Quantum Mechanics for Mathematicians

acting onH 1. Just as for energy and momentum, we can construct angular
momentum operators in the quantum field theory as quadratic field operators,
in this case getting

̂L=

∫

R^3

Ψ̂†(x)(x×(−i∇))Ψ(̂x)d^3 x (38.12)

These will generate the action of rotations on the field operators. For instance,
ifR(θ) is a rotation about thex 3 axis by angleθ, we will have

Ψ(̂R(θ)x) =e−iθ̂L^3 Ψ(̂x)eiθ̂L^3

The operatorsP̂andL̂together give a representation of the Lie algebra of
E(3) on the multi-particle state space, satisfying theE(3) Lie algebra commu-
tation relations

[−iP̂j,−iP̂k] = 0, [−iL̂j,−iP̂k] =jkl(−iP̂l), [−iL̂j,−iL̂k] =jkl(−iL̂l)

(38.13)

̂Lcould also have been found by the moment map method. Recall from

section 8.3 that, for theSO(3) representation on functions onR^3 induced from
theSO(3) action onR^3 , the Lie algebra representation is (forl∈so(3))

ρ′(l) =−x×∇

The action on distributions will differ by a minus sign, so we are looking for a
moment mapμsuch that

{μ,Ψ(x)}=x×∇Ψ(x)

and this will be given by

μ−x×∇=i

∫

R^3

Ψ(x)(−x×∇)Ψ(x)d^3 x

After quantization, this gives equation 38.12 for the angular momentum operator
̂L.

38.3.3 Spin^12 fields

For the case of two-component wavefunctions describing spin^12 particles satisfy-
ing the Pauli-Schr ̈odinger equation (see chapter 34 and section 37.5), the groups
U(1),U(n) (for multiple kinds of spin^12 particles) and theR^3 of translations

act independently on the two spinor components, and the formulas forN̂,X̂
andP̂are just the sum of two copies of the single component equations. As
discussed in section 34.2, the action of the rotation group on solutions in this
case requires the use of the double coverSU(2) ofSO(3), withSU(2) group
elements Ω acting on two-component solutionsψby

ψ(x)→Ωψ(R−^1 x)