34
3 Mathematical Framework
L D :=
i
2
¯
ψγ
a
∇ a ψ − ∇ a ¯
ψγ
a
ψ
− m ¯
ψψ
≡
i
2
¯
ψ γ
a
∂ a ψ − ∂ a ¯
ψ γ
a
ψ
− m ¯
ψψ −
1
2
ε abcd
abc S
d
, (3.2.75)
where we have used the identity
γ a γ b γ c ≡ −η ab γ c + η ca γ b − η bc γ a + iε abcd γ
d
γ
5
,
γ
5
:= iγ
0
γ
1
γ
2
γ
3
,
(3.2.76)
and
S
a
:=
1
2
¯
ψγ
a
γ
5
ψ ,
(3.2.77)
is the spin pseudo-current. Perhaps a more familiar expression for the latter is found
when its spatial part is expressed in terms of left/right-handed Weyl spinors
S
i
≡
1
2
ψ
†
L σ
i
ψ L + ψ
†
R σ
i
ψ R
,
(3.2.78)
where the σ
i are the Pauli matrices. Just as the electric current is defined through Eq.
(3.2.68), the spin current can be defined by varying the Dirac action with respect to
an independent spin connection
S abc := −2e cμ
δS D
δ ab
μ
= ε abcd S
d
.
(3.2.79)
As for the energy-momentum tensor
T ab := −
1
e
e
μ
b
δS D
δe aμ ≡ −
i
2
¯
ψγ a ∇ b ψ − ∇ b ¯
ψγ a ψ
−
1
2
∇ c S
c
ab + η ab L D , (3.2.80)
it is indeed symmetric when the equation of motion of ψ
iγ
a
∇ a − m
ψ = 0 ,
(3.2.81)
is satisfied. To see this, multiply the above equation with γ
[b
γ
c] from the left and use
(3.2.76) to find the relation
γ
[a
∇
b]
ψ =
i
2
mγ
[a
γ
b]
− ε
abcd
γ c γ
5
∇ d
ψ ,
(3.2.82)
with which it is trivial to show that the antisymmetric part of (3.2.80) vanishes.
Finally, let us come back to the connection with the action of the Poincaré group
in Minkowski space-time field theory. In that case there exists a gauge where e
a
μ =
δ
a
μ , thus effectively identifying the two types of indices. This gauge fixing is then
preserved under a subgroup of combined MDs and LLTs (which we express in terms
3 Mathematical Framework
L D :=
i
2
¯
ψγ
a
∇ a ψ − ∇ a ¯
ψγ
a
ψ
− m ¯
ψψ
≡
i
2
¯
ψ γ
a
∂ a ψ − ∂ a ¯
ψ γ
a
ψ
− m ¯
ψψ −
1
2
ε abcd
abc S
d
, (3.2.75)
where we have used the identity
γ a γ b γ c ≡ −η ab γ c + η ca γ b − η bc γ a + iε abcd γ
d
γ
5
,
γ
5
:= iγ
0
γ
1
γ
2
γ
3
,
(3.2.76)
and
S
a
:=
1
2
¯
ψγ
a
γ
5
ψ ,
(3.2.77)
is the spin pseudo-current. Perhaps a more familiar expression for the latter is found
when its spatial part is expressed in terms of left/right-handed Weyl spinors
S
i
≡
1
2
ψ
†
L σ
i
ψ L + ψ
†
R σ
i
ψ R
,
(3.2.78)
where the σ
i are the Pauli matrices. Just as the electric current is defined through Eq.
(3.2.68), the spin current can be defined by varying the Dirac action with respect to
an independent spin connection
S abc := −2e cμ
δS D
δ ab
μ
= ε abcd S
d
.
(3.2.79)
As for the energy-momentum tensor
T ab := −
1
e
e
μ
b
δS D
δe aμ ≡ −
i
2
¯
ψγ a ∇ b ψ − ∇ b ¯
ψγ a ψ
−
1
2
∇ c S
c
ab + η ab L D , (3.2.80)
it is indeed symmetric when the equation of motion of ψ
iγ
a
∇ a − m
ψ = 0 ,
(3.2.81)
is satisfied. To see this, multiply the above equation with γ
[b
γ
c] from the left and use
(3.2.76) to find the relation
γ
[a
∇
b]
ψ =
i
2
mγ
[a
γ
b]
− ε
abcd
γ c γ
5
∇ d
ψ ,
(3.2.82)
with which it is trivial to show that the antisymmetric part of (3.2.80) vanishes.
Finally, let us come back to the connection with the action of the Poincaré group
in Minkowski space-time field theory. In that case there exists a gauge where e
a
μ =
δ
a
μ , thus effectively identifying the two types of indices. This gauge fixing is then
preserved under a subgroup of combined MDs and LLTs (which we express in terms