A () quantum field theory
16. In problem 13 of Problem Set 4, you showed that the Lorentz transformation of the Dirac field is ψ(x)→ e−iλμνσμν/4ψ(Λ−1x) and in class we learned that the parity transformation is ψ(x, t) → eγψ(−x, t). Show from these rules that the Dirac bilinears, ψ̄(x)ψ(x), ψ̄(x)γ5ψ(x), ψ̄(x)γ ψ(x), ψ̄(x)γ5γ ψ(x), and ψ̄(x)σψ(x), transform under Lorentz and parity transformations as scalar, pseudoscalar, vector, axial vector, and second rank tensor fields respectively. Here ψ̄ ≡ ψ†γ0 is the Dirac adjoint. A useful first step in the solution is to prove that ψ̄(x) → ψ̄(Λ−1x)e+iλμνσμν/4 under Lorentz transformations. Also it is enough to specialize to infinitesimal Lorentz transformations. (The terms pseudoand axial refer to opposite than “normal” parity properties. For a normal 4-vector, the time component is even and the space component is odd under parity. A normal scalar is even under parity.)