3Discrete symmetries
III The Standard Model
3.2 Parity
Our objective is to figure out how
ˆ
P = W (P, 0)
acts on our different quantum fields. For convenience, we will write
x
µ
7→ x
µ
P
= (x
0
, −x)
p
µ
7→ p
µ
P
= (p
0
, −p).
As before, we will need to make some assumptions about how
ˆ
P
behaves.
Suppose our field has creation operator a
†
(p). Then we might expect
|pi 7→ η
∗
a
|p
P
i
for some complex phase η
∗
a
. We can alternatively write this as
ˆ
P a
†
(p) |0i = η
∗
a
a
†
(p
P
) |0i.
We assume the vacuum is parity-invariant, i.e.
ˆ
P |0i
=
|0i
. So we can write
this as
ˆ
P a
†
(p)
ˆ
P
−1
|0i = η
∗
a
a
†
(p
P
) |0i.
Thus, it is natural to make the following assumption:
Assumption.
Let
a
†
(
p
) be the creation operator of any field. Then
a
†
(
p
)
transforms as
ˆ
P a
†
(p)
ˆ
P
−1
= η
∗
a
†
(p
P
)
for some η
∗
. Taking conjugates, since
ˆ
P is unitary, this implies
ˆ
P a(p)
ˆ
P
−1
= ηa(p
P
).
We will also need to assume the following:
Assumption.
Let
φ
be any field. Then
ˆ
P φ
(
x
)
ˆ
P
−1
is a multiple of
φ
(
x
P
), where
“a multiple” can be multiplication by a linear map in the case where
φ
has more
than 1 component.
Scalar fields
Consider a complex scalar field
φ(x) =
X
p
a(p)e
−ip·x
+ c(p)
†
e
+ip·x
,
where
a
(
p
) is an annihilation operator for a particle and
c
(
p
)
†
is a creation
operator for the anti-particles. Then we have
ˆ
P φ(x)
ˆ
P
−1
=
X
p
ˆ
P a(p)
ˆ
P
−1
e
−ip·x
+
ˆ
P c
†
(p)
ˆ
P
−1
e
+ip·x
=
X
p
η
∗
a
a(p
P
)e
−ip·x
+ η
∗
c
c
†
(p
P
)e
+ip·x
Since we are integrating over all p, we can relabel p
P
↔ p, and then get
=
X
p
η
a
a(p)e
−ip
P
·x
+ η
∗
c
c
†
(p)e
+ip
P
·x
We now note that x · p
P
= x
P
· p by inspection. So we have
=
X
p
η
a
a(p)e
−ip·x
P
+ η
∗
c
c
†
(p)e
ip·x
P
.
By assumption, this is proportional to
φ
(
x
P
). So we must have
η
a
=
η
∗
c
≡ η
P
.
Then we just get
ˆ
P φ(x)
ˆ
P
−1
= η
P
φ(x
P
).
Definition
(Intrinsic parity)
.
The intrinsic parity of a field
φ
is the number
η
P
∈ C such that
ˆ
P φ(x)
ˆ
P
−1
= η
P
φ(x
P
).
For real scalar fields, we have
a
=
c
, and so
η
a
=
η
c
, and so
η
a
=
η
P
=
η
∗
P
.
So η
P
= ±1.
Definition
(Scalar and pseudoscalar fields)
.
A real scalar field is called a
scalar field (confusingly) if the intrinsic parity is
−
1. Otherwise, it is called a
pseudoscalar field.
Note that under our assumptions, we have
ˆ
P
2
=
I
by the composition rule
of the
W
(Λ
, a
). Hence, it follows that we always have
η
P
=
±
1. However, in
more sophisticated treatments of the theory, the composition rule need not hold.
The above analysis still holds, but for a complex scalar field, we need not have
η
P
= ±1.
Vector fields
Similarly, if we have a vector field V
µ
(x), then we can write
V
µ
(x) =
X
p,λ
E
µ
(λ, p)a
λ
(p)e
−ip·x
+ E
µ∗
(λ, p)c
†λ
(p)e
ip·x
.
where E
µ
(λ, p) are some polarization vectors.
Using similar computations, we find
ˆ
P V
µ
ˆ
P
−1
=
X
p,λ
E
µ
(λ, p
P
)a
λ
(p)e
−ip·x
P
η
a
+ E
µ∗
(λ, p
P
)c
†λ
(p)e
+ip·x
P
η
∗
c
.
This time, we have to deal with the
E
µ
(
λ, p
P
) term. Using explicit expressions
for E
µ
, we have
E
µ
(λ, p
P
) = −P
µ
ν
E
ν
(λ, p).
So we find that
ˆ
P V
µ
(x
P
)
ˆ
P
−1
= −η
P
P
µ
ν
V
ν
(x
P
),
where for the same reasons as before, we have
η
P
= η
a
= η
∗
c
.
Definition
(Vector and axial vector fields)
.
Vector fields are vector fields with
η
P
= −1. Otherwise, they are axial vector fields.
Dirac fields
We finally move on to the case of Dirac fields, which is the most complicated.
Fortunately, it is still not too bad.
As before, we obtain
ˆ
P ψ(x)
ˆ
P
−1
=
X
p,s
η
b
b
s
(p)u(p
P
)e
−p·x
P
+ η
∗
d
d
s†
(p)v
s
(p
P
)e
+ip·x
P
.
We use that
u
s
(p
P
) = γ
0
u
s
(p), v
s
(p
P
) = −γ
0
v
s
(p),
which we can verify using Lorentz boosts. Then we find
ˆ
P ψ(x)
ˆ
P
−1
= γ
0
X
p,s
η
b
b
s
(p)u(p)e
−p·x
P
− η
∗
d
d
s†
(p)v
s
(p)e
+ip·x
P
.
So again, we require that
η
b
= −η
∗
d
.
We can see this minus sign as saying particles and anti-particles have opposite
intrinsic parity.
Unexcitingly, we end up with
ˆ
P ψ(x)
ˆ
P
−1
= η
P
γ
0
ψ(x
P
).
Similarly, we have
ˆ
P
¯
ψ(x)
ˆ
P
−1
= η
∗
P
¯
ψ(x
P
)γ
0
.
Since γ
0
anti-commutes with γ
5
, it follows that we have
ˆ
P ψ
L
ˆ
P
−1
= η
P
γ
0
ψ
R
.
So the parity operator exchanges left-handed and right-handed fermions.
Fermion bilinears
We can now determine how various fermions bilinears transform. For example,
we have
¯
ψ(x)ψ(x) 7→
¯
ψ(x
P
)ψ(x
P
).
So it transforms as a scalar. On the other hand, we have
¯
ψ(x)γ
5
ψ(x) 7→ −
¯
ψ(x
P
)γ
5
ψ(x
P
),
and so this transforms as a pseudoscalar. We also have
¯
ψ(x)γ
µ
ψ(x) 7→ P
µ
ν
¯
ψ(x
P
)γ
ν
ψ(x
P
),
and so this transforms as a vector. Finally, we have
¯
ψ(x)γ
5
γ
µ
ψ(x) 7→ −P
µ
0
¯
ψ(x
P
)γ
5
γ
µ
ψ(x
P
).
So this transforms as an axial vector.