4Spinors
III Quantum Field Theory
4.6 Parity operator
So far, we’ve considered only continuous transformations, i.e. transformations
continuously connected to the identity. These are those that preserve direction
of time and orientation. However, there are two discrete symmetries in the full
Lorentz group — time reversal and parity:
T : (t, x) 7→ (−t, x)
P : (t, x) 7→ (t, −x)
Since we defined the spin representation via exponentiating up infinitesimal
transformations, it doesn’t tell us what we are supposed to do for these discrete
symmetries.
However, we do have some clues. Recall that we figured that the
γ
µ
trans-
formed like 4-vectors under continuous Lorentz transformations. So we can
postulate that
γ
µ
also transforms like a 4-vector under these discrete symmetries.
We will only do it for the parity transformation, since they behave interestingly
for spinors. We will suppose that our parity operator acts on the γ
µ
as
P : γ
0
7→ γ
0
γ
i
7→ −γ
i
.
Because of the Clifford algebra relations, we can write this as
P : γ
µ
7→ γ
0
γ
µ
γ
0
.
So we see that
P
is actually conjugating by
γ
0
(note that (
γ
0
)
−1
=
γ
0
), and this
is something we can generalize to everything. Since all the interesting matrices
are generated by multiplying and adding the
γ
µ
together, all matrices transform
via conjugation by γ
0
. So it is reasonable to assume that P is γ
0
.
Axiom. The parity operator P acts on the spinors as γ
0
.
So in particular, we have
Proposition.
P : ψ 7→ γ
0
ψ, P :
¯
ψ 7→
¯
ψγ
0
.
Proposition. We have
P : γ
5
7→ −γ
5
.
Proof.
The
γ
1
, γ
2
, γ
3
each pick up a negative sign, while
γ
0
does not change.
Now something interesting happens. Since
P
switches the sign of
γ
5
, it
exchanges P
+
and P
−
. So we have
Proposition. We have
P : P
±
7→ P
∓
.
In particular, we have
P ψ
±
= ψ
∓
.
As
P
still acts as right-multiplication-by-
P
−1
on the cospinors, we know
that scalar quantities etc are still preserved when we act by
P
. However, if we
construct something with
γ
5
, then funny things happen, because
γ
5
gains a sign
when we transform by P . For example,
P :
¯
ψγ
5
ψ 7→ −
¯
ψγ
5
ψ.
Note that here it is important that we view
γ
5
as a fixed matrix that does
not transform, and
P
only acts on
¯
ψ
and
ψ
. Otherwise, the quantity does not
change. If we make
P
act on everything, then (almost) by definition the resulting
quantity would remain unchanged under any transformation. Alternatively, we
can think of P as acting on γ
5
and leaving other things fixed.
Definition
(Pseudoscalar)
.
A pseudoscalar is a number that does not change
under Lorentz boosts and rotations, but changes sign under a parity operator.
Similarly, we can look at what happens when we apply
P
to
¯
ψγ
5
γ
µ
ψ
. This
becomes
¯
ψγ
5
γ
µ
ψ 7→
¯
ψγ
0
γ
5
γ
µ
γ
0
ψ =
(
−
¯
ψγ
5
γ
µ
ψ µ = 0
¯
ψγ
5
γ
µ
ψ µ 6= 0
.
This is known as an axial vector.
Definition
(Axial vector)
.
An axial vector is a quantity that transforms as
vectors under rotations and boosts, but gain an additional sign when transforming
under parity.
Type Example
Scalar
¯
ψψ
Vector
¯
ψγ
µ
ψ
Tensor
¯
ψS
µν
ψ
Pseudoscalar
¯
ψγ
5
ψ
Axial vector
¯
ψγ
5
γ
µ
ψ
We can now add extra terms to
L
that use
γ
5
. These terms will typically break
the parity invariance of the theory. Of course, it doesn’t always break parity
invariance, since we can multiply two pseudoscalars together to get a scalar.
It turns out nature does use
γ
5
, and they do break parity. The classic example
is a
W
-boson, which is a vector field, which couples only to left-handed fermions.
The Lagrangian is given by
L = ··· +
g
2
W
µ
¯
ψγ
µ
(1 − γ
5
)ψ,
where (1 − γ
5
) acts as the left-handed projection.
A theory which puts
ψ
±
on an equal footing is known as vector-like. Otherwise,
it is known as chiral.