Quantum Mechanics for Mathematicians

Note that infinitesimal boosts do not commute with infinitesimal time trans-
lation, so after quantization boosts will not commute with the Hamiltonian.
Boosts will act on spaces of single-particle wavefunctions in a relativistic the-
ory, and on states of a relativistic quantum field theory, but are not symmetries
in the sense of preserving spaces of energy eigenstates.

42.2 Irreducible representations of the Poincar ́e group

We would like to construct unitary irreducible representations of the Poincar ́e
group. These will be given by unitary operatorsu(a,Λ) on a Hilbert spaceH 1 ,
which will have an interpretation as a single-particle relativistic quantum state
space. In the analogous non-relativistic case, we constructed unitary irreducible

representations ofE(3) (or its double coverE ̃(3)) as

• The space of wavefunctions of a free particle of massm, with a fixed energy
  E(chapter 19). These are solutions to

Dψ=Eψ

where

D=−

1

2 m

∇^2

and theψare single-component wavefunctions. E(3) acts on wavefunc-

tions by

ψ→u(a,R)ψ(x) =ψ(R−^1 (x−a))

• The space of solutions of the “square root” of the Pauli-Schr ̈odinger equa-
  tion (see section 34.2). These are solutions to

Dψ=±

√

2 mEψ

where

D=−iσ·∇

and theψare two-component wavefunctions.E ̃(3) acts on wavefunctions

by

ψ→u(a,Ω)ψ(x) = Ωψ(R−^1 (x−a))

(whereRis theSO(3) element corresponding to Ω∈SU(2) in the double

cover).

In both cases, the group action commutes with the differential operator, or
equivalently one hasuDu−^1 =D, and this is what ensures that the operators
u(a,Ω) take solutions to solutions.