Mon., Nov. 22:
# Reiterated the realization of gauge-invariance in QM. See last week's minutes.
# Discussed energy in light of gauge invariance. The Hamiltonian is not gauge invariant...
rather we have (id_t -H')U = U(id_t - H), where U = exp(ie alpha(x,t)),
so its eigenvalues or expectation value  cannot always be the energy. In two gauges
related by a time-independent gauge parameter the scalar potential is the same,
only the vector potential changes, and  H' = UHU^-1. In this case H' and H are related
by a unitary transformation, so they have the same spectrum and expectation values
in corresponding states. The funny business comes when the gauge parameter
is time dependent. Then we must decide what we want to call the energy. For example,
in a constant electric field, in a time-independent gauge the electric field is minus the
gradient of the scalar potential. The Hamiltonian is time-independent, so its eigenvalues
and expectation values are conserved. In a gauge with the scalar potential zero and
the vector potential linear in time however, the Hamiltonian is time-dependent, hence not conserved.
The difference corresponds to whether or not the potential energy is included in the definition
of energy. If there EXISTS a gauge in which both the scalar and vector potentials are time independent,
then the Hamiltonian in this gauge defines a conserved energy. Moreover, any other gauge
in which the potentials are all time independent must be related by a gauge parameter of the
form a(x) + bt, where b is constant. Thus the scalar potential will differ only by a constant
shift, and H' is unitarily related to H up to a constant shift, H' = UHU^-1 - eb.
[By the way, in general relativity, energy is a source for gravity, so no ambiguity in the energy
can be tolerated. This is consistent with gauge symmetry, since the energy-momentum tensor
for matter fields is gauge-invariant, even thought the Hamiltonian is not.]
# Aharonov-Bohm effect revisited: we saw a while ago that the energy levels of a particle on a
ring can feel a magnetic flux through the ring: the energies are (n - Phi/Phi_0)^2 hbar^2/2mb^2.
I pointed out this is one way to have answered the homework problem from way back, involving
the unitarily inequivalent representation of the momentum operator on R^3 minus a cylinder.
That problem is identical to a situation in which there is a gauge field with vanishing magnetic
field but nontrivial loop integral around a closed curve: recall the ambiguity in the momentum
p_i was p_i -> p_i + f_i, with f_[i,j]=0. Identifying f_i with -(e/c)A_i, the two problems become
equivalent. Thus one can think of the quantization ambiguity as parametrized by the magnetic
flux that might be in the removed cylinder. As the ring example shows, this ambiguity has
physical consequences.
# The Aharonov-Bohm effect also operates on scattering amplitudes, not just energy levels.
This comes up in the operation of a SQUID (superconducting quantum interference device),
which is a lead into a current loop with two Josephson junctions, followed by a lead out.
To analyze the effect on the current  due to a magnetic field through the loop, we can use
gauge invariance. Consider the paths through the two sides of the loop, paths 1 and 2.
To a good approximation we can think of the electrons as coherently split between the
two paths, with no "back scattering". On each half of the loop the potential due to the additional
magnetic flux through the loop is pure gauge, so can be written as grad alpha_1,2. However,
there can be no single continuous gauge parameter that yields the vector potential on both paths.
If the input and output leads are at points p and q, and we choose alpha_1(p)=alpha_2(p), then
the magnetic flux though the loop is
\oint A.dl = \int_1 grad alpha_1.dl  - \int_2 grad alpha_2.dl  = alpha_1(q) - alpha_2(q).
That is, the discontinuity of alpha is the flux through the loop.
Now suppose the solution to the Schrodinger eqn without the extra field is a sum of pieces
that propagate around the two sides of the loop, psi_1 + psi_2. In the presence of the extra
flux, the Hamiltonian for the two sides is modified by the gauge transformation generated
by alpha_1,2 so the solution is modified by the phase factor exp(ie alpha_1,2) (with hbar=c=1).
Hence at q we have the solution
exp(ie alpha_1)psi_1 + exp(ie alpha_2)psi_2
= exp(ie alpha_2) [exp(i 2pi Phi/Phi_0) psi_1 + psi_2],
which exhibits an interference between the two paths that is modified by the extra flux enclosed,
modulo Phi_0.
# This effect leads to the fact that, in a squid, the maximum current oscillates with Phi as
 I_max = 2I_0 cos(2pi Phi/Phi_0). Since a small fraction of an oscillation can be resolved, a squid
can be a very sensitive magnetometer.

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Wed., Nov. 24:
# Zeeman effect: shift & splitting of energy levels of atoms in a uniform external magnetic field.
Except for super strong fields such as at the surface of a neutron star the diamagnetic term in the
Hamiltonian can be neglected for atomic electrons, so we have for B in the z-direction
H_Z = -mu.B = (mu_B B/hbar)[ L_z + 2 S_z]. This splits many of the otherwise degenerate
levels, since the external magnetic field breaks rotation invariance.
# Note that H_Z is gauge-invariant. Although when we identified it as the important term we used
a particular gauge (A = B x r/2), its gauge-invariance assures us that the fact that it is the important term
does not depend on the gauge we use.
# We saw previously that the terms in H_Z are of order a million smaller than one Rydberg for a magnetic
field of one Tesla, so the Zeeman effect is numerically small in an atom, but it is important since
experimenters can resolve it and can tune radio frequencies or laser beat frequencies to this resolution.
# It is fairly pointless to examine the effect of H_Z on the levels of a Coulomb potential, since physically
we also have fine and hyperfine structure which can be comparable to or larger than the Zeeman splitting.
So we'll come back to the Zeeman effect after we've treated these other effects.
# Magnetic monopoles: This topic is not essential, but it is interesting, and it plays an important role
in various theoretical developments of field theory,  and monopoles are predicted in some unified
gauge theory models. So let's take a look at it.
# Maxwell's eqn div B=0, if true in all space, implies via Gauss' theorem that the magnetic flux
through any surface vanishes, so that there are no magnetic monopoles. But what if there is a
singularity of some sort at the location of the monopole, so that div B is not zero there?
At first it seems that this is irrelevant, since we can argue that the magnetic flux though a
spherical surface vanishes without reference to the interior:
Divide the surface into northern and southern hemispheres, which meet at an equator.
Since B = curl A,  Stokes' theorem implies that  the northern flux is equal to the line integral of A
clockwise around the equator, while the southern flux is the line integral counterclockwise.
The sum of these vanishes, so the total flux must vanish.
# There is a way around this conclusion, first formulated by Dirac, and then given much later in a somewhat
less implausible form. In fact,  the two forms are actually equivalent. Let's first give Dirac's argument.
Dirac pointed out that the flux of the monopole could be fed in along an infinitesimally thin
flux tube---the Dirac string---that ends on the monopole.  In order not to be observable by the quantum mechanical phase the flux through the string must be an integer number of flux quanta Phi_0.
On the other hand, this flux is 4pi g, where g is the magnetic charge, so we have 4pi g = n Phi_0 = n hc/e.
Thus magnetic charge is quantized in units of e/(2 alpha) =~ (137/2) e.  This also implies that if there is even one magnetic pole, then all electric charges must also be quantized in units of e.
# The more plausible formulation of this abandons the string in favor of a non-globally defined vector potential. Going back to the northern and southern hemispheres, there is a northern vector potential
A_n and a southern vector potential A_s. The flux through the sphere is the line integral around the equator \oint (A_n - A_s).dl,  which would vanish if there were one global potential. All we really need to require however is that A_n and A_s are related by a gauge transformation, so A_n - A_s = grad alpha for some gauge parameter alpha. In this case, we have   Phi = \oint (A_n - A_s).dl = alpha(2pi) - alpha(0). If alpha is continuous this also vanishes, but  alpha need not be continuous.  Only the vector potential need be continuous. Hence it looks as if the flux Phi can be anything. But this disagrees with Dirac's quantization condition: we are leaving out the quantum charge, whose wave function also changes under the gauge transformation, by the phase factor exp(i(e/hbar c) alpha). The wave function, and hence this phase factor, must be continuous, hence the only allowable discontinuities of alpha are integer multiples of the flux quantum Phi_0 = hc/e. Therefore we arrive at the same conclusion as did Dirac: the flux of a magnetic monopole must be an integer multiple of Phi_0.
# Note that we need not even deal directly with the discontinuous field alpha. We can (and should) think of the gauge transformation as being specified by the continuous U(1) element (i.e. the phase factor)
U= exp(i(e/hbar c) alpha). We can think of U as the basic quantity, and of alpha as just an ambiguous logarithm of U. As derived in last week's minutes in the context of a general non-abelian
gauge transformation, the gauge transformation of A is given directly in terms of U by
A' = A + (- i hbar c/e) (grad U)  U^-1 = A + grad alpha.
# For any of this to be relevant there must be some place inside the sphere where div B is nonzero,
otherwise the flux of B can't but vanish by Gauss' theorem. Also, only if div B fails to vanish somewhere inside the sphere is it possible that there is no global vector potential for which B = curl A. As already mentioned, div B might not vanish if there is a singularity at the monopole, but this is not a physical model.
A more interesting case is when the electromagnetic U(1) is embedded in a larger Yang-Mills gauge group. Then the magnetic field is only part of a nonabelian gauge field that satisfies some generalization of Maxwell's equation. It can happen that there is an extended static solution to the Yang-Mills equations, a soliton, in which the other fields are nonzero and div B is nonzero but related to these other fields inside the soliton. The above argument shows that in this situation, the soliton may have magnetic charge, but that charge must be quantized in units of 4pi/Phi_0.

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