2Free field theory

III Quantum Field Theory



2.4 Complex scalar fields
What happens when we want to talk about a complex scalar field? A classical
free complex scalar field would have Lagrangian
L =
µ
ψ
µ
ψ µ
2
ψ
ψ.
Again, we want to write the quantized
ψ
as an integral of annihilation and
creation operators indexed by momentum. In the case of a real scalar field, we
had the expression
φ(x) =
Z
d
3
p
(2π)
3
1
p
2E
p
a
p
e
ip·x
+ a
p
e
ip·x
We needed the “coefficients” of
e
ip·x
and those of
e
ip·x
to be
a
p
and its
conjugate
a
p
so that the resulting
φ
would be real. Now we know that our
ψ
is a complex quantity, so there is no reason to assert that the coefficients are
conjugates of each other. Thus, we take the more general decomposition
ψ(x) =
Z
d
3
p
(2π)
3
1
p
2E
p
(b
p
e
ip·x
+ c
p
e
ip·x
)
ψ
(x) =
Z
d
3
p
(2π)
3
1
p
2E
p
(b
p
e
ip·x
+ c
p
e
ip·x
).
Then the conjugate momentum is
π(x) =
Z
d
3
p
(2π)
3
i
r
E
p
2
(b
p
e
ip·x
c
p
e
ip·x
)
π
(x) =
Z
d
3
p
(2π)
3
(i)
r
E
p
2
(b
p
e
ip·x
c
p
e
ip·x
).
The commutator relations are
[ψ(x), π(y)] = [ψ
(x), π
(y)] =
3
(x y),
with all other commutators zero.
Similar tedious computations show that
[b
p
, b
q
] = [c
p
, c
q
] = (2π)
3
δ
3
(p q),
with all other commutators zero.
As before, we can find that the number operators
N
c
=
Z
d
3
p
(2π)
3
c
p
c
p
, N
b
=
Z
d
3
p
(2π)
3
b
p
b
p
are conserved.
But in the classical case, we had an extra conserved charge
Q = i
Z
d
3
x (
˙
ψ
ψ ψ
˙
ψ) = i
Z
d
3
x (πψ ψ
π
).
Again by tedious computations, we can replace the
π
and
ψ
with their operator
analogues, expand everything in terms of
c
p
and
b
p
, throw away the pieces of
infinity by requiring normal ordering, and then obtain
Q =
Z
d
3
p
(2π)
3
(c
p
c
p
b
p
b
p
) = N
c
N
b
,
Fortunately, after quantization, this quantity is still conserved, i.e. we have
[Q, H] = 0.
This is not a big deal, since
N
c
and
N
b
are separately conserved. However, in
the interacting theory, we will find that
N
c
and
N
b
are not separately conserved,
but Q still is.
We can think of
c
and
b
particles as particle and anti-particle, and
Q
computes
the number of particles minus the number of antiparticles. Looking back, in the
case of a real scalar field, we essentially had a system with
c
=
b
. So the particle
is equal to its anti-particle.