Student Friendly Introduction to QFT · Problem 12.4

Renormalization: Putting It All Together

Physics — QFT · fermion-propagator-forms

Problem

From the form of the fermion propagator

S_F(p) = \frac{\not{p} + m_0}{p^2 - m_0^2 + i\varepsilon}

, derive the equivalent form

S_F(p) = \frac{1}{\not{p} - m_0 + i\varepsilon}.

Hint: You need use

(\not{p})^2 = \not{p}\not{p} = p_\mu \gamma^\mu p_\nu \gamma^\nu = \gamma^\mu \gamma^\nu p_\mu p_\nu = \frac{1}{2}(\gamma^\mu \gamma^\nu p_\mu p_\nu + \gamma^\nu \gamma^\mu p_\nu p_\mu) = \frac{1}{2}[\gamma^\mu, \gamma^\nu]_+ p_\mu p_\nu = g^{\mu\nu} p_\mu p_\nu = p^2

from Chap. 4 Appendix A. When you are all through, note that we have carried a 4

\times

4 identity matrix through at every step. That is, when we multiply two gamma matrices together (in spinor space) we get a 4

\times

4 matrix result. In terms of the Chap. 4 relation cited, with spinor matrices written out, it looks like

(\not{p})^2 = p_\mu \gamma^\mu_{\alpha\beta} p_\nu \gamma^\nu_{\beta\delta} = p^2 I_{\alpha\delta}

. The first two terms in the denominator of the second relation for the propagator then are actually

\not{p} - m_0 = p_\mu \gamma^\mu_{\alpha\delta} - m_0 I_{\alpha\delta}