I recently read an interesting paper [1] that uses holography to study decoherence in strongly-coupled systems. It relies on the fact that, in the case of a linear coupling between the subsystem and the environment, the Feynman-Vernon influence functional — which encapsulates the influence of the environment on the subsystem — can be viewed as the generating function for Schwinger-Keldysh propagators—that is, nonequilibrium Green’s functions. The latter are well-known objects of study in AdS/CFT, and can be computed on the gravity side for strongly-coupled field theories. Unfortunately, there’s a fatal flaw in the connection to holography that invalidates the specific application of this path-integral formalism in [1]. But the Feynman-Vernon/Schwinger-Keldysh approach itself is generally valid and particularly elegant. And in fact, there may be a means of salvaging the holographic hopes of [1]. Hence we shall proceed to discuss the analysis, and return to comment on the stumbling block at the end.

Let us very briefly summarize the notion of decoherence before we begin. The evolution of the entire system (i.e., universe) is of course unitary, but an initially pure subsystem can evolve to a mixed state via interaction with its complement, the environment. The effect of the latter is to select from the Hilbert space of the subsystem a basis of states which are most stable against further environmental perturbation. These are called pointer states, and correspond to the classical solutions in the limit . The quantum coherence between these pointer states is encoded in the off-diagonal elements of the reduced density matrix of the subsystem, and thus, loosely speaking, the disappearance of these off-diagonal elements characterizes the decoherence process.

(As a technical footnote, the adjective “loosely” is related to the qualifier “most” above: as noted in [1], the disappearance of the off-diagonal elements is a basis-dependent statement. But since the pointer states are not exactly stable away from the classical limit, they are not invariant under the decoherence process. Thus one needs a more careful mathematical characterization of decoherence, but we can safely ignore this technicality for the moment).

The reduced density matrix for the subsystem is defined in the usual manner, namely by tracing out the degrees of freedom of the environment:

where is the density matrix of the total system, which evolves unitarily according to some Hamiltonian ,

One can prepare the subsystem to be in an initially pure state, with

whereupon decoherence evolves from a quantum to classical state. This is the process we wish to study.

Since we shall work in the path-integral formalism, we need a Lagrangian, which we take to be of the form

where and denote the degrees of freedom of the subsystem and the environment, respectively, which are coupled only through the interaction term . The authors of [1] consider a strictly linear coupling of the form

where is an arbitrary functional of the fields , and is a dimensionful coupling (in Euclidean, , so will depend on the precise form of ). Unfortunately, as we will see shortly, the linear form of the coupling is actually necessary in order to view the influence functional as a source term. This will result in us having to put in certain other terms by hand. It is, as far as I know, an open question as to whether it’s possible to extend the formalism to allow more general couplings. However, despite this drawback, a powerful aspect of this approach is that the environment Lagrangian is entirely arbitrary (in principle, at least; practice is, as usual, another matter entirely). The idea is then that the subsystem itself can be used as a probe to study decoherence due to different environments. Accordingly, [1] considers the simple Lagrangian

The aforementioned influence functional, which we will define precisely below, can then “be regarded as the probe’s effective action, which is obtained after the environmental degrees of freedom are integrated out” [1]. This is the basic idea behind the Feynman-Vernon formalism.

To proceed, we must re-express the evolution equation (2) in the language of path integrals. The time-evolution operators then require that this integral is computed along a closed path, namely the Keldysh contour (basically, this amounts to a Lorentzian timefold in an otherwise Euclidean contour, thereby allowing one to insert unitary operators). For concreteness and convenience, [1] considers the thermostatic case , so that Euclidean time runs from to . Hence the contour first moves along the Lorentzian timefold from to and back, and then continues in Euclidean to .

With this picture in mind, we wish to compute the reduced density matrix at . Specifically, consider the amplitude

where we have used (1), with the bar denoting the values of the fields at . By inserting two resolutions of the identity, and using the evolution equation (2) with (3), this becomes

The reason for rewriting it in this way is that we can now isolate the propagators corresponding to the two (forward and backward) legs of the Lorentzian timefold:

Inserting these path-integral representations back into (8), and using the fact that the correlator factorizes, we have

Now, as mentioned above, the central feature of the Feynman-Vernon formalism is that it packages all information about the effects of the environment into the so-called influence functional, which we define as

where we have used (4). This is then used to define the propagation function

which describes the evolution of the subsystem. With this in hand, we can neatly express the correlator (10) as

Thus the main outstanding task is the computation of the influence functional (11). In general of course, this is prohibitively difficult; but here is where the linear form of the interaction (5) comes into play. First, observe that we can express the influence functional more compactly as the expectation value of the initial density matrix :

where is the path-ordering symbol with respect to the Keldysh contour, and

Note that, as foreshadowed above, this is tantamount to tracing out the environmental degrees of freedom . Such a trace would normally be written

for some operator . In this case however, we have to take into account the Keldysh contour that runs forward to and back. Thus, denoting fields on the forward and backward legs with the subscripts and , respectively, what we really want is something like . The trace then instructs us to integrate over all possible paths that interpolate between the two, subject to the boundary condition that at ; hence

which provides the link to (11). Now, given the form (14), we can view the probe field as a source for the environmental fields , whereupon this expression is precisely the generating function for the Schwinger-Keldysh propagator (i.e., real-time finite-temperature Green’s functions), defined as

where .

As an aside, note that the natural log appears because the Schwinger-Keldysh Green’s function is given by the connected 2-point function. That is, recall that given the partition function (where we’re temporarily reverting to standard QFT notation in which denotes the source), which generates both connected and disconnected Feynman diagrams via

the generating functional for the connected -point function is related (in the present, Euclidean convention) by

When working with the effective action instead of the action, only connected diagrams contribute—hence one uses rather than , and directly computes the connected -point function via

In the present case, since we’re after the Green’s function, we set and multiply by to fix conventions:

which gives (18) upon identifying with . I believe this is what the authors mean when they say that the influence functional may be regarded as an effective action for the system (having integrated out the environment degrees of freedom), though strictly speaking this is misleading: first, because obviously partition functions are not actions, and second, because the effective action is given by rather than itself.

Equation (18) thus provides the connection between the Feynman-Vernon and Schwinger-Keldysh formalisms. And while it clearly enables one to compute the latter given the former, reference [1] is interested in the opposite scenario: given the Schwinger-Keldysh propagator , which we can compute for strongly-coupled environments via AdS/CFT, how do we extract the influence functional that describes the decoherence effect of this environment on the subsystem?

For irrelevant couplings (), this is feasible if we approximate the influence functional to quadratic order. This is more clearly explained in [2] (beware however that this reference uses the term “influence functional” to mean ; they also appear to use a different sign convention. I suspect reference [1] is atypical in both regards, and that the notation in [2] is actually more standard). The basic idea is that for small coupling, we may evaluate the path integral perturbatively, and exponentiate the result to obtain the effective action. We first expand (14) (suppressing the time-ordering symbol for compactness):

This is the standard expansion of the exponential of the connected correlation function mentioned above, and hence we identify everything but the leading unit term on the r.h.s. with (up to factors of , depending on convention). Upon adding an appropriate counterterm to kill the tadpole (i.e., imposing that the vev of the 1-pt function vanishes, ), the effective action contains only the quadratic contribution (up to this order), and we may therefore write

where on the second line we have switched to the so-called “relative-average” basis

and re-expressed the Schwinger-Keldysh Green’s functions in terms of the standard advanced, retarded, and symmetric Green’s functions,

where

Note that by definition, and .

In principle, one can use (18) to study decoherence for general environments, but further simplifications are obtained by restricting to the thermostatic case mentioned above, where . In this case the trace in (15) becomes a thermal average, and the corresponding path integral runs from to (after the Keldysh contour, i.e., the Lorentzian timefold from to and back). At thermal equilibrium, the Green’s functions become periodic in imaginary time; that is, they satisfy the KMS condition

In momentum space, this translates into the relation

Here is simply the thermal distribution function. The utility of this relation is that it allows us to express (18) entirely in terms of , and therefore knowledge of the environment’s retarded Green’s function completely determines the dynamics of .

As alluded above, (18) is a rather powerful formula, since it applies for arbitrary environments. Traditionally however, computing the retarded Green’s function is intractable except in the case of free theories. The novel aspect of [1] is to use holography to compute for strongly-coupled theories; if this were valid, it would enable the study of decoherence for a whole new class of systems. The details are quite technical, and we refer the interested reader to the paper for details; here we will only sketch the most basic overview.

First, the authors show that the propagation function has the correct semi-classical limit, namely that it yields the Langevin equation for quantum brownian motion of the center of mass—in this case, for the “average” field . This is achieved by observing that, in the relative-average basis , can be thought of as a light field that encodes the random fluctuations around the heavy center-of-mass field . Integrating out the fast degree of freedom then leads to the classical equation of motion — the Langevin equation — for the slow field .

The authors then proceed to derive the master equation for . This entails explicitly evaluating the path integral in the relative-average basis, and then computing its time-derivative. One can then deduce the differential equation describing the dynamics of from (13).

In some sense, the master equation for is all one needs to describe the decoherence process. However, as mentioned above, the disappearance of the off-diagonal elements of the reduced density matrix is not a rigorous characterization of decoherence, since the basis of pointer states is itself evolving (away from the classical limit, ). (There is also the subtlety that since is a (continuous) field, is an infinite-dimensional matrix; but this can be circumvented by suitably coarse-graining the system). Thus, instead of examining the off-diagonal elements of , a better, basis-independent probe of decoherence is given by the Wigner function , which is the Fourier transform of the density matrix with respect to the fast field . In other words, the Wigner function is the quantum analogue of the distribution function over phase space. The key feature is that in general, is not positive-definite, but becomes so once the subsystem decoheres to a classical state. The negative contributions to therefore parameterize the degree of decoherence, and their rate of disappearance allows one to define the decoherence timescale.

Interestingly, as the authors of [1] observe, Rényi entropies — which are scalar quantities — also provide a basis-independent characterization of decoherence. Since the entanglement entropy is zero for a pure state and maximized for a completely mixed state, and decoherence evolves the subsystem from the former to the latter, entanglement or Rényi entropies are a natural way of tracking this process. This also allows the authors to compare the timescales for decoherence with those obtained in the study of local quantum quenches.

Unfortunately, as mentioned at the beginning of this post, there’s a basic flaw in the connection to holography that invalidates the application of this beautiful formalism. Specifically, the key observation of [1] was that the influence functional (14) has precisely the same form as the partition function in the extrapolate dictionary. They therefore identify the subsystem field with the source, and the environment field with the corresponding CFT operator. But the source isn’t a dynamical field on the boundary, and hence the splitting of the Lagrangian (4) does not make sense in this context. Another way to say this is that the (bulk) source and (boundary) operator are dual operators, and hence do not interact in the same Hilbert space. However, this implicitly assumes the standard choice of boundary conditions in holography, and it’s not clear whether a more sophisticated treatment would enable one to salvage this approach. [I am grateful to Billy Cottrell for discussions on this issue]. Given the fundamental importance of decoherence and (to?) holography, and the otherwise general elegance of the Feynman-Vernon approach, this is an interesting open question.