There is a symmetry in physics which we might call the
**Local Phase Symmetry** in quantum mechanics.
In this symmetry we change the phase of the (electron) wavefunction by a different
amount everywhere in spacetime.
To compensate for this change, we need to also make a
gauge transformation
of the electromagnetic potentials.
They all must go together like this.

The local phase symmetry requires that Electromagnetism exist and have a gauge symmetry so that we can keep the Schrödinger Equation invariant under this phase transformation.

We exploit the gauge symmetry in EM to show that, in **field free regions**,
the function
can be simply equal to a line integral of the vector potential
(if we pick the right gauge).

We use this to show that the magnetic flux enclosed by a superconductor is quantized.

We also show that magnetic fields can be used to change interference effects in quantum mechanics.
The **Aharanov Böhm Effect** brings us back to the two slit diffraction experiment
but adds magnetic fields.

The relative phase from the two slits depends on the flux between the slits. By varying the field, we will

Jim Branson 2013-04-22