Density matrix for QFT from the path integral?

In summary, the conversation discusses the use of the density matrix formalism for quantum fields and its ability to incorporate temperature and time dependence. One reference is mentioned for further reading and the conversation ends with a question about the other person's work on time correlation functions in such systems.
  • #1
wandering.the.cosmos
22
0
(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.
 
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  • #2
wandering.the.cosmos[QUOTE said:
](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.)

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 anyone 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

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



regards

sam
 
  • #3
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
Epicurus said:
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

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?
 
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  • #5
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.
 

1. What is a density matrix in quantum field theory?

A density matrix, also known as a density operator, is a mathematical representation of a quantum system that takes into account the probabilities of all possible states of the system. In quantum field theory, the density matrix is used to describe the state of a quantum field, which is a continuous field of quantum particles.

2. How is the density matrix for QFT derived from the path integral?

The density matrix for QFT is derived from the path integral, which is a mathematical tool used to calculate the probability amplitude for a quantum system to transition from one state to another. The path integral is used to integrate over all possible paths of the quantum field, taking into account the interactions between particles and their respective probabilities.

3. What is the significance of the density matrix in quantum field theory?

The density matrix is a powerful tool in quantum field theory, as it allows for the calculation of the expectation values of observables which cannot be directly measured. It also takes into account the effects of entanglement and decoherence, which are important phenomena in quantum systems.

4. Can the density matrix for QFT be used to describe entangled states?

Yes, the density matrix for QFT can be used to describe entangled states, where multiple quantum particles are connected in such a way that the state of one particle is dependent on the state of the other. The density matrix takes into account the correlations between entangled particles and can be used to calculate their joint probabilities.

5. Are there any limitations to using the density matrix for QFT?

Yes, there are limitations to using the density matrix for QFT. It is only applicable for systems in thermal equilibrium or with a well-defined temperature, and it assumes that the system is in a pure state. It also does not take into account the effects of gravity, which can be significant in some quantum systems.

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