Srednicki — Quantum Field Theory · Problem 39.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · canonical-quantization-spinor-fields-ii

Problem

Show that

U(\Lambda)^{-1}b_s^\dagger(\mathbf{p})U(\Lambda) = b_s^\dagger(\Lambda^{-1}\mathbf{p})\,,

U(\Lambda)^{-1}d_s^\dagger(\mathbf{p})U(\Lambda) = d_s^\dagger(\Lambda^{-1}\mathbf{p})\,, \tag{39.39}

and hence that

U(\Lambda)|p,s,q\rangle = |\Lambda p,s,q\rangle\,, \tag{39.40}

where

|p,s,+\rangle \equiv b_s^\dagger(\mathbf{p})|0\rangle\,,

|p,s,-\rangle \equiv d_s^\dagger(\mathbf{p})|0\rangle \tag{39.41}

are single-particle states.