- 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
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
Aμ 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
In this case the Dirac equation reads with the generalized momentum
pμ + e Aμ:
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
(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
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