Srednicki — Quantum Field Theory · Problem 70.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · index-adjoint-representation-sun

Problem

a) Use eqs. (70.12) and (70.26) to compute

T(\mathrm{A})

for

\mathrm{SU}(N)

. b) For

\mathrm{SU}(2)

, the adjoint representation is specified by

(T^a_{\mathrm{A}})^{bc} = -i\varepsilon^{abc}

. Use this to compute

T(\mathrm{A})

explicitly for

\mathrm{SU}(2)

. Does your result agree with part (a)? c) Consider the

\mathrm{SU}(2)

subgroup of

\mathrm{SU}(N)

that acts on the first two components of the fundamental representation of

\mathrm{SU}(N)

. Under this

\mathrm{SU}(2)

subgroup, the N of

\mathrm{SU}(N)

transforms as

2 \oplus (N-2)

1's. Using eq. (70.26), figure out how the adjoint representation of

\mathrm{SU}(N)

transforms under this

\mathrm{SU}(2)

subgroup. d) Use your results from parts (b) and (c) to compute

T(\mathrm{A})

for

\mathrm{SU}(N)

. Does your result agree with part (a)?