## QED

### Gauge Invariance

__Gauge Invariance in Classical Electrodynamics__

__Phase invariance of the Dirac field__

GLOBAL PHASE INVARIANCE

In the following the Lagrangian of a free Dirac field describing a particle with mass m
is considered:
It is possible to change the absolute phase of the Dirac field without changing
the physics, i.e. the Lagrangian is invariant under the transformation
Also the E.o.M., that is the Dirac equation, is invariant under such a transformation:

LOCAL PHASE INVARIANCE

What happens if one chooses a different phase at a different time-space coordinate?

The field is changed then by
and the derivative in the Dirac equation becomes
As a consequence, the last term spoils the invariance under a local phase transformation.

Local phase invariance can be achieved however by the following replacement:
where e is the charge of the particle described by the field ψ and the new field
A_{μ} transforms under the phase transformation as
which looks like a gauge transformation of the electromagnetic four-potential.

Hence, one has to pay a price in order to achieve local phase invariance. A price however
which is very welcome: Local phase invariance is not possible for a free particle field
but one needs to introduce an additional field which couples to the particle via the
(electric) charge as the electromagnetic field does.

This is an example how a fundamental interaction (here electrodynamics) can be introduced
by the dynamical principle of LOCAL PHASE INVARIANCE or LOCAL GAUGE INVARIANCE!

GAUGE INVARIANCE IN QUANTUM MECHANICS

One could have chosen another point of view and started with the equation describing
a Dirac particle with charge -e interacting with an external electromagnetic
four-potential.
In this case the Dirac equation reads with the generalized momentum
p^{μ} + e A^{μ}:
One can then change the four-potential applying a gauge transformation:
The Dirac equation is only consistent with electrodynamics, that is, invariant under
such a gauge transformation, if the Dirac field is simultaneously changed by a local
phase transformation:

(NON-)ABELIAN GAUGE GROUPS

The local phase transformations discussed so far are forming a group: U(1).

As a consequence, the order of two subsequent local phase transformations does not
matter, that is, they commutate. Hence, QED is said to be an ABELIAN LOCAL GAUGE
THEORY.

Weak interations and QCD are examples of NON-ABELIAN GAUGE THEORIES. The main
consequence of non-commutating local phase transformations is the fact that
the gauge fields (W,Z and gluons, respectively) are carrying charges themselves
resulting in self-interactions of the gauge fields.

__The Aharanov-Bohm effect__