this equation becomes
((σ·p)^2 − 2 mE)(
ψ ̃ 1 (p)
ψ ̃ 2 (p))
= (|p|^2 − 2 mE)(
ψ ̃ 1 (p)
ψ ̃ 2 (p))
= 0 (34.4)
and as in chapter 19 our solution space is given by distributions supported on
the sphere of radius
√
2 mE=|p|in momentum space which we will write as
(
ψ ̃ 1 (p)
ψ ̃ 2 (p))
=δ(|p|^2 − 2 mE)(
ψ ̃E, 1 (p)
ψ ̃E, 2 (p))
(34.5)
whereψ ̃E, 1 (p) andψ ̃E, 2 (p) are functions on the sphere|p|^2 = 2mE.
34.2 Solutions of the Pauli equation and repre-
sentations ofE ̃(3)
Since σ·p
|p|is an invertible operator with eigenvalues±1, solutions to 34.4 will be given by
solutions to
σ·p
|p|
(
ψ ̃E, 1 (p)
ψ ̃E, 2 (p))
=±
(
ψ ̃E, 1 (p)
ψ ̃E, 2 (p))
(34.6)
where|p|=
√
2 mE. We will write solutions to this equation with the + sign as
ψ ̃E,+(p), those for the−sign asψ ̃E,−(p). Note thatψ ̃E,+(p) andψ ̃E,−(p) are
each two-component complex functions on the sphere
√
2 mE=|p|(or, more
generally distributions on the sphere). Our goal in the rest of this section will
be to show
Theorem.The spaces of solutionsψ ̃E,±(p)to equations 34.6 provide irreducible
representations ofE ̃(3), the double cover ofE(3), with eigenvalue 2 mEfor the
first Casimir operator
|P|^2 = (σ·P)^2
and eigenvalues±^12
√
2 mEfor the second Casimir operatorJ·P.We will not try to prove irreducibility, but just show that these solution spaces
give representations with the claimed eigenvalues of the Casimir operators (see
sections 19.2 and 19.3 for more about the Casimir operators and general theory
of representations ofE(3)). We will write the representation operators asu(a,Ω)
in position space andu ̃(a,Ω) in momentum space, withaa translation, Ω∈
SU(2) andR= Φ(Ω)∈SO(3).
The translation part of the group acts as in the one-component case of
chapter 19, by the multiplication operator
u ̃(a, 1 )ψ ̃E,±(p) =e−ia·pψ ̃E,±(p)