 Solution

ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2024/2/a/solution

p_{μ} p_{ν}

kills the antisymmetric commutator of the gamma matrices. The Clifford algebra gives

\neq p \neq p = \frac{1}{2} p_{μ} p_{ν} {γ^{μ}, γ^{ν}} = p^{2} I .

Consequently

(\neq p)^{2 r} = (p^{2})^{r} I

and

(\neq p)^{2 r + 1} = (p^{2})^{r} \neq p

F (z) = \sum_{n} c_{n} z^{n}

, split its even and odd powers:

F (\neq p) = A (p^{2}) \neq p + B (p^{2}) I, A (z) = r \sum c_{2 r + 1} z^{r}, B (z) = r \sum c_{2 r} z^{r} .

This reduction assumes scalar coefficients; arbitrary noncommuting matrix coefficients would not give the stated scalar functions

A, B

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