Srednicki — Quantum Field Theory · Problem 70.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · antisymmetric-symmetric-representations-sun

Problem

a) For

\mathrm{SU}(N)

, we have

\mathrm{N}\otimes\mathrm{N} = \mathcal{A}_{\mathrm{A}} \oplus \mathcal{S}_{\mathrm{S}}, \tag{70.37}

where

\mathcal{A}

corresponds to a field with two antisymmetric fundamental

\mathrm{SU}(N)

indices,

\varphi_{ij} = -\varphi_{ji}

, and

\mathcal{S}

corresponds to a field with two symmetric fundamental

\mathrm{SU}(N)

indices,

\varphi_{ij} = +\varphi_{ji}

. Compute

D(\mathcal{A})

and

D(\mathcal{S})

. b) By considering an

\mathrm{SU}(2)

subgroup of

\mathrm{SU}(N)

, compute

T(\mathcal{A})

and

T(\mathcal{S})

. c) For

\mathrm{SU}(3)

, show that

\mathcal{A} = \bar{3}

. d) By considering an

\mathrm{SU}(3)

subgroup of

\mathrm{SU}(N)

, compute

A(\mathcal{A})

and

A(\mathcal{S})