Quasiparticle excitations in relativistic quantum field theory
QQuasiparticle excitations in relativistic quantum field theory
Daniel Arteaga
Departament de Física Fonamental and Institut de Ciències del Cosmos,Facultat de Física, Universitat de Barcelona,Av. Diagonal 647, 08028 Barcelona (Spain) e-mail: [email protected]
Abstract
We analyze the particle-like excitations arising in relativistic field theo-ries in states different than the vacuum. The basic properties characterizingthe quasiparticle propagation are studied using two different complemen-tary methods. First we introduce a frequency-based approach, wherein thequasiparticle properties are deduced from the spectral analysis of the two-point propagators. Second, we put forward a real-time approach, whereinthe quantum state corresponding to the quasiparticle excitation is explicitlyconstructed, and the time-evolution is followed. Both methods lead to thesame result: the energy and decay rate of the quasiparticles are determinedby the real and imaginary parts of the retarded self-energy respectively. Bothapproaches are compared, on the one hand, with the standard field-theoreticanalysis of particles in the vacuum and, on the other hand, with the mean-field-based techniques in general backgrounds. a r X i v : . [ h e p - ph ] D ec ontents
1. Introduction 3
2. Particles in the vacuum 13
3. Quasiparticles: the spectral approach 22
4. Quasiparticles: the real-time approach 27
5. Alternative mean-field-based approaches 35
6. Summary and discussion 39A. The closed time path method 42B. Gaussian states 46C. Contour integration of I ( t, t ; p ) . Introduction In this paper we examine the elementary particle-like excitations in generic quantumstates, as seen from the viewpoint of relativistic quantum field theory. We will refer tothose excitations as quasiparticles . Quasiparticles are one of the most ubiquitous con-cepts in physics, appearing in many different contexts with slightly different meanings,for instance in Bose-Einstein condensation [1], quantum liquids [2], superconductivity [3],and, more in general, in condensed matter field theory [4, 5], thermal field theory [6, 7]and N -body quantum mechanics (non-relativistic field theory) [8]. We shall not by anymeans attempt to review the concept of quasiparticle in this paper. Instead, we shalllimit ourselves to studying the dynamics of particle-like excitations arising in relativisticscalar field theories in states different than the vacuum, focusing on those propertieswhich can be extracted from the two-point correlation functions.In order to analyze the particle or quasiparticle properties from a second-quantizedperspective the standard procedure is to add an elementary particle-like excitation to thesystem, and to analyze the subsequent dynamical evolution of the field mode correspond-ing to the initial particle momentum. With non-relativistic excitations this procedure isclear, given that the particle number is preserved along the evolution and the particleconcept is well-defined for all times. In this context it can be readily obtained that theenergy and decay rate of the (quasi)particle excitations can be recovered from the realand imaginary parts of the poles of the two-point propagators respectively [8].With relativistic excitations the procedure is less direct since the number of relativisticparticles in a given field state fluctuates. Properly speaking, in presence of interactionsparticles are only well-defined in the asymptotic limit, where the interaction betweenthem can be neglected. Single particle excitations in the vacuum constitute a particularcase, since Lorentz symmetry is enough to fully characterize their properties [9]. It isthen a textbook result to show that in the vacuum the energy of a single particle particleis also given by the location of the pole of the Feynman propagator, similarly to thenon-relativistic case. For unstable particles the treatment is less rigorous since properlyspeaking no asymptotic states can be associated to them. Nevertheless by analogy withthe stable case, and appealing to the optical theorem, it can be additionally argued thatthe imaginary part of the pole of the propagator determines the lifetime of the particle.In non-vacuum states, where it is not possible to bring up symmetry considerations,it is a priori not completely obvious how the relativistic quasiparticles should be treatedwithin a second-quantized formalism. One possibility is to simply forget about quasipar-ticles and to study the dynamics of the mean field under arbitrary small perturbations,using the linear response theory [6–8,10]. The response of the mean field is characterizedby a frequency and a decay rate, which are respectively connected to the real and imag-inary parts of the retarded propagator. This constitutes the standard way of analyzingrelativistic excitations in non-vacuum states [6, 8, 11], although for typical particle-like3 . Introduction excitations the expectation value of the field vanishes, and therefore the excitations con-sidered by the linear response method do not correspond to elementary quasiparticles.In any case, the approach based on the linear response theory is appropriate to analyzeanother different regime, the hydrodynamic or fluid regime [12–14].Whenever single quasiparticle excitations are important, a second possibility is try-ing to develop an appropriate quasiparticle framework, following what is done either innon-relativistic field theory or in relativistic field theory in the vacuum. This will bethe main object of analysis of this paper. We will investigate several questions: first,which is the field state corresponding to quasiparticle excitations; second, whether quasi-particle excitations appear naturally in physical situations; third, how to extract theirproperties from the field-theoretic correlation functions, and, finally, what is the rangeof applicability of the quasiparticle formalism. We will however not discuss under whichconditions a given field theory in a given field state possesses quasiparticle excitations: itwill be simply assumed that the field interaction and state are such that a quasiparticleregime exists.There have been some partial results in this direction. Weldon [15] recognized that inthermal field theory the imaginary part of the pole of the retarded propagator is linked tothe decay rate of the elementary excitations. Donoghue et al. [16] discussed the role of thethe real part of the pole of the propagator as related to the energy of the quasiparticles,and the fact that it is not necessarily Lorentz-invariant. Narnhofer et al. [17] studied theproperties of the quasiparticles in thermal states. Weldon [18] and Chu and Umezawa[19] attempted a reformulation of thermal field theories in terms of what they calledstable quasiparticles, the latter using a thermo-field dynamics formalism. Also in thethermo-field dynamics context, Nakawaki et al. [20] studied quasiparticle collisions (asubject which will not be discussed in this paper). Finally, let us mention that Greinerand Leupold [21], among other things, studied the conditions for the existence of aquasiparticle regime and analyzed the fluctuations of the quasiparticle number.The kinetic approach to field theory by Calzetta and Hu [22] also goes beyond themean-field treatment. Calzetta and Hu focus on the dynamics of the propagator bystudying the two-particle irreducible effective action, which leads to equations of motionfor the correlation functions, from where the equations of motion for the quasiparticleoccupation numbers are extracted, thereby connecting the second-quantized formalismwith a first-quantized statistical description. In this paper we will strictly limit to asecond-quantized approach; see Ref. [23] for some comments on the connection with thefirst-quantized analysis.It should be noted that, since we shall work within a second-quantized formalism, wewill not study the quasiparticle themselves but rather the field modes corresponding tothe initial momentum of the quasiparticles. We will therefore not attempt to follow thequasiparticle trajectory. It will be important to bear in mind that the appearance of adissipative term does not necessarily mean a quasiparticle decay, but most of the timesit simply indicates a change of momentum of the quasiparticle.The paper is organized as follows. In the remaining of this section we introduce someof the tools and concepts that will be used in the paper. Of particular interest is the4 . Introduction analysis of the two-point propagators in general backgrounds. In Sec. 2 we recall how theparticle interpretation can be recovered in field theory in the vacuum, introducing the twotechniques that later on will be employed for the analysis of the quasiparticles. In Sec. 3we develop a spectral approach to the analysis of the quasiparticle excitations in fieldtheory, in parallel to the procedure in the vacuum. In Sec. 4 we present a complementaryreal-time analysis of the same problem, namely we study the time evolution of therelevant observables in the presence of quasiparticle excitations. In the process we discussthe form of the quantum states associated to quasiparticles, and analyze the appearanceof these states in physical situations. In Sec. 5 we recall the standard mean-field-basedtechniques and compare them with our methods and results. Finally, in Sec. 6 wesummarize and discuss the main points of the paper. Appendices contain backgroundreference material as well as technical details of some calculations.Throughout the paper we use a metric with signature ( − , + , + , +) , work with a systemof natural units with (cid:126) = c = 1 , denote quantum mechanical operators with a hat, anduse a volume-dependent normalization in the definition of the field modes [see Eq. (1.17)below]. The same symbol will be later used for a quantity and its Fourier transformwhenever there is no danger of confusion. For the purposes of this paper we consider that a quasiparticle is a particle-like excitationwhich travels in some background and which is characterized by the following properties:1. It has some characteristic initial energy E . The fluctuations of the energy aremuch smaller than this characteristic value: ∆ E (cid:28) | E | .2. It has some characteristic initial momentum p . The fluctuations of the momentumare much smaller than the characteristic energy: | ∆ p | (cid:28) | E | .3. It has approximately constant energy and momentum during a long period of time T = 1 / Γ , before it starts to decay. Here “long” means that the decay rate Γ hasto be much smaller than the de Broglie frequency of the quasiparticle: Γ (cid:28) | E | .4. It is elementary, meaning that it cannot be decomposed in the (coherent or inco-herent) superposition of two or more entities, having each one separately the samethree properties above.Notice that the third property is somewhat redundant, since by the time-energy uncer-tainty principle the energy fluctuations are at least given by the decay rate: ∆ E (cid:38) Γ .The quasiparticle is essentially characterized by the energy E , the momentum p andthe decay rate Γ . Besides that, the quasiparticle can be characterized by other quan-tum numbers such as the spin (although in this paper we shall only deal with scalarquasiparticles).If the initial background state has large momentum or energy fluctuations the per-turbed state inherits them, and therefore the requirement that the fluctuations of the5 . Introduction momentum and energy of the quasiparticles are small cannot be fulfilled. For thermaland, more generally, for Gaussian states, we will see that momentum fluctuations arecomparable to the average momentum when the occupation numbers are of order one.Therefore for bosonic systems the quasiparticle description of Gaussian states requiresrelatively small occupation numbers. When the occupation numbers are of order oneor larger, a quasiparticle description might not be very adequate and it might be moreuseful to move to a hydrodynamic description [12–14] (see Sec. 6 for a further analysisof this point).In any case, quasiparticle excitations may exist even in strongly correlated systems.The excitations in these systems are usually radically different to the original particleswhich constitute the medium. For instance, the spin and mass of the quasiparticles mighthave nothing to do with the original vacuum particles. The quasiparticles which bearno resemblance with the original particles constituting the medium, and which usuallyinvolve the system as a whole, are also called collective modes [8]. Sometimes in the samesystem the collective modes can coexist with the quasiparticles directly correspondingto the medium constituents.In this paper we will consider the quasiparticle excitations of a single scalar field φ ina given background state. This field can have two different interpretations depending onthe context. First, the field φ can have a straightforward interpretation as the fundamen-tal field whose excitations in the vacuum give rise to the particles, and whose excitationsin states different than the vacuum constitute the corresponding quasiparticles. Forinstance, this would be an appropriate interpretation when studying the behavior of apion in a thermal bath. But the field φ can have a second different interpretation as aneffective field whose excitations correspond to the quasiparticles propagating in the sys-tem, not necessarily having any direct correspondence with the fundamental fields. Thissecond interpretation is more general and can accommodate for the collective modes.For instance, the field φ could represent the field of sound wave excitations in a Bose-Einstein condensate. In any case, in this paper we will consider the scalar field φ as givenand will not investigate its relation with the fundamental constituents of the system. The dispersion relation is the expression of the energy of the quasiparticle as a functionof the momentum, namely E = E p , (1.1a)where E is the energy of the quasiparticle excitation, and p is the momentum, withthe assumption of small spreads. The effective mass is the value of the energy at zeromomentum, m eff = E p = . When the states are thermal, the effective mass is called the thermal mass . For a unstable system with energy E and decay rate Γ , we define thecomplex generalized energy E as E := E − iE Γ . The generalized dispersion relation isthe function giving the generalized energy of a quasiparticle in terms of the momentum: E = E p − iE p Γ p . (1.1b)6 . Introduction Notice that the imaginary part of the generalized dispersion relation places a lower boundon the uncertainty of the real part.In flat spacetime in the vacuum, the propagation of a stable particle is fully charac-terized by the physical mass m ph : E = m + p . (1.2a)If the particle is unstable, the generalized dispersion relation is determined by the phys-ical mass and the decay rate in the particle rest frame γ : E = m + p − im ph γ. (1.2b)The decay rate must be much smaller than the particle mass; otherwise one would speakof resonances rather than unstable particles. In any case, the generalized dispersion rela-tion can be extracted from the location of the poles in the momentum representation ofthe Feynman propagator. In other words, the Feynman propagator can be approximatedaround the particle energy as: G F ( ω, p ) ≈ − iZ − ω + p + m − im ph γ , for ω ∼ p + m . (1.3)Several differences with respect to the vacuum case arise when studying the propa-gation in a non-vacuum state. First, in general the dispersion relation needs not beof any of the forms (1.2), and therefore the effective mass does not determine by itselfthe dispersion relation. The background constitutes a preferred reference frame, thusbreaking the global Lorentz invariance, and, because of the interaction with the envi-ronment modes, the quasiparticle energy is affected by the background in a way suchthat the dispersion relation is no longer Lorentz invariant [24–27]. Moreover, even ifsome of the basic features of the particle propagation can be encoded in the form of adispersion relation, one should bear in mind that a much richer phenomenology appearsin general (including, for instance, scattering, diffusion and decoherence). Additionally,let us point out that it is not completely obvious how in general the dispersion relationsshould be extracted from the poles of a propagator, in a similar way to Eq. (1.3); wewill also address this point in this paper. In the vacuum the analysis of the Feynman propagator is usually sufficient. In a genericstate ˆ ρ this is not usually the case, and an analysis of the different propagators is inorder [28–30]. The Feynman propagator, positive and negative Whightman functionsand Dyson propagator, G ( x, x (cid:48) ) = G F ( x, x (cid:48) ) := Tr (cid:0) ˆ ρ T ˆ φ ( x ) ˆ φ ( x (cid:48) ) (cid:1) , (1.4a) G ( x, x (cid:48) ) = G + ( x, x (cid:48) ) := Tr (cid:0) ˆ ρ ˆ φ ( x ) ˆ φ ( x (cid:48) ) (cid:1) , (1.4b) G ( x, x (cid:48) ) = G − ( x, x (cid:48) ) := Tr (cid:0) ˆ ρ ˆ φ ( x (cid:48) ) ˆ φ ( x ) (cid:1) , (1.4c) G ( x, x (cid:48) ) = G D ( x, x (cid:48) ) := Tr (cid:0) ˆ ρ (cid:101) T ˆ φ ( x ) ˆ φ ( x (cid:48) ) (cid:1) , (1.4d)7 . Introduction appear in the closed time path (CTP) formalism (which is natural when dealing withnon-vacuum states; see appendix A), and can be conveniently organized in a × matrix G ab , the so-called direct basis : G ab ( x, x (cid:48) ) = (cid:18) G F ( x, x (cid:48) ) G − ( x, x (cid:48) ) G + ( x, x (cid:48) ) G D ( x, x (cid:48) ) (cid:19) . (1.5)We may also consider the Pauli-Jordan or commutator propagator, G ( x, x (cid:48) ) := Tr (cid:0) ˆ ρ [ ˆ φ ( x ) , ˆ φ ( x (cid:48) )] (cid:1) , (1.6a)and the Hadamard or anticonmutator function G (1) ( x, x (cid:48) ) := Tr (cid:0) ˆ ρ { ˆ φ ( x ) , ˆ φ ( x (cid:48) ) } (cid:1) . (1.6b)For linear systems the Pauli-Jordan propagator is independent of the state and carriesinformation about the system dynamics. Finally, one can also consider the retarded andadvanced propagators, G R ( x, x (cid:48) ) = θ ( x − x (cid:48) ) G ( x, x (cid:48) ) = θ ( x − x (cid:48) ) Tr (cid:0) ˆ ρ [ ˆ φ ( x ) , ˆ φ ( x (cid:48) )] (cid:1) , (1.7a) G A ( x, x (cid:48) ) = θ ( x (cid:48) − x ) G ( x, x (cid:48) ) = θ ( x (cid:48) − x ) Tr (cid:0) ˆ ρ [ ˆ φ ( x ) , ˆ φ ( x (cid:48) )] (cid:1) , (1.7b)which also do not depend on the the state for linear systems.By doing a unitary transformation of the propagator matrix it is possible to work inthe so-called physical or Keldysh basis [30, 31], G (cid:48) a (cid:48) b (cid:48) ( x, x (cid:48) ) = (cid:18) G (1) ( x, x (cid:48) ) G R ( x, x (cid:48) ) G A ( x, x (cid:48) ) 0 (cid:19) , (1.8)For linear systems, the off-diagonal components of the Keldysh basis carry informationon the system dynamics, whereas the non-vanishing diagonal component has informationon the state of the field.The correlation functions in momentum space are defined as the Fourier transformof the spacetime correlators with respect to the difference variable ∆ = x − x (cid:48) keepingconstant the central point X = ( x + x (cid:48) ) / : G ab ( p ; X ) = (cid:90) d x e − ip · ∆ G ab ( X + ∆ / , X − ∆ / . (1.9)For homogeneous and static backgrounds the Fourier-transformed propagator does notdepend on the central point X . The retarded and advanced propagators, which arepurely imaginary in the spacetime representation, develop a real part in the momentumrepresentation.Obviously not all propagators are independent: the complete set of propagators is de-termined by a symmetric and an antisymmetric function. From the Feynman propagator By linear systems we mean systems whose Heisenberg equations of motion are linear. These correspondeither to non-interacting systems or to linearly coupled systems. . Introduction the other Green functions can be derived, but, in contrast, the retarded propagator lacksthe information about the symmetric part of the correlation function.As in the vacuum, self-energies can be introduced for interacting systems. The self-energy has a matrix structure and is implicitly defined through the Schwinger-Dysonequation : G ab ( x, x (cid:48) ) = G (0) ab ( x, x (cid:48) ) + (cid:90) d y d y (cid:48) G (0) ac ( x, y )[ − i Σ cd ( y, y (cid:48) )] G db ( y (cid:48) , z ) , (1.10)where G (0) ab ( x, x (cid:48) ) are the propagators of the free theory, and G ab ( x, x (cid:48) ) are the propa-gators of the full interacting theory. The CTP indices a, b, c . . . are either 1 or 2, andwe use a Einstein summation convention for repeated CTP indices. The ab componentof the self-energy can be computed in the direct basis, similarly to the vacuum case, asthe sum of all one-particle irreducible diagrams with amputated external legs that beginand end with type a and type b vertices, respectively (cf. appendix A).A particularly useful combination is the retarded self-energy, defined as Σ R ( x, x (cid:48) ) =Σ ( x, x (cid:48) ) + Σ ( x, x (cid:48) ) . It is related to the retarded propagator through G R ( x, x (cid:48) ) = G (0)R ( x, x (cid:48) ) + (cid:90) d y d y (cid:48) G (0)R ( x, y )[ − i Σ R ( y, y (cid:48) )] G R ( y (cid:48) , z ) , (1.11)This equation can be regarded as a consequence of the causality properties of the retardedpropagator. A similar relation holds between the advanced propagator G A ( x, x (cid:48) ) andthe advanced self-energy Σ A ( x, x (cid:48) ) = Σ ( x, x (cid:48) ) + Σ ( x, x (cid:48) ) . Another useful combinationis the Hadamard self-energy, which is defined as Σ (1) ( x, x (cid:48) ) = Σ ( x, x (cid:48) ) + Σ ( x, x (cid:48) ) [or equivalently as Σ (1) ( x, x (cid:48) ) = − Σ ( x, x (cid:48) ) − Σ ( x, x (cid:48) ) ] and which is related to theHadamard propagator through [23] G (1) ( x, x (cid:48) ) = − i (cid:90) d y d y (cid:48) G R ( x, y )Σ (1) ( y, y (cid:48) ) G A ( y (cid:48) , x (cid:48) ) (1.12)if the system has a sufficiently dissipative dynamics and interaction is assumed to beswitched on in the remote past (otherwise the right hand side of the above equationwould incorporate an extra term). All self-energy combinations can be determined fromthe knowledge of the Hadamard self-energy and the imaginary part of the retarded self-energy.So far, all expressions to arbitrary background states ˆ ρ . For static, homogeneous andisotropic backgrounds, Eq. (1.11) can be solved for the retarded propagator by going tothe momentum representation: G R ( ω, p ) = − i − ω + E p + Σ R ( ω, p ) − p i(cid:15) . (1.13)Notice that in general the self-energy is a separate function of the energy ω and the 3-momentum p , and not only a function of the scalar p , as in the vacuum. The Hadamard9 . Introduction function admits the following expression [which can be derived from Eq. (1.12)]: G (1) ( ω, p ) = i | G R ( ω, p ) | Σ (1) ( ω, p ) = i Σ (1) ( ω, p )[ − ω + E p + Re Σ R ( ω, p )] + [Im Σ R ( ω, p )] . (1.14)From the retarded propagator and the Hadamard function one can show: G F ( ω, p ) = − i (cid:2) − ω + E p + Re Σ R ( ω, p ) (cid:3) + i Σ (1) ( ω, p ) / (cid:2) − ω + E p + Re Σ R ( ω, p ) (cid:3) + [Im Σ R ( ω, p )] , (1.15a) G D ( ω, p ) = i (cid:2) − ω + E p + Re Σ R ( ω, p ) (cid:3) + i Σ (1) ( ω, p ) / (cid:2) − ω + E p + Re Σ R ( ω, p ) (cid:3) + [Im Σ R ( ω, p )] , (1.15b) G − ( ω, p ) = i Σ (1) ( ω, p ) / R ( ω, p ) (cid:2) − ω + E p + Re Σ R ( ω, p ) (cid:3) + [Im Σ R ( ω, p )] , (1.15c) G + ( ω, p ) = i Σ (1) ( ω, p ) / − Im Σ R ( ω, p )[ − ω + E p + Re Σ R ( ω, p )] + [Im Σ R ( ω, p )] . (1.15d)The imaginary part of the self-energy can be interpreted in terms of the net decayrate Γ p ( ω ) for an excitation of energy ω — i.e. , decay rate Γ − p ( ω ) minus creation rate Γ + p ( ω ) [15, 32]: Im Σ R ( ω ; p ) = − ω [Γ − p ( ω ) − Γ + p ( ω )] = − ω Γ p ( ω ) (1.16a)In turn the Hadamard self-energy can be interpreted in similar terms as [32]: Σ (1) ( ω ) = − i | ω | [Γ − p ( ω ) + Γ + p ( ω )] = − i | ω | Γ p ( ω )[1 + 2 n ( ω )] , (1.16b)where n ( ω ) is the occupation number of the modes with energy ω . When the field stateis not exactly homogeneous, the expressions in this paragraph are still correct up toorder Lp , where L is the relevant inhomogeneity time or length scale.In most sections of this paper we discuss general properties of the propagators andself-energies, which do not depend on any perturbative expansion. In practice, however,many times the only way to evaluate the interacting propagators is through a perturba-tive expansion in the coupling constant. For physically reasonable states far ultravioletmodes of the field are not occupied. Therefore, there is an energy scale beyond whichthe field can be treated as if it were in the vacuum. In the case of thermal field theorythis scale is given by the temperature, and the Bose-Einstein function can be viewed asa natural soft cutoff to the thermal contributions. The counterterms which renormalizethe vacuum theory also make the theory ultraviolet-finite in any physically reasonablestate. This does not mean however that the renormalization process is not modified. Aswe shall see in the following sections, and as it is shown in Refs. [16, 32, 33], finite partsof the counterterms need be adjusted in a different way for each background if a physicalmeaning is to be attributed to the different terms appearing in the action.10 . Introduction The infrared behavior is more subtle. On the one hand, naive perturbation theorymay break down under certain circumstances. In the case of thermal field theory, if therelevant contribution to a given process comes from external soft momenta, in order todo perturbation theory in a meaningful way the tree level propagators must be replacedby Braaten and Pisarski’s resummed propagator [34–36], which incorporates the effectof the hard thermal loops [6, 37]. On the other hand, even if resummed propagators areused, infrared divergences may still arise when the masses of the fields are negligible.These divergences may show up in the final results of the calculations, or can be hiddenin the intermediate stages, leading to finite but incorrect results if they are not properlyregularized [38]. Additionally, as we shall comment later on, the infrared behavior maylead to a modifications to the ordinary quasiparticle decay law [39–42]. The investigationof the infrared divergences at finite temperature is still an open problem [37].
The mode corresponding to the propagating quasiparticle can naturally be regarded asan open quantum system [43–45]: naively, the field mode would constitute the reducedsystem and all the other modes would form the environment. However, since the mode-decomposed field operator is a complex quantity obeying the contraint φ p = φ ∗− p , insteadof focusing on a single mode, it proves more useful to choose as the system of interestany two modes with given opposite momentum, and as the environment the remainingmodes of the field, as well as the modes of any other field in interaction.The field φ can be decomposed in modes according to φ p = 1 √ V (cid:90) d x e − i p · x φ ( x ) , (1.17)where V is the volume of the space, a formally infinite quantity which plays no role atthe end. Given a particular momentum p , the system is composed by the two modes φ p and φ − p , and the environment is composed by the other modes of the field, φ q , with q (cid:54) = ± p . Should there be other fields in interaction of any arbitrary spin, the modes ofthese additional fields would also form part of the environment. The entire system is in astate ˆ ρ ; the state of the reduced system is ˆ ρ s = Tr env ˆ ρ , and the state of the environmentis ˆ ρ e = Tr sys ˆ ρ . Generally speaking, the state for the entire system is not a factorizedproduct state ( i.e. , ˆ ρ (cid:54) = ˆ ρ s ⊗ ˆ ρ e ).The action can be decomposed as S = S sys + S count + S env + S int , where S sys is therenormalized system action, S sys = (cid:90) d t (cid:16) ˙ φ p ˙ φ − p − E p φ p φ − p (cid:17) , (1.18a) S count is the appropriate counterterm action, S count = (cid:90) d t (cid:110) ( Z p −
1) ˙ φ p ˙ φ − p − (cid:2) Z p ( p + m ) − E p (cid:3) φ p φ − p (cid:111) , (1.18b)11 . Introduction and S env and S int are respectively the environment and interaction actions. Notice thatwe have allowed for an arbitrary rescaling of the field φ → φ/ Z / p and for an arbi-trary frequency of the two-mode system E p , which needs not be necessarily of the form ( p + m ) / . Since it is always possible to freely move finite terms from the system to thecounterterm action and vice versa , both the field rescaling and the frequency renormal-ization should be taken into account even if no infinities appeared in the perturbativecalculations. In this paper we will investigate a physical criterion in order to fix thevalues of these two parameters. Notice that this is not just a matter of notation: inparticular, the election of E p will determine the value of the energy of the quasiparticle.The environment and interaction actions depend on the particular field theory model,and in general not many things can be said about them. However, provided that the stateof the field is stationary, homogeneous and isotropic, under a Gaussian approximationthe real environment can be always equivalently replaced by a one-dimensional masslessfield and the real interaction can be replaced by an effective linear interaction withthis environment [32]. In other words, any scalar two-mode system can be equivalentlyrepresented in terms of a pair of quantum Brownian particles [45–47], this is to say, bya pair of quantum oscillator interacting linearly with a one-dimensional massless field.In this paper we will apply the Gaussian approximation, and it will prove useful for usto reason in terms of the effective coupling constant and the effective environment. Theexplicit details of the correspondence, which can be found in Ref. [32], will not be neededthough. 12 . Particles in the vacuum We begin by reviewing some aspects of the notion of particle in standard quantum fieldtheory in the Minkowski vacuum. While most results in this section can be found instandard quantum field theory textbooks (see for instance Refs. [9, 48–52]), we presentthem in some detail because, first, analogous steps will be followed when studying quasi-particle excitations in general backgrounds, and, second, in order to clarify some aspectswhich will prove relevant later on. Except where more detailed references are given, weaddress the reader to the aforementioned textbooks for the remaining of this section.Let us consider a scalar field theory whose degree of freedom is the scalar field operator ˆ φ . If the theory is free, the field operators are connected to the creation and annihilationoperators via: ˆ φ p = 1 (cid:112) E p (cid:0) ˆ a p + ˆ a †− p (cid:1) , (2.1)where E p = m + p and where ˆ a † p and ˆ a p are the creation and annihilation operatorsrespectively, which verify the following conmutation relations: [ˆ a p , ˆ a q ] = [ˆ a † p , ˆ a † q ] = 0 , [ˆ a p , ˆ a † q ] = δ qp = 1 V (2 π ) δ (3) ( p − q ) , (2.2)where we recall that V is the (formally infinite) space volume, If the theory is interacting,the same above relations hold in the interaction picture. The normalization is chosen sothat it closely resembles the quantum mechanical normalization with a finite number ofdegrees of freedom.The Hilbert space of the states of the theory has the structure of a Fock space. Fornon-interacing theories, the Fock space can be built with the aid of the creation and anni-hilation operators: | p · · · p n (cid:105) = ˆ a † p · · · ˆ a † p n | (cid:105) (this equation assumes that all momentaare different; if this is not the case, the right hand side should incorporate a factor /m ! for each repeated momentum). The states this way created have well-defined momentumand energy: ˆ p | p · · · p n (cid:105) = ( p + · · · + p n ) | p · · · p n (cid:105) , (2.3a) ˆ H | p · · · p n (cid:105) = ( E (0) + E p + · · · + E p n ) | p · · · p n (cid:105) , (2.3b)where the momentum and energy operators are: ˆ p = (cid:90) d k (2 π ) k ˆ a † k ˆ a k , ˆ H = (cid:90) d k (2 π ) E k (cid:18) ˆ a † k ˆ a k + 12 (cid:19) . (2.4)Any state of the field can be expanded in the Fock space: | Ψ (cid:105) = (cid:88) m (cid:89) p ,..., p m f ( p , . . . , p m ) | p · · · p m (cid:105) . Particles in the vacuum . Complementary to the Fock space expansion, the Hilbert space of the field also admitsa mode decomposition. Namely, every pure state of the field can be decomposed in thefollowing way: | Ψ (cid:105) = (cid:80) n k (cid:81) k c ( n k ) | n k (cid:105) . where the modes | n k (cid:105) verify ˆ p | n k (cid:105) = n k k | n k (cid:105) , ˆ H | n k (cid:105) = n k E k | n k (cid:105) (2.5)The situation gets more involved when the theory becomes interacting. In this case,the Hilbert space is still spanned by a set of eigenvectors of the momentum operator ˆ p and the the full Hamiltonian ˆ H . However, multiparticle states cannot be labeled bythe momentum of each particle in the state, because particles are interacting and themomentum of the particles (and even the number of them) changes. However, in theremote future and past particles are well separated and the interaction between themis negligible. Labeling by | p · · · p n (cid:105) in(out) the eigenstate of the full Hamiltonian thatcorresponds to a multiparticle state in the limit t → −∞ ( t → ∞ ), one has ˆ H | p · · · p n (cid:105) in(out) = ( E p + · · · + E p n ) | p · · · p n (cid:105) in(out) , (2.6)where in this case E p = m + p , with m ph being the physical mass of the particles,which differs in general from the bare mass present in the Lagrangian. Notice that theparticles appearing in the in or out states do not necessarily correspond to the particlesappearing in the corresponding free theory: any unstable particle will not appear inthe asymptotic states, and we will possibly have to add bound states to the asymptoticstates. (For simplicity our notation assumes just one particle species and does not takeinto account these possibilities.) Notice also that the in and out states are defined for alltimes (although they only have special properties in the asymptotic limits). Therefore,either the in or out Fock spaces built from those states can be chosen as a basis for theHilbert space of the interacting theory. In the remote past and future one can build afree theory that matches the properties of the interacting theory in these regimes. These“free states” correspond unitarily to the in and out states of the interacting theory inthe asymptotic limits (the correspondence being different in each case). The in and outstates are therefore also unitarily related (via the S matrix). See Refs. [9,51,53] for moredetails. An arbitrary correlation function G any ( x, x (cid:48) ) admits the following Källén-Lehmann spec-tral representation, G any ( x, x (cid:48) ) = (cid:90) ∞ d s ρ ( s ) G (0) any ( x, x (cid:48) ; s ) , (2.7) In order to properly define this correspondence one must work with wavepackets, so that there islocalization in time and space; otherwise the states of the Fock space are completely delocalized. SeeRef. [9] for more details. . Particles in the vacuum where G (0) any ( x, x (cid:48) ; s ) is the corresponding free function with squared mass parameter s and ρ ( s ) is the vacuum spectral function , which is defined as ρ ( − p ) := 12 π (cid:88) α (2 π ) δ (4) ( p − p α ) |(cid:104) | φ (0) | α (cid:105)| , (2.8)and which verifies, among others, the following properties: (i) ρ ( s ) ≥ , (ii) ρ ( s ) = 0 for s < and (iii) (cid:82) ∞ d s ρ ( s ) = 1 . In this last equation | α (cid:105) is a complete set of orthonormalstates, eigenstates of the four-momentum operator ˆ p = ( ˆ H, ˆ p ) , spanning the identity: ˆ1 = (cid:80) α | α (cid:105)(cid:104) α | , ˆ p µ | α (cid:105) = p µα | α (cid:105) . An example of such class of states are the in or out states just considered.The spectral representation can be also expressed in momentum space: G + ( p ) = 2 πρ ( − p ) θ ( p ) , (2.9a) G ( p ) = 2 πρ ( − p ) sign( p ) , (2.9b) G R ( p ) = (cid:90) ∞ − iρ ( s ) d s − ( p + i(cid:15) ) + p + s , (2.9c) G F ( p ) = (cid:90) ∞ − iρ ( s ) d sp + s − i(cid:15) . (2.9d)The first two equations show that in vacuum the Whightman functions and the Pauli-Jordan propagator essentially amount to the spectral function. The latter two equationsshow that the retarded propagator and the Feynman propagator have well-defined ana-lyticity properties when considered functions in the complex p plane. In fact, they alsohave analyticity properties in the complex energy plane, as shown in the following twoequivalent representations: G R ( p ) = (cid:90) d k (cid:20) iρ ( k − p ) p − k + i(cid:15) − iρ ( k − p ) p + k + i(cid:15) (cid:21) , (2.10a) G F ( p ) = (cid:90) d k (cid:20) iρ ( k − p ) p − k + i(cid:15) − iρ ( k − p ) p + k − i(cid:15) (cid:21) (2.10b)Taking into account the relation of the spectral function with the Pauli-Jordan propa-gator, given by Eq. (2.9b), one can also write G R ( p ) = (cid:90) d k π iG ( k , p ) p − k + i(cid:15) . (2.11)This last equation also follows directly from the definition of the retarded propagator. From all the states of the theory, let us single out the one-particle states correspondingto stable particles (assuming they exist), characterized by the momentum p and the15 . Particles in the vacuum physical mass m ph : ˆ p | p (cid:105) = p | p (cid:105) , ˆ H | p (cid:105) = E p | p (cid:105) , with E p = p + m . Consideringthose states explicitly, the spectral function can be developed as ρ ( − p ) = Zδ ( p + m ) + θ ( p + m ∗ ) σ ( − p ) , (2.12)where Z = |(cid:104) | φ (0) | p (cid:105)| is a positive constant (which does not depend on the momentumbecause of the Lorentz invariance of the theory), σ ( − p ) is the contribution from themultiparticle states, and m ∗ is the minimum rest energy of the multiparticle states. The Z constant is frequently renormalized to one by rescaling the field (which amounts toadding a suitable counterterm to the original action).According to Eq. (2.9d), the Feynman propagator in momentum space has the struc-ture: G F ( p ) = − iZp + m − i(cid:15) + (cid:90) ∞ m ∗ d s − iσ ( s ) p + s − i(cid:15) . (2.13)Thus, the Feynman propagator is analytic in the complex p plane, except for a poleat p = − m + i(cid:15) , and a branch cut starting at p = − m ∗ + i(cid:15) and running parallelto the real axis. There are additional poles if bound states can be formed; we shall nottake into account this possibility in the following. In presence of massless particles thepole and the cut may overlap. When doing perturbation theory this is reflected in theappearance of infrared divergences in the calculation of the residues of the pole. Fora discussion of this point in the case of QED see e.g. Ref. [52]. We will assume in thefollowing that the pole and the cut are well-separated.In perturbative field theory, the renormalized Feynman propagator can be resummedin the following way G F ( p ) = − ip + m + Σ( − p ) , (2.14)where m is the renormalized mass and Σ( − p ) is the self-energy , corresponding to thesum of all one-particle irreducible Feynman diagrams. The locus of the poles and cuts isgiven by the solution of the equation p + m + Σ( − p ) = 0 . The zeros of this equation lienext to the real axis as dictated by the spectral representation (2.9d). The lowest zero ofthe equation is the pole at p = − m . With the on-shell renormalization conditions therenormalized mass coincides with the physical mass, m = m , so that Σ( − m ) = 0 .Although it is usual to consider the Feynman propagator, the analytic structure ofother propagators can be studied as well. The retarded propagator will be of specialimportance: G R ( p ) = − iZp + m − i(cid:15)p + (cid:90) ∞ m ∗ d s − iσ ( s ) p + s − i(cid:15)p . (2.15)As shown in figure 2.1, in the complex energy plane (complex p plane), the retardedpropagator is analytic except for the two poles at p = ± E p − i(cid:15) and the two branchcuts. 16 . Particles in the vacuum Im p Re p q m + p − q m + p p m ∗ + p − p m ∗ + p Figure 2.1.: Analytic structure of the retarded propagator in the vacuum, as seen inthe complex energy plane. There are two poles corresponding to the stableparticle, and two branch cuts, whose branching points indicate the minimumenergy for the multiparticle states. Between the poles and the branchingpoints there might be as well other poles corresponding to bound states(not shown in the plot).
Recall that unstable particles are not asymptotic states and hence do not correspondto eigenstates of the Hamiltonian nor they belong to the 1-particle sector of the Fockspace. Rather, they are combinations of those multiparticle states corresponding to thedecay products of the unstable particle. In terms of the Källén-Lehmann representationof the propagator, this means that unstable states are represented by a branch cutrather than a pole. Hence the spectral function for the multiparticle states is given by ρ ( − p ) = θ ( p + m ∗ ) ρ ( − p ) , where m ∗ is in this case the minimum energy to create amultiparticle state.Unstable particles however develop a pole of the propagator in a second Riemannsheet of the complex p plane [48, 54–59]. The Feynman propagator can be resummedas G F ( p ) = − ip + m + Σ( − p ) = − ip + m + Re Σ( − p ) + i Im Σ( − p ) . (2.16)We now define the mass of the unstable particle as the lowest order solution of − m + m + Re Σ( m ) = 0 (2.17)and identify γ = − m ph Im Σ( m ) (2.18)as the decay rate in the particle rest frame, according to the optical theorem. Themass as defined above corresponds approximately to the rest energy of the particle (thisassertion will be checked later on), although it should be noted that the energy of anunstable particle fluctuates according to the time-energy uncertainty principle. Thus,17 . Particles in the vacuum the second Riemann sheet of the Feynman propagator in momentum space has a polein the region Im Σ( p ) < , whose real part corresponds to the approximate mass of theparticle and whose imaginary part corresponds to the decay rate: G F ( p ) ≈ − iZp + m − im ph γ + (cid:101) G ( p ) , (2.19)The function (cid:101) G ( p ) is analytic function in the vicinity of the pole, but it does develop asingular behavior when approaching the different particle creation thresholds. So far we have carried out the analysis in the energy-momentum representation. It willprove also illustrative to consider the time-momentum representation of the propagator, G F ( t, t (cid:48) ; p ) = (cid:90) d p π e − ip ( t − t (cid:48) ) G F ( p ) . The aim is to compute the behavior of the propagator for large time lapses. We shallconsider both the stable and unstable cases simultaneously (with γ = (cid:15) if the particle isstable).From Eq. (2.19), the time behavior of the pole can be easily derived: G F ( t, t (cid:48) ; p ) = − iZ (cid:113) p + m − im ph γ e − i q p + m − im ph γ | t − t (cid:48) | + (cid:101) G ( t, t (cid:48) ; p ) In order for the particle concept to be meaningful, the condition γ (cid:28) m ph must beverified (otherwise one would speak of wide resonances rather than particles). Underthese conditions, the above expression can be approximated as G F ( t, t (cid:48) ; p ) = − iZ E p e − iE p | t − t (cid:48) | e − Γ p | t − t (cid:48) | / + (cid:101) G ( t, t (cid:48) ; p ) (2.20)with E p = p + m , and where we have defined the decay rate in the laboratory restframe Γ p := mγE p = − E p Im Σ( m ) . (2.21)The behavior of remaining piece (cid:101) G ( t, t (cid:48) ; p ) remains to be determined. In general, itdepends on the precise value of the spectral function along the branch cut, which inturn depends on the multiparticle structure of the theory. However, by appealing to theRiemann-Lebesgue theorem, one can show [48] that an order of magnitude estimationof (cid:101) G ( t, t (cid:48) ; p ) is given by | (cid:101) G ( t, t (cid:48) ; p ) | ∼ M α +1 | t − t (cid:48) | α , (2.22)18 . Particles in the vacuum where M is the scale in which the leading multiparticle threshold starts, and where α issome positive coefficient which depends on the particular structure of the theory.For unstable particles three different time regimes can be therefore distinguished. Forshort times, of the order of the de Broglie size of the particle m − , transient effects dom-inate, and the behavior depends on the particular details of the field theory model. Forlarge time lapses ( | t − t (cid:48) | (cid:29) M − ), the behavior is dominated by the pole, which in thiscase is located off the real axis. The modulus of the propagator decreases exponentiallywith a rate Γ p . Transient effects are subdominant in this regime, since they decay ina much faster timescale M − . For extremely long times, however, “transient” effectsdominate again, since any power low decay dominates over an exponential decay forsufficiently long times. Experimentally the breakdown of the exponential law for verylong times is almost never observable, since power-law terms dominate again after manyparticle lifetimes, when the chances of observing the particle are almost null (as we willsee in the following). We have just seen that the 2-point correlation functions match to the 2-point correlationfunction of a free field plus an additional multiparticle contribution, which vanishes forlong times: G + ( t, t (cid:48) ; p ) = ZG (1p) + ( t, t (cid:48) ; p ) + (cid:101) G + ( t, t (cid:48) ; p ) −−−−−→ t − t (cid:48) →∞ ZG (1p) + ( t, t (cid:48) ; p ) (2.23a)or equivalently (cid:104) | ˆ φ p U ( t, t (cid:48) ) ˆ φ − p | (cid:105) −−−−−→ t − t (cid:48) →∞ Z E p (cid:104) p | U ( t, t (cid:48) ) | p (cid:105) (2.23b)where | (cid:105) and | p (cid:105) are the vacuum and 1-particle state of the interacting theory respec-tively, and where U ( t, t (cid:48) ) is the time evolution operator. Therefore, we can make theidentification | p (cid:105) ∼ = (cid:112) E p √ Z ˆ φ − p | (cid:105) . (2.24)The symbol ∼ = here means equivalence when evaluated in a matrix element in the limitof large time lapses. Physically, the field operator excites the one-particle state and themultiparticle sector, but multiparticle excitations are off the mass shell and they decayquickly.The heuristic argument given above can be connected to the fact that the particlecontent of the theory corresponds to that of a free theory in the asymptotic limits. Letus consider the asymptotic field operator ¯ φ , which by assumption obeys free equationsof motion, ( ∂ µ ∂ µ + m ) ¯ φ = 0 , − ¨¯ φ p + ( p + m ) ¯ φ p = 0 , (2.25a)and which corresponds to the field operator through ¯ φ ( x ) ∼ = Z − / φ ( x ) , ¯ φ p ∼ = Z − / φ p . (2.25b)19 . Particles in the vacuum Let us also consider the corresponding creation and annihilation operators, ˆ¯ a p = (cid:112) E p (cid:20) ˆ¯ φ − p + iE p ˆ¯ π − p (cid:21) , ˆ¯ a † p = (cid:112) E p (cid:20) ˆ¯ φ p − iE p ˆ¯ π p (cid:21) , (2.26)where ˆ¯ π p is the canonical momentum operator and E p = p + m . In and out Fockspaces can be constructed with the field ¯ φ [9,51,53]. With the asymptotic field operatorthe one-particle state can be represented as | p (cid:105) ∼ = (cid:112) E p ˆ¯ φ − p | (cid:105) in/out = ˆ¯ a † p | (cid:105) in/out (2.27)Comparing Eq. (2.24) and Eq. (2.27) we see that the distinction between the field φ andthe the asymptotic field ¯ φ may be blurred provided the constant Z is renormalized toone and assumes that in the asymptotic limit the field obeys effective free equations ofmotion. Under these assumptions one may simply write | p (cid:105) ∼ = (cid:112) E p ˆ φ − p | (cid:105) in/out ∼ = ˆ a † p | (cid:105) in/out (2.28)When particles are unstable the situation is less clear since strictly speaking a one-particle sector which also is an eigenstate of the Hamiltonian does not exist (the onlyeigenstates being the multiparticle states corresponding to the decay products of theunstable particles). However, if particles are long-lived one may think of approximate1-particle states (which in fact correspond to multiparticle state combinations). Weassume that those are also approximately given by Eq. (2.28). We have previously seen that in frequency space the poles of the propagator are con-nected with the energy and lifetime of the particle. Let us confirm this statement study-ing the time evolution of the expectation value of the Hamiltonian operator. In fieldtheory, strictly speaking, only asymptotic properties are completely well-defined. How-ever, as long as sufficiently large time lapses are considered (compared to the relevantinverse energy scales), approximate results for finite times can be obtained. We shall usethe asymptotic particle representation in terms of fields (described in the previous sub-section) for finite but sufficiently long time lapses, shall blurry the distinction betweenthe usual and asymptotic fields (supposing that the quantity Z has been renormalizedto one) and shall treat the stable and unstable cases simultaneously.According to Eq. (1.18a), the reduced Hamiltonian operator for the relevant two-pairmode is given by ˆ H sys = E p (cid:16) ˆ a † p ˆ a p + ˆ a †− p ˆ a − p + 1 (cid:17) . (2.29) One possible concern is the fact that there is an apparent contradiction between the two canonicalconmutation relations [ ˆ φ, ˆ π ] = i and [ ˆ¯ φ, ˆ¯ π ] = i . In reality there is no such contradiction becausethe two fields are equivalent in the asymptotic limits only when evaluated in matrix elements (weakoperator convergence) [51]. In any case, the reader must be warned that a fully rigorous analysis isusually not feasible for standard field theories. . Particles in the vacuum If in the remote past time t a particle is introduced into the system, the state for latertimes will be | p (cid:105) ∼ = a † p | (cid:105) . The evolution of the number of particles in this state is givenby E ( t, t ; p ) := (cid:104) p | ˆ H p ( t ) | p (cid:105) = E p (cid:104) | ˆ a p U ( t , t ) (cid:16) ˆ a † p ˆ a p + ˆ a †− p ˆ a − p + 1 (cid:17) U ( t, t )ˆ a † p | (cid:105) . (2.30)Introducing a resolution of the identity we obtain: E ( t, t ; p ) = E p (cid:88) α (cid:104) | ˆ a p U ( t , t )ˆ a † p | α (cid:105)(cid:104) α | ˆ a p U ( t, t )ˆ a † p | (cid:105) + E (0) , where E (0) = (cid:104) | ˆ H sys | (cid:105) is the vacuum energy. By energy and momentum conservation,only the vacuum survives from the above summation. Therefore: E ( t, t ; p ) = E (0) + 4 E p |(cid:104) | ˆ φ − p U ( t , t ) ˆ φ p | (cid:105)| , from where we obtain E ( t, t ; p ) = E (0) + 4 E p | G F ( t , t ; p ) | . (2.31)Introducing the explicit value of the propagator, given by Eq. (2.20), and neglecting theoff-shell contribution (cid:101) G ( t, t (cid:48) ; p ) we get the expected result E ( t, t ; p ) = E (0) + E p e − Γ p ( t − t ) . (2.32)Particles have energy E p and decay in a timescale Γ p in the e domain of validity of theexponential law.We recall once more that unstable particles do not correspond to any eigenstate ofthe Hamiltonian, and thus no asymptotic states can be associated to them. Therefore,strictly speaking, one-particle states are a linear combination of many multiparticle statesand one cannot make reference to energy conservation, since energy conservation is asso-ciated to asymptotic properties. However, if the lifetime of the particles is long enoughone can think of approximate asymptotic states and approximate energy conservation,so that the above calculation would be approximately valid.21 . Quasiparticles: the spectral approach In a general background the quickest and clearest way to derive the spectral repre-sentation is by simply recalling the definition of the propagators. From Eq. (1.7a) weimmediately obtain the spectral representation for the retarded propagator [28] G R ( p ; X ) = (cid:90) d k π iG ( k , p ; X ) p − k + i(cid:15) . (3.1a)This equation is identical to its vacuum counterpart, Eq. (2.11). The advanced propa-gator follows a similar representation, G A ( p ; X ) = (cid:90) d k π iG ( k , p ; X ) p − k − i(cid:15) . (3.1b)Hereafter the Pauli-Jordan function will also be called spectral function . The similaritieswith the vacuum case end here. The retarded and advanced propagators are the onlypropagators that can be expressed in terms of an integral of the spectral function. Noticealso that in general the spectral representation can only be expressed as an integral overthe energy, and not as an integral over the invariant mass.The Pauli-Jordan function verifies the following properties: G ( p ; X ) > , if p > , (3.2a) G ( − p ; X ) = − G ( − p ; X ) , (3.2b) (cid:90) d k π k G ( k ; X ) = 1 . (3.2c)The first two properties are a simple consequence of the definition of the propagator.The third property is a sum rule, consequence of the equal-time commutation relations, [ ˆ φ ( t, x ) , ∂ t ˆ φ ( t, y )] = iδ (3) ( x − y ) . For stationary backgrounds an explicit representation for the Pauli-Jordan functioncan be obtained, similarly to Eq. (2.8). The Pauli-Jordan function can be expressed inthe basis of eigenstates of the Hamiltonian as G ( t, t (cid:48) ; p ) = (cid:88) α ρ α (cid:104) α | [ ˆ φ − p ( t ) , ˆ φ p ( t (cid:48) )] | α (cid:105) = (cid:88) α ρ α (cid:104) α | ˆ φ − p e − i ˆ H ( t − t (cid:48) ) ˆ φ p e i ˆ H ( t − t (cid:48) ) | α (cid:105) − (c.c) , where ρ α = (cid:104) α | ˆ ρ | α (cid:105) . We have used the fact that the state is stationary, so that thedensity matrix operators diagonal in the basis of eigenstates of the Hamiltonian. Let us22 . Quasiparticles: the spectral approach now introduce a resolution of the identity | β (cid:105)(cid:104) β | : G ( t, t (cid:48) ; p ) = (cid:88) α,β ρ α (cid:104) α | ˆ φ − p e − i ˆ H ( t − t (cid:48) ) | β (cid:105)(cid:104) β | ˆ φ p e i ˆ H ( t − t (cid:48) ) | α (cid:105) − (c.c.) = (cid:88) α,β ρ α (cid:104) α | ˆ φ − p | β (cid:105)(cid:104) β | ˆ φ p | α (cid:105) e − i ( E β − E α )( t − t (cid:48) ) − (c.c.) = (cid:88) α,β ρ α |(cid:104) α | ˆ φ p | β (cid:105)| ( − i ) sin [( E β − E α )( t − t (cid:48) )] , where we have used that ˆ φ † p = ˆ φ − p Going to the frequency space we obtain the desiredexpression: G ( p ) = (cid:88) α,β ρ α |(cid:104) α | ˆ φ p | β (cid:105)| (cid:2) δ ( p + E α − E β ) − δ ( p − E α + E β ) (cid:3) . (3.3)For stationary background states the Pauli-Jordan propagator is proportional to theprobability for the field operator of momentum p to induce a transition to a state withhigher energy p , minus the probability to induce a transition to a state of lower energy p . Let us assume that around some range of energies the Pauli-Jordan propagator has thefollowing structure: G ( ω, p ) = Z p E p (cid:2) δ ( ω − E p ) − δ ( ω + E p )] , for | ω | ∼ E p . (3.4)This means that the field operator with momentum p creates an excitation whose energyis exactly E p . Since there is no spread in the energy, the excitation must be infinitelylived (must be stable). The corresponding retarded propagator is: G R ( ω, p ) = − iZ p − ω + E p − iω(cid:15) , for | ω | ∼ E p . (3.5)The excitation this way created would have exact energy E p and exact momentum p .Therefore it corresponds to a stable quasiparticle. Notice that, in contrast to the stableparticles in the vacuum, Z p can depend on the 3-momentum p and E p need not be ofthe form p + m .However, stable quasiparticles are an idealization, and do not correspond exactly toany physical situation, since dissipation is a generic feature of non-vacuum states aslong as there is interaction. We recall that by “dissipation” we do not necessarily meanthe quasiparticle decaying into a product of different quasiparticles: “dissipation” cansimply mean the quasiparticle changing its momentum. In any case, stable quasiparticlesmay be a good approximation in many situations in which the effective coupling to the23 . Quasiparticles: the spectral approach environment is very small (in the sense of the quantum Brownian motion correspondenceexplained in Ref. [32] and summarized in the introduction). However, this does not meanat all that the real coupling must be small, or that the approximation is limited to weaklyinteracting systems; as a matter of fact, in strongly coupled situations there might besituations in which assuming free quasiparticles might well be a good approximation.Anyway, quasiparticles are in general not stable, and instead decay with some rate Γ p . The generic form of the retarded propagator in terms of the self-energy is G R ( ω, p ) = − i − ω + E p + Σ R ( ω, p ) . (3.6)Following the vacuum analysis of subsection 2.1.3, whenever there is a long-lived quasi-particle the analytic continuation of the retarded propagator can be approximated by: G R ( ω, p ) = − iZ p − ω + E p − iω Γ p + (cid:101) G ( ω, p ) , (3.7)where (cid:101) G ( ω, p ) is an analytic function in the vicinity of the pole. This means that forunstable (but long-lived) quasiparticles, the Pauli-Jordan function, instead of being adelta function around the energy of the pole, it is an approximate Lorentzian function,whose width corresponds to the decay rate. The momentum-dependent functions E p and Γ p are given, in terms of the self-energy, by E p = m + p + Re Σ R ( E p , p ) , (3.8a) Γ p = − E p Im Σ R ( E p , p ) , (3.8b)under the assumption that Γ p is much smaller than E p . The spectral representationindicates that E p corresponds to the quasiparticle energy. The physical meaning of Γ p can be extracted from the interpretation of the imaginary part of the self-energy,Eq. (1.16a): it corresponds to the net decay rate of the quasiparticle.The structure of the retarded propagator (3.7) is completely analogous to that of thevacuum (see Sec. 2 and figure 3.1). There are however two important differences that itis worth commenting. First, in the vacuum the analysis can be equivalently performedwith the Feynman or the retarded propagators, while in generic backgrounds the spectralanalysis can only be applied to the retarded propagator, since in general the Feynmanpropagator has an additional explicit dependence on the background state of the field.Second, in the vacuum one can study the spectral structure either in terms of the energyor in terms of the squared four-momentum. In general, there is a preferred referenceframe and therefore the explicit Lorentz invariance of the results is broken, and, hence,only a spectral analysis based on the energy is meaningful. See Refs. [11,60] for a more indepth investigation of the analytic structure of the propagators in thermal field theory.It must be noted that in presence of massless particles, there are situations in whichthe decay rate appears to be infrared-divergent from a perturbative calculation. This canbe attributed to the fact that in presence of massless particles the retarded propagator24 . Quasiparticles: the spectral approach Im p Re p E p − E p − Γ p Figure 3.1.: Analytic structure of the retarded propagator for quasiparticles in generalbackgrounds, in the complex energy representation. The retarded propaga-tor has no poles but only has a cut parallel to the real axis. Its analyticcontinuation (or second Riemann sheet) has a pair of poles, whose real partcorresponds to the energy of the excitation and whose imaginary part cor-responds to the decay rate. The cut need not extend from −∞ to + ∞ : itmight be interrupted for some energy sectors.does not necessarily exhibit the form (3.7), because there is no threshold for the creationof massless excitations. For instance, when computing the lifetime of a quasiparticle ina very hot QED plasma it is found that the retarded propagator does not exhibit anysingularity near the particle resonance energy [40,41,60], although the propagator is stillstrongly peaked around this point. Although we will comment further on this possibility,in the following we will limit our calculations to those situations in which a well-defineddecay rate can be associated to the quasiparticles.When the retarded and advanced propagators can be approximated by G R ( ω, p ) ≈ − iZ p − ω + R p − iω Γ p , (3.9a) G A ( ω, p ) ≈ iZ p − ω + R p + iω Γ p , (3.9b)the Pauli-Jordan propagator (or spectral function) is given by G (1) ( ω, p ) ≈ | ω | Z p Γ p (cid:0) − ω + E p (cid:1) + ( ω Γ p ) . (3.9c)So far we have analyzed the propagators which are independent of the occupation num-ber. Other propagators depend on the occupation number. Approximating n ( | ω | ) bythe value at the pole, n p := n ( E p ) , we obtain from Eq. (1.16b) Σ (1) ( E p , p ) ≈ − i Im Σ R ( E p , p )(1 + 2 n p ) , (3.9d)25 . Quasiparticles: the spectral approach where n p can be interpreted as the occupation number of the mode with momentum p .Combining the above equation with Eqs. (1.15) yields G ( ω, p ) ≈ ωZ p Γ p (1 + 2 n p ) (cid:0) − ω + E p (cid:1) + ( ω Γ p ) . (3.9e)Finally, the Feynman and Dyson propagators and the Whightman functions can berepresented by G F ( ω, p ) ≈ − iZ p ( − ω + E p ) + | ω | Z p Γ p (2 n p + 1) (cid:0) − ω + E p (cid:1) + ( ω Γ p ) , (3.9f) G D ( ω, p ) ≈ iZ p ( − ω + E p ) + | ω | Z p Γ p (2 n p + 1) (cid:0) − ω + E p (cid:1) + ( ω Γ p ) , (3.9g) G + ( ω, p ) ≈ | ω | Z p Γ p [ n p + θ ( ω )] (cid:0) − ω + E p (cid:1) + ( ω Γ p ) , (3.9h) G − ( ω, p ) ≈ | ω | Z p Γ p [ n p + θ ( − ω )] (cid:0) − ω + E p (cid:1) + ( ω Γ p ) . (3.9i)26 . Quasiparticles: the real-time approach Even in absence of interaction it is not trivial to construct a quantum state which verifiesthe quasiparticle properties mentioned in the introduction. As we have seen, in a non-interacting theory in the vacuum the one-particle state is naturally represented by theaction of the creation operator on the vacuum, | p (cid:105) = ˆ a † p | (cid:105) , or equivalently by the actionof the field operator: | p (cid:105) = (cid:112) E p ˆ φ − p | (cid:105) . Over a homogeneous and stationary state ˆ ρ ,the positive-energy quasiparticle state can be represented by the action of the creationoperator: ˆ ρ (+) p := 1 n p + 1 ˆ a † p ˆ ρ ˆ a p = 1 n p + 1 (cid:88) n, { m } ρ n, { m } ( n + 1) | ( n + 1) p , { m }(cid:105)(cid:104) ( n + 1) p , { m }| , (4.1)where the second equality is written in a highly schematic notation in order to avoidcumbersome expressions. Here n p = Tr ( ˆ ρ ˆ a † p ˆ a p ) (4.2)represents the mean occupation number of the mode with momentum p , | n p , { m }(cid:105) isa basis of the Hilbert space with the p sector singled out, and ρ n, { m } is the diagonalrepresentation of the background state in this basis. Notice that the state is normalized: Tr ˆ ρ (+) p = 1 . The factors n + 1 on the second equality are a consequence of the Bose-Einstein statistics: the probability to create an additional particle increases with thenumber of already-existing particles.The Bose-Einstein statistics are responsible for the following surprising fact: it is asimple exercise to show that the expectation value of the particle number, (cid:104) ˆ N p (cid:105) (+) = Tr (cid:0) ˆ ρ (+) p ˆ N p (cid:1) = Tr (cid:0) ˆ ρ (+) p ˆ a † p ˆ a p (cid:1) = 1 n p + 1 (cid:104) ( ˆ N p + 1) (cid:105) , (4.3)is actually increased more than one with respect to the unperturbed value: (cid:104) ˆ N p (cid:105) (+) = n p + 1 + δn p n p =: n p + N (+) p , (4.4)where N (+) p = 1 + δn p / (1 + n p ) is the number of excitations and δn p = (cid:104) ( ˆ N p − n p ) (cid:105) = (cid:104) ˆ N p (cid:105) − n p ≥ (4.5)is the dispersion of the number of particles in the background state. For a particlenumber eigenstate δn p = 0 and N (+) p = 1 , and for a Gaussian state (see appendix B)27 . Quasiparticles: the real-time approach δn p = n p ( n p +1) and N (+) p = 1+ n p . Therefore, when the background is in a state whichis not an eigenstate of the Hamiltonian, ˆ ρ (+) p represents slightly more than one additionalquasiparticle. The reason for this is purely statistical: the highly occupied componentsof the state are more likely to become excited, and therefore they tend to gain statisticalweight, thereby increasing the particle number. In other words, the action of the creationoperator has two simultaneous effects: on the one hand, adding a quasiparticle to thesystem; on the other hand, increasing the statistical weight of the highly excited states.The statistical contribution is significant when the occupation numbers are large.The expectation value of the momentum operator is also affected by this statisticaleffect: (cid:104) ˆ P (cid:105) (+) = Tr (cid:0) ˆ ρ (+) p ˆ P (cid:1) = Tr (cid:0) ˆ ρ (+) p p ˆ a † p ˆ a p (cid:1) = p N (+) p . (4.6a)Anyway the momentum per excitation is p . The energy is similarly affected: (cid:104) ˆ H sys (cid:105) (+) = Tr (cid:0) ˆ ρ (+) H sys (cid:1) = Tr (cid:104) ˆ ρ (+) p (cid:16) ˆ a † p ˆ a p + a †− p ˆ a − p + 1 (cid:17)(cid:105) = E (0) + E QP N (+) p , (4.6b)where E (0) = Tr ( ˆ ρ ˆ H sys ) is the energy of the background and E QP = E p = (cid:112) p + m isenergy per excitation.Quasiparticles require small momentum spreads. The spread of the momentum perquasiparticle in the case of Gaussian states is (cid:104) ∆ p (cid:105) (+) QP := 1 N (+) p (cid:110) (cid:104) ˆ P (cid:105) (+) − [ (cid:104) ˆ P (cid:105) (+) ] (cid:111) / = | p | (4 n p + 5 n p ) / n p , (4.7)which is a small quantity if the occupation numbers are small. Likewise, the spreads inthe particle number and the energy have the same corresponding values. Therefore, inorder for the state ˆ ρ (+) p to adequately represent a quasiparticle occupation numbers mustbe small as compared to one.Since the statistical contribution looks awkward, one can imagine a different way ofconstructing the quasiparticle excitation: ˆ ρ (alt) p := (cid:88) n, { m } ρ n, { m } | ( n + 1) p , { m }(cid:105)(cid:104) ( n + 1) p , { m }| . (4.8)This state has essentially identical properties to ˆ ρ (+) p , except that it contains exactlyone additional particle. Therefore it looks like that it corresponds more closely to thequasiparticle concept we discussed before. However, notice that the above state cannotbe easily created from the background via the action of the creation and annihilationoperators. We shall argue in the next section that the state ˆ ρ (+) p , and not ˆ ρ (alt) p , appearsnaturally when studying quasiparticle creation processes.We shall consider that the state ˆ ρ (+) p represents exactly one additional real quasiparticleof momentum p and energy E p . The additional contribution to the particle expectationnumber will be called statistical quasiparticle contribution. The statistical quasiparticlecontribution can be interpreted as being a consequence of the increased knowledge ofthe background state which is derived from the creation of a real quasiparticle.28 . Quasiparticles: the real-time approach Another novelty is that negative energy excitations, or holes, can also be defined,represented by the action of the annihilation operator: ˆ ρ ( − ) p := 1 n p ˆ a − p ˆ ρ ˆ a †− p = 1 n p (cid:88) n, { m } ρ n, { m } n | ( n − − p , { m }(cid:105)(cid:104) ( n − − p , { m }| . (4.9)This hole state is also affected by the statistical considerations described above, as it ismanifest by showing the expectation value of the number operator: (cid:104) ˆ N p (cid:105) = Tr ( ˆ ρ ( − ) p ˆ N ) = n p − δn p n p =: n p − N ( − ) p , (4.10)where N ( − ) p = 1 − δn p /n p is the expected number of negative-energy excitations. There-fore the hole state contains at most one negative-energy excitation. This state hasmomentum p , (cid:104) ˆ P (cid:105) ( − ) = p N p , and has an additional amount of negative energy E p , (cid:104) ˆ P (cid:105) ( − ) = E (0) − E p N ( − ) p .For a particle number eigenstate N ( − ) p = 1 , but a is that for a Gaussian state thenumber of negative energy excitations is actually negative N ( − ) p = − n p . The reason forthis last surprising fact is that the annihilation operator, besides explicitly removing oneparticle from the state, also enhances the probability of the highly populated sectors ofthe state, and the last effect is actually dominating. We will consider that ˆ ρ ( − ) p representsa single real hole plus some contribution of statistical quasiparticles of opposite momen-tum. Again, the statistical quasiparticle contribution can be thought as a consequenceof the increased knowledge of the background state coming from the absorption of aquasiparticle (or creation of a hole). In any case, the fact that for Gaussian states thestatistical contribution always dominates is an indication that holes cannot be consid-ered true quasiparticles in bosonic systems. A confirmation of this fact is given by thespread in the momentum per quasiparticle, (cid:104) ∆ p (cid:105) ( − ) QP := 1 | N ( − ) p | (cid:110) (cid:104) ˆ P (cid:105) ( − ) − [ (cid:104) ˆ P (cid:105) ( − ) ] (cid:111) / = | p | (5 + 1 /n p ) / , (4.11)which is of the order of the expectation value of the momentum per quasiparticle, evenif the occupation numbers are small. Even if holes are not true quasiparticles in bosonicsystems, it will be important to remember that negative energy excitations are possiblewhen the states are different than the vacuum.So far we have seen that quasiparticles and holes are respectively created by thecreation and annihilation operators. The field operator ˆ φ p , being a linear combinationof creation and annihilation operators, creates a coherent superposition of quasiparticlesand holes. In effect, the state ˆ φ − p ˆ ρ ˆ φ p corresponds to the linear superposition of aquasiparticle and a hole: ˆ φ − p ˆ ρ ˆ φ p = 12 E k (cid:16) ˆ a † p + ˆ a − p (cid:17) ˆ ρ (cid:16) ˆ a p + ˆ a †− p (cid:17) . (4.12)For states characterized by low occupation numbers the quasiparticle contribution dom-inates over the hole contribution. 29 . Quasiparticles: the real-time approach Let us argue that only the states created with the creation and annihilation operators,or equivalently with the field operator, are physically meaningful. In a realistic situationquasiparticles are created with the interaction of the field with some external agent. Thisexternal agent can be modeled by a “quasiparticle emitter”—which essentially coincideswith the usual particle detector considered in the analysis of the Unruh effect [61,62]: anexternal device described by some quantum mechanical degree of freedom Q (which cancorrespond for instance to a harmonic oscillator or a two-level system). The device startsin an excited state. For simplicity, let us assume that the emitter is linearly coupled withone pair of field modes, g Q Q ( t )[ φ p ( t ) + φ − p ( t )] . The coupling constant g Q is very smallso that the external device, other than emitting quasiparticles, does not significantlyperturb the dynamics of the field. The initial state of the field plus detector is assumedto be ˆ ρ ⊗ | Q (cid:105)(cid:104) Q | . The aim is to find the final state for the field ˆ ρ (cid:48) when the detectoris measured in its unexcited state | (cid:105) .The time evolution of the entire system under the interaction is given by ˆ ρ total ( t ) = T e − i R tt g Q ˆ φ I ( s ) ˆ Q I ( s ) d s ˆ ρ ⊗ | Q (cid:105)(cid:104) Q | T e i R tt g Q ˆ φ I ( s ) ˆ Q I ( s ) d s , (4.13)where the subindex I indicates interaction picture, and φ I := φ p + φ − p . Since thecoupling is small, the above equation can be expanded as ˆ ρ total ( t ) = (cid:20) − i (cid:90) tt g Q ˆ φ I ( s ) ˆ Q I ( s ) d s (cid:21) ˆ ρ ⊗ | Q (cid:105)(cid:104) Q | (cid:20) i (cid:90) tt g Q ˆ φ I ( s ) ˆ Q I ( s ) d s (cid:21) , or, developing the expression ˆ ρ total ( t ) = ˆ ρ ⊗ | Q (cid:105)(cid:104) Q | − ig Q (cid:90) tt d s g Q (cid:2) ˆ φ I ( s ) , ˆ ρ (cid:3) ⊗ (cid:2) ˆ Q I ( s ) , | Q (cid:105)(cid:104) Q | (cid:3) + g Q (cid:90) tt d s (cid:90) tt d s (cid:48) ˆ φ I ( s ) ˆ ρ ˆ φ I ( s (cid:48) ) ⊗ ˆ Q I ( s ) | Q (cid:105)(cid:104) Q | ˆ Q I ( s (cid:48) ) . (4.14)The final state for the system, if the emitter is found unexcited at time t , is given bythe projection into the ground state of the emitter [59]: ˆ ρ (cid:48) = Tr emitter ( ˆ ρ total ( t ) | Q (cid:105)(cid:104) Q | )Tr total ( ˆ ρ total ( t ) | Q (cid:105)(cid:104) Q | ) . (4.15)Developing Eq. (4.14) we find ˆ ρ (cid:48) = N (cid:90) tt d s (cid:90) tt d s (cid:48) ˆ φ I ( s ) ˆ ρ ˆ φ I ( s (cid:48) ) ⊗ (cid:104) | ˆ Q I ( s ) | Q (cid:105)(cid:104) Q | ˆ Q I ( s (cid:48) ) | (cid:105) = N (cid:48) (cid:90) tt d s (cid:90) tt d s (cid:48) e i Ω( s − s (cid:48) ) ˆ φ I ( s ) ˆ ρ ˆ φ I ( s (cid:48) ) , Notice that in this simplified model the emitter cannot couple to a single mode of the field becausethat couping would not be momentum-conserving (or, from another point of view, would not beHermitian). . Quasiparticles: the real-time approach where Ω is the frequency of the harmonic oscillator and N and N (cid:48) are normalizationconstants. Expanding in terms of creation and annihilation operators we get: ˆ ρ (cid:48) = N (cid:48)(cid:48) (cid:90) tt d s (cid:90) tt d s (cid:48) e i Ω( s − s (cid:48) ) (cid:16) ˆ a † p ˆ ρ ˆ a p e − iE p ( s − s (cid:48) ) +ˆ a † p ˆ ρ ˆ a †− p e − iE p ( s + s (cid:48) ) + ˆ a − p ˆ ρ ˆ a p e iE p ( s + s (cid:48) ) +ˆ a − p ˆ ρ ˆ a †− p e iE p ( s − s (cid:48) ) +ˆ a †− p ˆ ρ ˆ a − p e − iE p ( s − s (cid:48) ) + ˆ a †− p ˆ ρ ˆ a † p e − iE p ( s + s (cid:48) ) +ˆ a p ˆ ρ ˆ a − p e iE p ( s + s (cid:48) ) +ˆ a p ˆ ρ ˆ a † p e iE p ( s − s (cid:48) ) (cid:17) . Let us assume that the frequency of the oscillator is tuned so that
Ω = E p . In thiscase, for sufficiently large time lapses the dominant value of the integral is given by thestationary value of the integral of the first term, which amounts to considering energyconservation. In that case ˆ ρ (cid:48) ≈ N (cid:48)(cid:48)(cid:48) (cid:0) ˆ a † p ˆ ρ ˆ a p + ˆ a †− p ˆ ρ ˆ a − p (cid:1) = 12 (cid:0) ˆ ρ (+) p + ˆ ρ (+) − p (cid:1) . (4.16)Therefore, upon deexcitation of the emitter, the system gets promoted to a superpositionof two quasiparticle states. The argument could be repeated with the same measuringdevice in the ground state, now interpreted as a particle detector. When the measuringdevice gets excited, the state of the field colapses to a superposition of the hole states ˆ ρ ( − ) p and ˆ ρ ( − ) − p . We have just seen that when no interaction is present, positive-energy quasiparticles arecreated by the action of the creation operator, and negative-energy holes are createdby the annihilation operator. If interaction is present and quasiparticles are stable,similarly to the vacuum case (see section 2.2.2), one can think of defining asymptoticquasiparticle fields and states, from which the two-point correlation functions can bereproduced. Namely, quasiparticle states correspond to ˆ ρ (+) p ∼ = 1¯ n p + 1 ˆ¯ a † p ˆ ρ ˆ¯ a p , (4.17a)and negative energy holes correspond to ˆ ρ ( − ) p ∼ = 1¯ n p ˆ¯ a − p ˆ ρ ˆ¯ a †− p , (4.17b)where ¯ n p = Tr ( ˆ ρ ˆ¯ a † p ˆ¯ a p ) is the expectation value of the number of asymptotic quasipar-ticles in the state with momentum p , and we recall the symbol ∼ = means equivalencewhen evaluated in matrix elements in the asymptotic limit. The asymptotic creationand annihilation operators ˆ¯ a and ˆ¯ a † are related to the asymptotic field operator through ˆ¯ φ p = 1 (cid:112) E p (cid:0) ˆ¯ a p + ˆ¯ a †− p (cid:1) , (4.18)31 . Quasiparticles: the real-time approach where E p is the quasiparticle energy. The asymptotic field, which obeys free equationsof motion, ¨¯ φ + ( m + E p ) ¯ φ = 0 , is connected to the usual field through ˆ¯ φ p ∼ = Z − / p ˆ φ p , (4.19)where Z p is defined in Eq. (3.4). See Refs. [17, 21] for comparable approaches.However, since quasiparticles are generically unstable and therefore not completelywell-defined from a strict point of view, it is not worth pursuing a very formal description.Therefore we will blurry the distinction between the usual and the asymptotic fields, andwork with the usual field operator but rescaled a factor Z − / p , i.e. , we will set Z p = Z p [see Eqs. (1.18)]. With this assumption equations (4.17) can be more simply restated: ˆ ρ (+) p ≈ n p + 1 ˆ a † p ˆ ρ ˆ a p , ˆ ρ ( − ) p ≈ n p ˆ a − p ˆ ρ ˆ a †− p . (4.20)Bear in mind that this representation is only valid when studying (approximately)asymptotic properties.When we presented the open quantum system analysis of the field modes (see theintroduction), we also introduced two free parameters, E p and Z p . At this point wehave, on the one hand, a criterion to fix their value, and, on the other hand, a physicalinterpretation for both of them. With respect to the former, E p must be fixed to thevalue of the real part of the pole of the propagator, and represents the physical energyof the quasiparticle excitation. With respect to the latter, Z p measures the probabilitythat the interacting field operator excites the quasiparticle state. Roughly speaking, byrescaling the field a factor Z p = Z p we ensure that creation and annihilation operatorscreate and destroy quasiparticles with the proper normalization. From now on we shallassume that such a scaling has been performed so that Z p = Z p . From a practical pointof view this amounts to setting Z p = 1 in equations (3.9). The aim is to find the time evolution of the expectation value of the Hamiltonian ofthe system. We shall make use of the open quantum system viewpoint presented inthe introduction. Let us start with the positive energy excitations, and focus on thestate with momentum p . The expectation value of the Hamiltonian operator in such aquantum system is given by E (+) ( t, t ; p ) := (cid:104) ˆ H sys ( t ) (cid:105) (+) = 1 n p + 1 Tr (cid:0) ˆ a † p ˆ ρ ˆ a p U ( t , t ) ˆ H sys U ( t, t ) (cid:1) , (4.21)where the system Hamiltonian is given by Eq. (2.29). Discarding the connected partof the 4-point correlation functions, as it is manifest in the following application of theWick theorem [see appendix B and in particular Eq. (B.8)]: E (+) ( t, t ; p ) ≈ n p + 1 (cid:110) Tr sys (cid:0) ˆ ρ s ˆ a p ˆ a † p ) Tr sys (cid:0) ˆ ρ s ˆ H sys (cid:1) + E p Tr sys (cid:2) ˆ ρ s ˆ a p U ( t , t )ˆ a † p U ( t, t ) (cid:3) Tr sys (cid:2) ˆ ρ s ˆ a † p U ( t, t )ˆ a p U ( t , t ) (cid:3)(cid:111) , . Quasiparticles: the real-time approach which can be rewritten as E (+) ( t, t ; p ) ≈ E (0) + E p n p + 1 (cid:12)(cid:12)(cid:12) Tr sys (cid:2) ˆ ρ s ˆ a p U ( t , t )ˆ a † p U ( t, t ) (cid:3)(cid:12)(cid:12)(cid:12) , where E (0) := Tr sys (cid:0) ˆ ρ s ˆ H sys (cid:1) = E p ( n p + n − p +1 / is the energy of the reduced subsystembefore the perturbation, or as E (+) ( t, t ; p ) ≈ E (0) + E p n p + 1 (cid:12)(cid:12)(cid:12)(cid:12) E p ( E p + i∂ t )( E p − i∂ t ) G + ( t, t ; p ) (cid:12)(cid:12)(cid:12)(cid:12) , (4.22)where we have used the expression of the creation and annihilation operators in termsof the asymptotic field and asymptotic canonical momentum, Eq. (2.26), and the “field”representation of the canonical momentum, ˆ π p ( t ) = ˙ˆ φ p ( t ) .Using translation invariance the energy of the perturbation can be rewritten as E (+) ( t, t ; p ) ≈ E (0) + E p n p + 1 | I ( t, t ; p ) | . (4.23a)where I ( t, t ; p ) = 12 E p ( E p + i∂ t ) G + ( t, t ; p ) (4.23b)Using Eq. (3.9h), the expression I ( t, t ; p ) can be written as: I ( t, t ; p ) = (cid:90) d ω π e − iω ( t − t ) | ω | ( ω + E p ) Γ p [ n p + θ ( ω )] (cid:0) − ω + E p (cid:1) + ( ω Γ p ) . (4.23c)The above integral is evaluated by integration in the complex plane in appendix C.Neglecting Γ p in front of E p it is found I ( t, t ; p ) ≈ E p ( n p + 1) e − Γ p ( t − t ) , and thereforethe result is E (+) ( t, t ; p ) ≈ E (0) + E p ( n p + 1) e − Γ p ( t − t ) . (4.24)The factor n p + 1 is the statistical factor (recall that for a Gaussian state the expectednumber of excitations is N (+) p = n p + 1 ). Discounting this statistical factor the energyof the quasiparticle thus evolves as: E (+) QP ( t, t ; p ) ≈ E p e − Γ p ( t − t ) . (4.25)For the hole excitations we find a similar result E ( − ) ( t, t ; p ) ≈ E (0) + E p n p e − Γ p ( t − t ) . (4.26)Recall that, because of statistical effects, for Gaussian states the initial expectation valuefor the energy of the hole excitations is given by E p n p .Equations (4.24) and (4.25) are the main results of this section. From these equations E p can be identified as the energy of the quasiparticles, in agreement to the results ofthe previous section, and Γ p as the decay rate of the quasiparticles, according to the33 . Quasiparticles: the real-time approach interpretation of the imaginary part of the self-energy as the decay rate—recall that E p and E p Γ p correspond to the real and imaginary parts of the self-energy respectively.Notice that four basic assumptions have been used in order to reach these results.First, we have introduced an approximately asymptotic representation for the field op-erators in terms of the creation and annihilation operators, which is valid if we are con-sidering the evolution over times much longer than the characteristic interaction time.Second, and related to this, we have used a near-on shell approximation for the propaga-tors, which is also correct if we are investigating long times (but not extremely long, asdiscussed in section 2.2). Third, we introduced a Gaussian approximation, which makesthe problem solvable in term of the two-point correlation functions. Finally, we have alsoassumed that the retarded propagator has the analytic structure given by Eq. (3.7). Aswe have already mentioned, in presence of massless excitations there can be situationsin which the analytic structure is different, leading to non-exponential modifications ofthe decay law (4.24), as in the case of hot QED plasmas [40–42, 63, 64].It could be argued that the derivation presented in this section is somewhat cyclic,because we are assuming from the beginning that E p is the frequency corresponding tothe 2-mode system. Following what is done in Ref. [23], one could instead assume thatthe frequency of the system has an unknown value E (cid:48) p . We would then find a rapidoscillatory behavior for the expectation value of the energy. Only when E (cid:48) p = E p theenergy follows a smooth exponential decay as is expected on physical grounds.We have computed the time evolution of the propagators [and hence the time evolutionof the energy, according to Eqs. (4.23a) and (4.23b)] by going to the frequency spaceand taking profit of the spectral analysis of the propagators carried out in the previoussection. Alternatively, it is possible to bypass the spectral analysis by considering theequation of motion followed by the propagators, which in general is an integro-differentialequation. Making a local approximation to the self-energy in terms of the energy an thedecay rate, the equation of motion turns into an ordinary differential equation whichcan readly be solved [65]. Both methods are physically equivalent and lead to identicalresults. 34 . Alternative mean-field-based approaches The standard way of analyzing the quasiparticle properties is with the aid of the linearresponse theory [6–8, 10]. In linear response theory the Hamiltonian H of a quantummechanical system is perturbed with some external perturbation V ( t ) at time t , so thatthe total Hamiltonian is H + V ( t ) for t > t . Given some quantum operator ˆ O ( t ) , it is asimple exercise to show that to first order in the potential the expectation value of thatoperator in is given by (cid:104) ˆ O ( t ) (cid:105) = (cid:104) ˆ O ( t ) (cid:105) + δ (cid:104) ˆ O ( t ) (cid:105) = (cid:104) ˆ O ( t ) (cid:105) − i (cid:90) ∞ t Tr (cid:0) ˆ ρ θ ( t − t (cid:48) ) (cid:2) ˆ O I ( t ) , ˆ V I ( s ) (cid:3)(cid:1) , (5.1)where I indicates interaction picture with respect to the external perturbation and (cid:104)·(cid:105) indicates the expectation value in absence of the external potential.Linear response theory is usually applied to a free or interacting scalar field theory,with the following identifications: ˆ V ( t ) = (cid:90) d x j ( t, x ) ˆ φ ( x ) , ˆ O ( t ) = ˆ φ ( x ) . Then, from Eq. (5.1), one obtains: δ (cid:104) ˆ φ ( t, x ) (cid:105) = − i (cid:90) d x G R ( x, x (cid:48) ) j ( x (cid:48) ) . (5.2)For the case of an impulsive perturbation, j ( x ) = j ( q ) e i q · x δ ( t ) , if the retarded propa-gator is approximated as G R ( ω, p ) ≈ iZ p (2 E p )( ω − E p + i Γ p / , ω ∼ E p , the following result for the dynamics of the expectation value of the scalar field is ob-tained [6]: δ (cid:104) ˆ φ ( t, x ) (cid:105) ≈ − ij ( q )2 E p θ ( t ) e i ( q · x − E p t ) e − Γ t/ . (5.3)The fluctuation oscillates with a frequency E p and decays with a rate Γ p / . Those areinterpreted as the energy and damping rates of the excitations. Notice that the energyof the oscillation decays with a rate Γ p .In non-relativistic N -body theory, linear response is usually applied to the fermiondensity, instead of the field itself, and the final result depends on the density correlationfunctions [8]. Beyond the field theory context, linear response has many different appli-cations in statistical mechanics [10], solid state physics [66], and even gravitation [67].35 . Alternative mean-field-based approaches Another possible method to study the quasiparticle properties has been the analysis ofthe effective dynamics of the mean field. This method has been previously used in theliterature by Weldon [18] and by Drummond and Hathrell [68] in a curved backgroundcontext (see also Ref. [69]). Let us briefly review it in the CTP context.Functionally differentiating the CTP effective action we get the effective equations ofmotion for the expectation value of the field, ϕ := Tr( ˆ ρ ˆ φ ) (see appendix A): δ Γ[ ϕ , ϕ ] δϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ = ϕ = 0 . (5.4)The effective action can always be expanded in the proper vertices Γ a ··· a r ( x , . . . , x r ) : Γ[ ϕ , ϕ ] = (cid:88) r r ! (cid:90) d x · · · d x r Γ a ··· a r ( x , . . . , x r ) ϕ a ( x ) · · · ϕ a r ( x r ) . (5.5)A straightforward generalization of the usual argument (see e.g. Ref. [49]) shows thatthis 2-point vertex corresponds to the inverse propagator, Γ ab ( x, y ) = i ( G − ) ab ( x, y ) . (5.6)The Schwinger-Dyson equation, which defines the self-energy Σ ab ( x, x (cid:48) ) , G ab ( x, y ) = G (0) ab ( x, y ) − i (cid:90) d z d w G (0) ac ( x, z )Σ cd ( z, w ) G cb ( w, y ) , can be manipulated to give ( G − ) ab ( x, y ) = A ab ( x, y ) + i Σ ab ( x, y ) , (5.7)where A ab ( x, y ) is the inverse of the free propagator, A ab ( x, y ) = [( G (0) ) − ] ab ( x, y ) = c ab i ( − (cid:3) + m ) δ (4) ( x − y ) , (5.8)with c ab = diag(1 , − . We see that the 2-point vertex can be expressed as Γ ab ( x, y ) = iA ab ( x, y ) − Σ ab ( x, y ) . (5.9)Hence the 2-point vertex essentially corresponds to the self-energy. Other proper verticesalso have similar interpretations in terms of one-particle irreducible diagrams.If the relevant vertex is the 2-particle vertex Γ ab , the effective equations of motion canbe expressed as δ Γ δϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ = ϕ = (cid:90) d y [Γ ( x, y ) + Γ ( x, y )] ϕ ( y ) = 0 , (5.10)36 . Alternative mean-field-based approaches which, taking into account Eqs. (5.8) and (5.9) can be expanded as ( − (cid:3) x + m ) ϕ ( x ) + (cid:90) d y Σ R ( x, y ) ϕ ( y ) = 0 , (5.11)where Σ R ( x, y ) = Σ ( x, y ) + Σ ( x, y ) is the retarded self-energy. Introducing theFourier transform around the mid point, the above equation can be rewritten as p + m + Σ R ( p ; X ) = 0 . (5.12)Eq. (5.12) amounts to finding the poles of the retarded propagator. Thus, in flat space-time the self-energy and effective action methods lead to the same dispersion relationprovided we use a CTP approach in both situations and we neglect vertices with threeexternal particles or more. However, let us see that the interpretation of the dispersionrelation is slightly different. Provided we can approximate, p + m + Σ R ( p ; X ) ≈ − ω + m + p + E p − iω Γ p the solution to (5.11) can be written as ϕ ( x ) ≈ (cid:90) d k (2 π ) (cid:104) A ( p ) e − i k · x + B ( p ) e i k · x (cid:105) e − iE p t e − Γ p t/ . (5.13)Therefore the real and imaginary parts of the poles have the role of the frequency of thedamping rate of the mean field excitation. The mean-field-based approaches are simpler than the methods that we have developedin this paper, highlight the importance of the retarded propagator, and quickly relatethe real and imaginary parts of the poles of the propagator to the energy and the decayrate respectively. However, the expectation value of the field operator is not a usualobservable in field theory. Moreover, the kind of perturbations considered does not cor-respond to quasiparticles, since the latter, which are of the form ˆ a † ˆ ρ ˆ a , have vanishingexpectation values for the field operator. Given all that, we believe that in relativisticfield theory a linear response-based approach based on the study of the dynamics of ex-pectation value of the field operator is not well-suited to study the dynamical propertiesof quasiparticles in the regime of validity of the quasiparticle description. It is adequatethough to describe the hydrodynamic regime.Relativistic field theory in the hydrodynamic limit [12, 13] can be understood as aneffective theory describing the dynamics of long wavelength and short frequency per-turbations [14]. It has been extensively studied in the literature: for instance, let usmention that the evaluation of the hydrodynamic transport coefficients has been carriedout by Jeon and Yaffe [70,71] and by Calzetta et al. [72], and that Aarts and Berges havestudied the non-equilibrium time evolution of the spectral function in this regime [73].37 . Alternative mean-field-based approaches The limit of long wavelengths and short frequencies corresponds to considering smallmomenta. For Gaussian systems other than the vacuum, small-momentum modes haveoccupation numbers of the order of one or larger, an therefore this regime is not ad-equately described by a quasiparticle description, as commented in the introduction.Therefore, the hydrodynamic and quasiparticle descriptions represent two different com-plementary descriptions valid in two different regimes. The classical-like perturbations atlow momenta are adequately described by a hydrodynamic description, and the individ-ual particle-like perturbations at high momenta can be analyzed within a quasiparticleformalism.In any case, we have seen that both the dynamics of the mean field and the quasi-particles can be characterized with the retarded self-energy. This can be understoodin several ways, one of them being the following: loosely speaking, when the Gaussianapproximation is introduced and higher order correlation functions are neglected, thesystem behaves as if it were effectively linear, and thus the dynamics of all relevantquantities is essentially determined by the solution of a corresponding stochastic prob-lem [21, 45, 74]. In this context it is not surprising that the elementary dynamics of boththe mean field and the quasiparticles can be described with the same elements.38 . Summary and discussion
In this paper we have presented two different approaches to the analysis of the quasipar-ticle properties in relativistic field theory: first, a frequency-based approach, in whichthe properties of the excitations have been deduced from the analysis of the spectralrepresentation and, second, a real-time approach, wherein the quantum states corre-sponding to the quasiparticle excitations have been explicitly constructed, and the timeevolution of the expectation value of the energy in those states has been studied. In bothapproaches it has been possible to show that the real and imaginary part of the self-energy determine the energy and the decay rate of the quasiparticles respectively—seeEqs. (3.8). Although previous evidences existed, to our knowledge this is the first sys-tematic corroboration that the real part of the self-energy determines the physical energyof quasiparticles in non-vacuum relativistic field theory. The dynamics of the quasipar-ticle excitation can be also encoded in the form of a generalized dispersion relation [seeEq. (1.1b)]: E = E p − iE p Γ p = m + p + Σ R ( E p , p ) , (6.1)which needs not be Lorentz-invariant.Several additional interesting points have been illustrated by the real-time approach.We have explored the open quantum system viewpoint for the quantum field modes,and have found a physical criterion to fix the values of the system frequency and fieldrenormalization parameters, equivalent to the on-shell renormalization condition in thevacuum. Using a very simplified model of a quasiparticle creation device, we havediscussed that quasiparticles are adequately described by the action of the creationoperator on the background state for long observation times—see Eq. (4.16). At thesame time, we have shown that the quantum state corresponding to the quasiparticlecontains actually more than one particle excitation. We have argued that this result isactually a statistical effect, due to the fact that sectors with higher occupation numbersare more likely to be excited, because of the Bose-Einstein statistics. We have also builtthe hole state [Eq. (4.9)], but have discussed that it cannot be properly considered aquasiparticle state for bosonic systems.Finally, by comparing our results with other mean-field-based approaches, we haveargued that the latter are more suited for the study of classical-like configurations in thehydrodynamic limit rather than to the study of the properties of individual quasiparti-cles.Most expressions in this paper have referred to homogeneous, isotropic and stationarybackgrounds. In fact, the main results depend, first, on the existence of the diago-nal relation (1.13) between the self-energy and the propagator in Fourier space (whichdemands homogeneity and stationarity); second, on the diagonalization of the densitymatrix in the basis of eigenstates of the Hamiltonian (which is implied by stationar-ity), and, third, on the equivalence of the p and − p modes (which requires isotropy).Therefore, it appears that the quasiparticle interpretation will be completely spoiled39 . Summary and discussion when the background becomes non-stationary, non-homogeneous or non-isotropic. Letus argue that this is not the case, and that a particle interpretation is still feasiblefor non-homogeneous and non-stationary backgrounds provided that the characteristicscales of variation of the background state are sufficiently large. In the following we shallassume that L is the typical length scale in which the background changes significantly,and T is the typical timescale of evolution of the background.When the backgrounds are non-homogeneous or non-stationary there is still an ap-proximate diagonal relation between the retarded propagator and retarded self-energy[Eq. (1.13)], provided L (cid:29) /E and T (cid:29) /E , where E is the typical energy involved.The retarded propagator will have the usual analytic structure, with the only differencethat now the location of the poles will be time and space dependent, and therefore thegeneralized dispersion relation will also be time and space dependent: E = m + p + Σ R ( E p ( t, x ) , p ; t, x) = 0 . (6.2)On the other hand, for timescales much smaller than T , the background can be consideredstationary. Hence, if one considers energies which correspond to those short timescales, E (cid:29) /T , the density matrix components corresponding to those energies are diagonalin the basis of eigenstates of the Hamiltonian. Therefore the properties which dependon the diagonalization, such as the spectral representation, are still valid. Finally, if theorigin of the anisotropy is the inhomogeneity of the state, for the relevant energy scalesthe anisotropy will also be negligible if the condition L (cid:29) /E is fulfilled. However, it ispossible that the system is homogeneous but anisotropic, and that the magnitude of theanisotropy is large. In this case some of the expressions we have written down will nolonger be quantitatively valid, since we have demanded the equivalence of the forwardand backward modes for a given momentum. However in Refs. [23, 26] it is shown witha particular example that the general picture is not essentially modified.Let us now comment the subtleties related to the construction of the quasiparticlestates. Recall that quantum state corresponding to the quasiparticle contains more thanone additional particle excitation. As we have already mentioned, this is a statisticaleffect related to the fact that the background state is not an eigenstate of the numberoperator. It can be illustrated with the following toy model. Let ˆ ρ = ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) / bethe initial state of a harmonic oscillator. When a “quasiparticle” is introduced into thesystem by the action of the creation operator, the state becomes ˆ ρ (cid:48) = ( | (cid:105)(cid:104) | + 3 | (cid:105)(cid:104) | ) / .The expected number of particles in the initial and final states is 1 and 5/2 respectively,and therefore the particle number is increased in 3/2, despite the fact that each termof the state is increased with only one particle. Clearly, the reason for this is linked tothe Bose-Einstein statistics: the higher occupation states are more likely to be excited.A similar phenomenon happens if the incoherent mixture is replaced by a coherentsuperposition of two particle eigenstates. See Ref. [75] for an analysis of the samephenomenon in another context, with a discussion of possible energy non-conservationissues.We have seen that the dispersion relations governing the particle dynamics can bealso obtained from simpler mean-field-based methods (linear response theory or effectiveaction), albeit with a slightly different interpretation. We believe that the effort put40 . Summary and discussion in the construction and analysis of the quasiparticle excitations was anyway useful fordifferent reasons. First, the quasiparticle and mean field approaches provide the sameanswer to two different physical questions: in the former case one studies the dynamics ofindividual quasiparticles, and in the latter, the dynamics of the mean fields. Second, andrelated to this, both descriptions have different complementary regimes of validity, theformer approach being suited for high-momentum perturbations and the latter beingsuited for low-momentum perturbations. Finally, the route to the construction andanalysis of the quasiparticle properties shed light on many intermediate results whichare interesting by themselves.A key element in our analysis has been the Gaussian approximation, which on the onehand has allowed manageable expressions, and on the other hand has provided physi-cal interpretations for those expressions, without having to restore to any perturbativeexpansion in the coupling constant. The Gaussian truncation leads to the most basicdescription of the dynamics of the quasiparticles, the description in terms of a dispersionrelation which we have investigated in this paper. Non-Gaussianities would appear in amore elaborate description of the quasiparticle dynamics. The Gaussian approximationcan be formally controlled with the large- N expansion in the number of fields [76, 77].In this paper we have limited ourselves to those systems which can be correctly de-scribed in terms of a pole in the retarded propagator, or, in other words, in terms adispersion relation. As we have commented, there are systems posessing massless excita-tions in which the analytic structure of the propagator differs, leading to non-exponentialdecay laws [40–42, 63] and to the so-called non-Fermi liquid behavior [64].A natural extension of the work in this paper would be to generalize the results fornon-zero spin fields. The analysis of fermion fields should reveal the emergence of holeexcitations as true quasiparticles, in contrast to the bosonic systems, in which we haveseen that hole states, although can be constructed, do not have the appropriate quasipar-ticle properties. Another direction in which the work of this paper could be extended isthe analysis of quasiparticle interactions. In the same way as the Källén-Lehmann spec-tral representation can be adapted to non-vacuum situations, the Lehmann-Symanzik-Zimmerman formalism of vacuum field theory could in principle also be extended tonon-vacuum situations. Some work has been already done in this direction in the con-text of thermofield dynamics [20].The results of this paper can also be generalized to study the propagation of interactingparticles or quasiparticles in curved backgrounds, the propagation in curved backgroundshaving many similarities with the propagation in a physical medium [78,79]. In Ref. [65]the main results of this paper are generalized to account for the adiabatic propagationof interacting particles in cosmological backgrounds. Acknowledgments
I am very grateful to Cristina Manuel, Guillem Pérez-Nadal, Renaud Parentani, AlbertRoura and Joan Soto for several helpful discussions and suggestions, and to Enric Verda-guer for his critical reading of the manuscript. This work is partially supported by theResearch Projects MEC FPA2007-66665C02-02 and DURSI 2005SGR-00082.41 . The closed time path method
In this appendix we give a brief introduction to the closed time path (CTP) method(also called in-in method, in contrast to the conventional in-out method, or Keldysh-Schwinger method), stressing those aspects relevant for this paper. For further detailswe address the reader to Refs. [28, 80–83].Let us consider a free or an interacting scalar field φ . The path-ordered generatingfunctional Z C [ j ] is defined as Z C [ j ] = Tr (cid:16) ˆ ρT C e i R C d t R d x ˆ φ ( x ) j ( x ) (cid:17) , (A.1)where ˆ φ ( x ) is the field operator in the Heisenberg picture, C is a certain path in thecomplex t plane, T C means time ordering along this path and j ( x ) is a classical externalsource. By functional differentiation of the generating functional with respect to φ ,path-ordered correlation functions can be obtained: G C ( x, x (cid:48) ) = Tr (cid:2) ˆ ρT C ˆ φ ( x ) ˆ φ ( x (cid:48) ) (cid:3) = − δ Z C δj ( x ) δj ( x (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) j =0 . (A.2)Introducing a complete basis of eigenstates of the field operator in the Heisenberg picture, ˆ φ ( t, x ) | φ, t (cid:105) = φ ( t, x ) | φ, t (cid:105) , as a representation of the identity, the generating functionalcan be expressed as: Z C [ j ] = (cid:90) (cid:101) d φ (cid:101) d φ (cid:48) (cid:104) φ, t i | ˆ ρ | φ (cid:48) , t i (cid:105)(cid:104) φ (cid:48) , t i | T C e i R C d t R d x ˆ φ ( x ) j ( x ) | φ, t i (cid:105) (A.3)The functional measures (cid:101) d φ and (cid:101) d φ (cid:48) go over all field configurations of the fields at fixedinitial time t . If the path C begins and ends at the same point t i , then the transitionelement of the evolution operator can be computed via a path integral: Z C [ j ] = (cid:90) (cid:101) d φ (cid:101) d φ (cid:48) (cid:104) φ, t i | ˆ ρ | φ (cid:48) , t i (cid:105) (cid:90) ϕ ( t i , x )= φ (cid:48) ( x ) ϕ ( t i , x )= φ ( x ) D ϕ e i R C d t R d x { L [ ϕ ]+ ϕ ( x ) j ( x ) } , (A.4)where L [ φ ] is the Lagrangian density of the scalar field.Let us consider the time path shown in Fig. A.1. If we define ϕ , ( t, x ) = ϕ ( t, x ) and j , ( t, x ) = j ( t, x ) for t ∈ C , , then the generating functional can be reexpressed as: Z [ j , j ] = (cid:90) (cid:101) d φ (cid:101) d φ (cid:48) (cid:101) d φ (cid:48)(cid:48) (cid:104) φ, t i | ˆ ρ | φ (cid:48) , t i (cid:105)× (cid:90) ϕ ( t f , x )= φ (cid:48)(cid:48) ( x ) ϕ ( t i , x )= φ ( x ) D ϕ e i R d x { L [ ϕ ]+ ϕ ( x ) j ( x ) } × (cid:90) ϕ ( t f , x )= φ (cid:48)(cid:48) ( x ) ϕ ( t i , x )= φ (cid:48) ( x ) D ϕ e − i R d x { L [ ϕ ]+ ϕ ( x ) j ( x ) } . (A.5)42 . The closed time path method Im t Re tt f t i C C Figure A.1.: Integration path in the complex-time plane used in the CTP method. Theforward and backward lines are infinitesimally close to the real axis.In the following it will prove useful to use a condensed notation where neither theboundary conditions of the path integral nor the integrals over the initial and finalstates are explicit. With this simplified notation the above equation becomes Z [ j , j ] = (cid:90) D φ D φ (cid:104) φ, t | ˆ ρ | φ (cid:48) , t (cid:105) e i R d x { L [ ϕ ] − L [ ϕ ]+ ϕ ( x ) j ( x ) − ϕ ( x ) j ( x ) } (A.6)An operator representation is also possible: Z [ j , j ] = Tr (cid:18) ˆ ρ (cid:101) T e − i R t f t i d t R d x ˆ φ ( x ) j ( x ) T e i R t f t i d t R d x ˆ φ ( x ) j ( x ) (cid:19) . (A.7)By functionally differentiating the generating functional the different correlation func-tions can be obtained: G ab ( x, x (cid:48) ) = − δ Zδj a ( x ) δj b ( x (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) j a = j b =0 . (A.8)We recall that lowercase roman indices may acquire the values 1 and 2 are raised andlowered with the “CTP metric” c ab = diag(1 , − . Higher order correlation functionscan be obtained in a similar way.At least formally, all expressions so far are valid either for interacting or free fieldtheories. However, explicit results can only be obtained when the theory is free andthe initial state is Gaussian. In this case the path integrals in Eq. (A.5) can be exactlyperformed, and one obtains: Z (0) [ j , j ] = Z (0) [0 , e − R d x d x (cid:48) j a ( x ) G (0) ab ( x,x (cid:48) ) j b ( x (cid:48) ) . (A.9)where the propagators G (0) ab verify c ab ( − ∂ µ ∂ µ + m ) G (0) ab = − iδ (4) ( x − x (cid:48) ) . As with theconventional in-out method, the perturbative expansion can be organized in terms ofFeynman diagrams. There are two kinds of vertices, type 1 and type 2, and four kindsof propagators linking the two vertices. The Feynman rules are those of standard scalar43 . The closed time path method field theory, supplemented by the prescription of adding a minus sign for every type 2vertex.From the generating functional the connected generating functional W [ j, j (cid:48) ] is defined W [ j, j (cid:48) ] := − i ln Z [ j, j (cid:48) ] (A.10)Next we introduce the following objects: ϕ a ( x ) = δWδj a ( x ) , (A.11)which must be understood as functionals of j and j even if this dependence is notexplicit. If j = j both φ and φ give the expectation value of the field under thepresence of a classical source j . Finally, the effective action Γ[ ϕ , ϕ ] is defined as theLegendre transform of the connected generating functional: Γ[ ϕ , ϕ ] := W [ j , j ] − (cid:90) d x j a ( x ) ϕ a ( x ) . (A.12)In this equation j and j must be understood as functionals of ϕ and ϕ , which canbe obtained by inverting Eq. (A.11). By functionally differentiating the effective actionwith respect to ϕ and setting ϕ = ϕ the equation of motion for the expectation valueof the scalar field is obtained: δ Γ δϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ =0 = j ( x ) . (A.13)In contrast to the conventional in-out treatment, the equations of motion obtained fromthe CTP generating functional are real and causal because they correspond to the dy-namics of true expectation values [84].Thermal field theory can be seen as a strict particular case of the CTP method, inwhich the state ˆ ρ happens to be ˆ ρ = e − β ˆ H Tr( e − β ˆ H ) , (A.14)where ˆ H is the Hamiltonian operator of the system. To apply the techniques presented inthis appendix the only requirement is to compute the free thermal propagators. However,it proves convenient to make a slight a adaptation of the formalism and modify thecomplex time path in order to connect with the usual approaches to thermal field theory.Noticing that the density matrix operator can be seen as the time translation operatorin the complex plane, e − β ˆ H = U ( t − iβ, t ) , Eq. (A.3) can be reexpressed as: Z C [ j ] = 1Tr( e − β ˆ H ) (cid:90) (cid:101) d φ (cid:104) φ, t i − iβ | T C e i R C d t R d x ˆ φ ( x ) j ( x ) | φ, t i (cid:105) , (A.15)where we have used the completeness relation (cid:82) (cid:101) d φ (cid:48) | φ (cid:48) , t i (cid:105)(cid:104) φ (cid:48) , t i | = 1 . This way we havemanaged to incorporate the information on the state on the dynamical evolution. Now,44 . The closed time path method PSfrag repla ementsIm t Re t(cid:0)(cid:27)(cid:0)(cid:12) tfti C1C2 C3C4
Figure A.2.: Integration contour in the complex-time plane used in the real-time ap-proach to thermal field theory. The choice σ = 0 + makes the formalismanalogous to the CTP method.if the path C starts at t i and ends at t i − iβ , a path-integral representation can beintroduced: Z C [ j ] = 1Tr( e − β ˆ H ) (cid:90) (cid:101) d φ (cid:90) ϕ ( t i , x )= φ ( x ) ϕ ( t i , x )= φ ( x ) D ϕ e i R C d t R d x { L [ ϕ ]+ ϕ ( x ) j ( x ) } , (A.16)which can be rewritten in a more compact form as Z C [ j ] = N (cid:90) D ϕ e i R C d t R d x { L [ ϕ ]+ ϕ ( x ) j ( x ) } , (A.17)where N is a normalization constant, and the boundary conditions φ ( t i , x ) = φ ( t i − iβ, x ) are assumed. Different elections for the path C lead to different approaches to thermalfield theory: a straight line from t i to t i − iβ leads to the imaginary-time formalism, andthe contour shown in Fig. A.2 leads to the real-time formalism. By choosing σ = 0 + thereal-time formalism is virtually identical to the CTP method, since the the path along C and C can be neglected if we are interested in real-time correlation functions and theboundary conditions of the path integral are properly taken into account.45 . Gaussian states In the paper we make extensive use of the Gaussian states. In the following we shallgive a brief description of the aspects relevant to us without entering into details. See e.g.
Refs. [45, 85, 86] for a more complete introduction. In this appendix N will alwaysrepresent the correct normalization constant, which can always be computed with theGaussian integration formula if desired.In general, Gaussian states are those whose density matrices have a Gaussian formin a coordinate representation, or equivalently those whose Wigner functions have aGaussian form. A generic Gaussian state with (cid:104) ˆ p (cid:105) = 0 and (cid:104) ˆ q (cid:105) = 0 (or equivalently with (cid:104) ˆ a (cid:105) = (cid:104) ˆ a † (cid:105) = 0 ) can be represented as: ˆ ρ = N exp (cid:0) − F ˆ a † ˆ a + G ˆ a ˆ a + G ∗ ˆ a † ˆ a † (cid:1) , (B.1)where F is real. In this paper we have exclusively used Gaussian states with zero mean(many times without making explicit mention). Gaussian stationary states are thosewhich conmute with the Hamiltonian, [ ˆ ρ, ˆ H ] = 0 . The stationary state of a harmonicoscillator with ˆ H = Ω (cid:0) ˆ a † ˆ a + 1 / (cid:1) must be of the form ˆ ρ = N exp ( − F ˆ a † ˆ a ) , and thereforecorresponds to a thermal state.For a scalar field, decomposed in modes φ p , the most general state for the two-modesystem ± p which is translation invariant, i.e. , which conmutes with the momentumoperator, [ ˆ ρ s , ˆ p ] = 0 , ˆ p = p (cid:0) ˆ a † p ˆ a p − ˆ a †− p ˆ a − p (cid:1) . (B.2)and which verifies (cid:104) ˆ a ± p (cid:105) = (cid:104) ˆ a †± p (cid:105) = 0 is given by ˆ ρ s = N exp (cid:2) − F (cid:0) ˆ a † p ˆ a p + ˆ a †− p ˆ a − p (cid:1) + 2 G ˆ a p ˆ a − p + 2 G ∗ ˆ a † p ˆ a †− p (cid:3) . (B.3)If we further impose stationarity with respect to the reduced Hamiltonian, [ ˆ ρ s , ˆ H sys ] = 0 , ˆ H sys = ˆ π p ˆ π − p + E p ˆ φ p ˆ φ − p = E p (cid:0) ˆ a † p ˆ a p + ˆ a †− p ˆ a − p + 1 (cid:1) , (B.4)the most general Gaussian state corresponds to a factorized thermal state: ˆ ρ s = N exp (cid:2) − F (cid:0) ˆ a † p ˆ a p + ˆ a †− p ˆ a − p (cid:1)(cid:3) = N exp (cid:0) − F ˆ a † p ˆ a p (cid:1) exp (cid:0) − F ˆ a †− p ˆ a − p (cid:1) . (B.5)A quantum mechanical or field theory system is Gaussian if its generating functionalis Gaussian. For a single degree of freedom this means (assuming vanishing expectationvalues for the position operator) Z [ j a ] = exp (cid:20) − (cid:90) d t d t (cid:48) j a ( t ) G ab ( t, t (cid:48) ) j b ( t ) (cid:21) (B.6) Do not confuse the physical momentum operator ˆ p , with the canonical momentum operator ˆ π p ,conjugate of the field operator. . Gaussian states Gaussian systems correspond either to Gaussian states following quadratic equations ofmotion (as in Ref. [32]), or alternatively to an approximation for general states followinggeneral equations of motion (as in this paper).For Gaussian systems, according to the Wick theorem [6, 8], the n -point correlationfunctions can be reduced to the two-point correlation functions. Instead of trying togive a general formulation of the Wick theorem let us simply show some particularapplications. Whenever the system is Gaussian, any time-ordered four-point correlationfunction can be expressed in terms of two-point correlation functions (we assume thatthe expectation value of the field operators vanishes): (cid:104) T ˆ q ( t )ˆ q ( t )ˆ q ( t )ˆ q ( t ) (cid:105) = (cid:104) T ˆ q ( t )ˆ q ( t ) (cid:105)(cid:104) T ˆ q ( t )ˆ q ( t ) (cid:105) + (cid:104) T ˆ q ( t )ˆ q ( t ) (cid:105)(cid:104) T ˆ q ( t )ˆ q ( t ) (cid:105) + (cid:104) T ˆ q ( t )ˆ q ( t ) (cid:105)(cid:104) T ˆ q ( t )ˆ q ( t ) (cid:105) (B.7)If the correlation function is a mixture of time- and antitime-ordered expressions, theequivalent expression goes as follows (cid:104) T ˆ q ( t )ˆ q ( t ) (cid:101) T ˆ q ( t )ˆ q ( t ) (cid:105) = (cid:104) T ˆ q ( t )ˆ q ( t ) (cid:105)(cid:104) (cid:101) T ˆ q ( t )ˆ q ( t ) (cid:105) + (cid:104) ˆ q ( t )ˆ q ( t ) (cid:105)(cid:104) ˆ q ( t )ˆ q ( t ) (cid:105) + (cid:104) ˆ q ( t )ˆ q ( t ) (cid:105)(cid:104) ˆ q ( t )ˆ q ( t ) (cid:105) (B.8)These expressions can be demonstrated by taking derivatives on the Gaussian generatingfunctional (B.6).In field theory, if the state is translation-invariant, momentum conservation can sim-plify the application of the Wick theorem. For instance, if k (cid:54) = q , (cid:104) T ˆ φ k ( t ) ˆ φ − q ( t ) ˆ φ q ( t ) ˆ φ − k ( t ) (cid:105) = (cid:104) T ˆ φ k ( t ) ˆ φ − k ( t ) (cid:105) + (cid:104) T ˆ φ q ( t ) ˆ φ − q ( t ) (cid:105) . (B.9)The two-point correlators can be also expressed as a function of the creation and anni-hilation operator. If the state is stationary, only those terms having a creation and anannihilation operator survive: (cid:104) ˆ φ k ( t ) ˆ φ − k ( t ) (cid:105) = 12 E p (cid:16) (cid:104) ˆ a †− k ( t )ˆ a − k ( t ) (cid:105) + (cid:104) ˆ a k ( t )ˆ a † k ( t ) (cid:105) (cid:17) . (B.10)47 . Contour integration of I ( t, t ; p ) When computing the time evolution of the propagator, the following integral appears[see eqs. (4.23b) and (4.23c)]: I ( t, t ; p ) = 12 E p (cid:90) d ω π e − iω ( t − t ) | ω | ( ω + E p ) Γ p [ n p + θ ( ω )] (cid:0) − ω + E p (cid:1) + ( ω Γ p ) (C.1)Apparently, this integral cannot be evaluated with complex plane techniques since theintegrand contains the factor and | ω | , which is non-analytic with the usual prescription | ω | = √ ωω ∗ , and the factor θ ( ω ) , which in principle is defined in the real axis. Let usdo a more careful analysis.We begin by extending the problematic terms to the complex plane in the followingway: θ ( ω ) → θ (Re ω ) , | ω | → ω sign(Re ω ) . Those terms continue to be non-analytic, but only on a branch cut located in the imag-inary axis. Therefore, with this prescription the integrand is analytic everywhere on thecomplex plane except on the branch cut and on the poles. Notice that the branch cut israther special, since the function is continuous across the branch cut.Second, notice that the following related contour integrals are well-defined for t > t I ( t, t ; p ) = 12 E p (cid:73) C d ω π e − iω ( t − t ) − ω ( ω + E p ) Γ p n p (cid:0) − ω + E p (cid:1) + ( ω Γ p ) ,I ( t, t ; p ) = 12 E p (cid:73) C d ω π e − iω ( t − t ) ω ( ω + E p ) Γ p ( n p + 1) (cid:0) − ω + E p (cid:1) + ( ω Γ p ) , where C and C are the closed anticlockwise paths at the boundaries of the left lowerand right lower quadrants respectivelyThird, note that with the above prescription I ( t, t ; p ) = I ( t, t ; p ) + I ( t, t ; p ) , sincethe path at infinity does not contribute, and the contribution from the path in theimaginary axis cancels (because the function is continuous at the branch singularity).Therefore, in practice, the integral I ( t, t ; p ) can be computed by residues as if therewere no branch cut singularity.Let us now compute the integrals I ( t, t ; p ) and I ( t, t ; p ) . We start by I ( t, t ; p ) .There is a pole in the lower right quadrant at ω ≈ E p − i Γ p / . Neglecting Γ p in frontof E p we find I ( t, t ; p ) ≈ E p (1 + n p ) e − iE p ( t − t ) e − Γ p / . With respect to the integral I ( t, t ; p ) , the contribution from the pole in the lower leftquadrant is very suppressed because of the factor ( ω + E p ) in the numerator. Therefore48 . Contour integration of I ( t, t ; p ) I ( t, t ; p ) (cid:28) I ( t, t ; p ) . Given all this we obtain the final result: I ( t, t ; p ) ≈ E p (1 + n p ) e − iE p ( t − t ) e − Γ p / . (C.2)Alternatively this integral can also be evaluated directly in the time domain [65].49 ibliography [1] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, “Theory of Bose-Einstein conden-sation in trapped gases,” Rev. Mod. Phys. , 463–512 (1999).[2] D. Pines and P. Nozières, Theory of Quantum Liquids (Perseus Books, 1989).[3] S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang, S. Jafarey and D. J. Scalapino,“Quasiparticle and phonon lifetimes in superconductors,”
Phys. Rev. B , 4854–4873(1976).[4] A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Methods of Quantum Field Theoryin Statistical Physics (Prentice-Hall, Englewood Cliffs, N. J., 1964).[5] A. Altland and B. Simons,
Condensed Matter Field Theory (Cambridge University Press,2006).[6] M. le Bellac,
Thermal Field Theory (Cambridge University Press, Cambridge, England,1996).[7] J. I. Kapusta,
Finite-Temperature Field Theory: Principles and Applications (CambridgeUniversity Press, Cambridge, England, 1994).[8] A. L. Fetter and J. D. Walecka,
Quantum Theory of Many-Particle Systems (McGraw-Hill,New York, 1971).[9] S. Weinberg,
The Quantum Theory of Fields (Cambridge University Press, Cambridge,1995).[10] L. E. Reichl,
A Modern Course in Statistical Physics (Wiley-Interscience, 1998).[11] H. A. Weldon, “Mass-shell behavior of the electron propagator at low temperature,”
Phys.Rev. D , 065002 (1999).[12] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Perseus Books, 1990).[13] S. de Groot, W. van Leeuwen and C. van Weert,
Relativistic Kinetic Theory: Principles andApplications (North-Holland; Elsevier, Amsterdam, 1980).[14] D. T. Son and A. O. Starinets, “Viscosity, black holes, and quantum field theory,”
Ann.Rev. Nucl. Part. Science , 95–118 (2007), .[15] H. A. Weldon, “Simple rules for discontinuities in finite-temperature field theory,” Phys.Rev. D , 2007–2015 (1983).[16] J. F. Donoghue, B. R. Holstein and R. W. Robinett, “Renormalization and radiative cor-rections at finite temperature,” Ann. Phys. (N.Y) , 233 (1985).[17] H. Narnhofer, M. Requardt and W. E. Thirring, “Quasi-particles at finite temperatures,”
Commun. Math. Phys. , 247 (1983).[18] H. A. Weldon, “Quasiparticles in finite-temperature field theory,” (1998), hep-ph/9809330 .[19] H. Chu and H. Umezawa, “Stable quasiparticle picture in thermal quantum field physics,” Int. J. Mod. Phys. A , 1703–1730 (1994).[20] Y. Nakawaki, A. Tanaka and K. Ozaki, “On quasiparticles at finite temperature,” Prog.Theor. Phys. , 498–511 (1989). ibliography [21] C. Greiner and S. Leupold, “Stochastic interpretation of Kadanoff-Baym equations and theirrelation to Langevin processes,” Annals Phys. , 328–390 (1998), hep-ph/9802312 .[22] E. Calzetta and B. L. Hu, “Nonequilibrium quantum fields: Closed-time-path effectiveaction, Wigner function and Boltzmann equation,”
Phys. Rev. D , 2878–2900 (1988).[23] D. Arteaga, “Particle propagation in non-trivial backgrounds: a quantum field theory ap-proach,” Ph.D. thesis, Universitat de Barcelona (2007), arXiv:0707.3899[hep-ph] .[24] I. Ojima, “Lorentz invariance vs. temperature in QFT,” Lett. Math. Phys. , 73–80 (1986).[25] D. Arteaga, R. Parentani and E. Verdaguer, “Propagation in a thermal graviton back-ground,” Phys. Rev. D , 044019 (2004).[26] D. Arteaga, R. Parentani and E. Verdaguer, “Retarded green functions and modified dis-persion relations,” Int. J. Theor. Phys. , 1665–1689 (2005).[27] M. Requardt, “Spontaneous symmetry breaking of Lorentz and (Galilei) boosts in (rela-tivistic) many-body systems,” (2008), .[28] K.-C. Chou, Z.-B. Su, B.-L. Hao and L. Yu, “Equilibrium and nonequilibrium formalismsmade unified,” Phys. Rept. , 1–131 (1985).[29] I. D. Lawrie, “Perturbative description of dissipation in nonequilibrium field theory,”
Phys.Rev. D , 3330 (1989).[30] A. Das, Finite temperature field theory (World Scientific, Singapore, 1997).[31] M. A. van Eijck, R. Kobes and C. G. van Weert, “Transformations of real-time finite-temperature feynman rules,”
Phys. Rev. D , 4097–4108 (1994).[32] D. Arteaga, “Quantum Brownian motion representation for the quantum field modes,”(2007), .[33] J. F. Donoghue and B. R. Holstein, “Renormalization and radiative corrections at finitetemperature,” Phys. Rev. D , 340–348 (1983).[34] E. Braaten and R. D. Pisarski, “Resummation and gauge invariance of the gluon dampingrate in hot QCD,” Phys. Rev. Lett. , 1338 (1990).[35] E. Braaten and R. D. Pisarski, “Soft amplitudes in hot gauge theories: A general analysis,” Nucl. Phys.
B337 , 569 (1990).[36] R. Pisarski, “Scattering amplitudes in hot gauge theories,”
Phys. Rev. Lett. , 1129–1132(1989).[37] U. Kraemmer and A. Rebhan, “Advances in perturbative thermal field theory,” Rept. Prog.Phys. , 351 (2004), hep-ph/0310337 .[38] A. Rebhan, “Comment on ‘High-temperature fermion-propagator: Resummation and gaugedependence of the damping rate’,” Phys. Rev. D , 4779–4781 (1992).[39] H. A. Weldon, “Unified treatment of the electron propagator near the mass shell in threetemperature regions,” Phys. Rev. D , 116007 (2002).[40] J.-P. Blaizot and E. Iancu, “Lifetime of quasiparticles in hot QED plasmas,” Phys. Rev.Lett. , 3080–3083 (1996), hep-ph/9601205 .[41] J.-P. Blaizot and E. Iancu, “Lifetimes of quasiparticles and collective excitations in hot QEDplasmas,” Phys. Rev. D , 973–996 (1997), hep-ph/9607303 .[42] D. Boyanovsky, H. J. de Vega, R. Holman, S. Prem Kumar and R. D. Pisarski, “Real-timerelaxation of condensates and kinetics in hot scalar QED: Landau damping,” Phys. Rev. D , 125009 (1998), hep-ph/9802370 .[43] E. B. Davies, Quantum theory of open systems (Academic Press, London, 1976). ibliography [44] H. P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford UniversityPress, Oxford, 2002).[45] C. W. Gardiner and P. Zoller,
Quantum Noise (Springer, Berlin, 2000).[46] A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,”
Physica , 587–616 (1983).[47] W. G. Unruh and W. H. Zurek, “Reduction of a wave packet in quantum Brownian motion,”
Phys. Rev. D , 1071 (1989).[48] L. S. Brown, Quantum Field Theory (Cambridge University Press, Cambridge, England,1992).[49] M. E. Peskin and D. V. Schroeder,
An Introduction to Quantum Field Theory (Addison-Wesley, Reading, Mass., 1998).[50] C. Itzykson and J.-B. Zuber,
Quantum Field Theory (McGraw-Hill, New York, 1980).[51] W. Greiner and J. Reinhardt,
Field Quantization (Springer, Berlin, 2006).[52] B. Hatfield,
Quantum field theory of point particles and strings (Addison-Wesley, RedwoodCity, CA, 1992).[53] R. Haag,
Local Quantum Physics (Springer-Verlag, Berlin, 1992).[54] P. T. Matthews and A. Salam, “Relativistic field theory of unstable particles,”
Phys. Rev. , 283–287 (1958).[55] P. T. Matthews and A. Salam, “Relativistic field theory of unstable particles. II,”
Phys.Rev. , 1079–1084 (1959).[56] R. Jacob and R. G. Sachs, “Mass and lifetime of unstable particles,”
Phys. Rev. , 350–356 (1960).[57] M. Veltman, “Unitarity and causality in a renormalizable field theory with unstable parti-cles,”
Physica , 186–207 (1967).[58] D. Cocolicchio, “Characterization of unstable particles,” Phys. Rev. D , 7251–7261 (1998).[59] A. Galindo and P. Pascual, Quantum Mechanics (Springer-Verlag, New York, 1991).[60] H. A. Weldon, “Analytic properties of finite-temperature self-energies,”
Phys. Rev. D ,076010 (2002), hep-ph/0203057 .[61] W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D , 870–892 (1976).[62] N. D. Birrell and P. C. W. Davies, Quantum fields in curved space (Cambridge UniversityPress, Cambridge, England, 1982).[63] D. Boyanovsky and H. J. de Vega, “Anomalous kinetics of hard charged particles: Dynamicalrenormalization group resummation,”
Phys. Rev. D , 105019 (1999), hep-ph/9812504 .[64] D. Boyanovsky and H. J. de Vega, “Non-Fermi liquid aspects of cold and dense QEDand QCD: Equilibrium and non-equilibrium,” Phys. Rev. D , 034016 (2001), hep-ph/0009172 .[65] D. Arteaga, “A field theory characterization of interacting adiabatic particles in cosmology,” Clas. Quant. Grav. (2008), .[66] N. W. Ashcroft and D. N. Mermin, Solid State Physics (Brooks Cole, 1976).[67] P. R. Anderson, C. Molina-Paris and E. Mottola, “Linear response, validity of semiclassicalgravity, and the stability of flat space,”
Phys. Rev. D , 024026 (2003), gr-qc/0209075 .[68] I. T. Drummond and S. J. Hathrell, “QED vacuum polarization in a background gravita-tional field and its effect on the velocity of photons,” Phys. Rev. D , 343 (1980). ibliography [69] G. M. Shore, “Quantum gravitational optics,” Contemp. Phys. , 503–521 (2003), gr-qc/0304059 .[70] S. Jeon, “Hydrodynamic transport coefficients in relativistic scalar field theory,” Phys. Rev.D , 3591–3642 (1995).[71] S. Jeon and L. G. Yaffe, “From quantum field theory to hydrodynamics: Transport coeffi-cients and effective kinetic theory,” Phys. Rev. D , 2901–2919 (1988).[72] E. Calzetta, B. L. Hu and S. A. Ramsey, “Hydrodynamic transport functions from quantumkinetic field theory,” Phys. Rev. D , 125013 (2000), hep-ph/9910334 .[73] G. Aarts and J. Berges, “Non-equilibrium time evolution of the spectral function in quantumfield theory,” Phys. Rev. D , 105010 (2001), hep-ph/0103049 .[74] E. Calzetta, A. Roura and E. Verdaguer, “Stochastic description for open quantum systems,” Physica , 188–212 (2003), quant-ph/0011097 .[75] W. G. Unruh and R. M. Wald, “What happens when an accelerating observer detects aRindler particle,”
Phys. Rev. D , 1047–1056 (1984).[76] F. Cooper and E. Mottola, “Initial-value problems in quantum field theory in the large- N approximation,” Phys. Rev. D , 3114–3127 (1987).[77] F. Cooper, S. Habib, Y. Kluger, E. Mottola and J. P. Paz, “Nonequilibrium quantum fieldsin the large- N expansion,” Phys. Rev. D , 2848–2869 (1994).[78] D. Arteaga, R. Parentani and E. Verdaguer, “Particle propagation in cosmological back-grounds,” Int. J. Theor. Phys. , 227–2241 (2007).[79] D. Arteaga, “Particle propagation in cosmological backgrounds,” J. Phys. A , 6901–6906(2007), .[80] E. Calzetta and B. L. Hu, “Closed-time-path functional formalism in curved spacetime:Applicaton to cosmological back-reaction problems,” Phys. Rev. D , 495–509 (1987).[81] A. Campos and B. L. Hu, “Nonequilibrium dynamics of a thermal plasma in a gravitationalfield,” Phys. Rev. D , 125021 (1998).[82] A. Campos and E. Verdaguer, “Semiclassical equations for weakly inhomogeneous cosmolo-gies,” Phys. Rev. D , 1861–1880 (1994).[83] S. Weinberg, “Quantum contributions to cosmological correlations,” Phys. Rev. D ,043514 (2005).[84] R. D. Jordan, “Effective field equations for expectation values,” Phys. Rev. D , 444–454(1986).[85] D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin, 1994).[86] G. Adam, “Density matrix elements and moments for generalized Gaussian state fields,”
J.Mod. Optics , 1311–1328 (1995)., 1311–1328 (1995).