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November 7, 2013 141
a real proof must rest on the way they form a representation of the rotation
group. The rotation group is, of course, a subgroup of the Lorentz group.
Now, we have argued that the vector p µ and the matrix /p contain exactly the
same information, for any vector p µ . Therefore, we must be able to find how /p
transforms under a Lorentz transformation. Let us define by Λ(p; q) the minimal
Lorentz transformation, that is it makes p µ go over in q µ while keeping any
vector r µ unchanged for which p · r = q · r = 0. Rotations are an example : in
that case p 0 = q 0 = 0, |⃗p| = |⃗q|, and ⃗r·⃗p = ⃗r·⃗q = 0. Since /p is a matrix, the effect
of a Lorentz transformation must be represented by a matrix transformation,
that is
Λ(p; q) : /p → Σ 1 /p Σ 2 . (5.81)
Since we must ensure that Dirac conjugation commutes with Lorentz transformation,
we must have Σ 2 = Σ 1 ; and in order to have matrix multiplication
commute with Lorentz transformations as well 23 we must have Σ 2 Σ 1 = 1. We
conlude that
Λ(p; q) : /p → Σ /p Σ , Σ Σ = 1 . (5.82)
The explicit form of Σ reads 24
(
Σ = C 1 + /q/p )
p 2
, |C| 2 =
You can simply check that this is indeed correct :
(
Σ Σ = |C| 2 /q/p + /p/q
1 +
p 2 + /q/p/p/q )
p 4
p 2
(p + q) 2 . (5.83)
(
= |C| 2 1 + 2(pq)
p 2 + p2 q 2 )
p 4 = 1 , (5.84)
and
(
Σ /pΣ = |C| 2 /q/p/p + /p/p/q
/p +
p 2 + /q/p/p/p/q )
p 4
(
= |C| 2 /p + 2/q + /q/p/q )
p 2 = /q , (5.85)
where we have used the anticommutation result /q/p/q = 2(pq)/q − /pq 2 . The other
requirements, Σ/qΣ = /p and Σ/rΣ = /r, are proven trivially. For general Clifford
elements Γ, we have now also ensured that
Γ → Σ Γ Σ (5.86)
23 So that we can either first mutiply /p 1 and /p 2 , and then Lorentz-transform them, or do
the Lorentz transform first and the multiplication afterwards.
24 This form tacitly assumes that under minimal Lorentz transforms the sign of p 2 and
(p + q) 2 are the same. This is not obvious ; however, for boosts and spatial rotations it does
hold.