3.2 Space-Time Fields
33
˜
V
μ
( ˜
x) =
μ
ν V
ν
(x) →
∂ f
μ
∂x ν V
ν
(x) ,
(3.2.71)
in the transformation rules of flat space-time. But the possibility of performing this
replacement relies crucially upon the fact that the indices of the Lorentz matrix are
space-time indices, i.e. that SO(1, 3) can be viewed as a subgroup of the diffeomorphism group. This is not the case of SL(2, C), so there is no such generalization
for the half-integer representations where the corresponding transformation matrix
U (() has spinor indices. Spinors therefore require the action of the Lorentz group,
be it global or local, and thus the presence of tetrads.
Here we will work for definiteness with Dirac spinors, because all types of spin1/2 particles can be expressed in this representation. We will consider a Dirac mass
in concrete examples, but the Majorana and massless cases can be easily obtained
with minor modifications. We will denote the Dirac indices by a, b, c, . . . , but most
of the time these will be kept implicit for simplicity, as is customary. Thus, a Dirac
spinor ψ
a
(x) is a set of scalars under MDs, while under an LLT with parameter
θ ab (x)
˜
ψ(x) = U (x) ψ(x) ,
U (x) := exp
−
1
4
θ ab (x) γ
a
γ
b
,
(3.2.72)
where the γ
a are the gamma matrices obey the Clifford algebra
{γ
a
, γ
b
} = −2η
ab
.
(3.2.73)
Now that the spinor indices are “internal” from the viewpoint of MDs, the local
action of the Lorentz group on ψ is qualitatively the same as the one of the SU(N )
group in Yang-Mills theory. The only difference is that the group dimension and
signature are related to the ones of the space-time manifold M.
The need for a tetrad becomes also obvious when trying to construct the kinetic
part of the general-relativistic Dirac Lagrangian. One must turn the diffeomorphism
index of the derivative ∂ μ into a Lorentz index in order to contract with γ
a , i.e. the
combination γ
a e
μ
a ∂ μ ≡ γ
a
∂ a .
5 Moreover, since the Lorentz group now acts locally,
one must consider the covariant derivative in the Dirac representation
∇ μ ψ :=
∂ μ +
1
4
abμ γ
a
γ
b
ψ .
(3.2.74)
The Dirac Lagrangian reads
5 Another common choice is to define instead the Dirac matrix fields γ μ (x) := γ a e
μ
a (x), which
therefore obey the modified Clifford algebra {γ μ , γ ν } ≡ −2g μν , so that γ a e
μ
a ∂ μ ≡ γ μ ∂ μ , but all
this is only a matter of interpretation.
33
˜
V
μ
( ˜
x) =
μ
ν V
ν
(x) →
∂ f
μ
∂x ν V
ν
(x) ,
(3.2.71)
in the transformation rules of flat space-time. But the possibility of performing this
replacement relies crucially upon the fact that the indices of the Lorentz matrix are
space-time indices, i.e. that SO(1, 3) can be viewed as a subgroup of the diffeomorphism group. This is not the case of SL(2, C), so there is no such generalization
for the half-integer representations where the corresponding transformation matrix
U (() has spinor indices. Spinors therefore require the action of the Lorentz group,
be it global or local, and thus the presence of tetrads.
Here we will work for definiteness with Dirac spinors, because all types of spin1/2 particles can be expressed in this representation. We will consider a Dirac mass
in concrete examples, but the Majorana and massless cases can be easily obtained
with minor modifications. We will denote the Dirac indices by a, b, c, . . . , but most
of the time these will be kept implicit for simplicity, as is customary. Thus, a Dirac
spinor ψ
a
(x) is a set of scalars under MDs, while under an LLT with parameter
θ ab (x)
˜
ψ(x) = U (x) ψ(x) ,
U (x) := exp
−
1
4
θ ab (x) γ
a
γ
b
,
(3.2.72)
where the γ
a are the gamma matrices obey the Clifford algebra
{γ
a
, γ
b
} = −2η
ab
.
(3.2.73)
Now that the spinor indices are “internal” from the viewpoint of MDs, the local
action of the Lorentz group on ψ is qualitatively the same as the one of the SU(N )
group in Yang-Mills theory. The only difference is that the group dimension and
signature are related to the ones of the space-time manifold M.
The need for a tetrad becomes also obvious when trying to construct the kinetic
part of the general-relativistic Dirac Lagrangian. One must turn the diffeomorphism
index of the derivative ∂ μ into a Lorentz index in order to contract with γ
a , i.e. the
combination γ
a e
μ
a ∂ μ ≡ γ
a
∂ a .
5 Moreover, since the Lorentz group now acts locally,
one must consider the covariant derivative in the Dirac representation
∇ μ ψ :=
∂ μ +
1
4
abμ γ
a
γ
b
ψ .
(3.2.74)
The Dirac Lagrangian reads
5 Another common choice is to define instead the Dirac matrix fields γ μ (x) := γ a e
μ
a (x), which
therefore obey the modified Clifford algebra {γ μ , γ ν } ≡ −2g μν , so that γ a e
μ
a ∂ μ ≡ γ μ ∂ μ , but all
this is only a matter of interpretation.