Standard Model and Beyond · Problem 3.10

Gauge Theories

Physics — Particle Physics · isospin-generators-pion-nucleon

Problem

In the free-field limit the isospin generators for the pion-nucleon fields in (3.82) can be written as

T^i = \int \frac{d^3\vec{p}}{(2\pi)^3 2E_p} \sum_s \left[b_c^\dagger(\vec{p},s)\left(\frac{\tau^i_{cb}}{2}\right)b_b(\vec{p},s) + d_c^\dagger(\vec{p},s)\left(-\frac{\tau^i_{bc}}{2}\right)d_b(\vec{p},s)\right] + \int \frac{d^3\vec{p}}{(2\pi)^3 2E_p} a_j^\dagger(\vec{p})(-i\epsilon_{ijk})a_k(\vec{p}),

where

b_a

and

d_a

are, respectively, the annihilation operators for nucleon

a = p

n

and for its antiparticle, and

a_i

is the annihilation operator for

\pi^i

. Show that the

T^i

satisfy the

SU(2)

Lie algebra and that they have the right commutation rules with the fields.