# A QFT at finite density

1. Dec 26, 2016

### ShayanJ

I need to take a look at some references about QFT at finite density but I can't find anything, or at least I don't know where to look. I should emphasize that what I need is QFT at zero temperature and finite density so it seems to me QFT in finite temperature books may not be what I need or maybe I'm not understanding what the title's representing.
I'd really appreciate it if anyone clears my confusion here and point me to a good reference.
Thanks

2. Dec 28, 2016

### dextercioby

One usually has QFT at finite temperature which is similar to non-specially relativistic QFT (many-body theory). I have never heard of QFT at finite density.

3. Dec 28, 2016

### ShayanJ

Well, to be honest, I'm not sure what I'm talking about. But let me try to explain and then we'll see whether I make sense or not.
What I have in mind is an interacting QFT at zero temperature that has field excitations even at its ground state. Yeah, field excitations at ground state may seem contradictory but I guess that can be understood by comparing this theory to the corresponding free theory.

4. Dec 29, 2016

### vanhees71

Well, of course there's QFT at $T=0$ but finite density, e.g., for cold nuclear matter. You get it as the limit of $T \rightarrow 0^+$ from the usual formalisms (Matsubara/imaginary time or Schwinger-Keldysh real-time). The vacuum by definition is the ground state, i.e., the state of lowest energy. There are by definition no excitations of anything if you have the system prepared in this state.

5. Dec 29, 2016

### ShayanJ

Yeah, but what if we approximate the interacting theory starting with the free theory? Then the approximate interacting ground state is actually a state with particles(free theory particles), right?

6. Dec 29, 2016

### vanhees71

Yes, and in finite-temperature/density perturbation theory you correct for both the state (i.e., the (grand-)canonical statistical opertor) and the dynamics of the fields. This is, because the (grand-)canonical state is treated as time evolution for imaginary times, and the time evolution is perturbatively described with the diagrammatic rules (including the KMS conditions for the fields).

7. Dec 29, 2016

### ShayanJ

Can I find such calculations on every finite temperature field theory book? Does any of them do such calculations using path integrals?

8. Dec 29, 2016

### vanhees71

Here's my manuscript (emphasizing the real-time formalism, using both operators and path integrals):

http://th.physik.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

Then there are three good textbooks:

J. I. Kapusta and C. Gale, Finite-Temperature Field Theory; Principles and Applications, Cambridge University Press, 2 ed., 2006.
M. LeBellac, Thermal Field Theory, Cambridge University Press, Cambridge, New York, Melbourne, 1996.
M. Laine and A. Vuorinen, Basics of Thermal Field Theory, vol. 925 of Lecture Notes in Physics, 2016.
http://dx.doi.org/10.1007/978-3-319-31933-9

For the real-time formalism, using path integrals, see the review article

N. P. Landsmann and C. G. van Weert, Real- and Imaginary-time Field Theory at Finite Temperature and Density, Physics Reports, 145 (1987), p. 141.
http://dx.doi.org/10.1016/0370-1573(87)90121-9 [Broken]

Last edited by a moderator: May 8, 2017