A. Zee, Quantum Field Theory in a Nutshell · Problem V.5.1

Field Theory Redux

Physics — QFT

Problem

Parallel to the discussion in chapter II.1 you can see easily that the space of 2 by 2 matrices is spanned by the four matrices II, γμ\gamma^\mu, and γ5≡γ0γ1=σ3\gamma^5 \equiv \gamma^0\gamma^1 = \sigma_3. (Note the peculiar but standard notation of γ5\gamma^5.) Convince yourself that 12(I±γ5)\frac{1}{2}(I \pm \gamma^5) projects out right- and left handed fields just as in (3+1)(3 + 1)-dimensional spacetime. Show that in the bilinear ψˉγμψ\bar{\psi}\gamma^\mu\psi left handed fields are connected to left handed fields and right handed to right handed and that in the scalar ψˉψ\bar{\psi}\psi and the pseudoscalar ψˉγ5ψ\bar{\psi}\gamma^5\psi right handed is connected to left handed and vice versa. Finally, note that under the transformation ψ→eiθγ5ψ\psi \to e^{i\theta\gamma^5}\psi the scalar and pseudoscalar rotate into each other. Check that this transformation leaves the massless Dirac Lagrangian (3) invariant.

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