Gauge Symmetry in Quantum Mechanics

Gauge symmetry in Electromagnetism was recognized before the advent of quantum mechanics. We have seen that symmetries play a very important role in the quantum theory. Indeed, in quantum mechanics, gauge symmetry can be seen as the basis for electromagnetism and conservation of charge.

We know that the all observables are unchanged if we make a global change of the
phase of the wavefunction,
.
We could call this **global phase symmetry.**All relative phases (say for amplitudes to go through different slits in a diffraction
experiment) remain the same and no physical observable changes.
This is a symmetry in the theory which we already know about.
Let's postulate that there is a bigger symmetry and see what the consequences are.

That is, we can change the phase by a different amount at each point in spacetime and the physics will remain unchanged. This

Its clear that this transformation leaves
the **absolute square of the wavefunction the same**,
but what about the
Schrödinger equation?
It must also be unchanged.
The **derivatives in the Schrödinger equation**
will act on
changing the equation
unless we do something else to cancel the changes.

A little calculation shows that the equation remains unchanged if we also transform the potentials

This is just the **standard gauge transformation of electromagnetism**, but, we now see that
local phase symmetry of the wavefunction requires gauge symmetry for the
fields and indeed
even requires the existence of the EM fields to cancel terms in the Schrödinger
equation.
Electromagnetism is called a **gauge theory** because the gauge symmetry actually
defines the theory.
It turns out that the **weak and the strong interactions are also gauge theories** and, in
some sense, have the next simplest possible gauge symmetries after the one in Electromagnetism.

We will write our **standard gauge transformation** in the traditional way to conform
a bit better to the textbooks.

There are measurable

Then the old vector potential is then given by

Integrating this equation, we can write the function in terms of .

If we choose so that , then we have a

We can derive
the quantization of magnetic flux
by calculating the line integral of
around a closed loop in a field free region.

A good example of a region is a

This is due to the pairing of electrons inside a superconductor.

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