Here are some ideas for the final project.
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In all suggestions, you can/should have a discussion (or several of them) with me to define the study object more carefully. For that we can arrange zoom meetings via email. These are just
suggestions. You (we) can modify them, or you can bring in your own suggestions as well.


1) Exact partition function on a complex time path with real time dimension

Show by a direct computation that the partition function of a free thermalized scalar field is exactly the one given in the lectures, for a complex time path that starts at $t = t_{in}$, then moves linarily to $t = t_{end} + i\sigma$ and finally  $t = t_{in} + i\beta$.  

This should be performed by a generalization of the method studied in the excercise (x.x). But one should also contrast the result against the insight gained from the study of the different complex time paths in the real time formalism (splcing of the density operator at will).

2) Thermal corrections to the energy and entropy degrees of freedom in the standard model.

In the cosmology of the early universe, it is important to know the radiation density $\rho$  and the entropy density $s$ as a function of the temperature. These are needed for example to precisely define the time-temperature relation $T= T(t)$, based on the adiabatic expansion of the universe. Usually one computes these quantities neglecting interactions. Here you should compute the thermal corrections to $\rho$, $P$ and $s$ the leading non-trivial order in the standard model couplings. 

Slight problem: most of the calculations needed are already done in the course content. How to extend to make more interesting?


3) Gauge dependence of the effective potential and Nielsen identities

During the course we pointed out that the effective potential is a gauge dependent quantity.  Explain carefully the origin of this gauge dependence.  Derive the Nielsen-Olesen identities that quantify the gauge dependence of the effective potential in the Lorentz and/or $R_\xi$ gauges. (References given upon request.)

4) Completion of the phase transition

A first order phase transition completes by nucleation and subsequent expansion of the bubbles of the broken phase. During the course we studied this phenomenon analytically in the simple potential function. This procedure led to defintion of three characteristic temperatures $T_0$, $T_c$ and $T_N$. A fourth characteristic temperature corresponds to the percolation temperature $T_p$, eg the temperature when the transition completes ($T_c > T_N > T_p > T_0$).  Go carefully through the material given in the course, extending the analysis to include the percolation phase and derive $T_p$. I will give more instructions over the zoom, if you are interested in this poroblem. One aspect of this problem could be to give a numrical implementation that computes each $T_i$ for an arbitrary 1-loop effective action (one-step transition in this problem)


5) The 2PI-effective action

Derive the 2PI-effective action. Refs. given upon request. Extra task: connect these equations to Kadanoff-Baym equations (this is a long problem, but most details are found in literature.)


6) The Kadanoff-Baym equations and the Boltzmann equation limit

During the course we have used Dyson equations to define the resummed thermal 2-point function in the real time formalism including interactions. Droppint the assumption that the self-energy function was thermal, one can turn the Dyson equation into equation of motion for the 2-point function. Performing a Wigner transformation, eg a Fourier transformation with respect to the relative coordinate in the 2-point function, one obtains the Kadanoff-Baym equations for the 2-point function, which contain an infinite number of gradient terms. Assuming that gradients are small and taking the limit of zero width (neglecting the complex (or anti-hermitean) parts) in the pole equations (equations for the retarded and advanced propagators), one can derive a non-equilibrium kinetic equations in the spectral limit. Assuming translational invariance, these equations reduce to the standard Boltzmann equations.