Quantum Mechanics for Mathematicians

(hereRis theSO(3) rotation corresponding to Ω). This action can be thought
of as an action on a tensor product ofC^2 and a space of functions onR^3 , with
the matrix Ω acting on theC^2 factor, and the action on functions the induced
action from rotations onR^3. On distributional fields, the action will be by the
inverse
Ψ(x)→Ω−^1 Ψ(Rx)

(where Ψ has two components).
TheSU(2) action on quantum fields will be given by a unitary operator
U(Ω) satisfying
U(Ω)Ψ(̂x)U−^1 (Ω) = Ω−^1 Ψ(̂Rx)

which will give a unitary representation on the multi-particle state space. The
Lie algebra representation on this state space will be given by the sum of two
terms
̂J=̂L+̂S

corresponding to the fact that this comes from a representation on a tensor
product. Here the operator̂Lis just two copies of the single component version
(equation 38.12) and comes from the same source, the induced action on solu-
tions from rotations ofR^3. The “spin” operatorŜcomes from theSU(2) action
on theC^2 factor in the tensor product description of solutions and is given by

̂S=

∫

R^3

Ψ̂†(x)

(

1

2

σ

)

Ψ(̂x)d^3 x (38.14)

It mixes the two components of the spin^12 field, and is a new feature not seen
in the single component (“spin 0”) theory.
It is a straightforward exercise using the commutation relations to show that
these operatorŝJsatisfy thesu(2) commutation relations and have the expected
commutation relations with the two-component field operators. They also com-
mute with the Hamiltonian, providing an action ofSU(2) by symmetries on the
multi-particle state space.
States of this quantum field theory can be produced by applying products of
operatorsa†j(p) for various choices ofpandj= 1,2 to the vacuum state. Note

that theE ̃(3) Casimir operator̂J·P̂does not commute with thea†j(p). If one

wants to work with states with a definite helicity (eigenvalue of̂J·P̂divided by
the square root of the eigenvalue of the operator|P̂|^2 ), one could instead write
wavefunctions as in equation 34.10, and field operators as

Ψ̂±(x) =^1

(2π)

(^32)

∫

R^3

eip·xa±(p)u±(p)d^3 p

Ψ̂†±(x) =^1

(2π)

(^32)

∫

R^3

e−ip·xa†±(p)u†±(p)d^3 p

Here the operatorsa±(p),a†±(p) would be annihilation and creation operators
for helicity eigenstates. Such a formalism is not particularly useful in the non-
relativistic case, but we mention it here because its analog in the relativistic
case will be more significant.