# Question on finite temperature field theory

In quantum field theory at zero temperature, the expectation values of operators are taken with respect to the vacuum. Is it the case that in quantum field theory at finite temperature, the expectation values of operators are taken with respect to a thermal state? If so, how is it connected to the time period of time of the usual finite temperature field theory and the thermal field theory? Is there any detailed representation of this state? Any good references?

• I'm not understanding this question. Are you asking for time-dependent + non-zero temperature field theory, or will you content with equilibrium statistical field theory (temperature but no time-dependence) ? In the later case, the book by Abrikosov, Gor'kov and Dzyaloshinski is a good (but old) reference. The expectation value are taken with respect to the ground state, which you need to properly identify. At zero-temperature the ground state is the vacuum of quasi-particles, at finite temperature you may assume the ground state to be in thermal equilibrium. – FraSchelle Dec 3 '14 at 8:24
• @FraSchelle I was thinking about time-independent quantum field theory at finite temperature. Thanks a lot for your recommendation! – Yue Huang Dec 3 '14 at 8:27
• In that case, the full references is A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinsky, Methods of Quantum Field Theory in Statistical Physics (Prentice Hall, 1963). It's re-edited by Dover, so you can find cheap printing of this book. Note nevertheless the mathematical language is a bit ageing now. You may like to complete with the nice introduction given in A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge University Press, 2010). I don't know your background, mine is clearly condensed matter, so I give you reference for this topic. – FraSchelle Dec 3 '14 at 8:40
• Thanks a lot for your patience! My study field focuses mainly on gravitation, but I think I'll find the books you recommended of great use! – Yue Huang Dec 3 '14 at 8:43
• If you prefer a more modern and systematic introduction, I guess the references cited on the wikipedia page en.wikipedia.org/wiki/Thermal_quantum_field_theory should be more interesting for you. I found the book by M. Le Bellac, Thermal Field Theory (Cambridge University Press, 1996) particularly easy to read on this topic, but I was already introduced to the books I cited in my previous comment, so perhaps this book is less pedagogical than I thought. – FraSchelle Dec 3 '14 at 8:44