# Thermal field theory

What is the basic idea and purpose of the thermal field theory? I don't need a full in depth description of it, not at the moment at least. I am just trying to understand how it is relevant in the calculation of angular distribution of temperature of CMB(Comic Microwave Background) over the sky.

## Answers and Replies

king vitamin
Science Advisor
Gold Member
"Usual" quantum field theory calculations are all done with respect to the ground state of the system; correlation functions are all taken with respect to the ground state. In other words, they are done at zero temperature, which is a perfectly valid approximation for many applications.

However, in some applications you are working at finite temperature, or maybe you explicitly want temperature-dependent observables like the specific heat. For example, if your system is in equilibrium with some temperature T, you need to sum over all excited states with each state weighted by a Boltzmann factor:
$$\langle A \rangle = \frac{1}{Z}\sum_n \langle n | A | n \rangle e^{-\beta E_n} = \frac{\mathrm{Tr}\left( A e^{- \beta H} \right)}{\mathrm{Tr}\left( e^{- \beta H} \right)}$$
Computing these expectation values using perturbation theory/diagrams is done using the Matsubara formalism, which you can find in many textbooks.

I'm not familiar enough with cosmology to know the specific application to the CMB which you mentioned.

• atyy
Thanks for the introduction. I was studying a review paper on the Cosmic Microwave Background. The temperature is written as a sum of Spherical Harmonics. I stumbled upon the thermal field theory when they mentioned something called the two point temperature function and I did not know what that meant. May be you could shed some light on this as well (?).

Many thanks for your time.