3Symmetries of the path integral

III Advanced Quantum Field Theory

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.

Suppose we had a variation

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

d−1

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

n−1

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!