4Spinors
III Quantum Field Theory
4.3 Properties of the spin representation
We have produced a representation of the Lorentz group, which acts on some
vector space V
∼
=
R
4
. Its elements are known as Dirac spinors.
Definition
(Dirac spinor)
.
A Dirac spinor is a vector in the representation
space of the spin representation. It may also refer to such a vector for each point
in space.
Our ultimate goal is to construct an action that involves a spinor. So we
would want to figure out a way to get a number out of a spinor.
In the case of a 4-vector, we had these things called covectors that lived in the
“dual space”. A covector
λ
can eat up a vector
v
and spurt out a number
λ
(
v
).
Often, we write the covector as
λ
µ
and the vector as
v
µ
, and then
λ
(
v
) =
λ
µ
v
µ
.
When written out like a matrix, a covector is represented by a “row vector”.
Under a Lorentz transformation, these objects transform as
λ 7→ λΛ
−1
v 7→ Λv
(What do we mean by
λ 7→ λ
Λ
−1
? If we think of
λ
as a row vector, then this is
just matrix multiplication. However, we can think of it without picking a basis
as follows —
λ
Λ
−1
is a covector, so it is determined by what it does to a vector
v. We then define (λΛ
−1
)(v) = λ(Λ
−1
v))
Then the result λv transforms as
λv 7→ λΛ
−1
Λv = λv.
So the resulting number does not change under Lorentz transformations. (Math-
ematically, this says a covector lives in the dual representation space of the
Lorentz group)
Moreover, given a vector, we can turn it into a covector in a canonical way,
by taking the transpose and then inserting some funny negative signs in the
space components.
We want to do the same for spinors. This terminology may or may not be
standard:
Definition
(Cospinor)
.
A cospinor is an element in the dual space to space of
spinors, i.e. a cospinor
X
is a linear map that takes in a spinor
ψ
as an argument
and returns a number
Xψ
. A cospinor can be represented as a “row vector” and
transforms under Λ as
X 7→ XS[Λ]
−1
.
This is a definition we can always make. The hard part is to produce some
actual cospinors. To figure out how we can do so, it is instructive to figure out
what S[Λ]
−1
is!
We begin with some computations using the γ
µ
matrices.
Proposition. We have
γ
0
γ
µ
γ
0
= (γ
µ
)
†
.
Proof. This is true by checking all possible µ.
Proposition.
S[Λ]
−1
= γ
0
S[Λ]
†
γ
0
,
where
S
[Λ]
†
denotes the Hermitian conjugate as a matrix (under the usual basis).
Proof. We note that
(S
µν
)
†
=
1
4
[(γ
ν
)
†
, (γ
µ
)
†
] = −γ
0
1
4
[γ
µ
, γ
ν
]
γ
0
= −γ
0
S
µν
γ
0
.
So we have
S[Λ]
†
= exp
1
2
Ω
µν
(S
µν
)
†
= exp
−
1
2
γ
0
Ω
µν
S
µν
γ
0
= γ
0
S[Λ]
−1
γ
0
,
using the fact that (
γ
0
)
2
=
1
and
exp
(
−A
) = (
exp A
)
−1
. Multiplying both sides
on both sides by γ
0
gives the desired formula.
We now come to our acclaimed result:
Proposition. If ψ is a Dirac spinor, then
¯
ψ = ψ
†
γ
0
is a cospinor.
Proof.
¯
ψ transforms as
¯
ψ 7→ ψ
†
S[Λ]
†
γ
0
= ψ
†
γ
0
(γ
0
S[Λ]
†
γ
0
) =
¯
ψS[Λ]
−1
.
Definition
(Dirac adjoint)
.
For any Dirac spinor
ψ
, its Dirac adjoint is given
by
¯
ψ = ψ
†
γ
0
.
Thus we immediately get
Corollary.
For any spinor
ψ
, the quantity
¯
ψψ
is a scalar, i.e. it doesn’t transform
under a Lorentz transformation.
The next thing we want to do is to construct 4-vectors out of spinors. While
the spinors do have 4 components, they aren’t really related to 4-vectors, and
transform differently. However, we do have that thing called
γ
µ
, and the indexing
by
µ
should suggest that
γ
µ
transforms like a 4-vector. Of course, it is a collection
of matrices and is not actually a 4-vector, just like
∂
µ
behaves like a 4-vector
but isn’t. But it behaves sufficiently like a 4-vector and we can combine it with
other things to get actual 4-vectors.
Proposition. We have
S[Λ]
−1
γ
µ
S[Λ] = Λ
µ
ν
γ
ν
.
Proof. We work infinitesimally. So this reduces to
1 −
1
2
Ω
ρσ
S
ρσ
γ
µ
1 +
1
2
Ω
ρσ
S
ρσ
=
1 +
1
2
Ω
ρσ
M
ρσ
µ
ν
γ
ν
.
This becomes
[S
ρσ
, γ
µ
] = −(M
ρσ
)
µ
ν
γ
ν
.
But we can use the explicit formula for M to compute
−(M
ρσ
)
µ
ν
γ
ν
= (η
σµ
δ
ρ
ν
− η
ρµ
δ
σ
ν
)γ
ν
= γ
ρ
η
σµ
− γ
σ
η
ρµ
,
and we have previously shown this is equal to [S
ρσ
, γ
µ
].
Corollary.
The object
¯
ψγ
µ
ψ
is a Lorentz vector, and
¯
ψγ
µ
γ
ν
ψ
transforms as a
Lorentz tensor.
In
¯
ψγ
µ
γ
ν
ψ
, the symmetric part is a Lorentz scalar and is proportional to
η
µν
¯
ψψ
, and the anti-symmetric part transforms as an antisymmetric Lorentz
tensor, and is proportional to
¯
ψS
µν
ψ.