Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 3 / iv

Table of contents
- Solution iv

Solution

Under parity,

\overline{ψ} (t, x) \mapsto \overline{ψ} (t, - x) γ^{0}

. Since

γ^{0} γ^{0} γ^{0} = γ^{0}

and

γ^{0} γ^{i} γ^{0} = - γ^{i}

,

J_{V}^{0} (t, x) \mapsto J_{V}^{0} (t, - x), J_{V}^{i} (t, x) \mapsto - J_{V}^{i} (t, - x) .

Moreover

γ^{0} γ^{5} γ^{0} = - γ^{5}

, so the axial current is a pseudovector:

J_{A}^{0} (t, x) \mapsto - J_{A}^{0} (t, - x), J_{A}^{i} (t, x) \mapsto J_{A}^{i} (t, - x) .

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