where
(
ψ 1 (q)
ψ 2 (q)
)
=
1
(2π)
(^32)
∫
R^3
δ(|p|^2 − 2 mE)(ψ ̃E,+(p) +ψ ̃E,−(p))eip·qd^3 p
=
1
(2π)
(^32)
∫
R^3
δ(|p|^2 − 2 mE)(αE,+(p)u+(p) +αE,−(p)u−(p))eip·qd^3 p
(34.10)
34.3 TheE ̃(3)-invariant inner product
One can parametrize solutions to the Pauli equation and write anE ̃(3)-invariant
inner product on the space of solutions in several different ways. Three different
parametrizations of solutions that can be considered are:
- Using the initial data at a fixed time
(
ψ 1 (q)
ψ 2 (q)
)
Here theE ̃(3)-invariant inner product is
〈(
ψ 1 (q)
ψ 2 (q)
)
,
(
ψ 1 ′(q)
ψ 2 ′(q)
)〉
=
∫
R^3
(
ψ 1 (q)
ψ 2 (q)
)†(
ψ′ 1 (q)
ψ′ 2 (q)
)
d^3 q
This parametrization does not make visible the decomposition into irre-
ducible representations ofE ̃(3).
- Using the Fourier transforms
(
ψ ̃ 1 (p)
ψ ̃ 2 (p)
)
to parametrize solutions, the invariant inner product is
〈(
ψ ̃ 1 (p)
ψ ̃ 2 (p)
)
,
(
̃ψ′
1 (p)
ψ ̃′ 2 (p)
)〉
=
∫
R^3
(
ψ ̃ 1 (p)
ψ ̃ 2 (p)
)†(
ψ ̃′ 1 (p)
ψ ̃′ 2 (p)
)
d^3 p
The decomposition of equation 34.5 can be used to express solutions of
energyEin terms of two-component functionsψ ̃E(p), with an invariant
inner product on the space of such solutions given by
〈ψ ̃E(p),ψ ̃′E(p)〉=
1
4 π
∫
S^2
ψ ̃E(p)†ψ ̃′
E(p) sin(φ)dφdθ
where (p,φ,θ) are spherical coordinates on momentum space andS^2 is
the sphere of radius
√
2 mE.
Theψ ̃E(p) parametrize not a single irreducible representation ofE ̃(3) but
two of them, including both helicities.