130_notes.dvi

This is just thestandard 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 Schr ̈odinger equation. Electromagnetism
is called agauge theorybecause the gauge symmetry actually defines the theory. It turns out that
theweak and the strong interactions are also gauge theoriesand, in some sense, have the
next simplest possible gauge symmetries after the one in Electromagnetism.

We will write ourstandard gauge transformationin the traditional way to conform a bit better
to the textbooks.

A~ → A~−∇~f(~r,t)

φ → φ+

1

c

∂f(~r,t)

∂t

ψ(~r,t) → e−i

̄hcef(~r,t)

ψ(~r,t)

There are measurablequantum physics consequencesof this symmetry. We can understand a
number of them by looking at thevector potential in a field free regions. IfB= 0 thenA~can
be written as the gradient of a functionf(~r). To be specific, take our gauge transformation of the
vector potential. Make a gauge transformation such thatA~′= 0. This of course is still consistent
withB~= 0.

A~′=A~−∇~f(~r) = 0

Then the old vector potential is then given by

A~=∇~f(~r).

Integrating this equation, we can write the functionf(~r) in terms ofA~(~r).

∫~r

~r 0

d~r·A~=

∫~r

~r 0

d~r·∇~f=f(~r)−f(~r 0 )

If we choosefso thatf(~r 0 ) = 0, then we have avery useful relation between the gauge
function and the vector potential in a field free region.

f(~r) =

∫~r

~r 0

d~r·A.~

We canderive(see section 20.5.8)the quantization of magnetic flux by calculating the line integral
ofA~around a closed loop in a field free region.

Φ =

2 nπ ̄hc

e

A good example of aB= 0 region is asuperconductor. Magnetic flux is excluded from the
superconducting region. If we have a superconducting ring, we have a B=0 region surrounding some
flux. We have shown then, that the flux going through a ring of superconductor is quantized.