### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Preliminaries
- 3 Retarded Green functions
- 4 Non-equilibrium action
- 5 The large distance limit
- 6 Keldysh–Green functions
- 7 Example: sliding interfaces
- 8 Conclusions
- Appendix A
- Appendix B
- Appendix C
- Appendix D
- Appendix E
- Acknowledgments
- References

The electromagnetic field in a typical geometry of the Casimir effect is described in the Schwinger–Keldysh formalism. The main result is the photon distribution function (Keldysh Green function) in any stationary state of the field. A two-plate geometry with a sliding interface in local equilibrium is studied in detail, and full agreement with the results of Rytov fluctuation electrodynamics is found.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Preliminaries
- 3 Retarded Green functions
- 4 Non-equilibrium action
- 5 The large distance limit
- 6 Keldysh–Green functions
- 7 Example: sliding interfaces
- 8 Conclusions
- Appendix A
- Appendix B
- Appendix C
- Appendix D
- Appendix E
- Acknowledgments
- References

In his seminal article 65 years ago, Casimir formulated a physical problem [1] which has had a tremendous influence on physics. His pioneering analysis of the physical consequences of field quantization under external, macroscopic boundary conditions is still of current relevance. Indeed, it turned out to be one of the most prolific ideas in modern theoretical physics. Having a pure quantum background and being closely related to classical physics, the Casimir effect is a universal and wide-spread phenomenon that can be found on all scales in the Universe.

One of the fascinating aspects of the Casimir effect is its simplicity, i.e., there are theoretical models that are amenable to exact solutions. Leaving aside many important features of modern Casimir physics [2], we consider the simple system of Ref. [1] with a slight generalization. Namely, our setting consists of two half-spaces of different materials with a plane-parallel gap in between. What we know about the two media are their reflection coefficients (a matrix in general) and macroscopic conditions like temperature distributions, macroscopic current densities, distance and relative motion parallel to the interfaces. We assume that these conditions are maintained stationary by external (generalized) forces. We want to compute correlation functions of the electromagnetic field in the gap. These yield for example the average energy density of the field and the pressure exerted on the two bodies as components of the stress (energy-momentum) tensor. We focus in particular on symmetrized correlations, also known as Keldysh-Green functions (KGF) in quantum kinetics, and are able to derive them under rather general circumstances in a stationary Casimir geometry. We thus establish a natural non-equilibrium extension of Casimir's results.

Our approach starts from a simple observation which is a common feature among all manifestations of the Casimir effect: the interaction between the two interfaces disappears when their distance becomes large enough. On quite general grounds, we may therefore conclude that the electromagnetic field which we observe in any Casimir system results from several fields that originate in the material of the different half-spaces and in the vacuum gap between them. The distance dependence of the Casimir interaction is such that these fields become independent (uncorrelated) as the interfaces are infinitely far away. This fact is the cornerstone of our analysis: we use the large-distance limit to express the KGF in a two-plate setting in terms of KGFs of independent half-spaces.

In this paper, we relax the assumption that the two plates share the same temperature and state of motion. This makes it impossible to describe the field between the plates as being in thermal equilibrium. The appropriate tool to calculate photon correlation functions is then the Schwinger-Keldysh technique[3, 4] of nonequilibrium processes. The application of the Keldysh formalism to the field of Casimir-Van der Waals interactions has a relatively short history. The first application, to our knowledge, is due to Janowicz, Reddig and Holthaus [5] in calculations of the electromagnetic heat transfer between two bodies at different temperature. Sherkunov used the non-equilibrium technique extensively for dispersion interactions involving excited atoms and excited media [6]. A general expression for the electromagnetic force on an atom, in terms of the KGFs of atom and radiation field, was found a few years ago by one of the present authors [7].

The physical processes behind these field-mediated interactions are the multiple reflections of photons between the interfaces and their tunneling from one body to the other, which follow from the boundary conditions for the electromagnetic field on the interfaces. To include these boundary conditions, we use an effective action in the Schwinger-Keldysh technique with auxiliary fields and evaluate generating functionals for KGFs by performing path integrals [8]. Path integral approaches for the Casimir effect were introduced by Bordag, Robaschik and Wieczorek [9]. Li and Kardar considered the interaction between bodies mediated by a fluid with long range correlations [10, 11]. They applied a path integral technique to include arbitrarily deformed bodies on which any kind of boundary condition can be implemented. This feature makes the approach amenable to a perturbative analysis of any deformed ideally conducting surfaces [12, 13]. Further on, Emig and Büscher [14] used the optical extinction theorem [15] to reformulate the boundary conditions for a vacuum-dielectric interface in integral form that depends only on the fields on the vacuum side (i.e., in the gap between two bodies). They then derived with path integral techniques an effective Gaussian action for the photon gas in the gap, providing the free energy and in particular the Casimir interaction of dielectric bodies with arbitrary shaped surfaces. A similar approach has been followed by Soltan et al. [16] for a dispersive medium between the bodies. Recently, Behunin and Hu applied path integrals to problems of the Casimir-Polder type (atom-surface interaction) in different nonequilibrium situations: in Ref. [17] is considered the Van der Waals interaction between an atom and a substrate in a stationary state out of global equilibrium; Ref. [18] is calculating atom-atom interactions in a quantized radiation field which is in a nonequilibrium state.

Our theoretical approach to the non-equilibrium Casimir interaction where the boundary conditions involve different, locally defined temperatures, is thus a synthesis between the Feynman path integral and the Keldysh-Schwinger formalism of nonequilibrium processes [19, 20].

The paper is organized as follows: after preliminary notations and introductions in Sec. 'Preliminaries', we introduce rather general expressions for retarded Green functions in the Casimir geometry of two planar boundaries (Sec. 'Retarded Green functions'). In Sec. 'Non-equilibrium action' is analyzed a Gaussian action in Schwinger-Keldysh space which yields the non-equilibrium correlations for the fields (Keldysh-Green functions). This action implements in particular the boundary conditions for the retarded Green functions. In Sec. 'The large distance limit' we discuss the retarded Green functions in the limit where the interfaces of the Casimir system are infinitely removed from each other. Using this limiting procedure, the Keldysh-Green function at any point in the gap is expressed via Keldysh functions of the single interface systems defined at the surfaces and via photon numbers in free space (Sec. 'Keldysh–Green functions'). As an application of the developed technique, we find in Sec. 'Example: sliding interfaces' the field correlation functions in a Casimir geometry with a sliding interface. This result is checked in Appendix D where the same problem is solved using Rytov fluctuation electrodynamics. We illustrate our results by analyzing the electromagnetic energy density between the plates in two typical non-equilibrium situations (Appendix E). Concluding remarks are given in Sec. 'Conclusions', while technical details are collected in the other Appendices.

### 8 Conclusions

- Top of page
- Abstract
- 1 Introduction
- 2 Preliminaries
- 3 Retarded Green functions
- 4 Non-equilibrium action
- 5 The large distance limit
- 6 Keldysh–Green functions
- 7 Example: sliding interfaces
- 8 Conclusions
- Appendix A
- Appendix B
- Appendix C
- Appendix D
- Appendix E
- Acknowledgments
- References

We have calculated the photon distribution function between two parallel plates (Casimir geometry) under rather general conditions: on the two plates, any arbitrary stationary nonequilibrium state is allowed for. Using the Schwinger-Keldysh formalism of nonequilibrium field theory, we could express the Keldysh-Green function of photons at any point in the gap between the plates via the Keldysh functions of single interface systems, defined at the surfaces, and via photon numbers in free space. As a cross check of the results, we consider one plate sliding relative to the other in local equilibrium, and we find full coincidence of the results with Rytov theory, without any restriction in relative velocities.

Our approach is flexible enough to allow for the zero-temperature limit to be taken. The example of the sliding plates of Sec. 'Example: sliding interfaces' then illustrates that one does *not* recover an equilibrium situation. The resulting frictional stress that opposes the relative motion is an interesting (and controversial [31, 33-35]) manifestation of an unstable vacuum state. For similar situations, we may quote the Klein paradox [36] (electron-positron pairs created by maintaining a static field configuration), and the Schwinger-Unruh effect (thermalization of an accelerated detector in vacuum). Indeed, one possible explanation for quantum friction involves the creation of particle pairs in the two plates [34, 35, 37], as pointed out in earlier work by Polevoi within the context of Rytov theory [26].

Finally, we suggest that the developed formalism is general enough to investigate generalizations beyond the standard fluctuation electrodynamics. The crucial assumptions of the latter are clearly spelled out: statistical independence of the sources localized on the macroscopic bodies. If this is relaxed, one has to establish the photon source strengths in some other way. Concepts from non-equilibrium kinetic theory like the Boltzmann equation or the balance of energy and entropy exchanges are likely to be instrumental here.