universe 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 inte- gral 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(w) and magnetic permeabilities mr(w), r = 1, ... , N. The bodies occupy the volumes Vr with surfaces Sr 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(w) and magnetic permeability m0(w). 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 j on body r is obtained by integrating the expec- tation value T of the Maxwell stress tensor h iji 1 1 T (x) = E (x)E (x) + H (x)H (x) d [ E (x)E (x) + H (x)H (x) ] (1) h ij i 4p 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 j = d s 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 xn) , h i n=0 G ¥ ˆ ˆ (HH) Hi(x)Hj(x0) = 2kBT ∑ 0 ij (x, x0; i xn) , (3) h i n=0 G where xn = 2pnkBT/¯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 (ab) h i formally divergent expression, since the Green functions (x, x ; i x ) are singular in the Gij 0 n Universe 2021, 7, 225 4 of 34 coincidence limit x = x0.