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.
Its clear that this transformation leaves
the absolute square of the wavefunction the same,
but what about the
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.
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.
We can derive
the quantization of magnetic flux
by calculating the line integral of
around a closed loop in a field free region.
The Aharanov Böhm Effect brings us back to the two slit diffraction experiment but adds magnetic fields.
Jim Branson 2013-04-22