# Density matrix for QFT from the path integral?

1. Aug 20, 2006

### wandering.the.cosmos

(1) How does one obtain the density matrix formalism for quantum fields from the path integral?

(2) Suppose I have a box containing interacting particles of different kinds. Is it possible to incorporate into the density matrix formalism both a non-zero temperature T as well as a time t dependence on the distribution f(T,t,p) of the various particles? (I've seen treatments on finite temperature field theory, where imaginary time is identified with inverse temperature, but I'm wondering how we can include both T and t.)

Thanks for any input.

2. Aug 21, 2006

### samalkhaiat

If you are talking in the contex of Thermo Field Dynamics (TFD), then you should look at (Matsumoto et al.);

Phys Rev (1983),D28,1931.
Phys. Lett (1984),140B,53.
Phys Rev (1984),D29,2838.
Phys Rev (1984),D29,1116.
$(1985),D31,1495.$###,429.

You will find the answer to your 2nd question in any one of the above references, which is Yes you can, but it turned out that the two formalsims are identical.

As for your 1st question, I suppose, you could always use the relation

$$<T{\phi(x_{1})....\phi(x_{n})}> = Tr[T{\phi(x_{1})...}\rho]$$

regards

sam

3. Aug 22, 2006

### Epicurus

wandering.the.cosmos, I would very much like to know how long you have had this idea for? I may be workin on a very similar thing. There has not been to much work on these things

4. Aug 23, 2006

### wandering.the.cosmos

I've had in my mind questions about such issues for perhaps 1-2 years, but have never really sat down to seriously work things out or see what has been done. One particular paper that I came across regarding these is

E. Calzetta and B.L. Hu
Nonequilibrium quantum fields: Closed-time-path effective action, Wigner function, and Boltzmann equation
Phys. Rev. D 37, 2878–2900 (1988)

But I am rather far from understanding it properly. They seemed to have obtained quantum Boltzmann kinetic equations from the path integral, including the BBGKY hierachy. My understanding is that the density matrix formalism ought to yield Boltzmann equations as a special case, so perhaps the above is what one needs to investigate such issues.

What have you been working on, Epicurus?

Last edited: Aug 23, 2006
5. Aug 23, 2006

### Epicurus

The path integral formulation of the thermal density matrix is able to give us the partition function for Boltzmann, Bosonic and fermionic systems. In this representation, particles are given as closed loops through a correspodance with imaginary time and inverse temperature. I have interested in time correlation functions in these systems.