1. Gauge Invariance and Conserved Quantities
“Noether's theorem” was proven by German mathematician, Emmy Noether in 1915 and published in Noether's theorem has become a fundamental tool of quantum field theory – and has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics".
Amalie Emmy Noether

2. Consider the charged scalar field,
and the following transformation on ,
where  is a constant.

3. =1
Now suppose that  can be varied continuously – so that we have
an infinite number of small, continuous  for which
 exp(i )  = ’ and L  L.
The set of all these transformations, U = exp(i ), form a group of
operators. It is called U(1), a unitary group (because U*U = 1)
Here the Hermitian conjugate is the complex conjugate
because U is not a matrix.

4. Now, we have a more astounding result: we can vary the (complex) phase
of the field operator, , everywhere in space by any continuous amount
and not affect the “laws of physics” (that is the L) which govern the system!
Note that everywhere in space the phase changes by the same .
This is called a global symmetry. ’ 
Remember Emmy Noether!

5. With the help of Emmy Noether, we can prove that
charge is conserved!

6. Deriving the conserved current and the conserved charge:
Euler-Lagrange equation
conserved current
But our Lagrangian density also contains a *, so we obtain additional terms
like the above, with  replaced by *. In each case the Euler –Lagrange equations
are satisfied. So, the remaining term is as follows:

7. The conserved current condition is
To within an overall constant the conserved current is :
a four – vector!

8. Now we need to evaluate .
The great advantage of  being a continuous constant is that there
are an infinite number of very small  which carry with them all the
physics of the “continuity”. That is, with no loss of rigor we can
assume  is small!

9. Finally, the conserved current operator (to within an overall constant)
for the charged, spin = 0, particle is
Since we may adjust the overall constant to reflect the charge
of the particle, we can replace  with q. The formalism gives
operator for charge, but not the numerical value.

10. The value of the charge is calculated from:
integrate
over time
S0(t)
incoming
particle
outgoing
particle
integrate over all space
One obtains a number! p
incoming
outgoing

11. In this calculation

12. Note Dirac delta function in k’ and p‘

13. Note Dirac delta function in k and p‘

14. Next, we can do the integrations over d3x
Next, we can do the integrations over d3x. Each gives a Dirac delta function in k and k’.

15. Note: + and/or – must be together.
The time disappears! Q is time independent.
The integration over k’ is done
with the Dirac delta function
from the d3x integration.
The remaining integration over k will be done with the Dirac delta functions
from the commutation relations.
Note: + and/or – must be together.