Past exam of the quantum field theory course of the University of Oxford / 2024 / 2 / a / Solution

Using Feynman slash notation, the symmetric tensor $p_{\mu }{p}_{\nu }$ kills the antisymmetric commutator of the gamma matrices. The Clifford algebra givesConsequently $(/p{)}^{2r}=(p^2{)}^{r}I$ and $(/p{)}^{2r+1}=(p^2{)}^r/p$. For a scalar-coefficient polynomial $F(z)={\sum }_n{c}_n{z}^n$, split its even and odd powers:This reduction assumes scalar coefficients; arbitrary noncommuting matrix coefficients would not give the stated scalar functions $A,B$.