Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system
Fernando C. Lombardo, Francisco D. Mazzitelli, Adrian E. Rubio Lopez, Gustavo J. Turiaci
aa r X i v : . [ qu a n t - ph ] A p r EPJ manuscript No. (will be inserted by the editor)
Non-equilibrium Lifshitz theory as a steady state of a fulldynamical quantum system
Fernando C. Lombardo , Francisco D. Mazzitelli , Adri´an E. Rubio L´opez , and Gustavo J. Turiaci Departamento de F´ısica
Juan Jos´e Giambiagi , FCEyN UBA and IFIBA CONICET-UBA, Facultad de Ciencias Exactas yNaturales, Ciudad Universitaria, Pabell´on I, 1428 Buenos Aires, Argentina Centro At´omico Bariloche and Instituto Balseiro, Comisi´on Nacional de Energ´ıa At´omica, R8402AGP Bariloche, Argentina Physics Department, Princeton University, Princeton, NJ 08544, USAe-mail: [email protected]
Received: date / Revised version: date
Abstract.
In this work we analyze the validity of Lifshitz’s theory for the case of non-equilibrium scenariosfrom a full quantum dynamical approach. We show that Lifshitz’s framework for the study of the Casimirpressure is the result of considering the long-time regime (or steady state) of a well-defined fully quantizedproblem, subjected to initial conditions for the electromagnetic field interacting with real materials. Forthis, we implement the closed time path formalism developed in previous works to study the case of twohalf spaces (modeled as composite environments, consisting in quantum degrees of freedom plus thermalbaths) interacting with the electromagnetic field. Starting from initial uncorrelated free subsystems, wesolve the full time evolution, obtaining general expressions for the different contributions to the pressurethat take part on the transient stage. Using the analytic properties of the retarded Green functions, weobtain the long-time limit of these contributions to the total Casimir pressure. We show that, in the steadystate, only the baths’ contribute, in agreement with the results of previous works, where this was assumedwithout justification. We also study in detail the physics of the initial conditions’ contribution and theconcept of modified vacuum modes, giving insights about in which situations one would expect a nonvanishing contribution at the steady state of a non-equilibrium scenario. This would be the case whenconsidering finite width slabs instead of half-spaces.
In this paper we are concerned with a first principles cal-culation of the non-equilibrium Casimir pressure betweenreal materials, for the particular geometry of half-spacesseparated by a vacuum gap of constant width. By non-equilibrium we refer to a situation in which the objectsare held at different (but fixed) temperatures.Lifshitz formula describes the Casimir pressure in asteady situation, in terms of the macroscopic propertiesof the materials. The original derivation [1] was based onthe use of stochastic Maxwell equations, and relied heav-ily on the thermal properties of the stochastic sources, as-suming thermal equilibrium. A few years later, the sameformula was obtained in Ref.[2] within the framework ofquantum field theory at finite temperature. Whatever thechosen approach, the final result can be expressed in termsof the permittivity of the materials (or alternatively interms of their reflection coefficients), and therefore ad-mits a natural generalization to the case in which thehalf-spaces are maintained at different temperatures, as-suming that the system reaches a steady state. This kindof approach has been followed in Ref.[3], where the non-equilibrium Casimir force has been considered for the first time. Nevertheless, the latter seems rather an extension ofthe original derivation than a full quantum developmentfrom first principles which is, today, a pending challengein non-equilibrium scenarios. Although there are remark-able works in the context of Casimir physics (as Ref.[4]),the present paper focuses on the dynamical aspects of thebuilding-up of the steady state of an electromagnetic (EM)field interacting with matter in a non-equilibrium scenariothat, as far as we know, has not been done before, and wethink that contains relevant physics.Beyond the non-equilibrium features, this work alsoenters another (not completely unrelated) issue that hasbeen the subject of an intense controversy: whether Lif-shitz theory is applicable or not to real metals. In the sim-plest approach, free (conduction) electrons in metals aredescribed by the Drude model, although its dissipation-less limit, the plasma model, seems to be in concordancewith the experimental results [5]. Some authors claim thatLifshitz formula cannot be applied for the Drude model[6], while others have reached the opposite conclusion [7],claiming that the Lifshitz-Matsubara formula is not cor-rect for the plasma model. At first glance, our work willnot address directly this important question, because thro-ughout the paper we consider a material without free car-
Fernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system riers, i.e., an insulator which permittivity function is givenby the bounded electrons only (see Ref.[6], for example).However, the present work enters the discussion about asystematic treatment that allows to understand the limi-tations of the Lifshitz formula, and how to figure out whichmodel is consistent with it.A first principles calculation involves a microscopicmodel for the quantum polarization degrees of freedom ofthe materials, interacting both with the electromagneticfield and thermal baths that fix the different temperaturesof the objects (see Ref.[8,9] for more details about phys-ical aspects of the material model). The evolution of thefully quantized system from adequate initial conditionsshould eventually show the emergence of a steady situa-tion in the long time limit, with Casimir pressure givenby a Lifshitz-like formula. From a technical point of view,this program can be developed in the framework of thetheory of quantum open systems (see Ref.[10]). The quan-tization procedure can be worked out through Heisenbergequations for the quantum operators (canonical formalism[11]) or path integral methods (closed time path formalism(CTP) [12]), but both are subject to initial conditions [13].For the latter, after integration of the material degrees offreedom and thermal baths, one obtains the influence ac-tion (Refs.[12,14]) for the electromagnetic field. Then, in-tegrating the field, one can construct the generating func-tional to calculate the correlation functions (see Ref.[15]).Using the correlation functions derived, one can computethe time evolution of the mean value of the energy momen-tum tensor, and therefore the Casimir pressure as a func-tion of time. The main question is therefore whether thismean value reaches a well defined limit at long times, andwhether this limit depends on the initial conditions or not.Note that the approaches that are usually implemented inCasimir physics can be considered as ‘steady’ quantiza-tion schemes in the sense that they assume without jus-tification that the system reaches a steady state and, inaddition, that the latter is of thermal equilibrium. Both re-quirements makes the ‘steadiness’ assumption physicallyreasonable and expected to treat equilibrium situations(see for example Refs.[13,16] for the ‘steady canonicalquantization’ schemes and Ref.[17] for the ‘steady pathintegral quantization’). Nonetheless, it is not clear how toextend it to non-equilibrium (steady) situations or how itis done from first principles. As we mentioned before, thefirst work in this direction was Ref.[3] which extends thestochastic approach, but a complete quantum approachwill bring to light the physics of the contributions of eachpart of the model to the electrodynamical quantities. Atthe end, the whole picture gives an appropriate and sys-tematic way to deal with non-equilibrium situations inCasimir physics.In previous works, some of us investigated preliminaryaspects of the problem at hand. In Ref.[18] the case of aquantum scalar field in the presence of an arbitrary mate-rial was studied. After developing the quantum open sys-tems approach for the particular situation of a scalar fieldinteracting with microscopic degrees of freedom and ther-mal baths, it was shown that the emergence of a steady state independent of the initial conditions is a nontrivialissue. While for a quantum field in bulk material such limitexists, this is not the case for slabs of material with finitewidth. The generalization to the electromagnetic field hasbeen considered in Ref.[19], where the CTP generatingfunctional for the electromagnetic field interacting witha composite environment has been described in detail.Specifically, the environment considered there consistedof the quantum polarization degrees of freedom at eachpoint of space, connected to thermal baths to fix the localtemperature. Formal expressions for the Maxwell tensorand the Poynting vector in terms of the Hadamard prop-agator were obtained, for the case of the electromagneticfield in bulk material, where, once again, the steady stateis independent of the initial conditions. Some technical is-sues regarding the gauge fixing procedure in the contextof CTP have also been discussed.In the present paper we will extend the results in [19]to the case of the electromagnetic field in the presenceof two half-spaces of real media separated by a vacuumgap. Considering the previous results mentioned above,it is not clear a priori wether the initial conditions willbe erased or not in the long time limit for this particulargeometry. Using the formalism of Ref.[19] we will obtainexplicit expressions for the Casimir pressure as a func-tion of time, assuming that the interaction of the vacuumfield with the materials starts at a given time t . TheCasimir pressure will receive three different contributions,coming from the field’s initial conditions, the polarizationdegrees of freedom, and from the thermal baths. We willshow that, in the long time limit, only the baths’ contri-bution survive. This is the contribution expected in theLifshitz approach. Therefore, we will provide a first prin-ciples derivation of Lifshitz formula for non-equilibriumsituations in the framework of quantum open systems.The paper is organized as follows: In the next Sec-tion we present the model and the influence action thatincludes noise and dissipation contributions to the calcu-lation of the Casimir force. Section III contains the ex-pression of the Casimir pressure in terms of the electro-magnetic correlation functions and, following this, the dis-cussion about the different contributions to the pressureappears in Sec. IV. The long-time limit of the completeproblem is included in Sec. V and finally, in Sec. VI weinclude our final remarks and conclusions. In order to include effects of dissipation and noise (fluc-tuations) in the calculation of the Casimir force betweentwo half-spaces of real material interacting with the EMfield, we will develop a full CTP approach to the problem.Therefore, a microscopic physical model for the material ofthe bodies will be introduced. Once the model is defined,one expects the material to show effective macroscopic EMproperties, that will be contained in the Feynman-Vernoninfluence functional included in the CTP formalism. ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 3
As in Ref. [19], we will consider a composite system, con-sisting in two parts: the EM field A µ (considered as amassless vector gauge field) and the real media, which ismodeled as a continuous set of quantum Brownian parti-cles representing the polarization density degree of free-dom P . These degrees of freedom (DOFs) are basicallythree-dimensional harmonic oscillators (with mass M [ j ] and frequency Ω [ j ] in each direction) coupled to the fieldat each point of space in a realistic way. The compos-ite system (EM field and DOFs) is also coupled to ex-ternal baths of harmonic oscillators in each direction as inthe well-known quantum Brownian motion theory (QBM).Therefore, the total action for the whole system is givenby: S [ A µ , P x , q n, x ] = S [ A µ ] + S [ P x ] + X n S [ q n, x ] (1)+ S Curr [ A µ , P x ] + X n S int [ P x , q n, x ] , where the first three terms are the free actions for eachpart of the total system, while the interaction action be-tween the field and the polarization DOFs is given bya current-type interaction which can be written in twoequivalent ways by defining the conserved current four-vector as J µ = ( ∇ · P , − ˙ P ) and the electric field E j = − ∂ j A − ∂ A j : S Curr [ A µ , P x ] = λ [( g J µ ) ∗ A µ ] = λ (cid:2)(cid:0) g P j (cid:1) ∗ E j (cid:3) , (2)where A ∗ B ≡ R d x A ( x , t ) B ( x , t ) and λ is the couplingconstant between them. It is worth noting that, when writ-ing the interaction actions, we have introduced the matterdistribution g ( x ), which takes binary values (1 or 0) de-pending whether or not there is material at the spatialpoint x .On the other hand, the interaction between the po-larization DOFs and each bath is simply linear couplingthrough the constants λ [ j ] n, x for each bath oscillator ( n ) ineach point ( x ) and direction ( j ): S int [ P x , q n, x ] = Z d x g ( x ) λ [ j ] n, x (cid:0) P j x · q jn, x (cid:1) , (3)where A · B ≡ R t f t i dt A ( t ) B ( t ), λ [ j ] n, x are coupling constantsbetween the DOFs and the baths, and [ j ] denotes that thissuperscript is not summed as in Einstein notation. It isclear that greek indices sum over 0 , , , , , β EM , β P x , β B, x - the material can also, inprinciple, be thermally inhomogeneous-), b ρ ( t i ) = b ρ EM ( t i ) ⊗ b ρ P x ( t i ) ⊗ b ρ { q n, x } ( t i ) . (4) As our goal is the study of the dynamics of the EM fieldcorrelation functions, we start by coarse-graining the bathsand the DOFs in order to obtain the exact EM field in-fluence action. Since we have quadratic actions and wechoose the initial states as Gaussian, the EM field influ-ence action will have a quadratic form, with dissipationand noise kernels acting over the EM field.It is well known from the QBM theory, coarse- grainingthe baths will result in QBM influence actions for eachcomponent of each DOF in each point of space, i. e., theresult will be S IF [ P j x , P ′ j x ] including the QBM dissipationand noise kernels. Therefore, at this point, the EM fieldinfluence action is defined as: e iS IF [ A µ ,A ′ µ ] = Z d P f Z d P i d P ′ i Z P ( t f )= P f P ( t i )= P i D P × Z P ′ ( t f )= P f P ′ ( t i )= P ′ i D P ′ e iλ R d x g ( x ) ( ∇ A · P + A · ˙ P −∇ A ′ · P ′ − A ′ · ˙ P ′ ) × e i ( S [ P ] − S [ P ′ ]+ S IF [ P , P ′ ] ) ρ P ( P i , P ′ i , t i ) . (5)Following Ref.[19], and considering an initial thermalstate for the DOFs, the EM field influence action is givenby: S IF [ A µ , A ′ µ ] = Z d x Z d x ′ ∆A µ ( x ) h − D µν ( x, x ′ ) × ΣA ν ( x ′ ) + i N µν ( x, x ′ ) ∆A ν ( x ′ ) i , (6)with ∆A µ = A ′ µ − A µ , ΣA µ = ( A µ + A ′ µ ) / D µν ( x, x ′ ) = Γ µν jk D jk , (7) N µν ( x, x ′ ) = Γ µν jk N jk , (8)which ensures gauge invariance thanks to the fact that thedifferential operator Γ µν jk ≡ δ µ δ ν ∂ jk ′ − δ µ δ ν k ∂ jt ′ − δ µ j δ ν ∂ tk ′ + δ µ j δ ν k ∂ tt ′ satisfies ∂ µ Γ jkµν = ∂ ′ ν Γ jkµν ≡
0. It is worth noting that the noise kernel presents twocontributions, one associated to the DOFs and the otherone associated to the baths: N jk ≡ N Pjk + N Bjk .All in all, the dissipation and noise kernels are givenby: D jk ( x, x ′ ) = δ jk δ ( x − x ′ ) g ( x ) λ , x G [ j ]Ret , x ( t − t ′ ) , (9) Fernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system N Pjk ( x, x ′ ) = δ jk δ ( x − x ′ ) g ( x ) λ , x M [ j ] x Ω [ j ] x coth β P j x Ω [ j ] x ! × h ˙ G [ j ]Ret , x ( t − t i ) ˙ G [ j ]Ret , x ( t ′ − t i )+ Ω [ j ]2 x G [ j ]Ret , x ( t − t i ) G [ j ]Ret , x ( t ′ − t i ) i , (10) N Bjk ( x, x ′ ) = δ jk δ ( x − x ′ ) g ( x ) λ , x G [ j ]Ret , x · N [ j ] x · h G [ j ]Ret , x i T , (11)where we are considering materials that can be inhomo-geneous and anisotropic (birefringent) [19]. N [ j ] x is the QBM noise kernel generated by the ther-mal bath over the DOF in the direction j and locatedat x . The Green functions G [ j ]Ret , x are the retarded Greenfunctions for each DOF, which satisfy G [ j ]Ret , x (0) = 0 and˙ G [ j ]Ret , x (0) = 1. In other words, they are the QBM retardedGreen functions having x and [ j ] as parameters. TheseGreen fuctions are defined once the type of baths are cho-sen at each point of space and direction. It is implicitthat the chosen directions defining the anisotropic prop-erties are the Fresnel’s principal axis’ basis (see Ref.[19])because, as we shall see, they will allow us to define a diag-onal permittivity tensor in this basis. In fact, we are con-sidering the same basis for every point filled with material.This is completely general since it is clear, for example,that disjoint bodies can have different Fresnel’s basis. Toinclude these features, it is necessary to include changesof basis between all the present basis. For simplicity, thiswill be omitted and, in fact, for isotropic materials this isunnecessary since the permittivity tensor is proportionalto the identity matrix.In the Lifshitz problem there are two different bodiesinteracting, which are assumed to be homogeneous andisotropic. Therefore a few simplifications arise. On the onehand, homogeneity implies that all the spatial subscriptsreduce to two labels associated to each of the two bodies.Considering two parallel plates separated by a distance l ,the homogeneity of each body means x → L, R dependingon the spatial point lays in the left or right body respec-tively. On the other hand, isotropy of the material in eachbody implies that superscripts [ j ] should be omitted be-cause there is no dependence with the direction. However,we will keep generality in the material properties for now. In the present section we will calculate the EM field CTP-generating functional and derive an expression for theHadamard propagator. We shall show how the Casimirpressure can be easily written in terms of this Hadamardpropagator.
Once we have calculated the EM field influence actiongenerated by the material, we proceed to calculate theEM field generating functional, which is defined by [19]: Z CTP [ J µ , J ′ µ ] = Z dA µ f Z dA µ i dA ′ µ i Z A µ ( t f )= A µ f A µ ( t i )= A µ i D A µ Z A ′ µ ( t f )= A µ f A ′ µ ( t i )= A ′ µ i D A ′ µ e i ( J µ ∗ A µ − J ′ µ ∗ A ′ µ ) e i ( S [ A µ ] − S [ A ′ µ ] ) × e iS IF [ A µ ,A ′ µ ] ρ EM ( A µ i , A ′ µ i , t i ) . (12)These integrals can be performed using the Faddeev-Popov procedure (adapted to the CTP formalism), to ex-tract the redundant sums over paths on the same gaugeclass. In [19], the gauge condition was introduced in theCTP-action as a typical gauge fixing term. Then, the CTPintegral is worked out by writing the paths as a sum of ahomogeneous solution A µ ( x ) (satisfying the initial condi-tions) plus a shift. The corresponding equations of motionfor this solution are the ones obtained from the EM CTPaction (including the EM field influence action), which inthe case of the temporal gauge reads: (cid:16) η µν (cid:3) − ∂ µ ∂ ν − α t µ t ν (cid:17) A ν ( x ) (13)+ 2 Z d x ′ D µν ( x, x ′ ) A ν ( x ′ ) = 0 , where α is the gauge fixing parameter and t µ is a time-like four-vector which can be taken as (1 , , ,
0) for thetemporal gauge. These are four equations for the four com-ponents of the EM field.Therefore, the functional integral can be calculated inthe Landau gauge (where the gauge fixing parameter α goes to 0). All in all, the generating functional reads: Z CTP [ Σ J , ∆ J ] = D e − i∆ J ∗ A E Σ A i ,Σ Π i e − i∆ J ∗G Ret ∗ Σ J × e − ∆ J ∗G Ret ∗ ( ∂ tt ′ N ) ∗G T Ret ∗ ∆ J , (14)where D ... E Σ A i ,Σ Π i = R dΣ A i R dΣ Π i ...W EM [ Σ A i , Σ Π i ,t i ], and W EM [ Σ A i , Σ Π i , t i ] is the initial EM field Wignerfunctional associated to the EM field initial density ma-trix b ρ EM ( t i ) in the temporal gauge (see Ref.[19] for itsdefinition).It is worth noting that A corresponds to the homo-geneous solution of the EM equations of motion after im-posing the temporal gauge, which is written as: A j ( x ) = − Z d x ′ ˙ G jk Ret ( x , x ′ , t − t i ) ΣA k i ( x ′ )+ Z d x ′ G jk Ret ( x , x ′ , t − t i ) ΣΠ k i ( x ′ ) . (15) ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 5 G Ret corresponds to the retarded Green tensor result-ing from Eq.(13) after imposing the temporal gauge con-dition in the Landau gauge, which results in a set of threeequations of motion plus a residual gauge condition ob-tained from the equation for A (which was erased by thegauge condition). The initial conditions for the retardedpropagator are, as usual, G jk Ret ( x , x ′ ,
0) = 0 , ˙ G jk Ret ( x , x ′ ,
0) = − δ jk δ ( x − x ′ ) . (16)Once we have calculated the EM field generating func-tional, we can functionally derive it in order to obtain theEM n − point functions and, in particular, the propagators.The calculation of the Wightman function is straight-forward, from which we can read the Hadamard propaga-tor: G jk H ( x , x ) ≡ D A j ( x ) A k ( x ) E Σ A i ,Σ Π i (17)+ h G Ret ∗ (cid:0) ∂ tt ′ N (cid:1) ∗ ( G Ret ) T i jk ( x , x ) . This expression holds for every initial state of the fieldand depends on the chosen gauge. Note that the Hadamardpropagator has two separated contributions. The first termis entirely associated to the field’s effective dynamics andthe initial state. The other contribution is associated tothe material degrees of freedom represented by the noisekernel N , which also splits in two contributions due to thecomposite nature of the material (DOFs plus bath in eachpoint of space).However, all the dynamics is up to the retarded Greentensor in the temporal gauge. As it was shown in Ref.[19],the choice of the Landau gauge naturally implies that A ≡ ∇ · (cid:20)Z tt i dt ′ ∂ t (cid:16) ←→ ε ( t − t ′ , x ) (cid:17) · A ( x , t ′ ) (cid:21) = 0 , (18)where the permittivity tensor for the inhomogeneous andanisotropic material is given by: ε mr ( t − t ′ , x ) ≡ δ mr (cid:16) δ ( t − t ′ ) + λ , x g ( x ) G [ m ]Ret , x ( t − t ′ ) (cid:17) . (19)Since the tensor is diagonal, the expression is given inthe Fresnel’s principal axes basis. The residual conditionin Eq.(18) is a generalization of the condition consideredin Ref.[16]. It was shown in Ref.[19] that in the case ofisotropic and non-dissipative material, Eq.(18) reduces tothe generalized Coulomb condition of Ref.[16] ∇ · [ ε ( x ) A ( x , t )] = 0 , where the permittivity tensor has been replaced by a singlefunction.Also in Ref.[19] it is proved that, in the general case,the equations of motion for the spatial components of theEM field in the temporal gauge can be written as: ∂ A ∂t + ∇ × ( ∇ × A ) + λ , x g ( x ) A ( x , t ) (20)+ λ , x g ( x ) Z tt i dt ′ ¨ ←→ G Ret , x ( t − t ′ ) · A ( x , t ′ ) = 0 , where (cid:16) ←→ G Ret , x (cid:17) mk = δ mk G [ m ]Ret , x . Again, from the factthat we could write this diagonal tensor (associated tothe retarded Green functions), it is clear that the chosenbasis is the Fresnel’s principal axes basis (in general, thetensor would be non-diagonal). It is also remarkable theappearance of the third term, which constitutes a finiterenormalization position-dependent mass term for the EMfield as the one found in the scalar case in Ref.[18]. As weshall see, this term will be irrelevant in the determinationof the Green tensor.Considering the equation of motion for the EM field,the retarded Green tensor ←→G Ret ( x , x ′ , t ) can be definedas:0 = ∂ ←→G Ret ∂t + ∇ × ∇ × ←→G Ret + λ , x g ( x ) ←→G Ret ( x , x ′ , t )+ λ , x g ( x ) Z t dt ′ ¨ ←→ G Ret , x ( t − t ′ ) · ←→G Ret ( x , x ′ , t ′ ) , (21)with the initial conditions given in Eq.(16).Laplace-transforming the equation, we easily obtain: ∇ × ∇ × ←→G Ret + s ←→ ε ( s, x ) · ←→G Ret ( x , x ′ , s ) = − I δ ( x − x ′ ) , (22)where the Laplace transform of the permittivity tensor ofEq.(19) is ←→ ε ( s, x ) = I + λ , x g ( x ) ←→ G Ret , x ( s ) . It is clear that including dissipation it is not possi-ble to define refractive indexes in the time domain. How-ever, since the Laplace transform of the permittivity ten-sor turns out to be diagonal in this basis, the refractiveindexes can be defined in the Laplace variable’s domain.In each direction j , we can define the complex refractiveindexes n [ j ]2 x = 1 + λ , x g ( x ) G [ j ]Ret , x ( s )in such a way that the Fresnel’s ellipsoid is a useful pictureto describe the material’s anisotropy.From Eq.(22) we can see that the Laplace transformof the EM retarded Green tensor turns out to be theFeynman’s Green tensor associated to the differential op-erator ∇ × ∇ × + s ←→ ε ( s, x ) · , and therefore satisfies G ij Ret ( x , x ′ , s ) = G ji Ret ( x ′ , x , s ). It is also worth to note that Fernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system the reality of the EM retarded Green tensor in the timedomain implies that G ij Ret ( x , x ′ , s ) = G ∗ ij Ret ( x , x ′ , s ∗ ) . We would like to write the pressure in terms of the Hadam-ard propagator. In Ref.[19], an expression for the expec-tation values of the components of the Maxwell tensoris given in terms of the Hadamard propagator, using thepoint-splitting technique. As we are interested in the cal-culation of the Casimir force in the Lifshitz problem, asimpler expression than the one in Ref.[19] can be achieved.Considering the symmetry of the configuration, theforce between the bodies will be given only by the pressurein the perpendicular direction to the surfaces, i. e., alongthe direction parallel to the separation distance l , whichwe will call the z axis. Therefore, in the Lifshitz problem,the pressure will be given directly by the zz − componentof the Maxwell tensor, which can be written for a fieldpoint x inside the gap as in Ref.[3]: b T zz ( x ) = − Λ ij π h b E i ( x ) b E j ( x ) + b B i ( x ) b B j ( x ) i , (23)where the electric field is given by E i = − ∂ A j , whilethe magnetic field is B i = ( ∇ × A ) i . Λ ij is the diagonalmatrix Λ = Λ = 1 = − Λ .Then, using the point-splitting technique and typicalrelations between the different propagators (see [12]), theexpectation value of the zz − component, can be writtenin terms of the Hadamard propagator: P Cas ( x ) ≡ D b T zz ( x ) E = − Λ ij π lim x → x h δ is δ jm ∂ t ∂ t + ǫ irs ǫ jlm ∂ r ∂ l i G sm H ( x , x ) , (24)once the propagator is renormalized in the coincidencelimit.It is worth noting that, unlike the one found in Ref.[3]which is proposed to correspond to the steady situation,the last expression actually is the Casimir pressure whichemerges at the initial time t i . Therefore, it comprises allthe transient dynamics of the pressure in the way to reachits final value at the steady situation. The pressure candepend on time but also on space during the transientstage, until finally achieves the steady situation, where itsvalue results to be time and space-independent: P Cas ( x ) → P ∞ Cas . (25)As a final remark, given the splitting of the contribu-tions to the Hadamard propagator in Eq.(17), and alsothat N = N P + N B (depending each one on its tempera-ture), it is clear that the total pressure can be written interms of three contributions: P Cas ( x ) = P IC ( x ) + P DOFs ( x , β P x ) + P B ( x , β B, x ) . (26)Then, each part of the total system will contributeto the Casimir pressure at a given space-time point. Themain subject of study of the next sections will be to de-termine which contributions will survive in the steady sit-uation.As we are dealing with an initial conditions problem,every time variable is defined in the interval [ t i , + ∞ ).Therefore, we can Laplace transform in each time variable t , t inside the coincidence limit of Eq.(24). Introducinga Mellin’s formula for each variable, the second term inEq.(24) is easily written in terms of the retarded Greentensor’s double Laplace transform (note that this termonly involves spatial derivatives):lim x → x ǫ irs ǫ jlm ∂ r ∂ l G sm H ( x , x ) = (27)= Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi e ( s + s )( t − t i ) × lim x → x h ǫ irs ǫ jlm ∂ r ∂ l G sm H ( x , s ; x , s ) i , where in the r.h.s. the coincidence limit was taken for thetime variables and where α , are taken to define verticallines in the s , − complex planes in such a way that all thepoles of the integrands taken as functions of s and s areat the left of these lines.Writing each time variable of the first term of Eq.(24)in terms of Laplace transforms, we find: ∂ t ∂ t G sm H ( x , x ) == Z α + i ∞ α − i ∞ ds πi e s ( t − t i ) L h ∂ t ∂ t G sm H ( x , x ) i = Z α + i ∞ α − i ∞ ds πi e s ( t − t i ) h s ∂ t G sm H ( x , s ; x ) − ∂ t G sm H ( x , t i ; x ) i = Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi e s ( t − t i ) e s ( t − t i ) × h s L (cid:16) ∂ t G sm H ( x , s ; x ) (cid:17) − L (cid:16) ∂ t G sm H ( x , t i ; x ) (cid:17)i = Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi e s ( t − t i ) e s ( t − t i ) × h s s G sm H ( x , s ; x , s ) − s G sm H ( x , s ; x , t i ) − s G sm H ( x , t i ; x , s ) + G sm H ( x , t i ; x , t i ) i . (28)The second term between brackets does not depend on s . Therefore, for that term, the integral over s have theintegrand e s ( t − t i ) , which is analytic in all the s − complexplane, so the integral over any contour vanishes for t > t i .The same happens for the third and last terms between ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 7 brackets. All in all, we have proved that for the first termin Eq.(24):lim x → x δ is δ jm ∂ t ∂ t G sm H ( x , x ) == Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi e ( s + s )( t − t i ) s s δ is δ jm × lim x → x G sm H ( x , s ; x , s ) . (29)Finally, the Casimir pressure can be re-written as: P Cas ( x ) = − π Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi e ( s + s )( t − t i ) lim x → x h Θ sm ( s , s ) G sm H ( x , s ; x , s ) i , (30)where we have defined the operator Θ sm ( s , s ) ≡ Λ ij (cid:16) s s δ is δ jm + ǫ irs ǫ jlm ∂ r ∂ l (cid:17) . Last expression for the Casimir pressure seems for-mally the same to the one found in Ref.[3], however thereare subtle differences all related to the statement of bothproblems. Eq.(30) is not a steady-situation expression (asthe one in Ref.[3]), but it also comprises all the informa-tion about the transient evolution of the zz − componentand describes the building up of the Casimir pressure inthe long time regime. In fact, in Ref.[3], the pressure iscalculated from the electric field correlation, which resultsto be proportional to a Dirac δ -function in the frequen-cies’ difference, imposed by the steady situation formalism(based on the fluctuation-dissipation theorem) in stochas-tic electrodynamics (SED). Thus, the double integrationis automatically reduced to one integration by the sourcecorrelation. In the present work, we calculate the pressurefrom the Hadamard propagator, which is related to theEM quantum field correlation and, as we shall see in nextsections, during the transient evolution it is not necessar-ily proportional to a Dirac δ -function. Finally, all thesepoints are also reflected in the definition of the Θ opera-tor. In Ref.[3], it only depends on one frequency variablewhich appears as a denominator due to the fact that thepressure is calculated from the electric field correlation.The present operator is a function of two Laplace vari-ables which appears as multiplicative factor, since we arecalculating the full time evolution of the pressure from theEM field correlation.In the next sections, we will calculate all the contribu-tions to the pressure and study its time evolution, startingfrom the expression in Eq.(30). The formal expression for the Casimir pressure obtained inEq.(30), is written in terms of the Hadamard propagator’s double Laplace transform. From Eq.(17) it is clear that ithas two separated contributions, one associated to the EMfield’s initial conditions (the first term) and the other oneassociated to the material (the second term). In fact, thenoise kernel also splits into two parts ( N = N P + N B ), oneassociated to the DOFs ( N P ) and the other one associatedto the baths ( N B ). Then, the Hadamard propagator’s dou-ble Laplace transform reads: G jk H ( x , s ; x , s ) ≡ D A j ( x , s ) A k ( x , s ) E Σ A i ,Σ Π i (31)+ L , h(cid:16) G Ret ∗ (cid:0) ∂ tt ′ N (cid:1) ∗ ( G Ret ) T (cid:17) jk i ( x , s ; x , s ) . Each contribution will be analyzed separately.
Provided that the homogeneous solution is given by Eq.(15) and the initial conditions for the Green tensor aregiven in Eq.(16), the Laplace transform of the homoge-neous solution results: A j ( x , s ) = Z d x ′ G jl Ret ( x , x ′ , s ) (cid:0) ΣΠ l i ( x ′ ) − s ΣA l i ( x ′ ) (cid:1) , (32)where G jl Ret ( x , x ′ , s ) is the Laplace transform of the EMretarded Green tensor.Therefore, the first term of the double Laplace trans-form of the Hadamard propagator in Eq.(31) results: D A j ( x , s ) A k ( x , s ) E Σ A i ,Σ Π i = (33)= Z d x ′ Z d x ′′ G jl Ret ( x , x ′ , s ) G km Ret ( x , x ′′ , s ) × Dh ΣΠ l i − s ΣA l i i ( x ′ ) h ΣΠ m i − s ΣA m i i ( x ′′ ) E Σ A i ,Σ Π i . It is clear that calculating the brackets average overinitial configurations will introduce the EM field’s initialstate through its Wigner functional. Therefore, this con-tribution clearly depends on the initial state for the EMfield. However, instead of proceeding directly to the cal-culation of the average, it is more convenient to calculatethe brackets as a quantum expectation value. As the fieldis free at the initial time t i , the quantum expectation val-ues in the operator formalism and these averages (usingthe homogeneous solution) are closely related. In fact, asa particular case of the theory developed in previous sec-tions, it is easy to show that a free theory (without inter-actions) verifies: Dh ΣΠ l i − s ΣA l i i ( x ′ ) h ΣΠ m i − s ΣA m i i ( x ′′ ) E Σ A i ,Σ Π i ≡≡ Dn b Π l i ( x ′ ) − s b A l i ( x ′ ); b Π m i ( x ′′ ) − s b A m i ( x ′′ ) oE , (34) Fernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system where the r.h.s. corresponds to the quantum expectationvalue of the anticommutator containing quantum free EMfield operators in the temporal gauge. In the free EM fieldcase, the temporal gauge also implies the Coulomb gaugecondition ( ∇ · A = 0) over the remaining components.Therefore, the EM free field operators are: b A ji ( x ) = Z d k p ω k (2 π ) X λ ε j ( k , λ ) hb a k ,λ e − i ( ω k t i − k · x ) + b a † k ,λ e i ( ω k t i − k · x ) i , (35) b Π ji ( x ) = Z d k p ω k (2 π ) X λ iω k ε j ( k , λ ) hb a k ,λ e − i ( ω k t i − k · x ) − b a † k ,λ e i ( ω k t i − k · x ) i , (36)where λ sums over the transverse electric (TE) and mag-netic (TM) polarizations, b a k ,λ , b a † k ,λ are the free EM fieldphoton annihilation and creation operators, ε j ( k , λ ) is the j − component of the polarization vectors and ω k = | k | .Considering the completeness relation for the polariza-tion vectors: X λ =TE , TM ε l ( k , λ ) ε m ( k , λ ) = δ lm − k l k m ω k , (37)and an initial thermal state for the EM field we get Db a k ,λ b a † k ′ ,λ ′ E = δ λλ ′ δ ( k − k ′ ) (1 + N ( ω k )) , (38) Db a † k ,λ b a k ′ ,λ ′ E = δ λλ ′ δ ( k − k ′ ) N ( ω k ) , (39) Db a k ,λ b a k ′ ,λ ′ E = Db a † k ,λ b a † k ′ ,λ ′ E = 0 , (40)where N ( ω k ) = ( e β EM ω k − ) is the photon occupation num-ber for the initial thermal state of temperature β EM . Then,after a change of variables, Eq.(34) can be written as: Dh ΣΠ l i − s ΣA l i i ( x ′ ) h ΣΠ m i − s ΣA m i i ( x ′′ ) E Σ A i ,Σ Π i == Z d k ω k (2 π ) (cid:18) δ lm − k l k m ω k (cid:19) coth (cid:18) β EM ω k (cid:19) ×× e i k · ( x ′ − x ′′ ) (cid:0) s s + ω k (cid:1) . (41)Therefore, the average over the initial conditions inEq.(33) results: D A j ( x , s ) A k ( x , s ) E Σ A i ,Σ Π i = (42)= Z d k ω k (2 π ) (cid:20) δ lm − k l k m ω k (cid:21) coth (cid:20) β EM ω k (cid:21) (cid:2) s s + ω k (cid:3) × (cid:18)Z d x ′ G jl Ret ( x , x ′ , s ) e i k · x ′ (cid:19) × (cid:18)Z d x ′′ G km Ret ( x , x ′′ , s ) e − i k · x ′′ (cid:19) . Having this result, we can obtain an expression for thecontribution of the initial conditions to the Casimir pres-sure. Replacing G sm H in Eq.(30) by the initial conditions’contribution D A j ( x , s ) A k ( x , s ) E Σ A i ,Σ Π i it is straight-forward that: P IC ( x , β EM ) = − π Z d k ω k (2 π ) (cid:20) δ lm − k l k m ω k (cid:21) × coth (cid:20) β EM ω k (cid:21) Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi × e ( s + s )( t − t i ) (cid:0) s s + ω k (cid:1) lim x → x " Θ jk ( s , s ) × (cid:18)Z d x ′ G jl Ret ( x , x ′ , s ) e i k · x ′ (cid:19) × (cid:18)Z d x ′′ G km Ret ( x , x ′′ , s ) e − i k · x ′′ (cid:19) . (43)Remarkably, this expression is quite general. In fact, itcomprises the full time evolution of the contribution of theinitial conditions to the Casimir pressure at any time forany point of space in a vacuum region, since the bound-aries appear at the initial time t i . This is the reason why itis physically expected that the contribution depends notonly in time but also in space until reaching the steadysituation. The information about the specific time evolu-tion of the problem is encoded in the analytical propertiesof the integrand as a function of s and s , i.e. in the polesand the branch cuts present, as we shall see below.Moreover, it is important to note that this expressionis valid for any geometry of the boundaries, including atleast one vacuum region (where we calculate the pressure).The information about the boundaries is contained in theLaplace transforms of the retarded Green tensors whichhave to be calculated in a specific situation in order toobtain a complete result. In next sections, we will showhow this works for the Lifshitz problem, deriving the long-time limit of this contribution. Let us consider the second term in the r.h.s. of Eq.(31),which is associated to the material. Due to the fact that ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 9 N = N P + N B , the contribution splits into two contri-butions, one associated to the DOFs and the other oneassociated to the baths. However, the first step in Laplace-transforming the contribution is the same. As the contri-bution reads L , h(cid:16) G Ret ∗ (cid:0) ∂ tt ′ N (cid:1) ∗ ( G Ret ) T (cid:17) jk i ( x , s ; x ,s ) and the retarded Green tensors depend on the timedifferences, i. e., G ij Ret ( x, x ′ ) = G ij Ret ( x , x ′ , t − t ′ ), then theproducts ∗ involve convolutions in the time variables be-tween the noise kernel ∂ tt ′ N and one of the retarded Greentensors. Laplace-transforming is straightforward: L , h(cid:16) G Ret ∗ (cid:0) ∂ tt ′ N (cid:1) ∗ ( G Ret ) T (cid:17) jk i ( x , s ; x , s ) == Z d x Z d x ′ G jl Ret ( x , x , s ) L , (cid:2) ∂ tt ′ N lm (cid:3) ( x , s ; x ′ , s ) × G km Ret ( x , x ′ , s ) . (44)A few simplifications arise by considering that the noisekernels satisfy ∂ tt ′ N lm ( x, x ′ ) = δ lm δ ( x − x ′ ) g ( x ) ∂ tt ′ N [ l ] x ( t, t ′ )(see Eqs.(10) and (11) ). Therefore, taking into accountthe δ -functions: L , h(cid:16) G Ret ∗ (cid:0) ∂ tt ′ N (cid:1) ∗ ( G Ret ) T (cid:17) jk i ( x , s ; x , s ) == Z d x g ( x ) G jl Ret ( x , x , s ) L , h ∂ tt ′ N [ l ] x i ( s , s ) × G kl Ret ( x , x , s ) , (45)where the presence of g denotes the fact that this contri-bution comes from the material regions exclusively.Considering the definitions in Eqs.(10) and (11), Lapla-ce-transforming in both time variables the EM noise kernel(taking into account that N [ l ] x , B ( t i , t i ) = 0 = N [ l ] x , B ( s , t i ) = N [ l ] x , B ( t i , s ) due to the causality of G [ l ]Ret , x ), we obtain: L , h ∂ tt ′ N [ l ] x i ( s , s ) = s s N [ l ] x , P ( s , s ) − s N [ l ] x , P ( s , t i ) − s N [ l ] x , P ( t i , s ) + N [ l ] x , P ( t i , t i )+ s s N [ l ] x , B ( s , s ) , (46)where each Laplace transform is given by: N [ l ] x , P ( s , s ) = λ , x M [ j ] x Ω [ j ] x coth β P j x Ω [ j ] x ! (cid:16) s s + Ω [ l ]2 x (cid:17) × G [ l ]Ret , x ( s ) G [ l ]Ret , x ( s ) , (47) N [ l ] x , P ( s n , t i ) = N [ l ] x , P ( t i , s n ) (48)= λ , x M [ l ] x Ω [ l ] x coth β P l x Ω [ l ] x ! s n G [ l ]Ret , x ( s n ) , N [ l ] x , P ( t i , t i ) = λ , x M [ l ] x Ω [ l ] x coth β P l x Ω [ l ] x ! , (49) N [ l ] x , B ( s , s ) = λ , x G [ l ]Ret , x ( s ) N [ l ] x ( s , s ) G [ l ]Ret , x ( s ) . (50)The last term in the r.h.s. of Eq.(46) is the only oneassociated to the baths. Moreover, it is important to notethat the second and third terms only depend on one of theLaplace variables, while the fourth term does not dependon them.Is clear that in order to continue the explicit calcula-tion, one has to define the type of baths which are consid-ered in each direction and point of space for the specificproblem. However, we can take a further step in the calcu-lation without loosing generality. From the QBM theory,is clear that for any type of bath, the QBM noise de-pends on the time differences, i. e., in the time domainwe have that N [ l ] x ( t, t ′ ) = N [ l ] x ( t − t ′ ). Therefore, althougheach time variable is defined in the interval [ t i , + ∞ ), itsdifference results defined in ( −∞ , + ∞ ). Assuming thatthe kernel verifies the convergence requirements (this canbe easily implemented by introducing a cut-off function inthe QBM noise kernel), it can be written in terms of itsFourier transform: N [ l ] x ( t − t ′ ) = Z + ∞−∞ dω π e − iω ( t − t ′ ) N [ l ] x ( ω ) , (51)where N [ l ] x ( ω ) contains the dependencies on the tempera-tures of the baths β x , B .Using this general form for the QBM noise kernel, itsLaplace transform is given by Eq.(50): N [ l ] x , B ( s , s ) = λ , x G [ l ]Ret , x ( s ) G [ l ]Ret , x ( s ) Z + ∞−∞ dω π × N [ l ] x ( ω )( s + iω )( s − iω ) . (52)Once we have calculated each term of Eq.(46), theEq.(30) gives expressions for each contribution to the Casi-mir pressure in Eq.(26). Therefore, we can write: P DOFs ( x , β P l x ) = − π Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi × e ( s + s )( t − t i ) lim x → x " Θ jk ( s , s ) Z d x g ( x ) λ , x M [ l ] x Ω [ l ] x × coth " β P l x Ω [ l ] x s G [ l ]Ret , x ( s ) − i h s G [ l ]Ret , x ( s ) − i + Ω [ l ]2 x s s G [ l ]Ret , x ( s ) G [ l ]Ret , x ( s ) (cid:17) × G jl Ret ( x , x , s ) G kl Ret ( x , x , s ) , (53) P B ( x , β x , B ) = − π Z + ∞−∞ dω π Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi × s s e ( s + s )( t − t i ) ( s + iω )( s − iω ) lim x → x h Θ jk ( s , s ) Z d x g ( x ) λ , x × G jl Ret ( x , x , s ) G [ l ]Ret , x ( s ) G [ l ]Ret , x ( s ) N [ l ] x ( ω ) × G kl Ret ( x , x , s ) i . (54)Note that this expression includes an integral over theFourier frequencies.Last results are valid in the most general case, which isan inhomogeneous and anisotropic material having localinitial temperatures at each point. However, these expres-sions are simplified when considering the interaction be-tween n different but homogeneous and isotropic materialbodies of volume V n , at homogeneous temperatures. Inthis case, all the spatial subscripts x (denoting inhomo-geneity) and superscripts [ l ] (denoting anisotropy) haveto be replaced by a single index n denoting the materialbody. Therefore the integral over the space splits as a sumof integrals over each volume V n , and both expressions canbe written as: P DOFs ( x , β P n ) = − π X n λ ,n M n Ω n coth (cid:18) β P n Ω n (cid:19) × Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi e ( s + s )( t − t i ) × h (cid:0) s G Ret , n ( s ) − (cid:1) (cid:0) s G Ret , n ( s ) − (cid:1) + Ω n s s G Ret , n ( s ) G Ret , n ( s ) i × lim x → x " Θ jk ( s , s ) Z V n d x G jl Ret ( x , x , s ) × G kl Ret ( x , x , s ) , (55) P B ( x , β n, B ) = − π X n λ ,n Z + ∞−∞ dω π N n ( ω ) Z α + i ∞ α − i ∞ ds πi × Z α + i ∞ α − i ∞ ds πi s s e ( s + s )( t − t i ) ( s + iω )( s − iω ) G Ret , n ( s ) G Ret , n ( s ) × lim x → x h Θ jk ( s , s ) Z V n d x G jl Ret ( x , x , s ) × G kl Ret ( x , x , s ) i . (56)In this case, the quantitylim x → x h Θ jk ( s , s ) Z V n d x G jl Ret ( x , x , s ) G kl Ret ( x , x , s ) i is common to both contributions. However, the Mellin’sintegrals are the ones that define the time evolution and steady state of these contributions, and are very differentdue to the analytical properties of each integrand.Moreover, is important to have in mind that in theseexpressions, it is implicit that at every point of the ma-terial bodies, the basis of Fresnel principal axis is thesame. Each point defines a Fresnel’s ellipsoid having itsaxis along the directions in which each component of theDOFs fluctuates. This is, in principle, a limitation in thepresent model, although it could be easily fixed by consid-ering different basis in each body (or even at each point ofspace). As we mentioned before, in order to simplify thecalculations but without losing the anisotropy properties,we will keep an unique basis for all the material points.This limitation disappears when considering isotropic ma-terials, as in the Lifshitz problem analyzed below.However, it is important to recognize that the ellip-soid’s picture is valid for real refractive indexes. In ourcase, as we shall see, due to the inclusion of dissipationin the problem, the three refractive indexes are complex,albeit the Fresnel’s ellipsoid results useful as a conceptualand practical picture.As final remarks, on the one hand, it is worth notingthat considering this case for the initial conditions’ contri-bution has no formal simplification over Eq.(43) since theinitial field modes and the spatial integrals are over thewhole space ( g ( x ) does not appear in those integrals). Onthe other hand, it is important to take into account thatall the expressions containing a coincidence limit have tobe regularized before calculating the limit. For the caseof the Lifshitz problem, this will be done after integratingover the space. With the general expressions for each contribution of theCasimir pressure generated by different bodies of homoge-neous and isotropic material at different temperatures ina vacuum region, we can study the Lifshitz problem con-sisting in two half-spaces by letting n be L or R for theleft and right plate respectively, as we have mentioned atthe end of Sect. 2.2. From Eqs.(43), (55) and (56) the time evolution and steadystate of each contribution will be governed by the analyti-cal properties of the integrands as functions of the Laplacevariables s , s . Therefore, for a given problem is critical toknow the analytical properties of both Laplace transformsof the QBM’s retarded Green function G Ret , n ( s ) and EMretarded Green tensor G ij Ret ( x , x ′ , s ).As the DOFs are considered as quantum Brownianparticles, irrespective of what type of environment is con-sidered the Laplace transform of the retarded Green func-tion can be easily obtained from the equation of motionof the QBM’s theory. In the present case, the result is the ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 11 same as the one found in Ref.[18]. Then, we can easilywrite: G Ret , n ( s ) = 1 (cid:16) s + Ω n − D n ( s ) (cid:17) , (57)where D n ( s ) is the Laplace transform of the QBM’s re-tarded Green function for the environment (baths) of theplate n .Thus, given a spectral density for the baths, the lo-cation of the poles will define the time evolution and theasymptotic behavior of the retarded Green function. How-ever, causality implies, by Cauchy’s theorem, that thepoles of G Ret , n should be located in the left-half of thecomplex s − plane. In fact, assuming that Ω n = 0 and thatthe baths include cutoff functions in frequencies, the realparts of the poles are negative (see Ref.[18]).On the other hand, the EM retarded Green tensor isdefined by the equations of motion obtained from the EMCTP action of Eq.(13) after imposing the temporal gauge.In the inhomogeneous and anisotropic case, by solvingEq.(22) for a given permittivity tensor defining boundariesand material bodies, we can obtain the Laplace transformof the EM retarded Green tensor and calculate the Casimirpressure through Eqs.(43), (53) and (54). However, thesolution in these cases is very complicated.Therefore, we turn to the simpler case of consideringthe Lifshitz problem, where two different homogeneousand isotropic parallel half-spaces are separated by a vac-uum gap of length l along the z direction. The origin of thecoordinate system is defined in the middle of the gap. Inthis case, each contribution to the Casimir pressure has tobe evaluated from Eqs.(43), (55) and (56), where the lasttwo equations are simplified versions of the general inho-mogeneous and anisotropic case, but the first one remainsunaltered. In this case, the Fresnel’s ellipsoid turns out tobe a sphere and the permittivity tensor results a functiontimes the identity matrix, allowing any basis to describeeach material body. Therefore, as we anticipated before,we omit the superscripts [ j ] and replace x by the indices L, R related to each material body. All in all, Eq.(22) sim-plifies to: (cid:16) ∇×∇× + s ε ( s, z ) (cid:17) ←→G Ret ( x , x ′ , s ) = − I δ ( x − x ′ ) , (58)where the refraction index only depends on z and it isgiven by n ( s, z ) = ε ( s, z ) = 1 + λ ,L Θ (cid:18) − l − z (cid:19) G L Ret ( s )+ λ ,R Θ (cid:18) z − l (cid:19) G R Ret ( s ) . It is remarkable that this equation and the one solvedin Ref.[3] are basically the same, but this time the solu-tion gives the Laplace transform of the retarded EM Greentensor. Two differences are in order: the equation is for-mally the same by replacing the Fourier variable in the solutions given in Ref.[3] by is ; on the other hand, afterthe replacement, the r.h.s. of the Green function equationin Ref.[3] is the r.h.s. of Eq.(58) times 4 πs , because itis compensating the differences in the definitions of theoperator Θ sm in the pressure expression of Eq.(30) as itwas commented at the end of Sect.3.2. As the Laplacevariable appears as a parameter in the equation, we caneasily obtain the Laplace transform of the EM retardedGreen tensor by dividing the one given in Ref.[3] by 4 πs .Due to translational invariance in the parallel coordinates,the solution is given in terms of the Fourier transform inthose coordinates: G ij Ret ( x , x ′ , s ) = Z d Q (2 π ) e i Q · ( x k − x ′k ) G ij Ret ( z, z ′ , Q , s ) , (59)where Q = ( Q x , Q y , x, ˇ y the parallel directions.For a field point inside the gap ( − l < z < l ), thetransform of the EM Green tensor depends on the valueof the source point z ′ . For z ′ < − l we have: G ij Ret ( z, z ′ , Q , s ) = − q (1) z X µ =TE , TM t µ D µ h e µ,i [+] e − q z ( z − l )+ e µ,i [ − ] r µ e q z ( z − l ) e − q z l i e (1) µ,j [+] e q (1) z ( z ′ − l ) , (60)and for z ′ > l : G ij Ret ( z, z ′ , Q , s ) = − q (2) z X µ =TE , TM t µ D µ h e µ,i [ − ] e q z ( z − l )+ e µ,i [+] r µ e − q z ( z − l ) i e (2) µ,j [ − ] e − q z l e − q (2) z ( z ′ − l ) , (61)while for a point source inside the gap − l < z ′ < l thetensor splits in bulk and scattered contributions ( G ij Ret = G ij Ret , Bu + G ij Ret , Sc ): G ij Ret , Bu ( z, z ′ , Q , s ) = − δ iz δ jz s δ ( z − z ′ ) (62) − q z X µ =TE , TM h e µ,i [+] e µ,j [+] e − q z ( z − z ′ ) Θ ( z − z ′ )+ e µ,i [ − ] e µ,j [ − ] e q z ( z − z ′ ) Θ ( z ′ − z ) i , G ij Ret , Sc ( z, z ′ , Q , s ) = − q z X µ =TE , TM D µ h e µ,i [+] e µ,j [+] × r µ r µ e − q z ( z − z ′ +2 l ) + e µ,i [+] e µ,j [ − ] r µ e − q z ( z + z ′ − l ) + e µ,i [ − ] e µ,j [+] r µ e q z ( z + z ′ − l ) + e µ,i [ − ] e µ,j [ − ] r µ r µ × e q z ( z − z ′ − l ) i . (63)Here we are considering the notation given in Ref.[3]adapted to our case. Therefore, the EM (complex) wavevector in the medium n is given by: q ( n ) [ ± ] = Q ± iq ( n ) z ˇ z , (64)with n = L, R corresponding to each plate (while omit-ting the indices for the vacuum region) and where the sign+ ( − ) corresponds to an upward (downward) wave. Thevector Q is always a real vector and appears as the pro-jection of the wave vector q ( n ) [ ± ] on the interface, whilethe z − component is given by: q ( n ) z = p ε n ( s ) s + Q . (65)The wave vector q ( n ) [ ± ] lies in the plane of incidencedefined by ˇ Q and ˇ z . We can introduce the tranverse elec-tric (TE) and magnetic (TM) polarization vectors: e ( n )TE [ ± ] = ˇ Q × ˇ z , (66) e ( n )TM [ ± ] = e ( n )TE [ ± ] × ˇ q ( n ) [ ± ] = Q ˇ z ∓ iq ( n ) z ˇ Q p ε n ( s ) is . (67)The reflection ( r ) and transmission ( t ) Fresnel coeffi-cients for each single surface and components are givenby: r TE n = q z − q ( n ) z q z + q ( n ) z , r TM n = ε n q z − q ( n ) z ε n q z + q ( n ) z , (68) t TE n = 2 q ( n ) z q z + q ( n ) z , t TM n = 2 p ε n ( s ) q ( n ) z ε n q z + q ( n ) z , (69)while the multiple reflections (which does not enter in thebulk part) are described by the denominator: D µ = 1 − r µ r µ e − q z l . (70)Beyond the apparent complicated expressions for theLaplace transform of the EM retarded Green tensor, itsanalytical properties can be analyzed relatively easy. Eachexpression presents several poles and branch cuts. Thepoles basically are the ones associated to D µ and there isone pole located at the origin s = 0.The poles coming from D µ should be calculated in or-der to obtain the time evolution of the system. However,causality implies that all these poles must lie on the left-half of the complex s − plane in order to give convergentcontributions in the evolution. It may happen that somepoles have vanishing real part. These poles would con-tribute to the long-time behaviour of the EM retardedGreen tensor (and therefore to the pressure). However, ifwe set s = i Im( s ) (with Im( s ) = 0) it can be easily shownthat there is no poles of this type.The pole at the origin appears explicitly in the firstterm of Eq.(62). However, in the second term and in Eqs.(60), (61) and (63), the pole is only associated to the termswith µ = TM, since in any region the TM polarization vectors provide the pole through its denominators, as isclear in Eq.(67).Besides the poles, the Laplace transforms in each re-gion present several branch cuts, which provide to thetransient time evolution of the EM Green tensor. In thepresent case, by inspection, the branch cuts are given bythe square roots q s + k k and q ε n ( s ) s + k k . The extrabranch cut p ε n ( s ) that appears in the expression of thetransmission coefficient for the TM terms in Eq.(69) thatenter the Laplace transforms in Eqs.(60) and (61) shouldnot be considered, since it cancels with the same squareroot contained in the denominator of the polarization vec-tors e ( n )TM ( ± ) of Eq.(67), that also enter the same Laplacetransforms.Summarizing, the analytical properties of the Laplacetransform of the EM retarded Green tensor are very sim-ple, despite of its complicated expressions. These proper-ties are important in order to compute the full time evo-lution of the problem and determining the long-time limitof the different physical quantities of interest. Eqs.(60)-(63) contain the Laplace-Fourier transforms forthe EM retarded Green tensor for a field point inside thegap in the Lifshitz problem, from which we can calculatethe corresponding Casimir pressure in the steady state.For all the three contributions of Eqs.(43), (55) and(56), in order to completely solve its time evolutions wehave to calculate both inverse-Laplace transforms throughthe residue theorem and the analytical properties associ-ated to branch cuts.However, as our aim is to determine the long-time limitof the contributions, the work is simpler and related to asubtle issue of the long-time regime. The ‘steadiness’ of thephysical quantities as the pressure or energy is related tothe fact that they do not depend on time in the long-timeregime. In other words, usually for the systems of interest,due to physical/intuitive reasons, the complete system isexpected (or assumed) to be in a steady situation in thelong-time limit. In many cases, this allows one to face andstudy the steady regime directly, without a rigorous (andunnecessary) derivation from a full dynamical problem.In the present case, we are in the opposite situation. Wehave solved the full dynamical problem in terms of doubleinverse-Laplace transform, that should be calculated usingthe residue theorem. But we cannot do that completelydue to the technical impossibility of locating the poles ofthe integrands in the complex planes analytically (we onlyknow a few properties related to causality of the retardedGreen tensors).Nevertheless, to obtain the long-time limit of quanti-ties given in terms of double inverse-Laplace transformsdepending on e ( s + s )( t − t i ) is relatively easier.As the time evolution depends on the configurationof poles and branch cuts that the integrand presents, wehave to distinguish which analytical properties contributeto the steady state of the quantities. As it is stressed in ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 13 Ref.[20], we can show that the typical branch cuts that ap-pear in the Laplace transform of the EM retarded Greentensor do not contribute to the long-time regime. Then,the steady situation must be given by some of the poles,which always result in exponential temporal behavioursdepending on the poles’ values. For example, consider-ing a typical Laplace transform (i.e., depending only onone Laplace variable), if a given pole have a non-zero realpart, through causality we know that it must be a negativenumber. Therefore, the resulting behaviour associated tothat pole will be an exponential decaying with time. Onthe other hand, if a given pole has vanishing real part (i.e., the pole is purely imaginary), the resulting behaviourwill be oscillating in time with a frequency given by theimaginary part of the pole. As a last case, it turns outthat if the pole is 0, the resulting behaviour will be of atime-independent constant.However, in our case, the presence of e ( s + s )( t − t i ) in the double inverse-Laplace transforms makes the polesassociated to s to combine with those associated to s in order to determine the time evolution of the quantity.Thus, the case of a pole at s = 0 is not the only waywe can obtain time-independent behaviours in our quan-tities of interest. In other words, if we solve both Laplaceintegrals, each term of the full time evolution of the quan-tity will be a combination of two of the poles, one associ-ated to the Laplace integral over s (for example s ∈ C )and the other one to s (consider it at s = s ). Then, ifwe combine two poles satisfying s = − s , there will beno exponential associated to that term in the time evo-lution, obtaining a time-independent term. In particular,is clear that the case of both poles equal to 0 satisfiesthe required condition, giving indeed a time-independentterm. However, if we take a pole with negative real part,for example s in such a way that Re[ s ] <
0, the re-quired condition to obtain a time-independent term would s = − s . As this implies Re[ s ] >
0, which is forbiddenby the causality property, then for all physical quantitiesbuilded from causal quantities, all the poles presentingnegative real part cannot be combined in order to obtaina time-independent term.As a last possibility, let us consider the poles with realpart equal to 0. In these cases, considering a pole i s with s ∈ R , we can verify the required condition with a pole i s = − i s in such a way that both oscillating evolutionscancel out in the exponential giving a time-independentterm.Following this train of thought, the time-independentterms will be so from the very beginning. The steady sit-uation rises when the transient term vanish, i. e., when allthe time-dependent terms resulting from poles combina-tion and branch cuts go to 0. Thus, the relaxation time ofthe system will be defined by the last term that vanishesbetween all the transient terms. Moreover, the relaxationtime will be equal to the smallest non-zero real part ofthe combination of poles or branch points in the threecontributions. All in all, in order to determine the steady state ofeach contribution it is necessary to study the combinationof the poles present in both integrands over s and s . We can start by considering one of the parts of the mate-rial’s contribution to the Casimir pressure for the Lifshitzproblem, which is associated to the DOFs’ contributiongiven by the Eq.(55) where, as we anticipated, n = L, R .The quantitylim x → x h Θ jk ( s , s ) Z V n d x G jb Ret ( x , x , s ) G kb Ret ( x , x , s ) i , which is present in this contribution as well as in the baths’one can be simplified in the present case, since in the lastSection we have given the Laplace transform of the EMretarded Green tensor for the Lifshitz problem and itsanalytical properties.Taking into account that for this problem the parallelcoordinates can be Fourier-transformed as in Eq.(59), theintegrations over the parallel coordinates x k can be easilydone, obtaining: Z V n d x G jb Ret ( x , x , s ) G kb Ret ( x , x , s ) == Z d Q (2 π ) e i Q · ( x k − x k ) Z V n dz G jb Ret ( z , z, Q , s ) × G kb Ret ( z , z, − Q , s ) , (71)where in the r.h.s. the integration over V n implies the in-tegration over each half-space. Then, we can write Z V n dz = ( − n Z ( − n ∞ ( − n l dz, where we associate n = L ( n = R ) on the l.h.s. with n = 1( n = 2) on the r.h.s. of this equality.By considering Eqs.(60) and (61), is clear that the re-maining integration over z involves exponential functionsin the second argument, being easily calculated, obtaining: Z V n d x G jb Ret ( x , x , s ) G kb Ret ( x , x , s ) = (72) × Z d Q (2 π ) e i Q · ( x k − x k ) 1 (cid:16) q ( n ) z ( s , Q ) + q ( n ) z ( s , Q ) (cid:17) × G jb Ret ( z , ( − n l/ , Q , s ) G kb Ret ( z , ( − n l/ , − Q , s ) . Therefore, from Eq.(55), we finally obtain the full timeevolution of the contribution to the Casimir pressure: P DOFs ( x , β P n , l ) = − π X n =L , R λ ,n M n Ω n coth (cid:20) β P n Ω n (cid:21) × Z d Q (2 π ) Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi e ( s + s )( t − t i ) (73) × h (cid:0) s G Ret , n ( s ) − (cid:1) (cid:0) s G Ret , n ( s ) − (cid:1) + Ω n s s G Ret , n ( s ) G Ret , n ( s ) i lim x → x " Θ jk ( s , s ) e i Q · ( x k − x k ) × G jb Ret ( z , ( − n l/ , Q , s ) G kb Ret ( z , ( − n l/ , − Q , s ) (cid:16) q ( n ) z ( s , Q ) + q ( n ) z ( s , Q ) (cid:17) , where we have stressed the fact that the contribution tothe pressure depends on the plates separation l .Now, in order to determine the long-time behaviourof this expression we have to analyze the integrand’s an-alytical properties in terms of s and s simultaneously.For a given n , all the terms will not contribute necessarily.Taking in account the analytical properties of the retardedGreen functions G Ret , n and the EM retarded Green tensor ←→G Ret (which splits into TE and TM terms) commentedin the last Section, by inspection it turns out that the onlypoles of the whole two-Laplace variables integrand result-ing in a time-independent term are the ones located at theorigin simultaneously for both variables. In other words,the poles located at the origin for both Laplace variables( s = 0 = s ) associated to the products TM × TM re-sulting from G jb Ret ( z , ( − n l/ , Q , s ) G kb Ret ( z , ( − n l/ , − Q , s ) . (74)However this is not so straightforward because this prod-uct is also multiplied by[( s G Ret , n ( s ) − s G Ret , n ( s ) − Ω n s s G Ret , n ( s ) G Ret , n ( s )]and Θ jk ( s , s ), containing linear factors s and s thatcan eventually prevent of s = 0 or s = 0 of being polesin a given term. At the end, we find poles of first andsecond order depending of which terms are considered.As we mentioned before, the poles at the origin arerelated to the terms associated to the TM polarizationvectors of Eq.(67). Therefore, from Eqs.(60) and (61), wecan see that the term TM × TM of the product in Eq.(74)contains 1 /s and 1 /s as factors.For example, terms associated to the combinations s G Ret , n ( s ) (or s G Ret , n ( s )) will cancel out the denomi-nators 1 /s (or 1 /s ) and therefore those terms will haveno pole at s = 0 ( s = 0). Those terms contribute onlyto the transient regime.On the other hand, the term independent of s and s resulting from( s G Ret , n ( s ) − s G Ret , n ( s ) − Ω n s s G Ret , n ( s ) G Ret , n ( s )will give poles of first order for both Laplace variablesthrough it combination with the first term of Θ jk ( s , s )(associated to the electric field’s contribution), and polesof second order for both variables in the combination withthe second term of Θ jk ( s , s ) (associated to the magneticfield’s contribution).Finally, the term Ω n s s G Ret , n ( s ) G Ret , n ( s ) will onlypresent poles of first order when combined with Θ jk ( s , s )for the product with the second term (the combinationwith the first term will not have poles at the origin).Therefore, the contribution to the pressure in Eq.(73)can be rewritten by grouping the terms according to theorder of the pole at the origin in both variables: P DOFs ( x , β P n , l ) = (Terms with second order poles at 0)+ (Terms with first order poles at 0)+ (Terms without poles at 0) . (75)In first place, it is clear that the last term has no con-tribution to the steady situation, taking part only in thetransient stage.In second place, it is easy to show that the second or-der poles at the origin result in terms that are directlyzero or that diverge in the long-time limit ( t i → −∞ ).This is because, in the residue calculation of the secondorder poles, it is necessary to differentiate the integrandwith respect to the Laplace variable in consideration (ei-ther s or s ). At this point, it is important to considerthat the residue in these terms can be analyzed separatelyin each variable and that the integrands have the form of e s ( t − t i ) times a function which depend on p s + Q and p ε n ( s ) s + Q . First, differentiating the exponential re-sults in ( t − t i ) e s ( t − t i ) times the same function dependingon p s + Q and p ε n ( s ) s + Q . The evaluation of thison s = 0 gives the mentioned term that diverge in the long-time limit. However, all these terms always have anotherterm which cancels them, giving no contribution. Sec-ondly, differentiation the function depending on p s + Q and p ε n ( s ) s + Q gives terms that appear accompaniedby s/ p s + Q or [ ε ′ n ( s ) s + 2 ε n ( s )] s/ p ε n ( s ) s + Q ,which evaluated on s = 0 are the mentioned vanishingterms.In conclusion, the second order poles at the origin donot contribute to the steady situation.The remaining terms are the ones containing first orderpoles at the origin. These terms, in the temporal and spa-tial coordinates’ domain, are associated to a time deriva-tive of one of the Heaviside functions Θ ( t − t i ) whichappear in the definition of the retarded Green functionsor the EM retarded Green tensor due to its causal be-haviour. Mathematically, deriving a Heaviside functiongives a Dirac δ -function. Therefore the expression is pro-portional to δ (0). However, this is caused by the suddenbeginning of the interaction. Physically, this is an approx-imation to the fact that the interaction rises in a very ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 15 short time. If instead, for example, we replace the Heavi-side function by a smooth function going from 0 to 1 in fi-nite time, approaching this value asymptotically, we wouldavoid this problem, keeping the convergent behaviour ofthe whole expression at the long-time limit. In summary,the first order poles at 0 are the result of deriving a Heav-iside function representing the sudden switching-on of theinteraction and hence they are not physical. Therefore,they will not have contribution to the long-time regime.This agrees with the limit for an analog contribution con-sidered in Refs.[18,19], where the same contributions inthe coordinates’ domain are discarded by the same rea-sons.Then, we have proved that for the contribution of theDOFs to the pressure, the long-time limit ( t i → −∞ ) isgiven by: P DOFs ( x , β P n , l ) −→ P ∞ DOFs ( β P n , l ) = 0 . (76) Now we proceed to calculate the contribution resultingfrom the baths.We recall the general expression of Eq.(56) found forthe baths’ contribution to the pressure in a scenario ofhomogeneous and isotropic bodies, considering as in thelast case n = L , R.As in the DOFs’ contribution, the pressure containsthe expression Z V n d x G jb Ret ( x , x , s ) G kb Ret ( x , x , s ) . For the case of half-infinite parallel plates, Eq.(72) pro-vides a simplification for the expression and then, the pres-sure reads: P B ( x , β n, B , l ) = − π X n =L , R λ ,n Z + ∞−∞ dω π N n ( ω ) Z d Q (2 π ) × Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi s s e ( s + s )( t − t i ) ( s + iω )( s − iω ) × G Ret , n ( s ) G Ret , n ( s ) (cid:16) q ( n ) z ( s , Q ) + q ( n ) z ( s , Q ) (cid:17) lim x → x h Θ jk ( s , s ) × e i Q · ( x k − x k ) G jb Ret ( z , ( − n l/ , Q , s ) × G kb Ret ( z , ( − n l/ , − Q , s ) i . (77)Despite the branch cuts, in order to determine thelong-time behaviour of this contribution we have to con-sider again combinations of poles in the Laplace variables s and s in such a way to obtain a time-independentterm.As a first step, as in the DOFs’ contribution, we haveto analyze the pole at s = 0 = s coming from the TM × TM terms. However, due to the presence of a factor s s in the integrand, the only term of Θ jk ( s , s ) that canhave a contribution through this pole is the second one Λ pm ǫ prj ǫ mlk ∂ r ∂ l (associated to the magnetic fields),giving first order poles for both Laplace variables. On theother hand, the first term of Θ jk ( s , s ) (associated to theelectric fields) contains an extra factor s s which finallycancels the denominators in the terms TM × TM whichprovide the pole at s = 0 = s , having no pole for thefirst term.The calculation of the pole implies summing the in-dexes in the expressions. Calculating explicitly all the sumsand taking into account the coincidence limit, it is easy tosee that the numerator also vanish for s = 0 = s . Then,when the denominator vanishes, the numerator vanishestoo. Using L’Hopital rule one obtains a finite value, andtherefore, s = 0 = s it is not a pole.Besides the point s = 0 = s , by inspection is clearthat this contribution also presents another combinationof poles on s and s that produces a time-independentterm. Due to the fact that the QBM noise kernel dependson the time difference (and then it can be written in termsof its Fourier transform), the denominators ( s + iω ) and( s − iω ) appear in this contribution denoting the non-damped dynamics of the baths. Therefore, the pole at s = − iω combined with the pole s = iω gives a time-independent term.All in all, the long-time contribution of the baths tothe Casimir pressure can be written as: P ∞ B ( β n, B , l ) = − π X n =L , R λ ,n Z + ∞−∞ dω π ω N n ( ω ) Z d Q (2 π ) × G Ret , n ( − iω ) G Ret , n ( iω ) (cid:16) q ( n ) z ( − iω, Q ) + q ( n ) z ( iω, Q ) (cid:17) lim x → x h Θ jk ( − iω, iω ) × e i Q · ( x k − x k ) G jb Ret ( z , ( − n l/ , Q , − iω ) × G kb Ret ( z , ( − n l/ , − Q , iω ) i . (78)This is the contribution of the baths to the Casimirpressure in a general non-equilibrium context. An impor-tant connection with previous works can be established byemploying a fluctuation-dissipation-type relation into theresult. Besides that it is well-known that the fluctuation-dissipation theorem is valid for systems in thermal equi-librium, and the total system in this case is out of equi-librium, we can use a fluctuation-dissipation-type relationfor the dissipation and noise kernel of the QBM ( D n and N n respectively), i. e., the kernels generated by the bathsin each point of space acting over each DOF. Therefore,we can write, for the Fourier transforms of the kernels, thefluctuation-dissipation relation: N n ( ω ) = coth (cid:18) β n, B ω (cid:19) Im (cid:2) D n ( ω ) (cid:3) , (79)where D n ( ω ) is the Fourier transform of the QBM’s dis-sipation kernel.From the definition of the permittivity tensor in Eq.(19), it can be proved that for the present case, the Fourier transform of the permittivity function is given by ε n ( ω ) = 1 + λ ,n G Ret , n ( ω ) , having that G Ret , n ( − ω ) = G ∗ Ret , n ( ω ) for the reality ofthe QBM Green function. On the other hand, since theLaplace transform of the QBM G Ret , n ( s ) of Eq.(57) is as-sumed, by causality, to have poles with negative real parts,then it is verified that G Ret , n ( − iω ) = G Ret ( ω ) and thesame happens for the QBM dissipation kernel D n ( − iω ) = D n ( ω ) (in fact, the connection between the Laplace andFourier transforms applies for every causal function whichis 0 for null value of its variable). Therefore, it is straight-forward to prove that:Im [ ε n ( ω )] = Im (cid:2) D n ( ω ) (cid:3) (cid:12)(cid:12) G Ret , n ( ω ) (cid:12)(cid:12) (80)= Im (cid:2) D n ( ω ) (cid:3) G Ret , n ( − iω ) G Ret , n ( iω ) . Then, introducing Eq.(79) into Eq.(78) and using Eq.(80), we obtain: P ∞ B ( β n, B , l ) = − π X n =L , R Z + ∞−∞ dω π ω coth (cid:20) β n, B ω (cid:21) × Im [ ε n ( ω )] Z d Q (2 π ) (cid:16) q ( n ) z ( − iω, Q ) + q ( n ) z ( iω, Q ) (cid:17) (81) × lim x → x h Θ jk ( − iω, iω ) e i Q · ( x k − x k ) ×G jb Ret ( z , ( − n l/ , Q , − iω ) G kb Ret ( z , ( − n l/ , − Q , iω ) i . At this point, we use Eq.(72) in order to go back tothe spatial integral having s = − iω and s = iω . Then,we can firstly employ the property associated to the real-ity of the EM retarded Green tensor in the time domainfor the last factor G ij Ret ( x , x ′ , iω ) = G ∗ ij Ret ( x , x ′ , − iω ), fol-lowed by the property of being a Feynman propagator tothe last Laplace transform of the EM retarded Green ten-sor G ∗ ij Ret ( x , x ′ , − iω ) = G ∗ ji Ret ( x ′ , x , − iω ). Finally, for bothLaplace transforms of the EM retarded Green tensor weuse the connection between the Laplace and Fourier trans-form ensured by the causality behaviour to obtain: P ∞ B ( β n, B , l ) = − π X n =L , R Z + ∞−∞ dω π ω coth (cid:20) β n, B ω (cid:21) × Im [ ε n ( ω )] lim x → x h Θ jk ( ω ) Z V n d x G jb Ret ( x , x , ω ) × G ∗ bk Ret ( x , x , ω ) i , (82)which is exactly the result obtained in Ref.[3] as the to-tal pressure for the Lifshitz problem, where we have set Θ jk ( ω ) ≡ Θ jk ( − iω, iω ) from its definition.All in all, we have proved that the baths’ contribu-tion in the long-time regime gives exactly the result found in Ref.[3] for the steady situation of the Lifshitz prob-lem. However, following this procedure, it is worth notingthat this result can be extended for the case of inhomoge-neous and anisotropic materials since both purely imagi-nary poles are always present as it can be seen from thegeneral expression of Eq.(54) provided one can calculatethe Laplace transform of the EM retarded Green tensorfor a given problem, which is the main difficulty in mostcases. Then, we can in general write: P ∞ B ( β x , B ) = − π Z + ∞−∞ dω π ω lim x → x h Θ jk ( ω ) Z d x g ( x ) × G jl Ret ( x , x , ω ) coth (cid:20) β x , B ω (cid:21) Im h ε [ ll ] x ( ω ) i × G ∗ lk Ret ( x , x , ω ) i , (83)which is the full generalization of the steady result foundin Ref.[3]. After determining the contribution of the material to theCasimir pressure, the remaining contribution is the oneassociated to the EM field’s initial conditions.From Eq.(43), it is worth noting that unlike the othercontributions this one does not simplify when consider-ing homogeneous and isotropic bodies. In fact, it does notdepend directly on the material properties and bound-aries as the material’s contributions do. The dependenceon the boundaries and the materials is encrypted by theEM retarded Green tensor but there is no more explicitdependence.Nevertheless, for the particular case of the Lifshitzproblem we can perform a Fourier transform in the paral-lel coordinates through Eq.(59) and then integrate them,obtaining: P IC ( x , β EM ) = − π Z d k ω k (2 π ) (cid:20) δ bm − k b k m ω k (cid:21) × coth (cid:20) β EM ω k (cid:21) Z α + i ∞ α − i ∞ ds πi Z α + i ∞ α − i ∞ ds πi (cid:0) s s + ω k (cid:1) × e ( s + s )( t − t i ) lim x → x " Θ jk ( s , s ) e i k k · ( x k − x k ) × Z dz ′ G jb Ret ( z , z ′ , k k , s ) e ik z z ′ Z dz ′′ G km Ret ( z , z ′′ , − k k , s ) e − ik z z ′′ , (84)which is a simplification that can be done when the bound-aries are parallel surfaces.The integrals over z ′ and z ′′ are over the whole axis( −∞ , + ∞ ). Then, as the EM retarded Green tensor is ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 17 given by Eqs.(60)-(63) for each region, the integration sep-arates into four integrals. For example, for the first integralwe have: Z dz ′ G jb Ret ( z , z ′ , k k , s ) e ik z z ′ == Z − l/ −∞ dz ′ G jb Ret ( z , z ′ , k k , s ) e ik z z ′ + Z l/ − l/ dz ′ G jb Ret , Bu ( z , z ′ , k k , s ) e ik z z ′ + Z l/ − l/ dz ′ G jb Ret , Sc ( z , z ′ , k k , s ) e ik z z ′ + Z + ∞ l/ dz ′ G jb Ret ( z , z ′ , k k , s ) e ik z z ′ , (85)where the first and the last are the integrations havingthe source point in each plate, while the other two are theintegrations having the source point inside the gap.As all the transforms of the EM retarded Green ten-sors are given in term of exponentials, the integration isstraightforward and can be written compactly as: Z dz ′ G jb Ret ( z , z ′ , k k , s ) e ik z z ′ = − δ j δ b s e ik z z + X n =1 X µ ( ( − q z [ q z + ( − n − ik z ] e µ,j (cid:2) ( − ) n − (cid:3) × e µ,b (cid:2) ( − ) n − (cid:3) e ( − n q z z h e [ ( − n − q z + ik z ] z − e [ − q z +( − n ik z ] l i + 1 D µ h e µ,j (cid:2) ( − ) n − (cid:3) r µn e q z [( − n z − l ] + e µ,j [( − ) n ] e q z [ ( − n − z − l +( − n l ] i e µ,b (cid:2) ( − ) n − (cid:3) r µ − n × h e [ q z +( − n − ik z ] l − e − [ q z +( − n − ik z ] l i! − e [( − n ik z − [( − n +1] q z ] l q ( n ) z h q ( n ) z + ( − n − ik z i t µn D µ e ( − n q ( n ) z l e ( n ) µ,b (cid:2) ( − ) n − (cid:3) × h e µ,j (cid:2) ( − ) n − (cid:3) e ( − n q z [ z − l ] + e µ,j [( − ) n ] r µ − n × e ( − n − q z [ z − l ] e [( − n − q z l i) . (86)It is clear that in order to obtain the result for theother integral R dz ′′ G km Ret ( z , z ′′ , − k k , s ) e − ik z z ′′ , we onlyhave to make the replacements z → z , s → s , j → k , b → m , k k → − k k and k z → − k z in Eq.(86).Therefore, studying the analytical structure (poles andbranch cuts) of the results of both integrals will give usthe transient evolution and also the steady regime of theinitial conditions’ contribution to the Casimir pressure.The branch cuts present in the integrands can be consid-ered by pairs (see Ref.[20]) and the regions in the complex plane where the integrands in each variable are multival-uated can be reduced to vertical intervals of finite lengthcharacterized by non-positive real parts. From Eq.(86) wesee that the branch cut interval with null real part cor-responds to the straight line s = i Im(s) with Im(s) ∈ ( − k k , k k ), although it does not contribute to the steadystate, it is important to know that it can be always con-sidered as a finite-lenght interval in the complex plane.Then, as for the material contributions, the key pointto determine the long-time regime of the present contri-bution is to combine the poles in each variable in sucha way that e ( s + s )( t − t i ) gives no temporal dependence.Therefore, we must consider complementary poles.At first sight, the pole at the origin in each variable(provided by the analytical structure of the retarded Greentensor) satisfies the requirement to give a steady term.Now, to see which terms present this pole in both vari-ables, we have to consider the combinations of the termsof Θ jk ( s , s ) and ( s s + ω k ). As it happened before,the terms TM × TM in the integrals product in Eq.(84)are the ones which give the poles at the origin in eachLaplace variable. The different combinations finally giveterms without poles at the origin and terms with poles offirst and second order in both variables.As for the material contributions, the terms withoutpoles do not contribute to the residue calculation, whilethe terms with poles of second order at the origin resultin terms which cancel out or directly vanish.On the other hand, as it happened before, the poles offirst order, in the temporal domain, correspond to deriva-tives of the Heaviside functions which gives Dirac δ -functi-ons. This is associated to the sudden beginning of the in-teraction and then these terms give no additional physicalinformation. They are only mathematical consequences ofthe description introduced for the switching-on of the in-teraction. Therefore, the poles of first order do not con-tribute either.All in all, the pole at the origin for both variables doesnot contribute to the pressure associated to the initialconditions.However, in this case, the poles at the origin are notthe only ones that may contribute. The spatial integrationover z provides additional poles which are not present inthe material’s contribution. This poles provide the (3+1)EM generalization for the modified modes which are con-sidered as an ansatz in Refs.[8,21] and fully demonstratedin Ref.[18], for the scalar case. In the present case, fromEq.(86), it is clear that integration over z results in addi-tional denominators given by q z + ( − n − ik z and q ( n ) z +( − n − ik z depending if the integration is carried out overthe vacuum gap or over the material plates respectively.In fact, for the terms associated to the integrationsover the material plates, the roots provided by the de-nominator q ( n ) z + ( − n − ik z , due to the presence of ε n ( s )inside q ( n ) z , present negative real parts, in agreement withthe causality properties and the fact that the field dissi-pates in such regions. For the cases where these roots areeffectively poles of the integrand (which is not necessarilytrue as we will see below), This implies that the calcula- tion of the residue will give exponential decays in time,resulting in terms with vanishing long-time limit.On the other hand, it can be immediately seen that forthe terms associated to integrations over the vacuum gap,the roots provided by q z + ( − n − ik z are s = ± iω k for n = 1 and k z < n = 2 and k z >
0, having null realparts due to the free propagation of the field inside thegap. This shows that s = ± iω k could be poles in general,because the multivalued region for the integrand is givenby the finite-lenght interval s = i Im(s) with Im(s) ∈ ( − k k , k k ), and these poles are always located outside itsince we always have ω k > k k for every k . Moreover, aswe anticipated before, these poles are associated to thementioned modified modes for the EM field.However, beyond the general case, for the present onewe only have shown that s = ± iω k are roots of the denom-inator of certain terms for n = 1 and k z < n = 2and k z >
0. Moreover, for the Lifshitz problem, the givenform of the Laplace transform of the retarded Green tensormakes that s = ± iω k are also roots of the numerator forthe same cases of n = 1 and k z < n = 2 and k z > ± iω k are indeterminate limits.Applying L’Hopital rule we can compute the limit givinga finite result. This means that, for the Lifshitz problem,despite s = ± iω k are roots of the denominator, they arenot poles. Therefore, there is no residue for this case and,moreover, there is no steady nor transient contribution for s = ± iω k .All in all, we have proved that for the Lifshitz problemthe initial conditions’ contribution to the Casimir pressurevanish at the steady state ( t i → + ∞ ): P IC ( x , β EM , l ) −→ P ∞ IC ( β EM , l ) = 0 . (87)This result means that there are no modified modesfor this problem that contribute to the long-time regimeof the Lifshitz problem. Physically, the result expressesa relaxation process where the field dissipates during thetransient stage until reaching a steady state in the long-time regime. Moreover, for the initial conditions’ contri-bution, the process is given by the competition betweenthe dissipation of the field on the material bodies presentand the free fluctuation in the dissipationless regions (seenext Section for further conclusions about these features).For the case of the Lifshitz problem, as the material re-gions are two half-spaces while the vacuum gap has finitelength, dissipation wins over free fluctuation and there areno modified modes in the long-time regime. When interac-tions begin, the transient stage take place and the initialfree field vacuum modes start to adapt to the presence ofthe material bodies (which has also a transient dynam-ics). During this process, the free fluctuations inside thegap attenuate due to the dissipation exerted by the ma-terial plates and no modified modes can raise, to finallysettle at the steady state.Finally, after calculating each contribution of Eqs.(76),(82) and (87), we can write the Casimir pressure in thesteady state ( t i → −∞ ) as: P Cas ( x ) −→ P ∞ Cas = P ∞ B ( β B , n , l ) , (88)i.e., the bath contribution at the long-time regime is theone that gives the total Casimir pressure for the compositesystem. Having solved the Lifshitz problem from a well-defined ini-tial condition problem, and achieved expressions for thethree contributions to the total pressure for all times, wehave shown that in the long-time limit t i → −∞ , theCasimir pressure is given only by the baths’ contribution.We also proved that this corresponds to the only contri-bution considered in Ref.[3], using an approach based onthe macroscopic Maxwell equations and the source the-ory. Thus, we have presented a first principles calculationin the context of non-equilibrium quantum open systemsthat shows that the initial conditions are erased in thesteady state.Although this looks absolutely natural at first glance,the results in the previous works [18,19] suggest that thereshould be important modifications for slabs of finite width.Indeed, for plates of finite width and for a δ -plate in 1 + 1dimensions, the contributions of the initial conditions arepresent in the steady state [18] and agree with the oneobtained from the modified modes of Refs.[8,21]. Lookingback to the situations analyzed in Refs.[18,19], and consid-ering the results of the present paper, the whole picturebecomes clear. It follows that a non-vanishing contribu-tion of the initial conditions at the steady state is relatedto the existence of dissipationless regions of infinite size.The physical interpretation is the following. When thereare regions of infinite size where the field dissipates, ithappens that the free fluctuations in the dissipationlessregions vanish in the long time limit because the dampinggenerated by the dissipative regions overcome the free fluc-tuations. Therefore, the initial conditions’ contribution tothe pressure also vanishes. On the contrary, for situationswhere the infinite regions are the ones without dissipation(while the dissipative regions are of finite size), the freefield fluctuations are damped only in the dissipative re-gions, resulting in modified modes at the steady situation(as it happens in Ref.[18] for the appropriate cases).In fact, these modes are associated to the poles ± iω k that result from the spatial integrations over the directionorthogonal to the boundaries. As we mentioned before,the contribution of the initial conditions can be matchedwith a contribution associated to homogeneous solutionsin ‘steady’ quantization schemes [8,17,21], but here we areshowing that the modified modes of that contribution arethe result of the dynamical adaptation of the free field vac-uum modes to the material boundaries. Therefore, on onehand, the modified modes become associated to the cre-ation and annihilation operators of the initial free field. Onthe other hand, from the perspective of a ‘steady’ canoni-cal quantization scheme, the Hilbert space of the modifiedmodes in the non-equilibrium situation is spanned by the ernando C. Lombardo et al.: Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system 19 same creation and annihilation operators that for a freefield.All in all, we have successfully proved the foundationsof Lifshitz theory in non-equilibrium scenarios. To reachthis goal, we have set up a first principles quantum dy-namical problem of the EM field interacting with matterand subjected to uncorrelated initial conditions. We thensolved the full dynamical problem and derived Lifshitztheory as the steady state by taking the long-time limit.The physical understanding of the contributions that enterin the full dynamical problem allowed us to gain power-ful insights about the occurrence of the steady state indifferent systems involving the interaction between quan-tum fields and material bodies, that commonly appear inCasimir physics. It would be of high interest to general-ize the results of the present work to situations of slabsof finite width, in order to quantify the relevance of thecontribution of the initial conditions to the Casimir pres-sure in the long time limit. Work in this direction is inprogress.As a final comment, regarding the material controversyaround the Drude and plasma models for the conductionelectrons, we can say that the first principle microscopicapproach could give fruitful results to bring light to thediscussion. As we mentioned before, here we have provedthe non-equilibrium Lifshitz foundations for insulator ma-terial plates, with permittivity functions that result fromthe bounded electrons modeled as polarization degreesof freedom. Including conduction electrons at the micro-scopic model requires an action for representing their dy-namics. This may result in different analytical propertiesfor the retarded Green functions, giving new contributionsto the steady state depending on the material model con-sidered and exposing in an explicit way the limitations ofthe Lifshitz formula. This is left as pending future work. We would like to thank Matias Leoni and Alan Garbarz foruseful comments and discussions about complex analysis. Thiswork is supported by CONICET, UBA, and ANPCyT, Ar-gentina.
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