3Symmetries of the path integral

3 Symmetries of the path integral
From now on, we will work with quantum field theory in general, and impose no
restrictions on the dimension of our universe. The first subject to study is the
notion of symmetries.
We first review what we had in classical field theory. In classical field theory,
Noether’s theorem relates symmetries to conservation laws. For simplicity, we
will work with the case of a flat space.
δφ = εf(φ, φ)
of the field. The most common case is when
f
(
φ, φ
) depends on
φ
only locally,
in which case we can think of the transformation as being generated by the
vector
V
f
=
Z
M
d
d
x f(φ, φ)
δ
δφ(x)
acting on the “space of fields”.
If the function
S
[
φ
] is invariant under
V
f
when
ε
is constant, then for general
ε(x), we must have
δS =
Z
d
d
x j
µ
(x)
µ
ε.
for some field-dependent current
j
µ
(
x
) (we can actually find an explicit expression
for j
µ
). If we choose ε(x) to have compact support, then we can write
δS =
Z
d
d
x (
µ
j
µ
) ε(x).
On solutions of the field equation, we know the action is stationary under
arbitrary variations. So δS = 0. Since ε(x) was arbitrary, we must have
µ
j
µ
= 0.
So we know that j
µ
is a conserved current.
Given any such conserved current, we can define the charge
Q
[
N
] associated
to an (oriented) co-dimension 1 hypersurface N as
Q[N] =
Z
N
n
µ
j
µ
d
d1
x,
where
n
µ
is the normal vector to
N
. Usually,
N
is a time slice, and the normal
points in the future direction.
Now if
N
0
and
N
1
are two such hypersurfaces bounding a region
M
0
M
,
then by Stokes’ theorem, we have
Q[N
0
] Q[N
1
] =
Z
N
0
Z
N
1
n
µ
j
µ
d
n1
x =
Z
M
0
(
µ
j
µ
) d
n
x = 0.
So we find
Q[N
0
] = Q[N
1
].
This is the conservation of charge!