- Directly solving for G
^{0}.- One way to proceed is to substitute for G
^{0}and also for the delta function and then to project out a givenk . Remember that the non-interacting system is uniform. - If you don't write these down by inspection, you're working too hard. Remember that the delta function is zero in each of those regions.
- Try integrating the equation from tau-tau' = -epsilon to +epsilon and then taking epsilon to zero. What survives?

- One way to proceed is to substitute for G
- The Beachball Diagram.
- Actually, I did most of this in class by mistake! So I guess the assignment is to write it up coherently.
- Here you need to do some contour integrals. Please come see me if you forget how to do these.
- First you need to identify the differences between one and
three dimensions in the Feynman rules and in the non-interacting
G
^{0}. Then just carry it out. You can compare the energy per particle rather than the energy density, if you prefer.

- Vanishing Diagrams at Third Order.
- So here I'm just checking that you can do it correctly. I realize this is tedious, but if you label your diagrams clearly it should go quickly.
- Even if you want to do this by hand as well, I think it would be worthwhile to try out the notebook. Remember to save the package file with the name "deltasimplify.m" in the save directory as the Mathematica notebook "spinsums1.nb".
- Hint: Of the five third-order diagrams, three are zero. The explanation for one of them is similar to the second-order case, but you may need to work harder to explain the other two! (Do not assume they are equal to zero for the same reason!)

- Three-body forces.
- Lowest-order means the smallest number of three-body and two-body vertices, with a least one three-body vertex. Hint: the diagram is analogous to the leading-order two-body diagram (the "bowtie" diagram).
- Apply the Feynman rules from the class notes in either momentum or coordinate space. There's not much difference! (So try both!)

- Evaluation of the Grand Potential from the One-Particle Green's
Function.
- The goal here is
*not*to derive the result (which is derived, more or less, in the class notes) but to*verify*that it works up to third order. That means identifying the combination of diagrams for the self-energy and for G that contribute at first, second, and third order, and to check that the correct overall factor is reproduced. - Note that there is an implicit trace over spin indices in the integral (see the class notes for a more explicit expression.)

- The goal here is

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