#### In Coordinate Space

The Green’s function of a fermion can be written as the sum of all connected diagrams

(1)

(2)

With the help of Wick’s theorem, the first order of the Green’s function is

(3)

The corresponding Feynman diagram is in Figure 1.

Interchange and will give the same value since the potential is symmetric under this substitution. Then we can omit the in Eq.3 and keep one of these term.

(4)

And the Feynman diagram is in Figure 2.

The rules for the Feynman diagrams of the nth-order contribution to is

- Draw all topologically distinct connected diagrams with interaction lines and directed Green’s function . Every vertex has only one interaction. Each of these diagrams represent permutations of among .
- Label each vertex with a four-dimensional space-time point .
- Each solid line represents a Green’s function running from to .
- Each wavy line represents an interaction .
(5)

- Integrate all internal variables over space and time.
- There’s a spin matrix product along each continuous fermion line, including the potential at each vertex.
- Affix a sign factor to each term, where is the number of closed fermion loopsin the diagram.
- Assign a factor to each th order term.
- A Green’s with equal time variable must be interpreted as .

With these rules, the second order Feynman diagram for is in Figure 3.

#### In Momentum Space

The Green’s function has two position arguments. For a uniform and isotropic system, we can have a Fourier transformation, in which has only one argument

(6)

(7)

where

(8)

(9)

In addition, we assume the interaction depends only on the coordinate difference

(10)

(11)

The Feynman diagram in momentum space is the graphical notation what simplify the expression of in terms of and .

Take the first order of for example, that is Eq.4. After the Fourier transformation, it’s component is

(12)

Notice that the double integral in Eq.\eqref{green4} has simplified into a single integral. The Feynman diagrams is show in Figure 4.

The corresponding Feynman rules for the th order contribution to is

- Daw all topologically distinct connected diagrams with interaction lines and directed Green’s functions.
- Assign a direction to each interaction line; associate a directed four-momentum with each line and conserve four-momentum at each vertex.
- Each Green’s function corresponds to a factor
(13)

- Each interaction corresponds to a factor .
- Perform a spin summation along each continuous particle line including the potential at each vertex.
- Integrate over the independent internal four-momenta.
- Affix a factor where is the number of closed fermion loops.
- Any single-particle line that forms a closed loop as in Figure 4(a) or that is linked by the same interaction line as in Figure 4(b) is interpreted as , where at the end of the calculation.

#### References

[1] Fetter, Alexander L., and John Dirk Walecka. Quantum theory of many-particle systems. Courier Corporation, 2012.

[2] Peskin, Michael Edward. An introduction to quantum field theory. Westview press, 1995.

[3] Coleman, Piers. Introduction to many-body physics. Cambridge University Press, 2015.