A. Zee, Quantum Field Theory in a Nutshell · Problem II.1.1

Field Theory Redux

Physics — QFT

Problem

Show that the following bilinears in the spinor field ψˉψ\bar{\psi}\psi, ψˉγμψ\bar{\psi}\gamma^\mu\psi, ψˉσμνψ\bar{\psi}\sigma^{\mu\nu}\psi, ψˉγμγ5ψ\bar{\psi}\gamma^\mu\gamma^5\psi, and ψˉγ5ψ\bar{\psi}\gamma^5\psi transform under the Lorentz group and parity as a scalar, a vector, a tensor, a pseudovector or axial vector, and a pseudoscalar, respectively. [Hint: For example, ψˉγμγ5ψ→ψˉ[1+(i/4)ωσ][γμγ5][1−(i/4)ωσ]ψ\bar{\psi}\gamma^\mu\gamma^5\psi \to \bar{\psi}[1 + (i/4)\omega\sigma][\gamma^\mu\gamma^5][1 - (i/4)\omega\sigma]\psi under an infinitesimal Lorentz transformation and →ψˉγ0γμγ5γ0ψ\to \bar{\psi}\gamma^0\gamma^\mu\gamma^5\gamma^0\psi under parity. Work out these transformation laws and show that they define an axial vector.]

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