Thermodynamic Equilibrium as a Symmetry of the Schwinger-Keldysh Action
Extended quantum systems can be theoretically described in terms of the Schwinger-Keldysh functional integral formalism, whose action conveniently describes both dynamical and static properties. We show here that in thermal equilibrium, defined by the validity of fluctuation-dissipation relations, the action of a quantum system is invariant under a certain symmetry transformation and thus it is distinguished from generic systems. In turn, the fluctuation-dissipation relations can be derived as the Ward-Takahashi identities associated with this symmetry. Accordingly, the latter provides an efficient test for the onset of thermodynamic equilibrium and it makes checking the validity of fluctuation-dissipation relations unnecessary. In the classical limit, this symmetry reduces to the well-known one which characterizes equilibrium in the stochastic dynamics of classical systems coupled to thermal baths, described by Langevin equations.
In recent years, the question under which conditions and how a quantum many-body system thermalizes has received ever-growing attention. This interest has been primarily triggered by the increasing ability to prepare and manipulate such systems, which might be either isolated Polkovnikov et al. (2011); Yukalov (2011); Eisert et al. (2015) — as it is typically the case in experiments with cold atoms Lamacraft and Moore (2012); Bloch et al. (2008) — or in contact with an environment (open), and therefore subject to losses and driving.
After an abrupt perturbation, isolated systems are generically expected to thermalize in the sense that expectation values of local quantities at long times can be determined on the basis of suitable statistical ensembles Polkovnikov et al. (2011); Eisert et al. (2015). However, this might not be the case because of the presence of an extensive amount of conserved quantities induced by integrability Jaynes (1957); Kinoshita et al. (2006); Rigol et al. (2007); Kollar et al. (2011); Caux and Konik (2012) or because of a breaking of ergodicity due to the occurrence of many-body localization Basko et al. (2006); Pal and Huse (2010); Serbyn et al. (2013); Vosk and Altman (2013). Although it is possible to define a variety of effective temperatures based on the static Rossini et al. (2009); Mitra and Giamarchi (2011) and dynamic properties Foini et al. (2011, 2012) under such circumstances, the lack of thermal behavior is witnessed by the fact that these temperatures do not necessarily take all the same thermodynamic value.
Examples of open systems include exciton-polaritons in semiconductor heterostructures Carusotto and Ciuti (2013); Byrnes et al. (2014), arrays of microcavities Hartmann et al. (2008); Houck et al. (2012), trapped ions Blatt and Roos (2012), as well as optomechanical setups Marquardt and Girvin (2009). In general it is unclear, a priori, by which physical mechanism an effective temperature is possibly established in these systems and, in case, what determines its value. Recent work, however, suggests possible mechanisms where an effective temperature can occur as a consequence of the competition between driven-dissipative and coherent dynamics Mitra et al. (2006); Diehl et al. (2008, 2010); Dalla Torre et al. (2010, 2012, 2013); Chiocchetta and Carusotto (2013); Sieberer et al. (2013); Täuber and Diehl (2014). Irrespective of its cause, effective thermalization often affects only the low-energy degrees of freedom Mitra et al. (2006); Diehl et al. (2008); Dalla Torre et al. (2010); Diehl et al. (2010); Dalla Torre et al. (2012, 2013); Chiocchetta and Carusotto (2013); Sieberer et al. (2013); Täuber and Diehl (2014); Wouters and Carusotto (2006); Mitra and Giamarchi (2011, 2012); Öztop et al. (2012); Maghrebi and Gorshkov ().
All these examples show clearly that the presence of effective thermodynamic equilibrium (which might be established only in a subsystem or within a specific range of frequencies) in the steady state of a system is often by no means obvious. Hence, before addressing the question of whether the time evolution of a certain system leads to thermalization or not, it is imperative to identify criteria which allow a clear-cut detection of thermodynamic equilibrium conditions in the stationary state. In this regard, it is important to consider not only the static properties of the density matrix of the system, which describes its stationary state, but also the dynamics of fluctuations: being encoded, e.g., in two-time correlation and response functions, it might or might not be compatible with equilibrium. As a fundamental difference between static and dynamic properties, the latter necessarily involves the generator of time evolution, while the former does not.
In this work we consider the following operative definition of thermal equilibrium: a system is in thermal equilibrium at a certain temperature if expectation values of arbitrary products of operators, evaluated at different times, are connected by quantum fluctuation-dissipation relations (FDRs) involving the temperature . These FDRs were shown Chou et al. (1985); Wang and Heinz (2002); Jakobs et al. (2010) to be equivalent to a combination of the quantum-mechanical time-reversal transformation Messiah (1965) and the Kubo-Martin-Schwinger (KMS) condition Kubo (1957); Martin and Schwinger (1959). Heuristically, the latter condition expresses the fact that the Hamiltonian ruling the time evolution of a system is the same as that one determining the density matrix of the canonical ensemble, which characterizes the system when it is weakly coupled to a thermal bath. In both the generalized FDRs and the KMS condition the temperature appears as a parameter.
From the theoretical point of view, static and dynamical properties of statistical systems (both classical and quantum) are often conveniently studied in terms of dynamical functionals, which are used in order to generate expectation values of physical observables in the form of functional integrals over a suitable set of fields. Then, it is natural to address the issue of the possible equilibrium character of the stationary state by investigating the properties of the corresponding dynamical functional. In the case of classical statistical systems evolving under the effect of an external stochastic noise of thermal origin, this issue has been discussed to a certain level of detail in the past Janssen (1976); Bausch et al. (1976); Janssen (1979, 1992); Aron et al. (2010, ), and it was found that the dynamical functional acquires a specific symmetry in thermodynamic equilibrium. As in the case of the FDRs and the KMS condition, the (inverse) temperature enters as a parameter in this symmetry transformation. Remarkably, classical FDRs can be derived as a consequence of this symmetry. For quantum systems, instead, we are not aware of any analogous derivation based on the symmetries of the corresponding dynamical functional, which takes the form of a Schwinger-Keldysh action (see, e.g., Refs. Schwinger, 1961; Bakshi and Mahanthappa, 1963a, b; Mahanthappa, 1962; Keldysh, 1965; Kamenev, 2011; Altland and Simons, 2010; Stoof, 1999).
The aim of the present work is to fill in this gap by showing that also the Schwinger-Keldysh dynamical functional of a quantum system in thermal equilibrium is characterized by a specific symmetry, i.e., it is invariant under a certain transformation . This symmetry may be considered as the generalization of the classical one mentioned above, to which it reduces in a suitable classical limit Altland et al. (2010). In addition, can be written as a composition of the quantum-mechanical time reversal expressed within the Schwinger-Keldysh formalism — reflecting a property of the generator of dynamics — and of the transformation which implements the KMS conditions, associated with a property of the state in question. The existence of this symmetry was already noticed in Ref. Altland et al., 2010 for mesoscopic quantum devices, where it was used to derive fluctuation relations for particle transport across them. However, to our knowledge, the connection between this symmetry and the presence of equilibrium conditions has not yet been established.
The rest of the presentation is organized as follows: the key results of this work are anticipated and summarized in Sec. II; in Sec. III, we specify the symmetry transformation , provide its various representations, and list a number of properties which are then detailed in Sec. IV. In particular, we discuss the invariance of unitary time evolution in Sec. IV.1, while in Sec. IV.2 we consider possible dissipative terms which are invariant under . We discuss how the quantum symmetry reduces in the limit to the one known in classical stochastic systems in Sec. IV.3. As we discuss in Sec. V, the symmetry can be interpreted as a practical implementation of the KMS condition on the Schwinger-Keldysh functional integral. Finally, Sec. VI presents applications of the equilibrium symmetry: in Sec. VI.1 we derive the FDR for two-point functions while in Sec. VI.2 we show that the steady states of a quantum master equation explicitly violate the symmetry. The case of a system driven out of equilibrium by a coupling with two baths at different temperature and chemical potential is considered in Sec. VI.3; Sec. VI.4 briefly touches upon a number of other applications of the symmetry.
Ii Key results
The invariance under of the Schwinger-Keldysh action is a sufficient and necessary condition for a system to be in thermal equilibrium.
As mentioned in Sec. I, we consider a system to be in thermal equilibrium if all the FDRs are satisfied with the same temperature or, equivalently Chou et al. (1985); Wang and Heinz (2002); Jakobs et al. (2010), if the KMS condition (combined with time reversal) is satisfied. In Sec. V, we show that these conditions imply the thermal symmetry of the Schwinger-Keldysh action corresponding to the stationary state of the system. Conversely, the fluctuation-dissipation relations can be derived as consequences of the symmetry, proving their equivalence.
A different perspective: thermal equilibrium as a symmetry.
A key conceptual step forward we take in this work is to provide a compact formulation of thermal equilibrium conditions of a quantum system — i.e., the KMS condition (or, alternatively, of the equivalent hierarchy of FDRs) — in terms of a single symmetry , which can be considered as the fundamental property of quantum systems in thermal equilibrium. This perspective is especially fruitful within the field-theoretical formalism, where various tools have been developed to work out the consequences of the symmetries of the action of a given system. In this context, for example, the hierarchy of generalized quantum FDRs can be derived straightforwardly as the Ward-Takahashi identities associated with the thermal symmetry (see Secs. V and VI.1). In addition, the Schwinger-Keldysh formalism provides a convenient framework to take advantage of very powerful and efficient renormalization-group techniques for studying the possible emergence of collective behaviors and for monitoring how the effective description of a statistical system depends on the length and time scale at which it is analyzed. The possible scale dependence of the restoration/violation of the equilibrium symmetry could shed light on the mechanism underlying the thermalization of extended systems.
As we mentioned above, the idea of viewing thermal equilibrium as a symmetry is certainly not new. However, while previous studies were primarily concerned with classical statistical physics Janssen (1976); Bausch et al. (1976); Janssen (1979, 1992); Aron et al. (2010, ), here we generalize this idea to the quantum case.
Unification of the quantum and classical description of equilibrium systems.
As pointed out in Ref. Altland et al., 2010, the equilibrium symmetry reduces, in the classical limit, to a known symmetry which characterizes thermal equilibrium in open classical systems Janssen (1976); Bausch et al. (1976); Janssen (1979, 1992); Aron et al. (2010, ). In Sec. IV.3 we review the classical limit of the Schwinger-Keldysh action for a system coupled to a thermal bath Kamenev (2011); Altland and Simons (2010) and we discuss in detail how the classical equilibrium symmetry is recovered. The comparison with the classical symmetry highlights some remarkable differences with the quantum case: in fact, in classical systems, thermal equilibrium can be regarded as a consequence of detailed balance, which, in turn, is related to the property of microreversibility of the underlying microscopic dynamics. In fact, the classical equilibrium symmetry is derived by requiring the dynamical functional to satisfy these properties Janssen (1979, 1992); Aron et al. (2010). For quantum system, instead, an analogous satisfactory definition of detailed balance and microreversibility is seemingly still missing, leaving open the important question about the very nature of thermal equilibrium of quantum systems.
Efficient check for the presence of thermodynamic equilibrium conditions.
The symmetry is of great practical value, as it reduces answering the question about the possible presence of thermodynamic equilibrium to verifying a symmetry of the Schwinger-Keldysh action instead of having to check explicitly the validity of all FDRs. In particular, we show in Sec. VI.2 that the Markovian quantum dynamics described by a Lindblad master equation Kossakowski (1972); Lindblad (1976) explicitly violates the symmetry. This reflects the driven nature of the system: indeed, the Lindblad equation may be viewed as resulting from the coarse graining of the evolution of an underlying time-dependent system-bath Hamiltonian, with a time dependence dictated by coherent external driving fields.
Moreover, in Sec. VI.3 we consider a bosonic mode coupled to two baths at different temperatures and chemical potentials: in this case, the resulting net fluxes of energy or particles drive the system out of equilibrium with a consequent violation of the symmetry.
A new perspective on the construction of the Schwinger-Keldysh action.
At the conceptual level, the existence of the symmetry provides a new perspective on the construction of Schwinger-Keldysh functional integrals. In particular, as customary in quantum field theories, one may consider the symmetry as the fundamental principle: indeed, it is explicitly present for any time-independent (time-translation invariant) Hamiltonian which generates the dynamics of a system at the microscopic scale. Then, requiring the symmetry to hold for the full effective Keldysh action at a different scale fixes the admissible dissipative terms so as to satisfy FDRs between response and correlation functions of arbitrary order; translating back into the operator language, this provides a concrete hint why stationary density matrices of the form are favored over arbitrary functions for the description of static correlation functions.
Iii Symmetry transformation
As we anticipated above, a convenient framework for the theoretical description of the time evolution of interacting quantum many-body systems is provided by the Schwinger-Keldysh functional integral formalism Kamenev (2011); Altland and Simons (2010). It offers full flexibility in describing both non-equilibrium dynamics and equilibrium as well as non-equilibrium stationary states, which is out of reach of the finite-temperature Matsubara technique Lifshitz and Pitaevskii (1980). In addition, it is amenable to the well-established toolbox of quantum field theory. The simplest way to illustrate the basic ingredients of the Schwinger-Keldysh formalism is to consider the functional integral representation of the so-called Schwinger-Keldysh partition function . For a system with unitary dynamics generated by the Hamiltonian and initialized in a state described by a density matrix , this function is given by . (Note that, as it stands, ; however, it is instructive to focus on its structure independently of its actual value.) In this expression, time evolution can be interpreted as occurring along a closed path: starting in the state described by , the exponential to the left of corresponds to a “forward” evolution up to the time , while the exponential to its right corresponds to an evolution going “backward” in time. The trace connects, at time , the forward with the backward branch of the time path and therefore it produces a closed-time-path integral. Along each of these two branches, the temporal evolution can be represented in a standard way as a functional integral of an exponential weight over suitably introduced (generally complex) integration variables, i.e., fields, and on the forward and backward branches, respectively. These fields are associated with the two sets of coherent states introduced as resolutions of the identity in-between two consecutive infinitesimal time evolutions in the Trotter decomposition of the unitary temporal evolution along the two branches Kamenev (2011); Altland and Simons (2010). The resulting Schwinger-Keldysh action is a functional of and it is generally obtained as a temporal integral along the close path in time of a Lagrangian density. (Explicit forms of will be discussed further below, but they are not relevant for the present discussion.) By introducing different (time-dependent) sources for the fields on the two branches, the partition function is no longer identically equal to 1 and its functional derivatives can be used in order to generate various time-dependent correlation functions (see, e.g., Refs. Kamenev, 2011; Altland and Simons, 2010; Stoof, 1999).
As we show further below in Sec. V, a system is in thermodynamic equilibrium at a temperature , if the corresponding Schwinger-Keldysh action is invariant under a certain transformation which acts on the fields along the closed time path. In order to specify the form of , we focus on the dynamics of a single complex bosonic field, which is sufficiently simple but general enough to illustrate conveniently all the basic ideas. In this case, the transformation turns out to be composed of a complex conjugation 111In Ref. Altland et al., 2010, the symmetry is stated in terms of the real phase variables of complex fields. Then, the complex conjugation in Eq. (1) should be replaced by a change of sign. of the field components with , an inversion of the sign of the time variable, and a translation of the time variable into the complex plane by an amount , i.e.,
For convenience and future reference we provide an alternative compact representation of the action of both in the time and real space domain as well as in the frequency-momentum domain . The convention for the Fourier transforms of the fields, conveniently collected into two spinors , is the following:
In this relation, is the spatial dimensionality of the system, and the field spinors in the frequency-momentum domain are defined as . Accordingly, we can write the symmetry transformation in the form
where we introduced the Pauli matrix . The transformation in real time requires evaluating the fields for complex values of the time argument, which in principle is not defined; however, the complementary representation in Fourier space indicates how this can be done in practice: in frequency space, the shift of time by an imaginary part amounts to a multiplication by a prefactor .
As usual within the Schwinger-Keldysh formalism, it is convenient to introduce what are known as classical and quantum fields. These are defined as the symmetric and antisymmetric combinations, respectively, of fields on the forward and backward branches:
Combining these fields into spinors — where the index distinguishes classical and quantum fields — the transformation takes the following form, which we report here for future reference:
can be written as a composition of a time-reversal transformation and an additional transformation , which we will identify in Sec. V.3 as the implementation of the KMS condition within the Schwinger-Keldysh functional integral formalism.
is not uniquely defined, due to a certain freedom in implementing the time-reversal transformation within the Schwinger-Keldysh functional integral formalism, as discussed in Sec. V.2. However, without loss of generality, we stick to the definition provided by Eq. (1) and we comment on the alternative forms in Sec. V.2.
The various forms of the transformation presented above apply to the case of a system of bosons with vanishing chemical potential . In the presence of , Eq. (1) becomes
with a consequent modification of Eq. (3), which can be easily worked out. After a transformation to the basis of classical and quantum fields according to Eq. (4), this modification amounts to shifting the frequency in the arguments of the hyperbolic functions in Eq. (5), i.e., to .
In taking the Fourier transforms in Eqs. (3) and (5) one implicitly assumes that the initial state of the system was prepared at time , while its evolution extends to . In the following we will work under this assumption, commenting briefly on the role of an initial condition imposed at a finite time in Sec. IV.3.
Iv Invariance of the Schwinger-Keldysh action
As we demonstrate further below in Sec. V, a system is in thermodynamic equilibrium if its Schwinger-Keldysh action is invariant under the transformation , i.e.,
where, for convenience of notation, collects all the fields introduced in the previous section into a single vector. The tilde in indicates that all the parameters in which are related to external fields have to be replaced by their corresponding time-reversed values (e.g., the signs of magnetic fields have to be inverted), while in the absence of these fields the tilde may be dropped.
According to the construction of the Schwinger-Keldysh functional integral outlined at the beginning of the previous section, the action corresponding to the unitary dynamics of a closed system is completely determined by its Hamiltonian . The initial state of the dynamics enters the functional integral as a boundary condition: if the system was prepared in the state at the time , the matrix element , where are coherent states, determines the (complex) weight of field configurations at the initial time with . In Sec. IV.1, we demonstrate the invariance of the Schwinger-Keldysh action associated with a time-independent Hamiltonian dynamics under the transformation . In particular, this invariance holds for for any value of Interestingly enough, the Schwinger-Keldysh action associated with a Hamiltonian of a simple non-interacting system — which can be diagonalized in terms of single-particle states — turns out to be invariant under an enhanced version of this transformation, involving possibly different values of for each of the single-particle states (see Sec. IV.1.3). A constraint on the value of , however, comes from the inclusion of the boundary condition for the functional integral which specifies the initial state . Here we are interested in the stationary state of the system, which is generically reached a long time after its preparation in the state . Hence, we assume that this was done in the distant past, i.e., at , and that the evolution of the system extends to (cf. point 6 in Sec. III). In the construction of the Schwinger-Keldysh functional integral for a system in thermodynamic equilibrium Kamenev (2011); Altland and Simons (2010), a convenient alternative approach for specifying the appropriate boundary conditions corresponding to the initial equilibrium state of the system, consists in adding infinitesimal dissipative contributions to the action. Usually Kamenev (2011); Altland and Simons (2010), the form of these contributions is determined by the requirement that the Green’s functions of the system are thermal with a specific temperature , i.e., that they obey a fluctuation-dissipation relation; once these terms are included, any reference to may be omitted. We demonstrate in Sec. IV.2.1, that these dissipative contributions are invariant under with exactly the same . Hence, the thermal symmetry provides a different perspective on the construction of the Schwinger-Keldysh functional integral for a system in thermal equilibrium: while the unitary contributions are fixed by the Hamiltonian of the system, the requirement of invariance under the symmetry transformation can be taken as the fundamental principle for specifying the structure of the dissipative terms which can occur in the action if the system is in thermodynamic equilibrium at temperature . We emphasize that only the simultaneous presence in the Schwinger-Keldysh action of both the Hamiltonian and the dissipative contributions yields a well-defined functional integral: the dissipative terms in the microscopic action are taken to be infinitesimally small as for an isolated system, where they merely act as a regularization which renders the functional integral finite and ensures that the bare response and correlation functions satisfy a FDR; on the other hand, if the isolated system is composed of a small subsystem of interest and a remainder which can be considered as a bath, then finite dissipative contributions emerge in the Schwinger-Keldysh action of the subsystem after the bath has been integrated out. This scenario is considered in Secs. IV.3 and VI.2. Moreover, the system can act as its own bath: in fact, one expects the effective action for the low-frequency and long-wavelength dynamics of the system to contain dissipative contributions which are due to the coupling to high-frequency fluctuations. In Sec. IV.2, we explicitly construct dissipative terms which comply with the thermal symmetry . In particular, we find that the noise components associated with these dissipative terms must necessarily have the form of the equilibrium Bose-Einstein distribution function, as appropriate for the bosonic fields which we are presently focussing on.
iv.1 Invariance of Hamiltonian dynamics
The Schwinger-Keldysh action associated with the dynamics generated by a time-independent Hamiltonian can be written as the sum of a “dynamical” and a “Hamiltonian” part, and , respectively,
where we used the shorthand , while is the Pauli matrix. This structure of the Schwinger-Keldysh action results from the construction of the functional integral outlined at the beginning of Sec. III. In particular, the Hamiltonians are matrix elements of the Hamiltonian operator in the basis of coherent states , i.e., , where the amplitudes of the coherent states are just the integration variables in the functional integral Kamenev (2011); Altland and Simons (2010). Henceforth we focus on the case of a bosonic many-body system with contact interaction, i.e., with Hamiltonians in Eq. (10) given by
Here is the mass of bosons, the chemical potential, and parametrizes the strength of the -wave two-body interaction. We consider this case because it is sufficiently general for the purpose of illustrating all basic concepts associated with the thermal symmetry and, in addition, in the classical limit it allows a direct comparison with classical stochastic models Hohenberg and Halperin (1977); Folk and Moser (2006), where plays the role of a bosonic order parameter field. This point is elaborated in Sec. IV.3.
Below we show that the invariance of the Schwinger-Keldysh action under is intimately related to the structure of the action, i.e., to the fact that it can be written as the sum of two terms containing, separately, only fields on the forward and backward branches.
iv.1.1 Dynamical term
To begin with, we show that the dynamical contribution to the Schwinger-Keldysh action given in Eq. (9), is invariant under , i.e., that . To this end, it is convenient to express the original fields in the so-called Keldysh basis, which is formed by the classical and quantum components introduced in Eq. (4). For the sake of brevity, we arrange these fields into the vector . Rewriting in these terms and in frequency-momentum space, we obtain ()
The combination with is an odd function of , whereas is even, and therefore the integral over the product of these terms vanishes. Then, with some simple algebraic manipulation, the first two terms in Eq. (12) are recognised to be nothing but , from which the invariance of follows straightforwardly. Note that this property holds independently of the value of the parameter in the transformation .
iv.1.2 Hamiltonian contribution
We consider now the transformation of the Hamiltonian contribution in Eq. (10) under . First, we argue that the strictly local terms (i.e., those which do not involve spatial derivatives) in the Hamiltonian (11) are invariant under ; then, we extend the argument to the case of quasilocal terms such as the kinetic energy contribution or non-local interactions. Consider a contribution to of the form
where is a generic local contribution to the Hamiltonian and is an integer. In particular, for we obtain the term proportional to the chemical potential in Eq. (11), while for , is just the contact interaction. Since is real, under the transformation [see Eq. (6)] only its time argument is shifted according to and, taking the Fourier transform with respect to time of this relation, one eventually finds
Accordingly, the vertex (13) is invariant under : in fact, being local in time, its diagrammatic representation — where the fields and are represented by ingoing and outgoing lines, respectively — satisfies frequency conservation for in- and outgoing lines, as can be seen by taking the Fourier transform of each of the fields in individually. In particular, the frequency in Eq. (14) corresponds to the difference between the sums of the in- and outgoing frequencies and only the component contributes to Eq. (13). (As stated above, we assume that the time integrals in Eqs. (9), (10), and therefore (14) extend over all possible real values, i.e., we focus on the stationary state of the dynamics.) This component, however, is invariant under as follows directly from Eq. (14), and hence the same is true for the vertex, for which .
Clearly, the invariance of the vertex and of the dynamical term in Eq. (9) relies on the fact that vertices, which are local in time, obey frequency conservation. (Note that, as in Sec. IV.1.1, this invariance holds independently of the value of the parameter in .) Accordingly, one concludes that any contribution to the Hamiltonian, which is local in time and does not explicitly depend on time, is invariant. In particular, the proof of invariance presented here for the vertex in Eq. (13) can be straightforwardly extended to expressions containing spatial derivatives such as the kinetic energy in Eq. (11) and even to interactions which are not local in space, as long as they are local in time, as anticipated above. Note, however, that these considerations do not rule out the possible emergence upon renormalization or coarse-graining of terms which are non-local in time, as long as they are invariant under . This case is discussed further below in Sec. IV.2.
iv.1.3 Enhanced symmetry for non-interacting systems
The equilibrium transformation presented in Sec. III involves a single parameter . While this form is appropriate for the Gibbs ensemble describing the thermal equilibrium state of the interacting many-body system with the Hamiltonian in Eq. (11), an enhanced version of the symmetry is realized in non-interacting systems. Since these systems can be diagonalized in terms of single-particle states, they are trivially integrable. Statistically, integrable systems are described by a generalized Gibbs ensemble Rigol et al. (2007); Iucci and Cazalilla (2009); Jaynes (1957); Barthel and Schollwöck (2008); Goldstein and Andrei (); Pozsgay et al. (2014); Mierzejewski et al. (2014); Wouters et al. (2014); Essler et al. (), constructed from the extensive number of conserved quantities (with possible exceptions, see, e.g., Refs. Goldstein and Andrei, ; Pozsgay et al., 2014; Mierzejewski et al., 2014; Wouters et al., 2014). In the case of non-interacting systems which we consider here (or, more generally, for any system that can be mapped to a non-interacting one), these integrals of motion are just the occupation numbers of single-particle states. Below we provide an example, in which the Lagrange multipliers associated with these conserved occupations enter as parameters in a generalization of the equilibrium transformation Eq. (3): more specifically, these multipliers play the role of effective inverse temperatures of the individual single-particle states. On the other hand, in non-integrable cases, the eigenstates of the Hamiltonian are not single-particle states. Then one generically expects the stationary state of the system to be in thermal equilibrium at a temperature , which is determined by the initial conditions of the dynamics of the system. Accordingly, the enhanced symmetry that is present in the stationary state of the non-interacting integrable system breaks down and the corresponding Schwinger-Keldysh action is invariant under a single , only for that specific value of . This shows that the transformation can be generalized in order to account for the appearance of a generalized Gibbs ensemble in the trivial case of a system that can be diagonalized in terms of single-particle states. However, the question whether the generalized Gibbs ensemble emerging in the stationary states of generic integrable systems is characterized by a symmetry involving the Lagrangian multipliers associated with the respective integrals of motion as parameters, is beyond the scope of the present work.
As an example, let us consider bosons on a -dimensional lattice with nearest-neighbour hopping and on-site interaction (i.e., the Bose-Hubbard model Fisher et al. (1989)), with Hamiltonian
where is the annihilation operator for bosons on the lattice site , is the hopping matrix element between site and its nearest-neighbours , while determines the strength of on-site interactions. We first consider the case , which is trivially integrable: the kinetic energy contribution to the Hamiltonian is diagonal in momentum space and the corresponding single-particle eigenstates are the Bloch states. These are labelled by a quasi-momentum , and in terms of creation and annihilation operators for particles in Bloch states, and respectively, the kinetic energy can be written as
Let us now consider a Schwinger-Keldysh functional integral description of the stationary state of the system. Then, the kinetic energy in Eq. (16) yields a contribution to the corresponding action which reads
where and are the fields on the forward and backward branches of the closed time path respectively, expressed in the basis of Bloch states. is invariant under the transformation of the fields
where, as in Eq. (3), we arrange the fields in a spinor . The crucial point is that can be chosen to depend on the quasi-momentum , indicating that to each eigenstate of the system we can assign an individual “temperature” such that the corresponding mean occupation number is determined by a Bose distribution with precisely this “temperature.”
Let us now consider the opposite limit in which the hopping amplitude vanishes while the interaction strength is finite. The interaction energy in Eq. (15) is diagonal in the basis of Wannier states localized at specific lattice sites and the occupation numbers of these sites are conserved, rendering the system integrable. The generalized symmetry transformation appropriate for this case can be obtained from Eq. (18) by replacing the quasi-momentum by the lattice site index and by introducing a set of “local (inverse) temperatures” () instead of ().
In the generic case, when both the hopping and the interaction are non-zero, the system is not integrable. Then, neither the generalized transformation Eq. (18) nor its variant with local “temperatures” are symmetries of the corresponding Schwinger-Keldysh action, showing that this case eventually admits only one single global temperature, which determines the statistical weight of individual many-body eigenstates of the system.
iv.2 Dissipative contributions in equilibrium
The functional integral with the action in Eq. (8), as it stands, is not convergent but it can be made so by adding to an infinitesimally small imaginary (i.e., dissipative) contribution Kamenev (2011); Altland and Simons (2010). Within a renormalization-group picture, this infinitesimal dissipation may be seen as the “initial value,” at a microscopic scale, of finite dissipative contributions, which are eventually obtained upon coarse graining the original action and which result in, e.g., finite lifetimes of excitations of the effective low-energy degrees of freedom. The precise form of the corresponding effective low-energy action and, in particular, of the dissipative contributions which appear therein, is strongly constrained by the requirement of invariance under of the starting action at the microscopic scale: in fact, terms which violate this symmetry will not be generated upon coarse-graining. In the discussion below we identify those dissipative contributions to the Schwinger-Keldysh action which are invariant under . This allows us to anticipate the structure of any low-energy effective action possessing this a symmetry. Note, however, that finite dissipative terms may appear even at the microscopic scale because of, e.g., the coupling of the system to an external bath. Below we consider two instances of this case: in Sec. VI.2 we show that cannot be a symmetry of the action if the system is coupled to Markovian baths and driven — a situation described by a quantum master equation. Another specific example, in which the equilibrium symmetry is realized, is the particle number non-conserving coupling of the Schwinger-Keldysh action Eq. (8) to an ohmic bath. This situation, which we discuss in Sec. IV.3, is of particular interest, because its classical limit renders what is known as the dynamical model A Hohenberg and Halperin (1977) with reversible mode couplings (termed model in Ref. Folk and Moser, 2006); this correspondence allows us to establish a connection with the known equilibrium symmetry of the generating functional associated with this classical stochastic dynamics.
Below we discuss dissipative terms of the action invariant under , which involve first single particles (being quadratic in the fields of the Schwinger-Keldysh action) in Sec. IV.2.1, and then their interactions in Sec. IV.2.2.
iv.2.1 Single-particle sector
Dissipative contributions to the single-particle sector of the Schwinger-Keldysh action which are invariant under take the form
with an arbitrary real function which transforms under time reversal as . When such dissipative terms are introduced in order to regularize the Schwinger-Keldysh functional integral, a typical choice for is Kamenev (2011); Altland and Simons (2010) with . This ensures that the Green’s functions in the absence of interactions satisfy a fluctuation-dissipation relation (we postpone the detailed discussion of such relations to Sec. VI.1). The FDR for non-interacting Green’s functions, together with the invariance of interactions under the transformation shown in Sec. IV.1.2, guarantees that the FDR is satisfied to all orders in perturbation theory Jakobs et al. (2010).
While there are no restrictions on the form of the function , the hyperbolic cotangent appearing in the last term of is uniquely fixed by the requirement of invariance under , as can be verified by following the line of argument presented in Appendix A. In particular, with a certain value of in the argument of is invariant under if and only if . This shows that, remarkably, the appearance of the thermodynamic equilibrium Bose distribution function at a temperature in , can be traced back to the fact that is a symmetry of the action. Note that for simplicity we considered here only the case of vanishing chemical potential, . For finite , the frequency in the argument of the hyperbolic cotangent in Eq. (19) should be shifted according to , as we discussed in point 5 in Sec. III.
iv.2.2 Dissipative vertices
The dissipative contributions discussed in the previous section are quadratic in the field operators and they naturally occur, e.g., when the system is coupled to a thermal bath by means of an interaction which is linear in those fields. However, this type of coupling necessarily breaks particle number conservation. The number of particles is conserved if instead the system-bath interaction term commutes with the total number of particles of the system, , where is the local density. In other words, to ensure particle number conservation, it is necessary that the coupling terms are at least quadratic in the system operators. Accordingly, dissipative vertices appear in the Schwinger-Keldysh action after integrating out the bath degrees of freedom. Then, the requirement of invariance of these terms under allows us to infer a priori their possible structure. In particular, we find that a frequency-independent number-conserving quartic vertex (i.e., the dissipative counterpart to the two-body interaction in the Hamiltonian (11)) is forbidden by the thermal symmetry.
A generic quartic vertex, which conserves the number of particles and which is local in time, can be parameterized as
where are real functions; in order to simplify the notation we do not indicate the (local) spatial dependence of the fields, which is understood together with the corresponding integration in space. Conservation of particle number is ensured by the invariance on each contour separately, with generic phases , while the overall -function on the frequencies guarantees locality in time. Restrictions on the functions in the generic dissipative vertex in Eq. (20) follow from the requirements of causality Kamenev (2011), according to which to the action must vanish for , and invariance of the dissipative vertex under the equilibrium transformation. These conditions are studied in detail in Appendix B. In particular, we find that they cannot be satisfied if are constant, i.e., do not depend on the frequencies. One particular choice of these functions that is compatible with the constraints is given by
with the Bose distribution function . It is interesting to note that, in the basis of classical and quantum fields, this corresponds to a generalization of Eq. (19) with , in which the fields are replaced by the respective densities defined as and . Another notable property of this solution is that for we have and , i.e., these limits of vanishing frequencies are finite. This implies that the form of with given by Eq. (21) is to some extent universal: indeed, it should be expected to give the leading dissipative contribution to the Schwinger-Keldysh action of any number-conserving system in the low-frequency limit. At higher frequencies, other less universal solutions might also be important and one cannot make a general statement.
iv.3 Classical limit, detailed balance and microreversibility
A transformation analogous to — which becomes a symmetry in equilibrium — was previously derived for the stochastic evolution of classical statistical systems in contact with an environment, within the response functional formalism Martin et al. (1973); Janssen (1976); Bausch et al. (1976); Janssen (1979, 1992); De Dominicis, C. (1976); De Dominicis (1978); Täuber (2014). This formalism allows one to determine expectation values of relevant quantities as a functional integral with a certain action known as response functional, which can also be derived from a suitable classical limit of the Schwinger-Keldysh action for quantum systems Kamenev (2011); Altland and Simons (2010). In these classical systems, the environment acts effectively as a source of stochastic noise over which the expectation values are taken.
Here, we show that the classical limit of Altland et al. (2010) yields exactly the transformation which becomes a symmetry when the classical system is at equilibrium Aron et al. (2010). In order to consider this limit within the Schwinger-Keldysh formalism, it is convenient to express the Schwinger-Keldysh action in Eq. (8) in terms of the classical and quantum fields and , respectively, defined in Eq. (4), and to reinstate Planck’s constant according to Kamenev (2011); Altland and Simons (2010)
Then, the action can be formally expanded in powers of in order to take the classical limit , and the classical part of the Schwinger-Keldysh action is given by the contribution which remains for . Note that the limit considered here is formally equivalent to approaching criticality in equilibrium at finite temperature , for which , where is the energy gap, which can be read off from the retarded Green’s function (see, e.g., Ref. Sieberer et al., 2014). This equivalence conforms with the expectation that quantum fluctuations generically play only a subdominant role in determining the critical behavior of quantum systems at finite temperature. In order to see the emergence of a stochastic dynamics driven by incoherent (thermal) noise from a quantum coherent dynamics, we supplement the Schwinger-Keldysh action in Eq. (8) (describing the latter) with dissipative terms arising from its coupling to a bath. For simplicity, we assume this bath to be characterized by an ohmic spectral density, while the system is assumed to have the Hamiltonian in Eq. (11). Deferring to Sec. VI.2 the discussion of the theoretical description of such a system-bath coupling, we anticipate here that the resulting contribution to the Schwinger-Keldysh action can be written as in Eq. (64), under the assumption that is linear in the frequency, i.e., and by choosing , with the thermal bath acting independently on each momentum mode Altland and Simons (2010). Then, in the classical limit , we find
This action has the form of the response functional of the equilibrium dynamical models considered in Ref. Hohenberg and Halperin, 1977: it includes both a linear and a quadratic contribution in the quantum field , but no higher-order terms. After having transformed the quadratic term into a linear one via the introduction of an auxiliary field (which is eventually interpreted as a Gaussian additive noise), this quantum field can be integrated out and one is left with an effective constraint on the dynamics of the classical field , which takes the form of a Langevin equation; here:
where is a (complex) Gaussian stochastic noise with zero mean and correlations
Equation (24) describes the dynamics of the non-conserved (complex scalar) field without additional conserved densities, which is known in the literature as model A Hohenberg and Halperin (1977). However, as can be seen from the complex prefactor of the time derivative on the left-hand side of Eq. (24), the dynamics is not purely relaxational as in model A, but it has additional coherent contributions, also known as reversible mode couplings Täuber (2014). The fact that the simultaneous appearance of dissipative and coherent dynamics can be described by a complex prefactor of the time derivative is specific to thermal equilibrium: in fact, dividing Eq. (24) by , the reversible and irreversible parts of the resulting Langevin dynamics are not independent of each other and in fact their coupling constants share a common ratio Janssen (1979, 1992); Gardiner (2004). Under more general non-equilibrium conditions, however, these reversible and irreversible generators of the dynamics have different microscopic origins and no common ratio generically exists. In the present equilibrium context, however, the action Eq. (23) corresponds to model A in the notion of Ref. Folk and Moser, 2006, and the form of the classical transformation appropriate for this case which becomes a symmetry in equilibrium was given in Ref. Sieberer et al., 2014. This transformation emerges as the classical limit of discussed in the previous sections Altland et al. (2010). In fact, for and neglecting the contribution of the quantum fields in the transformation of the classical fields (i.e., at the leading order in ), Eq. (5) becomes
after a transformation back to the time and space domains. Upon identifying the classical field with the physical field and with the response field , according to , Eq. (27) takes the form of the classical symmetry introduced in Ref. Aron et al., 2010. Note, however, that the transformation (27) is not the only form in which the equilibrium symmetry in the classical context can be expressed. In fact, the transformation of the response field can also be expressed Janssen (1979, 1992) in terms of a functional derivative of the equilibrium distribution rather than of the time derivative of the classical field as in Eq. (27). The existence of these different but equivalent transformations might be related to the freedom in the definition of the response field, which is introduced in the theory as an auxiliary variable in order to enforce the dynamical constraint represented by the Langevin equation Zinn-Justin (2002); Janssen (1979, 1992); Täuber (2014) such as Eq. (24). This implies Zinn-Justin (2002) that the related action acquires the so-called Slavnov-Taylor symmetry. As far as we know, the consequences of this symmetry have not been thoroughly investigated in the classical case and its role for quantum dynamics surely represents an intriguing issue for future studies.
We emphasize the fact that the derivation of the symmetry in the classical case involves explicitly the equilibrium probability density Janssen (1979, 1992). Indeed, the response functional contains an additional contribution from the probability distribution of the value of the fields at the initial time, after which the dynamics is considered. This term generically breaks the time-translational invariance of the theory Janssen (1979, 1992), unless the initial probability distribution is the equilibrium one. Accordingly, when the classical equilibrium symmetry is derived under the assumption of time-translational invariance, its expression involves also the equilibrium distribution. In the quantum case discussed in the previous sections, instead, time-translational symmetry was implicitly imposed by extending the time integration in the action from to , which is equivalent to the explicit inclusion of the initial condition (in the form of an initial density matrix) and makes the analysis simpler, though with a less transparent interpretation from the physical standpoint.
Although in classical systems this equilibrium symmetry takes (at least) two different but equivalent forms due to the arbitrariness in the definition of the response functional mentioned above, it can always be traced back to the condition of detailed balance Janssen (1979, 1992); Aron et al. (2010). Within this context, detailed balance is defined by the requirement that the probability of observing a certain (stochastic) realization of the dynamics of the system equals the probability of observing the time-reversed realization, and therefore it encodes the notion of microreversibility. This condition guarantees the existence and validity of fluctuation-dissipation relations, which can be proved on the basis of this symmetry. In addition, detailed balance constrains the form that the response functional can take as well as the one of the equilibrium probability distribution for this stochastic process.
The situation in the quantum case appears to be significantly less clear. In fact, a precise and shared notion of quantum detailed balance and quantum microreversibility is seemingly lacking. The first attempt to introduce a principle of quantum detailed balance dates back to Ref. Agarwal, 1973, where it was derived from a condition of microreversibility in the context of Markovian quantum dynamics described by a Lindblad master equation. The mathematical properties of these conditions were subsequently studied in detail (see, e.g., Refs. Alicki, 1976; Kossakowski et al., 1977; Frigerio and Gorini, 1984; Majewski, 1984; Alicki and Lendi, 2007) and were shown to constrain the form of the Lindblad super-operator in order for it to admit a Gibbs-like stationary density matrix. However, even when this occurs, these operators are not able to reproduce the KMS condition and the fluctuation-dissipation relations because of the underlying Markovian approximation, as we discuss in Sec. VI.2.
The notion of microreversibility in quantum systems appears to have received even less attention, as well as its connection with some sort of reversibility expressed in terms of the probability of observing certain “trajectories” and their time-reversed ones. The definition proposed in Ref. Agarwal, 1973 (also discussed in Ref. Carmichael and Walls, 1976) appears to be a natural generalization of the notion in the classical case, as it relates the correlation of two operators evaluated at two different times with the correlation of the time-reversed ones. However, to our knowledge, the relationship between this condition and thermodynamic equilibrium has never been fully elucidated. Although addressing these issues goes well beyond the scope of the present paper, they surely represent an interesting subject for future investigations.
V Equivalence between the symmetry and the KMS condition
In this section we show that the invariance of the Schwinger-Keldysh action of a certain system under (as specified in Sec. IV) is equivalent to having multi-time correlation functions of the relevant fields which satisfy the KMS condition Kubo (1957); Martin and Schwinger (1959). As the latter can be considered as the defining property of thermodynamic equilibrium, this shows that the same applies to the invariance under the equilibrium symmetry.
The KMS condition involves both the Hamiltonian generator of dynamics and the thermal nature of the density matrix which describes the stationary state of the system: heuristically this condition amounts to requiring that the many-body Hamiltonian which determines the (canonical) population of the various energy levels is the same as the one which rules the dynamics of the system. The equivalence proved here allows us to think of the problem from a different perspective: taking the invariance under as the fundamental property and observing that any time-independent Hamiltonian respects it, we may require it to hold at any scale, beyond the microscopic one governed by reversible Hamiltonian dynamics alone. In particular, upon coarse graining within a renormalization-group framework, only irreversible dissipative terms which comply with the symmetry (such as those discussed in Sec. IV.2) can be generated in stationary state and the hierarchy of correlation functions respect thermal fluctuation-dissipation relations. The validity of KMS conditions (and therefore of the symmetry ) hinges on the whole system being prepared in a canonical density matrix . Accordingly, if the system is described by a microcanonical ensemble, the KMS condition holds only in a subsystem of it, which is expected to be described by a canonical reduced density matrix. Equivalently, this means that, in a microcanonical ensemble, only suitable local observables satisfy this condition. In the case of quantum many-body systems evolving from a pure state, an additional restriction on the class of observables emerges due to the fact that, if thermalization occurs as conjectured by the eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki (1994); Rigol et al. (2007), the microcanonical ensemble is appropriate only if the observable involves the creation and annihilation of a small number of particles (low order correlation functions). This was shown to be also the case for FDRs Srednicki (1999); Khatami et al. (2013): however, as pointed out above, the thermal symmetry implies the validity of FDRs involving an arbitrary number of particles, which leads to the conclusion that it does not apply to an isolated system thermalizing via the ETH. In other words, the thermal symmetry implies that the whole density matrix takes the form of a Gibbs ensemble, while in thermalization according to the ETH, only finite subsystems are thermalized by the coupling to the remainder of the system, which acts as a bath. Thus we see how Hamiltonian dynamics favors thermal stationary states (with density matrix proportional to ) over arbitrary functionals . One explicit technical advantage of this perspective based on symmetry is that it allows us to utilize the toolbox of quantum field theory straightforwardly and to study the implications of being a symmetry; this is exemplified here by considering the associated Ward-Takahashi identities and by showing the absence of this symmetry in dynamics described by Markovian quantum master equations in Sec. VI.2. We also note that the presence of this symmetry provides a criterion for assessing the equilibrium nature of a certain dynamics by inspecting only the dynamic action functional, instead of the whole hierarchy of fluctuation-dissipation relations. In addition, this symmetry may be present in the actions of open systems with both reversible and dissipative terms.
In the following, we consider a quantum system with unitary dynamics generated by the (time-independent) Hamiltonian , which is in thermal equilibrium at temperature and therefore has a density matrix . The KMS condition relies on the observation that for an operator in the Heisenberg representation , one has
(for simplicity we do not include here a chemical potential, but at the end of the discussion we indicate how to account for it). This identity effectively corresponds, up to a translation of the time by an imaginary amount, to exchanging the order of the density matrix and of the operator and therefore, when Eq. (28) is applied to a multi-time correlation function, it inverts the time order of the involved times, which can be subsequently restored by means of the quantum-mechanical time-reversal operation. Hence, the quantum-mechanical time reversal naturally appears as an element of the equilibrium symmetry , while external fields have to be transformed accordingly, as indicated in Eq. (7). The application of time reversal yields a representation of the KMS condition which can be readily translated into the Schwinger-Keldysh formalism, as was noted in Refs. Chou et al., 1985; Wang and Heinz, 2002; Jakobs et al., 2010. In particular, it results in an infinite hierarchy of generalized multi-time quantum fluctuations-dissipation relations which include the usual FDR for two-time correlation and response functions of the bosonic fields as a special case (see Ref. Jakobs et al., 2010 and Sec. VI.1). One of the main points of this work is that these FDRs can also be regarded as the Ward-Takahashi identities associated with the invariance of the Schwinger-Keldysh action under the discrete symmetry 222Here we used the notion of “Ward-Takahashi identity” in the slightly generalized sense which encompasses the case of identities between correlation functions resulting from discrete symmetries (such as Eq. (3) in Sec. V.4, which leads to, c.f., Eqs. (47) and (48)) beyond the usual case of continuous symmetries Zinn-Justin (2002). and that, conversely, the full hierarchy of FDRs implies the invariance of under .
The argument outlined below, which shows the equivalence between the KMS condition and the thermal symmetry, involves several steps: as a preliminary we review in Secs. V.1 and V.2 how time-ordered and anti-time-ordered correlation functions can be expressed using the Schwinger-Keldysh technique and we specify how these correlation functions transform under quantum mechanical time reversal. We apply these results to the KMS condition in Sec. V.3: first we discuss its generalization to multi-time correlation functions and then we translate such a generalization into the Schwinger-Keldysh formalism. This part proceeds mainly along the lines of Ref. Jakobs et al., 2010, with some technical differences. Finally, we establish the equivalence between the resulting hierarchy of FDRs and the thermal symmetry at the end of Sec. V.3.
v.1 Multi-time correlation functions in the Schwinger-Keldysh formalism
Two-time correlation functions.
Let us first consider a two-time correlation function
between two generic operators and (in the following we are particularly interested in considering the case in which and are the field operators or at positions and ) evaluated at different times and , respectively, in a quantum state described by the density matrix . We assume that the dynamics of the system is unitary and generated by the Hamiltonian . Then, the Heisenberg operator at time is related to the Schrödinger operator at a certain initial time via
with an analogous relation for .
The two-time correlation function can be represented within the Schwinger-Keldysh formalism as (see Appendix C)
irrespective of the relative order of the times and . Here, the functional integral is taken over the fields , and the exponential weight with which a specific field configuration contributes to the integral is determined by the Schwinger-Keldysh action . In the following, by we indicate that a certain operator has been evaluated in terms of the fields defined on the forward/backward branch of the temporal contour associated with the Schwinger-Keldysh formalism (see, e.g., Refs. Kamenev, 2011; Altland and Simons, 2010).
Multi-time correlation functions.
We define multi-time correlation functions in terms of time-ordered and anti-time-ordered products of operators
for and , where is an arbitrarily chosen largest time. Here, are bosonic field operators. The specific sequence of time arguments in and (increasing and decreasing from left to right, respectively) leads to a time-ordering on the Schwinger-Keldysh contour: indeed, as we show in Appendix C, the multi-time correlation function can be expressed as a Schwinger-Keldysh functional integral in the form
Anti-time-ordered correlation functions.
Not only time-ordered correlation functions such as Eq. (33) can be expressed in terms of functional integrals, but also correlation functions which are anti-time-ordered and which, e.g., are obtained by exchanging the positions of and on the l.h.s. of Eq. (33). The construction of the corresponding functional integral can be accomplished with a few straightforward modifications to the procedure summarized in Appendix C (and presented, e.g., in Refs. Kamenev, 2011; Altland and Simons, 2010). In a stationary state one has and all the Heisenberg operators on the l.h.s. of Eq. (33) can be related to the Schrödinger operators at a later time . Then one finds
where the action describes the backward evolution and it is related to the action which enters the forward evolution in Eq. (33) simply by a global change of sign .
v.2 Quantum-mechanical time reversal
In this section we first recall some properties of the quantum-mechanical time reversal operation Messiah (1965) and then discuss its implementation within the Schwinger-Keldysh formalism. is an antiunitary operator, i.e., it is antilinear (such that for ) and unitary (). Scalar products transform under antiunitary transformations into their complex conjugates, i.e., , where we denote by and the state and the Schrödinger operator obtained from the state and the operator , respectively, after time reversal. Accordingly, expressing the trace of an operator in a certain basis , one finds
In the last equality we used the fact that, due to the unitarity of , also the time-reversed set forms a basis. For future convenience, we shall define the Heisenberg representation of time-reversed operators such that it coincides with the Schrödinger one at time , i.e., we set
Note that this is distinct from the Heisenberg representation defined in Eq. (30), which coincides with the Schrödinger one only at time . In order to simplify the notation, we shall not distinguish these two different Heisenberg representations, assuming implicitly that the latter and the former are used, respectively, for operators and their time-reversed ones, such that while .
Let us now study the effect of time reversal on the generic multi-time correlation function in Eq. (33). Due to translational invariance in time, the time arguments of the operators and can be shifted by without affecting the correlation function. Then, by using Eqs. (35) and (36), one has
where the subscript in indicates that the expectation value is taken with respect to the time-reversed density operator , which is time-independent. The expectation value on the r.h.s. of Eq. (37) is anti-time ordered and therefore it can be rewritten as a Schwinger-Keldysh functional integral by using Eq. (34). The l.h.s., instead, is time-ordered and therefore it can be expressed as in Eq. (33), such that Eq. (37) becomes
where the subscript in indicates that the sign of the action which describes the Hamiltonian evolution on the r.h.s. of this relation has been reversed, as explained below Eq. (34). The time-reversed action differs from the action associated with which enters (implicitly, cf. Eq. (31)) Eq. (33) because in the time evolution is generated by , the initial state is the time-reversed density matrix , and the integration over time extends from to . This latter difference becomes inconsequential as and .
Let us now consider the case in which and are products of the bosonic field operators and , such that and involve the corresponding products of and their complex conjugates. As there are no further restrictions on and , the l.h.s. of Eq. (38) can be generically indicated as , where is the product of various fields on the Schwinger-Keldysh contour corresponding to which, according to the notation introduced in Sec. III, are collectively indicated by