r × ∇ Now we prove that the scattering matrix is given by the matrix elements of the operator ˆ r of body r deﬁned in Equation (47), taken between two regular partial waves. To see this, K (r) one notes that for any two points (x, x0) outside the sphere S , it is legitimate to replace ˆ(0) in Equation (47) by its partial wave expansion Equation (A68). This substitution results G in the expansion:

(αβ) s( ) (α out) (β out) ↔ β 2 | Γ r (x, x0) = ( 1) λ ∑ ∑ Φplm| (x Xr) Φp l m (x0 Xr) − − − ⊗ 0 0− 0 − plm p0l0m0 (µν) s(µ) 3 3 (µ reg) ↔ (ν reg) ( 1) d y d y0Φ | (y Xr) r (y, y0) Φ | (y0 Xr) . (A78) ∑ pl m p0l0m0 × µν − ZVr ZVr − − · K · −

A comparison of Equation (A76) with Equation (A78) gives the desired formula:

(r) s(µ) 3 3 = ( 1) λ d y d y0 plm,p0l0m0 ∑ T µν − ZVr ZVr (µν) (µ reg) ↔ (ν reg) Φpl | m (y Xr) r (y, y0) Φp l| m (y0 Xr) , (A79) × − − · K · 0 0 0 − (r) which indeed shows that is the matrix element of the operator ˆ r between two Tplm,p0l0m0 K (r) partial waves. Note that in the T-operator approach, is expressed by an integral Tplm,p0l0m0 of the body’s T-operator Tˆr over the body’s volume Vr, while in the surface approach it is an integral of the surface operator Mˆ 1 over the boundary Σ of the body. − r− r Now we deﬁne the translation matrices (ij)(d). Consider two points X and X in U 1 2 space that differ by a displacement of magnitude d along the z-axis:

X X = d zˆ . (A80) 2 − 1 Universe 2021, 7, 225 33 of 34

The translation matrix (21)(d) is deﬁned such that at all points x whose distances U from X and X are both smaller than d (i.e., x X < d and x X < d) it holds: 1 1 | − 2| | − 1| (E out) (21) (E reg) Φ | (x X2) = (d) Φ | (x X1) , (A81) p0l0m0 ∑ p0l0;pl plm − pl U − (E out) (12) (E reg) Φ | (x X1) = (d) Φ | (x X2) , (A82) p0l0m0 ∑ p0l0;pl plm − pl U −

It can be shown that the relation holds:

(21)(d) = (12)†(d) . (A83) U U By applying the operator = (i/κ) to both members of the Equation (A82), L − ∇× and recalling the deﬁnition of the magnetic partial waves Equation (A74) one ﬁnds:

(H out) (21) (H reg) Φ | (x X2) = (d) Φ | (x X1) , (A84) p0l0m0 ∑ p0l0;pl plm − pl U − (H out) (12) (H reg) Φ | (x X1) = (d) Φ | (x X2) (A85) p0l0m0 ∑ p0l0;pl plm − pl U −

which shows that the translation matrices of the magnetic partial waves coincide with those for the electric partial waves.

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