31. Schwinger-Keldysh: brief review (Nonequilibrium diagrammatic methods 1)

Today I will start here reviewing diagrammatics of non-equilibrium QFT. This discussion is mainly based on the excellent book by Kadanoff and Beym and lectures by Alex Kamenev.

Why in QFT problems one could be left unhappy with standard Feynman diagrammar and need something else? The reason is the ideology of S-matrix. In equilibrium QFT you calculate vacuum-vacuum matrix elements. You pick a vacuum state (i.e., the one without particles) at of non-interacting Hamiltonian, adiabatically switch interaction (this is necessary because otherwise initial state gets perturbed and for example cannot be considered as a pure state after some time), then the physical event of interaction happens during very small amount of time, you adiabatically switch the interaction off and end up with vacuum state at .

The point is that vacuum of the system does not change but just acquires a phase due to the interaction event. This phase is what is called S-matrix.

Now, all really interesting (means complicated) physical problems do not have simple features like that:

1. interaction term in the Hamiltonian is important at all times
2. you simply do not have enough time to wait until the system reaches its asymptotic out-state or
3. this out state is unknown to you such as in the case when there is an intrinsic instability in your problem
4. interaction is not switched on and off adiabatically (or cannot be considered as such)
5. particle production is important in the problem under consideration

The problems of this type include cosmology (where QFT is important – inflation, reheating, early Universe), turbulence, heavy ion collisions (if you are interested in their dynamics), quantum kinetics, process of measurement in quantum mechanics etc. etc. It turns out that if you are trying to apply QFT to study and predict market behavior, most likely you end up with non-equilibrium QFT.

So, what to do if you do not have information about out-state of the system? The prescription was first introduced by Schwinger back in 1950s: instead of calculating matrix elements you calculate matrix elements (although you do not know final state, you do know initial state!)

The crucial diffence between the equilibrium diagrammatic (Feynman) methods and the Schwinger-Keldysh diagrammar is that amplitudes and partition functions in the latter case are calculated along a closed path in the complex plane of time which goes from to some fixed moment of time in the future and back to the infinite past.

Let us for suppose for simplicity that we have a theory with a single scalar field . The major consequence of calculating amplitudes along the closed time path is the effective doubling of degrees of freedom. In Schwinger-Keldysh diagrammatic technique, instead of a single field we introduce two Keldysh fields and defined on the lower and upper sides of the closed time path correspondingly. These two fields are not independent (as it would be in the case of the upper and lower parts of the contour being unconnected); in particular, the correlator is not zero.

After introducing the Keldysh indices , the generating functional for the Green functions has the form

and, as usual, the correlation functions of fields are constructed by differentiating with respect to sources and setting them to zero in the result of differentiating. For example, one has for the two-point correlation function of + and – Keldysh fields

Not all of the four posible two-point Green functions are independent. Due to causality constraints one has the identity

.

It is therefore possible to simplify the analysis of perturbation theory by doing the Keldysh rotation

,

since only the Green functions and are non-zero, where and are retarded and advanced Green functions, correspondingly, while is called the Keldysh Green function. The Green function remains zero non-perturbatively.

The field is usually denoted as “classical” while the field – as quantum, since among saddle points of the effective action there is always one such that and satisfies the classical equations of motion.

Next time I will discuss what information is carried by these Green functions.