Srednicki — Quantum Field Theory · Problem 36.5

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · symmetries-fermion-fields

Problem

Symmetries of fermion fields. (Prerequisite: 24.) Consider a theory with

N

massless Weyl fields

\psi_j

\mathcal{L} = i\psi_j^\dagger\sigma^\mu\partial_\mu\psi_j , \tag{36.74}

where the repeated index

j

is summed. This lagrangian is clearly invariant under the U(

N

) transformation,

\psi_j \to U_{jk}\psi_k , \tag{36.75}

where

U

is a unitary matrix. State the invariance group for the following cases: a)

N

Weyl fields with a common mass

m

\mathcal{L} = i\psi_j^\dagger\sigma^\mu\partial_\mu\psi_j - \tfrac{1}{2}m(\psi_j\psi_j + \psi_j^\dagger\psi_j^\dagger) . \tag{36.76}

N

massless Majorana fields,

\mathcal{L} = \tfrac{i}{2}\Psi_j^\mathrm{T}\mathcal{C}\gamma^\mu\partial_\mu\Psi_j . \tag{36.77}

N

Majorana fields with a common mass

m

\mathcal{L} = \tfrac{i}{2}\Psi_j^\mathrm{T}\mathcal{C}\gamma^\mu\partial_\mu\Psi_j - \tfrac{1}{2}m\Psi_j^\mathrm{T}\mathcal{C}\Psi_j . \tag{36.78}

N

massless Dirac fields,

\mathcal{L} = i\overline{\Psi}_j\gamma^\mu\partial_\mu\Psi_j . \tag{36.79}

N

Dirac fields with a common mass

m

\mathcal{L} = i\overline{\Psi}_j\gamma^\mu\partial_\mu\Psi_j - m\overline{\Psi}_j\Psi_j . \tag{36.80}