Srednicki — Quantum Field Theory · Problem 86.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spontaneously-broken-nonabelian-gauge-theory

Problem

Let

\varphi_i

be a complex scalar field in a complex representation R of the gauge group. Under an infinitesimal gauge transformation, we have

\delta\varphi_i = -i\theta^a(T^a_{\rm R})_i{}^j\varphi_j

. Let us write

\varphi_i = \frac{1}{\sqrt{2}}(\phi_i + i\phi_{i+d(\rm R)})

, where

\phi_i

is a real scalar field with the index

i

running from 1 to

2d(\rm R)

. Then, under an infinitesimal gauge transformation, we have

\delta\phi_i = -i\theta^a(\mathcal{T}^a)_{ij}\phi_j

. a) Express

\mathcal{T}^a

in terms of the real and imaginary parts of

T^a_{\rm R}

. b) Show that the

\mathcal{T}^a

matrices satisfy the appropriate commutation relations.