Thursday, March 3, 2016

Do you really need to use Keldysh Green's functions?

Or are you cutting butter with a chainsaw? Or using a helicopter to cross the road?

Previously I posted that Green's functions are just a technique.
Here I have a new point. When using them one has to make a choice about how one treats the time variable in the complex plane. This is rather technical, subtle, and confusing. The choices available include imaginary time (Matsubara), retarded and advanced, Kadanoff-Baym, Keldysh, ....

Matsubara Green's functions can only describe equilibrium properties. They reflect a beautiful and profound connection between imaginary time and temperature. However, due to the very powerful fluctuation-dissipation relation (embodied in the Kubo formula), they can also be used to calculate transport properties in the linear response regime. They are also the easiest to use, both analytically and computationally. Although for some numerical methods such as Quantum Monte Carlo the analytic continuation to real frequencies can be a can of worms.

Kadanoff-Baym and Keldysh are constructed to allow description of non-equilibrium states. A beautiful and profound application of them is the derivation of Boltzmann-type transport equations. I think when Kadanoff and Baym did this it was a major conceptual advance that further buttressed Fermi liquid theory.
A nice accessible introduction to Keldysh is the review by Rammer and Smith. Nevertheless, keeping track of the contours and the relation between Keldysh, retarded and advanced components of the Greens functions can quickly become overwhelming.

There is no doubt that far from equilibrium, Keldysh is necessary and good. However, my concern is that sometimes people do a lot of formalism with Keldysh and then in the end they just do some linear response calculation that they could have just done with Matsubara. I conjecture this applies to some of the examples considered by Rammer and Smith.

In my Ph.D I used Keldysh to consider the non-linear interaction of zero sound with order parameter collective modes in superfluid 3He-B. Much later I realised that I could probably have done the same calculations with Matsubara.


  1. I think the structure of Keldysh diagrammatics is often more transparent than Matsubara, so for calculations that would otherwise rely on dodgy analytic continuations it seems a defensible strategy (even when both methods in principle give the right answer).

  2. The other place that Keldysh is very nice is for problems involving disorder averaging. You don't need to do a replica calculation in which you take the limit of replicas going to zero, which is hard to understand.
    The calculations in disordered problems are often messy whether you use replicas or Keldysh, but at least with Keldysh it is more transparent, as noted by Zed.

  3. Using Keldysh gf eliminates the need for an analytical continuation (especially in numerics, where otherwise one would have to rely on uncontrollable methods), so there might be a spot even in equilibrium for them