# Thermal Field Theory: Calculating Angular Distribution of CMB

• A
• shahbaznihal
In summary, thermal field theory is a method used to calculate observables at finite temperatures, taking into account the effects of excited states and temperature-dependent quantities. This is achieved through the use of the Matsubara formalism, which involves summing over all excited states weighted by a Boltzmann factor. This method is particularly relevant in the calculation of the angular distribution of temperature of the Cosmic Microwave Background (CMB) over the sky. The "two point function" mentioned in the conversation refers to a two-point cross-correlation function, which is used to study the correlation between different points on the CMB.

#### shahbaznihal

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.

"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 (?).