Quantum Mechanics for Mathematicians

{μX,Ψj(x)}=−XjkΨj(x), {μX,Ψj(x)}=XkjΨk(x)

After quantization these become the operator relations 38.6 and 38.7. Note that
the factor ofiin the expression forμXis there to make it a real function forX∈
u(n). Quantization of this would give a self-adjoint operator, so multiplication

by−imakes the expression forX̂skew-adjoint, and thus a unitary Lie algebra
representation.
When, as for the free particle case of equation 38.4, the Hamiltonian is
invariant underU(n) transformations of the fields Ψj,Ψj, then we will have

[X,̂Ĥ] = 0

Energy eigenstates in the multi-particle state space will break up into irreducible
representations ofU(n) and can be labeled accordingly.

38.3 Spatial symmetries

We saw in chapter 19 that the action of the groupE(3) on physical space
R^3 induces a unitary action on the spaceH 1 of solutions to the free particle
Schr ̈odinger equation. Quantization of this phase space with this group action
produces a multi-particle state space carrying a unitary representation of the
groupE(3). There are several different actions of the groupE(3) that one needs
to keep track of here. Given an element (a,R)∈E(3) one has:

• An action onR^3 , by
  x→Rx+a


• A unitary action onH 1 induced by the action onR^3 , given by

ψ(x)→u(a,R)ψ(x) =ψ(R−^1 (x−a))

on wavefunctions, or, on Fourier transforms by

ψ ̃(p)→u ̃(a,R)ψ ̃(p) =e−ia·R−^1 pψ ̃(R−^1 p)

Recall from chapter 19 that this is not an irreducible representation of

E(3), but an irreducible representation can be constructed by taking the

space of solutions that are energy eigenfunctions with fixed eigenvalue

E=|p|

2

2 m.

• E(3) will act on distributional fields Ψ(x) by

Ψ(x)→(a,R)·Ψ(x) = Ψ(Rx+a) (38.8)

This is because elements ofH 1 can be written in terms of these distribu-

tional fields as

Ψ(ψ) =

∫

R^3

Ψ(x)ψ(x)d^3 x