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November 7, 2013 135
5.3 Dirac particles
5.3.1 Dirac spinors
The requirements on the object T (p) that we have gathered so far are that it
be a member of the Clifford algebra, and that
T (p) 2 = T (p) , T (p) = T (p) , (5.57)
although by a renormalization we may relax the first requirement into a proportionality.
Now, it must be remembered that any modification of the propagator
may be compensated for by a transformation of the vertices : so, if there is a
Clifford-algebra object Σ such that
then, effectively, the propagator
ΣΣ = ΣΣ = 1 ,
Σ T (p) Σ
is equivalent to T (p) itself. We may then perform a search 18 through all inequivalent
possibilities for T . The upshot is that there are precisely four projection
operators, for a choice of two Minkowski vectors k µ and s µ such that
k · k = 1 , s · s = −1 , k · s = 0 , (5.58)
and they read
Π(λ 1 , λ 2 ) = 1 4
( ) ( )
1 + λ 1 /k 1 + λ 2 γ 5 /s
, (5.59)
where λ 1,2 = ±1. We have
and
Π(λ 1 , λ 2 ) = Π(λ 1 , λ 2 ) (5.60)
Π(λ 1 , λ 2 )Π(λ ′ 1, λ ′ 2) = δ λ1,λ ′ 1 δ λ 2,λ ′ 2 Π(λ 1, λ 2 ) (5.61)
and also we conclude that, since there are precisely 4 projection operators,
we can settle for N = 4 for the Dirac matrices 19 . Since for on-shell particles
18 This is a quite tedious task, in particular the unearthing of the necessary Σ matrices.
This is relegated to Appendix 8, based on the efforts of J. de Groot.
19 This presupposes that a four-dimensional choice of dirac matrices is actually possible.
This is the case, witness the so-called Pauli representation :
⎛
γ 0 = ⎝
⎛
γ 2 = ⎝
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1
0 0 0 −i
0 0 i 0
0 i 0 0
−i 0 0 0
⎞
⎛
⎠ , γ 1 = ⎝
⎞
⎛
⎠ , γ 3 = ⎝
0 0 0 1
0 0 1 0
0 −1 0 0
−1 0 0 0
0 0 1 0
0 0 0 −1
−1 0 0 0
0 1 0 0
Any other representation will do as well: that is the whole point of it !
⎞
⎠ ,
⎞
⎠ . (5.62)