Variational principle for theories with dissipation from analytic continuation
The analytic continuation from the Euclidean domain to real space of the one-particle irreducible quantum effective action is discussed in the context of generalized local equilibrium states. Discontinuous terms associated with dissipative behavior are parametrized in terms of a conveniently defined sign operator. A generalized variational principle is then formulated, which allows to obtain causal and real dissipative equations of motion from the analytically continued quantum effective action. Differential equations derived from the implications of general covariance determine the space-time evolution of the temperature and fluid velocity fields and allow for a discussion of entropy production including a local form of the second law of thermodynamics.
Methods of quantum field theory are widely used to solve problems of elementary particle physics, nuclear physics, condensed matter physics or cosmology. This includes problems involving a few particles or excitations around the conventional vacuum but also thermal equilibrium states or arbitrary out-of-equilibrium dynamics. Problems of the latter type are usually treated in terms of the Schwinger-Keldysh closed time path Schwinger (); Keldysh (), see for example KadanoffBaym (); Danielewicz:1982kk (); Chou:1984es (); Calzetta:1986cq (); Berges:2004yj (); Rammer:2007zz (); Kamenev () for introductions and overviews. This formalism is particularly useful when a coupling parameter is small such that perturbative methods can be used. Theoretical techniques that are often employed are the mapping to an (effective) kinetic theory description or resummation schemes such as the two-particle irreducible one.
An advantage of the theoretical methods based on the closed time path is that they usually provide quite detailed microscopic information and that they are capable to follow the time evolution of arbitrary density matrices out-of-equilibrium, at least in principle. There are also no restrictions concerning the time and length scales one can consider, apart from those that are set by the range of validity of perturbative expansion schemes or other additional approximations (for example the classical statistical approximation which is sometimes used for problems involving large occupation numbers of bosonic particles). On the other side, this implies also a rather high complexity of the formalism and oftentimes high numerical effort for the concrete solution of a given problem.
For some theoretical problems, one is actually not interested in the full microscopic dynamics but rather in the behavior of the theory for large length and time scales. If it is not hindered by conservation laws or integrable dynamics, the long-time limit is usually dominated by thermalization and the corresponding equilibrium states. The thermal states have lost all memory of the initial state except for the conserved charges, in particular energy and momentum which determine the temperature and fluid velocity and conserved particle numbers which determine corresponding chemical potentials. The theoretic description of thermal states is quite universal as it only involves the thermodynamic equation of state and the standard thermodynamic formalism.
In addition to the thermal equilibrium states, also the long time dynamics is actually governed by the conservation laws to a large extend. Intuitively speaking, those modes that are fully or approximately preserved by conservation laws from relaxing to some equilibrium configuration dominate the long-time dynamics, while other modes relax on shorter time scales KadanoffMartin (); Hohenberg:1977ym (). The traditional formalism to describe this is the one of hydrodynamics, or fluid dynamics in modern terminology. In addition to the conserved currents, such as the energy momentum tensor or conserved number currents, also long-range fields and order parameters such as e. g. the electromagnetic field or the magnetization need to be included in the theoretical description Hohenberg:1977ym ().
The fluid dynamic description is not quite as universal as thermodynamics because it needs additional information about transport properties, for example the shear and bulk viscosity or dissipative damping terms for the long range fields and order parameters. The range of applicability of a fluid dynamic description is set by the time (and associated length) scale of thermalization. The thermalization time is shorter, and therefore the range of applicability of a fluid dynamic description larger, when interaction effects are stronger.
It is possible to obtain fluid dynamics as the long time limit of kinetic theory. Theoretical methods such as the Chapman-Enskog expansion ChapmanCowling () or Grad’s method of moments (see e. g. Denicol:2012es (); Denicol:2012cn ()) describe the mapping in detail. In terms of these methods one can also determine the transport properties by microscopic calculations based on kinetic theory. While this approach is fine for the situations where it is applicable (such as dilute gases) it has the conceptual disadvantage that it needs both a weak coupling assumption for the applicability of kinetic theory and a form of a strong coupling assumption for the applicability of a fluid dynamic description. One may therefore wonder whether there exists a shortcut in the theoretical setup that leads more directly from the formulation of a quantum field theory to a fluid dynamic type of formalism. Such a “shortcut formalism” should be capable of describing local equilibrium situations and the corresponding dynamics but without relying on the weak coupling assumptions underlying kinetic theory. For thermodynamic equilibrium, such a formalism exists of course in the form of the imaginary time or Matsubara formalism and it is heavily used also for non-perturbative calculations (e. g. in the setup of lattice gauge theory).
It is plausible that there should also be a general quantum field theoretic formalism which is applicable out-of-equilibrium, however not in the sense of arbitrary, far-from-equilibrium states but rather for states that can be described by an approximate local equilibrium. Such a situation may be coined as close-to-equilibrium. One formalism of this type is linear response theory. It is restricted to small excitations around global equilibrium states but within this limitation it allows for non-perturbative statements (in the sense of the expansion in a coupling constant), for example the fluctuation-dissipation theorem or the Green-Kubo relations. One may also argue that for strongly interacting theories with a gravitational dual, such a close-to-equilibrium formalism is used already at least implicitly in many calculations based on the AdS/CFT correspondence.
It should then also be possible to construct such a close-to-equilibrium formalism using solely the methods of quantum field theory without relying on perturbative assumptions and the mapping to kinetic theory. These considerations are part of the motivation for the present work.
The formalism we aim to develop here is a direct generalization of equilibrium quantum field theory. Instead of using the full machinery of the Schwinger-Keldysh closed time path, we will use analytic continuation111The analytic continuation technique has a long history as an approach to non-equilibrium problems, see for example refs. KadanoffMartin (); Abrikosov ().. This has the advantage that part of the formal complexity of the closed time path formalism can be avoided. For example, we will not need a full doubling of the fields corresponding to the two branches of the time contour. However, as will be discussed in detail below, we will have to introduce a specific sign operator that allows to parametrize the branch cuts in the analytic (generalized) frequency plane associated with dissipative behavior. Because the sign operator allows to distinguish between the two sides of the branch cut, and therefore retarded and advanced correlation functions, a few aspects of the resulting formalism work in practice similar as for the closed time path formalism. In particular, and that is the main result of this paper, it is thereby possible to obtain real and causal equations of motion including dissipative effects for field expectation values from the variation of an analytically continued version of the one-particle irreducible quantum effective action.
The fact that our formalism is based on an effective action implies that not only the field equations or equations of motion can be studied, but also very directly various correlation functions which follow from functional derivatives of the analytic action. This is in contrast to the traditional formulation of fluid dynamics which starts directly with the equations of motion. The correlation functions obtained in this way describe generalizations of thermal equilibrium fluctuations in various fields. In ref. Landau:1980mil (), such fluctuations are called quasi-stationary.
The analytic continuation we will study starts from an Euclidean description of local equilibrium states. In the presence of additional field expectation values or order parameters, these local equilibrium states should be understood in the sense of generalized Gibbs ensembles Hayata:2015lga (). One should also keep in mind that local equilibrium is typically only an approximation to the full dynamics (although it can be a rather good approximation). The formalism assumes implicitly that the theory is probed on length and time scales that are large compared to the thermalization dynamics, or, in other words, that approximate local thermalization is efficient enough for those degrees of freedom that make up the bath. There are, of course, also situations where this assumption is not justified.
We do not assume, however, that dissipative processes are absent. In contrast, their proper description will be one of the main points of the present paper. A particularly nice feature of the close-to-equilibrium formalism is that it allows a straight-forward discussion of entropy production. This includes entropy production due to dissipative processes concerning a (quantum) field expectation value as well as shear and bulk viscous dissipation.
As will be discussed in detail below, the temperature and fluid velocity do not have the same status as quantum field expectation values or order parameter fields . In particular, the analytic effective action is not stationary with respect to variations of the temperature or fluid velocity. These fields should rather be seen as certain parameter fields for which the dynamics does not directly follow from the equations of motion but rather somewhat implicitly from general coordinate invariance or the closely connected conservation laws for energy and momentum. More details are discussed below.
The formalism presented here allows to derive the equations of motion of fluid dynamics from the variation of an effective action. This is discussed for a fluid without conserved charges apart from energy and momentum within the so-called first order approximation in section 7. The extension to more general fluids as well as the second order formalism will be discussed elsewhere. Attempts to construct a variational principle that yields the equations of fluid dynamics have a long history, see for example Andersson:2006nr () for an overview. Recently, fluid dynamics was discussed extensively in the context of effective field theory Dubovsky:2005xd (); Nicolis:2011ey (); Dubovsky:2011sj (); Dubovsky:2011sk (); Nicolis:2011cs (); Bhattacharya:2012zx (); Endlich:2012vt (); Grozdanov:2013dba (); Andersson:2013jga (); Nicolis:2013lma (); Haehl:2013hoa (); Ballesteros:2014sxa (); Geracie:2014iva (); Delacretaz:2014jka (); Kovtun:2014hpa (); Haehl:2014zda (); Haehl:2015pja (); Harder:2015nxa (); Crossley:2015evo (). While actions for ideal fluid dynamics can be constructed in different ways, it is more difficult to treat dissipation, for a recent discussion see Crossley:2015evo ().
The formalism developed here differs from earlier works in several ways. Most prominently, it is based on the analytic continuation of the one-particle irreducible effective action. The effect of quantum and thermal fluctuations or noise is already included in this object. No functional integrals over fluid variables are part of this formalism (except possibly to describe additional initial state fluctuations). The effect of quantum and thermal fluctuations is taken into account by the functional integrals that define the quantum effective action in the Euclidean domain, i. e. before analytic continuation.
This setup implies in particular that the dissipative equations of motion derived from the analytic effective action for field expectation values are not subject to further renormalization effects by quantum and thermal fluctuations. In the presence of initial state fluctuations in the fluid fields, another level of statistical description is added which may of course be described by correspondingly defined expectation values and correlation functions as well as renormalized equations of motion, see for example Blas:2015tla ().
The one-particle irreducible effective action is very closely related to the partition function in the grand canonical ensemble. It was recently discussed how one can obtain constraints for the non-dissipative terms of fluid dynamics from investigations of the partition function Banerjee:2012iz (); Jensen:2012jh (); Bhattacharyya:2013lha (); Bhattacharyya:2014bha (); Becattini:2015tpl (). The formalism developed here could be used to generalize these considerations such that also dissipation is taken into account.
The present paper is organized as follows. We start in section 2 with an introductory discussion of the simple, damped harmonic oscillator. Some of the main elements of our the formalism can already be introduced there. In section 3 we turn then to quantum field theory. In particular, the quantum effective action for a situation with locally varying temperature and fluid velocity is defined first in the Euclidean domain in subsection 3.1. In order to understand properly the analytic structure of the quantum effective action, we discuss the analytic structure of two-point functions in subsection 3.2 and of more general correlation functions in subsection 3.3. The general structure of the analytically continued quantum effective action (or analytic effective action for short) is discussed in subsection 3.4 and the relation to other actions, such as the often considered time-ordered or Feynman effective action is considered in subsection 3.5.
In section 4 we discuss a generalized variational principle that allows to obtain dissipative equations of motion directly from the analytic effective action. We discuss there also the concept of a retarded functional derivative and why this leads to causal equations of motion. In section 5 we discuss the important issue of energy momentum conservation as well as entropy production. More specific, subsection 5.1 discusses how one can obtain the expectation value of the energy-momentum tensor from the analytic effective action, and subsection 5.2 discusses general coordinate covariance and the closely related issue of energy-momentum conservation. We obtain there differential equations that can be used to fix the space-time evolution of the temperature and fluid velocity. Particularly interesting is an equation one can derive in that way for entropy production. This is discussed in subsection 5.3 and also a local form of the second law of thermodynamic is stated there.
The different theoretical concepts and equations are then discussed for a particular example of an effective action that describes a scalar field with symmetry in section 6 as well as for an action describing viscous relativistic fluid dynamics in section 7. Finally, some conclusions are drawn in section 8. Appendix A contains a compilation of useful relations about various two-point correlation functions as they can be derived via linear response theory, for example the fluctuation-dissipation relation, Onsager’s relations or the spectral representation.
2 The damped harmonic oscillator
The equation of motion of a simple, damped harmonic oscillator is
which can also we written as
with the undamped frequency and the so-called damping ratio . Without the damping term one can obtain the equation of motion from the variation of the action
In the last equation we introduced the Fourier transform
and from it follows that .
To find an action for the damped harmonic oscillator one might be tempted to replace simply the kernel on the right hand side of (3) by . However, the additional term linear in would simply cancel out – it is odd with respect to while is even. In the time representation such a term would be a total derivative and would therefore not contribute to the field equations.
The kernel of the Fourier representation on the right hand side of eq. (3) plays the role of the inverse propagator. It has zero-crossings at corresponding to the two poles of the propagator or the two independent solutions of the equation of motion (2) for . The general expectation for an effective propagator in the presence of dissipative mechanisms is that poles are broadened and become branch cuts. For the inverse propagator this implies that zero-crossings are avoided by additional discontinuous terms. Consider for example the inverse propagator
Formally, the zero crossings of (5) are at
However, this equation has no solutions because the imaginary terms on both sides always have opposite signs. In other words, for it appears as if the zero-crossing of (5) were at and vice versa.
One could, however, also say that the expression in (5) has several zero-crossings on different Riemann sheets. On the Riemann sheet that corresponds to the analytic continuation of the upper half of the complex -plane, the zero-crossings are at , while on the Riemann sheet that is analytically connected to the lower half of the complex -plane they are at . As will be discussed below, the zero-crossings on the two Riemann sheets correspond to the dissipative equations of motion for forward and backward time evolution.
One may now try for the effective action of the damped harmonic oscillator the expression (we use conventions where the effective action differs from in overall sign)
In the second line we have gone back to the time representation and replaced
Note that due to the symbol in the first line and in the second line of (8) the damping terms are formally even under the symmetry in the frequency representation and cannot be written as total derivatives in the time representation.
The variation of the action (8) gives up to boundary terms
We have used here that is an odd function in in order to perform partial integration. One sees from (10) that the principle of stationary action leads to the right equation of motion for forward time propagation if one demands that . Similarly, the principle of stationary action for variations with leads to the equation of motion with reversed time direction. One could also write the variation of the action as
which gives the correct equation of motion when one sets for forward time evolution and for backward time evolution after the variation. More general, for forward time evolution, one has to set if is to the right of this operator and if is to the left of the operator . This will be discussed in more detail below.
One can also obtain the equations of motion for forward or backward time evolution, respectively, from variation of the action (8) in frequency representation. A summary of the correct variations to obtain the dissipative equations of motion is given in table 1.
|forward time ev.||with||with|
|backward time ev.||with||with|
We have also added here a term that accounts for the grand canonical potential of the bath degrees of freedom that are not described explicitly by the action (12) but have been “integrated out” already222To see that this is indeed the correct expression one considers the time integral along the Matsubara contour from some time to . In general, local thermal equilibrium is described by a manifold with periodicity in complex space-time coordinates such that bosonic (fermionic) fields satisfy where the inverse temperature vector is determined by the fluid velocity and the temperature ..
One can actually use partial integration to bring (12) to the alternative form
For this one considers to be an odd function of the argument and shows that a similar partial integration as above is actually possible for any odd function.
From the variation of with respect to one can obtain the expectation value of the 0+1-dimensional analog of the energy-momentum tensor,
The dissipative term does not contribute to this; the formal reason will be discussed below. For the variation of the grand canonical potential term one has to write and keep fixed. Generally, in local thermal equilibrium, the temperature is related to the inverse temperature vector or and should be kept fixed when the metric is varied. The inverse temperature does not play the role of a dynamical field but should rather be seen as an external parameter field which determines the local equilibrium state. It’s time dependence is fixed by the conservation laws of energy and momentum.
Using and , where is the thermal energy of the bath variables, gives for the total energy associated with (12) (setting now )
Although in the presence of viscous damping the mechanical energy and the thermal energy are not conserved separately, their sum is. Within a local equilibrium assumption one can actually use the energy conservation law to determine the time-dependence of the temperature from the time dependence of the mechanical energy which, in turn, follows via the dissipative equations of motion.
3 The analytic quantum effective action
After the introductory discussion of the simple damped harmonic oscillator, let us now turn to field theory. For quantum field theories that respect the principles of unitarity, dissipative effects are not present at the level of the fundamental theory described by the microscopic action. However, they can arise in specific sense on the level of the quantum effective action. This will be discussed in detail below. To gain some intuitive understanding how it is possible that effective dissipative terms arise in the transition from the microscopic action to the quantum effective action consider the following two examples.
The first example is the decay of an unstable particle or resonance. For concreteness take a muon that can decay into an electron and two neutrinos. The theory is unitary on the microscopic level, but the decay width appearing in the effective muon propagator can also be seen as a dissipative property. One can formally “integrate out” the fields corresponding to the decay products. This corresponds to a partial trace over the corresponding part of the full density matrix. In the remaining sector describing the muon, the dynamics is described by a reduced density matrix and can appear as dissipative. In particular, energy and momentum do not have to be conserved there. The situation is similar for strong electromagnetic fields with field strengths above the Schwinger threshold of electron-positron pair production. In a description where electrons and positrons are “integrated out” or “traced out”, this appears as a dissipative effect.
The second example is a thermal situation where many degrees of freedom play a role such that a complete microscopic description becomes intractable. A statistical description at non-zero temperature leads to modifications of the quantum effective action (and the corresponding equations of motion) and in particular additional dissipative terms can appear. Energy and information can be transferred to the heat bath and lead to an increase of the thermal energy and entropy.
In the following we will develop a theoretical formalism that can account for the physics of these effects which is based on the analytically continued quantum effective action, or analytic effective action for short. Alternative formulations could be based on the Schwinger-Keldysh closed time path effective action, see for example Calzetta:1986cq (). The two formalisms differ somewhat in practical terms. They also address different types of problems. While the closed time path effective action allows to address general far-from-equilibrium situations, the formalism discussed below is a direct generalization of the equilibrium formalism and therefore more suited for close-to-equilibrium situations.
3.1 Generating functionals in the Euclidean domain
We start the constructions by discussing different generating functionals and in particular the quantum effective action in Euclidean space. It can then be analytically continued in the subsequent step. One may take as a starting point the partition function of a theory described in terms of fundamental fields in the functional integral representation. In principle, might have bosonic and fermionic components, the latter being represented by Grassmann fields. For the present study we shall assume at some places bosonic fields for simplicity but the generalization to Grassmann fields is straight forward with some additional care concerning minus signs from the interchanging of fields.
In the presence of corresponding source terms , the partition function is
The configuration space for the Euclidean action depends on the vacuum state of the theory. We will be interested here in (approximate) local equilibrium states that can be described by a temperature and a fluid velocity . They will enter the construction in terms of the combination . For convenience we will use a general coordinate system with metric . Global thermal equilibrium is included as a special case and in particular also the conventional vacuum with vanishing temperature .
From the similarity between the local density matrix (we use metric with signature (-,+,+,+))
with the translation operator
one is lead to the representation of the partition function in terms of a functional integral on a geometry with periodicity in imaginary space direction such that bosonic (fermionic) fields satisfy .
The microscopic Euclidean action that enters (16) is defined by analytic continuation from the space-time with Minkowski signature. More specific, one has
where is the determinant of the metric (with Minkowski signature) and is the Lagrangian. We use here a formulation where the metric is not modified by the analytic continuation but the space-time differential is complex. More specific, in a coordinate system where the fluid velocity points in the direction, , one has where and the are real. In other coordinate systems, all components are in general complex.
To make this more concrete, we consider a time-like hypersurface which can be defined in terms of a scalar time function as the manifold with . To each position on this base manifold we associate a Matsubara circle in imaginary direction and write the position on the product Manifold as where . Correspondingly, we decompose the dimensional differential volume element into the integral over a dimensional hypersurface and the Matsubara integral in the imaginary flow time direction . Indeed, if the hypersurface is defined in terms of a time function as the manifold with , one has
and for any direction that is not orthogonal to one can write
This shows that the volume element in the Euclidean domain contains a factor .
The Lagrangian is of the standard form, for example for a scalar field with symmetry,
Note, however, that similar to the differential also the derivatives are complex. In a coordinate system where points in direction one has whereas the other derivatives are real. For Minkowski metric, the kinetic term in (22) becomes which explains in which sense the analytically continued theory is indeed of Euclidean form. More general, the derivatives are complex.
where similar as for the volume integral is over the imaginary, compact direction and the space-like hypersurface . Functional derivatives are defined such that for example (recall that contains a factor )
We use here a -dimensional functional derivative where the space-time argument in (24) consists of a (real) base point and an imaginary part which parametrizes the position on the Matsubara axis, . Accordingly, on the right hand side of eq. (24) the integral over the Matsubara circle is reduced to a sinlge point.
For some purposes, one actually needs a dimensional functional derivative that acts at a specific position on the surface but does not reduce the integral along the Matsubara circle to a single point. This is in particular needed to evaluate thermal expectation values of composite fields such as the energy-momentum tensor. In the present work, we will use the -dimensional functional derivatives in such cases but add an integral along the Matsubara direction afterwards by hand. For this prescription, auxiliary fields like the metric are formally extended to complex coordinates , as well, although final expressions will always be evaluated for configurations where is independent of the Matsubara time .
To repeat, the action in the Euclidean domain lives on a configuration space which associates to every base point one imaginary direction which is compactified at finite temperature. The other directions parametrize a space-like hypersurface of the complete space-time. The volume integral on the right hand side of (19) goes along the hyper surface and once around the loop in the compactified imaginary direction. This construction can be made more explicit in terms of the ADM formalism when needed Banerjee:2012iz (); Hayata:2015lga (). For the present purpose, we will not need further details.
The positioning of the surface in time direction can be varied freely if one assumes that local equilibrium in the sense of a generalized Gibbs ensemble holds everywhere. It is usually convenient to place the surface such that it intersects with a space-time point where expectation values should be calculated. The local temperature and fluid velocity , that describe a local equilibrium configuration are at this point not specified further. Ultimately, they will be fixed by general covariance and the closely related covariant conservation of energy and momentum as well as initial values. One should keep in mind that a local equilibrium picture is based on physical assumptions which hold in general at best approximately. In particular, the picture assumes that the processes that drive thermalization are efficient enough such that the local equilibrium picture can be applied on the time- and length scales one wishes to study. In a very homogeneous situation, or on very large space and time scales, this is typically the case and thermal equilibrium corresponds to the unique state of the theory for given conserved quantities, in particular energy and momentum.
It is important to emphasize that local equilibrium differs in general from a complete, global equilibrium. Deviations from the latter are parametrized by gradients of temperature and fluid velocity but also by other fields that deviate from their equilibrium configuration. In the formalism we develop, these additional, non-equilibrated fields are followed explicitly and are in this sense not part of the “thermal bath”. In this situation one might understand the local equilibrium configuration in terms of a generalized Gibbs ensemble.
By taking functional derivatives of with respect to the sources one can obtain expectation values and correlation functions corresponding to the state described by . It is useful to introduce also the Schwinger functional in the Euclidean domain by the definition
Repeated functional derivatives of yield connected correlation functions while a single functional derivative yields the expectation value of ,
The quantum effective action is a functional of field expectation values. Formally it is defined in the Euclidean domain as the Legendre transform of ,
Repeated functional derivatives of give one-particle irreducible correlation functions. This may seem to be a quite formal statement but it has very interesting implications: A complete connected correlation function can be determined by adding only tree-level expressions but with effective vertices and propagators obtained from the functional derivatives of instead to those from the functional derivatives of the microscopic action .
The effective action in the Euclidean domain is also subject to the following field equation, which can be directly derived from eq. (27),
For vanishing source, , this equation resembles the classical equation of motion that follows from Hamilton’s principle for a classical field theory, . However, so far the effective action is defined in the Euclidean domain where it accounts for the trace over a density matrix. Only statistical expectation values and correlation functions on a space-like hypersurface can be calculated directly from this. Note that the action and similarly the Schwinger functional and effective action in the Euclidean domain depend formally also on the choice of the space-like hypersurface . For example, one can calculate correlation functions for fields at different positions on this surface from the functional derivatives of . Correlation functions between points that are not on the surface cannot be calculated, yet.
Before we obtain a dynamical equation of motion for the field expectation value from we need to study analytic continuation and the analytic structure of . By the latter we mean here the analytic structure of the correlation functions that follow from the functional derivatives of as a function of the space-time arguments. The notion will become more clear in due course.
The result of this analysis will be an analytically continued effective action from which one can derive correlation functions or Greens functions that go beyond the expectation values in local thermal equilibrium on a fixed space-like hypersurface. In particular, the analytically continued effective action will allow to derive dynamical dissipative equations of motion for field expectation values . We will start to investigate the analytic structure of in the sector where it is best understood, namely the sector of two-point correlation functions.
3.2 Analytic continuation of two-point functions
We discuss now the analytic structure of connected two-point correlation functions which correspond to the second functional derivatives of the Schwinger functional and the effective action . For this subsection we consider global equilibrium states with temperature and fluid velocity in flat space with Minkowski metric . The description is simplest in the coordinate system where the fluid is at rest, i.e. and we will adopt this choice for the time being. Some results can later be taken over to more general local equilibrium states and curved space within the corresponding approximations.
The analytic structure of two-point correlation functions in global equilibrium states is rather well understood, see for example Abrikosov (). A comprehensive compilation of the main classical results such as the fluctuation-dissipation relation, Onsagers relations and the spectral representation including their derivation is provided in appendix A. Here we discuss the main elements needed for analytic continuation.
From the Schwinger functional in eq. (25) one can obtain the two-point function in the Euclidean domain as
The arguments and consist here of (real) spatial positions and imaginary times on the Matsubara torus. We write this as with etc. When evaluated for vanishing source and because of translational symmetry in space and imaginary time, the correlation function in (29) is a function of only,
It is useful to consider the Fourier representation defined by
where is the Matsubara frequency with values for bosonic fields , and for fermionic fields.
As discussed in appendix A, the Matsubara correlation function is actually a special case of a more general correlation function which is defined for general complex frequency argument. More specific, on the axis of imaginary Matsubara frequencies one has
and is the unique analytic continuation of to the full plane of complex frequencies which is analytic everywhere except for possible poles and brach cuts on the real frequency axis . The complex argument Greens function has a Källen-Lehmann spectral representation which makes its analytic structure with respect to directly apparent,
see also eq. (148).
Other correlations functions such as retarded, advanced or Feynman propagator functions follow from the complex frequency Greens function by evaluating it in particular regions in the complex frequency plane,
see also eq. (149). These relations show that the complex argument Greens function is actually a very useful object.
One can actually directly extend the definition of the Schwinger functional by analytic continuation in such a way that the functional derivatives yield the complex argument Green’s function Floerchinger:2011sc (). In momentum space,
Obviously, when evaluated in the appropriate regions of the complex frequency plane, this definition agrees with (30) but it has the advantage that one can also directly obtain other correlation functions, e. g. the retarded one, from the general expression (35).
So far we have worked in the reference frame where the fluid is at rest, . Obviously, objects like the complex argument Green’s function can also be evaluated in other frames. In that case one should take the combination to be complex in general, and the poles and brach cuts will be on the axis where is real. The momentum components orthogonal to the fluid velocity are real. In the following we will work in such a more general frame.
Let us now turn to the effective action as defined in eq. (27) and in particular its second functional derivative. From the definition as a Legendre transform in the Euclidean domain and assuming that is strictly convex, it follows that
The expectation value is related to the source by the field equation (28). When evaluated for homogeneous field , one can write the second functional derivative of in momentum space as
A priori this is only defined in the Euclidean domain, i. e. for imaginary Matsubara frequencies (as well as real momenta orthogonal to the fluid velocity). However, similar to (35) one can extend this to general complex frequencies by analytic continuation.
Due to the relation (36), the object satisfies
and is therefore called the inverse complex argument Green’s function. Because of the analytic structure of , the eigenvalues of cannot have any zero crossings or branch cuts except on the axis of real . One can decompose the inverse complex-argument two-point function
where . Both functions and are regular when crossing the real frequency axis. However, the sign changes, which leads to a branch cut behavior for the function . The function parametrizes the strength of the branch cut. From the definition (37) one obtains the symmetry properties (assuming bosonic fields)
So far, the analytic continuation of correlation functions was done in momentum and frequency space by analytic continuation from the discrete, imaginary Matsubara frequencies to the plane of complex frequencies . One would like to have also a position space representation which is not only defined in the configuration space with compact imaginary time direction but can be directly evaluated in real space. To that end one can define the real space representations of the functions and by
where the momentum and frequency integrals go along the real directions. No ambiguities arise here because and are regular at the real frequency axis. Note that we have included a factor in the definition of for convenience. In real position space one can now write the decomposition of the inverse Green’s function (39) as
We have also replaced here
Typically and will be a polynomial in combinations of momenta and accordingly the position space representations and will consist of combinations of distributions such as and derivatives thereof.
We define now also an analytically continued effective action in real position space via a deformation of the time integration contour. More specific, in the Euclidean domain, the time was integrated along the (compact) imaginary direction , see the discussion around (21). This contour can be deformed such that it goes along the (non-compact) real direction . The infinitesimal volume element is then real, as well. It is convenient to divide by the factor that has been introduced in (23) and to define the analytically continued action in real space as
For the effective action in real space we will use standard functional derivatives which differ formally from the functional derivatives in the Euclidean domain by a factor . However, this should not lead to any confusion is practice.
As an example, the part of the effective action that is quadratic in the fields is of the form
where the integrals over and are now in real coordinate space. While the first term in (45) is of the standard form, the second term is less common. It parametrizes the discontinuity of the inverse propagator along the real frequency axis.
If one considers to be an odd function of the argument, one can actually perform partial integration to transfer the operator between the different terms. This has been done in the second line of (45). In the present context, partial integration is possible because we work in cartesian coordinates and because is constant. In a more general coordinate system with differential volume element one can see that partial integration according to
is possible for . This could imply that relations like
could only be used for . However, the sign function is unchanged if the argument is rescaled by a positive factor and one can therefore replace by with chosen such that . This shows that is in this sense well defined also in such more general situations.
A more clear geometric picture is obtained if one replaces
or where and are Lie derivatives in the direction and , respectively. When acting on scalar fields one has but the Lie derivative is also well defined for vector, tensor or spinor valued expressions. Moreover, it does not depend on the metric connection. Working with might sometimes have advantages for calculating variations of fields with fixed .
Note that according to (34) and because of (36) one can obtain the operators inverse to the retarded and advanced Green’s functions in position space from (45) by specific choices of the sign . One has
so that these the two combinations of and on the left hand side of (49) can be understood as the inverse retarded and advanced propagators, respectively.
Let us now come to the implications of the field equation (28) after analytic continuation. To study its implications we expand the effective action in powers of fields
where contains terms of cubic and higher order in . We assume here that expectation value vanishes for vanishing source which implies that no linear term is present in eq. (50). We have also dropped a possible constant term as irrelevant for the following discussion. The field equation (28) reads now formally
To define the objects in (51), we have used here analytic continuation from the Euclidean domain to the Minkowski signature domain, such that the frequency is real. However, this leaves open what has to be taken for . From causality arguments, the linear part of the field equation for forward time evolution should be inverse to the retarded propagator (and for backward time evolution inverse to the advanced propagator). This suggests that one has to take for forward time evolution variations with and for backward time evolution . This agrees with the generalized variational principle introduced in sect. 2. The result is
It is straight forward to translate these relations to position space. The dissipative equations of motion will be discussed in more general context also in section 4.
3.3 Analytic structure of higher order correlation functions
So far we have studied the analytic structure of two-point functions or the corresponding sector of the effective action . In general, the analytic structure of higher order correlation functions is much less understood.
It is, however, nevertheless possible to generalize the method to somewhat more general situations. Consider an effective action of the following form (we take here and to be bosonic fields for simplicity)
where . The matrix can be decomposed as in eq. (39) and similarly . The effective vertex and the terms in are assumed not to contain any discontinuous terms involving (where is some combination of the involved momenta). The equations of motion following from (53) for forward (backward) time evolution are obtained by following the general principle described above, as
as well as
Because equation (55) is linear in it can formally be solved,
This solution, in turn, can be used in eq. (54) to give
Interestingly, in this form, the equation of motion for has now a non-linear term with a discontinuity. The field equation is such that the matrix