Quantum Mechanics for Mathematicians

onn-component wavefunctions, solutions to a wave equation (hereSis

anndimensional representation of the Lorentz group). These will be the

unitary representations classified in this chapter.

• The space of single-particle wavefunctions can be used to construct a quan-
  tum field theory, describing arbitrary numbers of particles. This will come
  with an action on the state space by unitary operatorsU(a,Λ). This will
  be a unitary representation, but very much not irreducible.

For the corresponding Lie algebra actions, we will use lower case letters (e.g.,
tj,lj) to denote the Lie algebra elements and their action on Minkowski space,
upper case letters (e.g.,Pj,Lj) to denote the Lie algebra representation on

wavefunctions, and upper case hatted letters (e.g.,P̂j,L̂j) to denote the Lie
algebra representation on states of the quantum field theory.

42.1 The Poincar ́e group and its Lie algebra

Definition(Poincar ́e group).The Poincar ́e group is the semi-direct product

P=R^4 oSO(3,1)

with double cover
P ̃=R^4 oSL(2,C)

The action ofSO(3,1)orSL(2,C)onR^4 is the action of the Lorentz group on
Minkowski space.

We will refer to both of these groups as the “Poincar ́e group”, meaning by
this the double cover only when we need it because spinor representations of
the Lorentz group are involved. The two groups have the same Lie algebra, so
the distinction is not needed in discussions that only involve the Lie algebra.
Elements of the groupPwill be written as pairs (a,Λ), witha∈R^4 and
Λ∈SO(3,1). The group law is

(a 1 ,Λ 1 )(a 2 ,Λ 2 ) = (a 1 + Λ 1 a 2 ,Λ 1 Λ 2 )

The Lie algebraLieP=LieP ̃has dimension 10, with basis

t 0 ,t 1 ,t 2 ,t 3 ,l 1 ,l 2 ,l 3 ,k 1 ,k 2 ,k 3

where the first four elements are a basis of the Lie algebra of the translation
group, and the next six are a basis ofso(3,1), with theljgiving the subgroup of
spatial rotations, thekjthe boosts. We already know the commutation relations
for the translation subgroup, which is commutative so

[tj,tk] = 0

We have seen in chapter 40 that the commutation relations forso(3,1) are

[l 1 ,l 2 ] =l 3 , [l 2 ,l 3 ] =l 1 , [l 3 ,l 1 ] =l 2