An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 15.5

Non-Abelian Gauge Invariance — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · non-abelian-gauge-invariance

Problem

Casimir operator computations. An alternative strategy for computing the quadratic Casimir operator is to compute

C(r)

in the formula

\text{tr}[t^a_r t^b_r] = C(r)\delta^{ab}

by choosing

t^a

and

t^b

to lie in an

SU(2)

subgroup of the gauge group. (a) Under an

SU(2)

subgroup of a general group

G

, an irreducible representation

r

G

will decompose into a sum of representations of

SU(2)

r \to \sum j_i,

where the

j_i

are the spins of

SU(2)

representations. Show that

3C(r) = \sum_i j_i(j_i+1)(2j_i+1).

(b) Under an

SU(2)

subgroup of

SU(N)

, the fundamental representation

N

transforms as a 2-component spinor (

j = \frac{1}{2}

) and

(N-2)

singlets. Use this relation to check the formula

C(N) = \frac{1}{2}

. Show that the adjoint representation of

SU(N)

decomposes into one spin 1,

2(N-2)

spin-

\frac{1}{2}

's, plus singlets, and use this decomposition to check that

C(G) = N

. (c) Symmetric and antisymmetric 2-index tensors form irreducible representations of

SU(N)

. Compute

C_2(r)

for each of these representations. The direct sum of these representations is the product representation

N \times N

. Verify that your results for

C_2(r)

satisfy the identity for product representations that follows from Eqs. (15.100) and (15.101).