Feynman Diagrams

In Coordinate Space

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

(1)   \begin{equation*} iG_{\alpha\beta}(x,y) =\sum_{m=0}^{\infty}\left(\frac{-i}{\hbar}\right)^m \frac{1}{m!} \int_{-\infty}^{\infty}dt_1\cdots\int_{-\infty}^{\infty}dt_m \langle \Phi_0| T\left[H_I(t_1)\cdots H_I(t_{\nu})\hat{\psi}_{\alpha}(x)\hat{\psi}^{\dagger}_{\beta}(y)\right] |\Phi_0\rangle_{\text{connected}} \end{equation*}

with the interaction

(2)   \begin{equation*}  H_I(t_i)=\frac{1}{2}\sum_{\lambda\lambda'\mu\mu'}\int d\mathbf{x}_id\mathbf{x'}_i \hat{\psi}^{\dagger}_{\lambda}(x_i) \hat{\psi}^{\dagger}_{\mu}(x'_i) U(x_i,x'_i)_{\lambda\lambda',\mu\mu'} \hat{\psi}_{\mu'}(x'_i) \hat{\psi}_{\lambda'}(x_i) \end{equation*}

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

(3)   \begin{align*}  \begin{split} G^{(1)}_{\alpha\beta}(x,y) &=\frac{i}{2\hbar}\int d^4x_1d^4x_1' \left\{ (-1)G^0_{\alpha\lambda}(x,x_1) U(x_1,x_1')_{\lambda\lambda',\mu\mu'} G^0_{\lambda'\beta}(x_1,y) G^0_{\mu'\mu}(x_1',x_1') \right.\\ &\qquad\qquad\qquad\qquad (-1)G^0_{\alpha\mu}(x,x_1') U(x_1,x_1')_{\lambda\lambda',\mu\mu'} G^0_{\mu'\beta}(x_1',y) G^0_{\lambda'\lambda}(x_1,x_1)\\ &\qquad\qquad\qquad\qquad +G^0_{\alpha\lambda}(x,x_1) U(x_1,x_1')_{\lambda\lambda',\mu\mu'} G^0_{\lambda'\mu}(x_1,x_1') G^0_{\mu'\beta}(x_1',y)\\ &\qquad\qquad\qquad\qquad \left.+ G^0_{\alpha\lambda}(x,x_1') U(x_1,x_1')_{\lambda\lambda',\mu\mu'} G^0_{\lambda'\mu}(x_1',x_1) G^0_{\mu'\beta}(x_1,y) \right\} \end{split} \end{align*}

The corresponding Feynman diagram is in Figure 1.

Interchange x_i and x_i' will give the same value since the potential is symmetric under this substitution. Then we can omit the \frac{1}{2} in Eq.3 and keep one of these term.

(4)   \begin{align*}  \begin{split} G^{(1)}_{\alpha\beta}(x,y) &=\frac{i}{\hbar}\int d^4x_1d^4x_1' \left\{ (-1)G^0_{\alpha\lambda}(x,x_1) U(x_1,x_1')_{\lambda\lambda',\mu\mu'} G^0_{\lambda'\beta}(x_1,y) G^0_{\mu'\mu}(x_1',x_1') \right.\\ &\qquad\qquad\qquad\qquad \left.+G^0_{\alpha\mu}(x,x_1) U(x_1,x_1')_{\lambda\lambda',\mu\mu'} G^0_{\mu'\lambda}(x_1,x_1') G^0_{\lambda'\beta}(x_1',y) \right\} \end{split} \end{align*}

And the Feynman diagram is in Figure 2.

The rules for the Feynman diagrams of the nth-order contribution to G_{\alpha\beta}(x,y) is

  1. Draw all topologically distinct connected diagrams with n interaction lines U and 2n+1 directed Green’s function G^0. Every vertex has only one interaction. Each of these diagrams represent n! permutations of among (x_1,x_1')\cdots(x_n,x_n').
  2. Label each vertex with a four-dimensional space-time point x_i.
  3. Each solid line represents a Green’s function G^0_{\alpha\beta} running from y to x.
  4. Each wavy line represents an interaction .

    (5)   \begin{equation*} U(x_i,x'_i)_{\lambda\lambda',\mu\mu'} =V(\mathbf{x_i,x'_i})_{\lambda\lambda',\mu\mu'}\delta(t_x-t_{x'}) \end{equation*}

  5. Integrate all internal variables over space and time.
  6. There’s a spin matrix product along each continuous fermion line, including the potential at each vertex.
  7. Affix a sign factor (-1)^F to each term, where F is the number of closed fermion loopsin the diagram.
  8. Assign a factor (-i)(-i/\hbar)^n(i)^{2n+1}=(i/\hbar)^n to each nth order term.
  9. A Green’s with equal time variable must be interpreted as G^0_{\alpha\beta}(\mathbf{x}t,\mathbf{x'}t^+).

With these rules, the second order Feynman diagram for G_{\alpha\beta}(x,y) is in Figure 3.

In Momentum Space

The Green’s function G^0_{\alpha\beta}(x,y) has two position arguments. For a uniform and isotropic system, we can have a Fourier transformation, in which G_{\alpha\beta} has only one argument

(6)   \begin{equation*} G_{\alpha\beta}(x,y) =(2\pi)^{-4}\int d^4k e^{ik\cdot(x-y)}G_{\alpha\beta}(k) \end{equation*}

(7)   \begin{equation*} G^0_{\alpha\beta}(x,y) =(2\pi)^{-4}\int d^4k e^{ik(x-y)}G^0_{\alpha\beta}(k) \end{equation*}

where

(8)   \begin{equation*} d^4k=d^3kd\omega \end{equation*}

(9)   \begin{equation*} k\cdot x =\mathbf{k\cdot x}-\omega t \end{equation*}

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

(10)   \begin{equation*} U(x,x')=V(\mathbf{x-x'})\delta(t-t') \end{equation*}

(11)   \begin{equation*} U(x,x')_{\lambda\lambda',\mu\mu'} =(2\pi)^{-4}\int d^4k e^{ik(x-y)}U(k)_{\lambda\lambda',\mu\mu'} \end{equation*}

The Feynman diagram in momentum space is the graphical notation what simplify the expression of G_{\alpha\beta}(k) in terms of G^0_{\alpha\beta}(k) and U(k)_{\lambda\lambda',\mu\mu'}.

Take the first order of G_{\alpha\beta}(x,y) for example, that is Eq.4. After the Fourier transformation, it’s k component is

(12)   \begin{align*} \begin{split} G^{(1)}_{\alpha\beta}(k) &=\frac{i}{\hbar}\int d^4k_1 \left\{ (-1)G^0_{\alpha\lambda}(k) U(0)_{\lambda\lambda',\mu\mu'} G^0_{\lambda'\beta}(k) G^0_{\mu'\mu}(k_1)e^{\i\omega_1\eta} \right.\\ &\qquad\qquad\qquad \left.+G^0_{\alpha\mu}(k) U(k-k_1)_{\lambda\lambda',\mu\mu'} G^0_{\mu'\lambda}(k_1) G^0_{\lambda'\beta}(k)^{\i\omega_1\eta} \right\} \end{split} \end{align*}

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 nth order contribution to G_{\alpha\beta}(k) is

  1. Daw all topologically distinct connected diagrams with n interaction lines and 2n+1 directed Green’s functions.
  2. Assign a direction to each interaction line; associate a directed four-momentum with each line and conserve four-momentum at each vertex.
  3. Each Green’s function corresponds to a factor

    (13)   \begin{align*} G^0_{\alpha\beta}(\mathbf{k},\omega) =\delta_{\alpha\beta}G^0(\mathbf{k},\omega) =\delta_{\alpha\beta}\left[ \frac{\theta(|\mathbf{k}|-k_F)} {\omega-\omega_{\mathbf{k}+i\eta}} +\frac{\theta(k_F-|\mathbf{k}|)} {\omega-\omega_{\mathbf{k}-i\eta}} \right] \end{align*}

  4. Each interaction corresponds to a factor U(1)_{\lambda\lambda'\mu\mu'}.
  5. Perform a spin summation along each continuous particle line including the potential at each vertex.
  6. Integrate over the n independent internal four-momenta.
  7. Affix a factor (i\hbar)^n(2\pi)^{-4}(-1)^F where F is the number of closed fermion loops.
  8. 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 e^{i\omega}G_{\alpha\beta}(\mathbf{k},\omega), where \eta\to 0^+ 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.

Leave a Reply

Your email address will not be published. Required fields are marked *