What do we mean when we say the QM wave function is a section of the $U(1)$ bundle?

Question

I have a couple questions here. To keep the discussion simple lets stick to the following case: what is the quantum mechanics of a single particle in the presence of a background EM field, such as that of a monopole, where multiple coordinate patches are required to define the vector field everywhere without singularity. I have learned about the geometric formulation of EM - where the field $B$ of a monopole is described as a real-valued two-form that is closed but not exact. I also know about using the vector potential as a connection for the complex wavefunctions, using the usual covariant derivative.

The questions I have in mind are

1. What (complex line) bundle is the wave function a section of?
2. What are the bundle data (i.e. transition functions) that specify this?
3. In what way can we see the monopole vector potential as a connection on this bundle?
4. People seem to talk about the monopole bundle only with the monopole connection in mind - what would it look like to have a different connection instead?

On a similar note, I've seen physicists refer to the classification of bundles, which there is one for every integer (based on the homology of $\mathbf{R}^3-\{0\}$), and call that integer the monopole number. But I've also seen people call the flux integral of the magnetic field two form over some closed surface, which relates instead to the curvature of some connection on the bundle - which should be quantized by Gauss-Bonnet theorem - the monopole number. Is there some reason that these are the same?

I was reading a great set of lecture notes around 10 pages which explained the need to use a section of a bundle for the wave function and the relation to the Dirac quantization condition, but I can no longer find them. Links to related resources would be very welcome.

Answer 1

This $U(1)$ is just the gauge symmetry of electromagnetism. For a wave function of a charged particle, the $U(1)$ simply acts by changing the phase of the wave function. The "section" means that we actually want the phase at each point to be determined. But because there's the $U(1)$ symmetry, the phase is largely arbitrary. However, an arbitrary position-dependent change of the phase doesn't really create an equivalent physical state because the variations are remembered in the changes of the gauge field – in mathematical terminology, the information is stored in the "connection on the fiber bundle".

In the presence of magnetic monopoles, the vector potential $\vec A$ can't be globally defined because $\vec B={\rm curl}\,\vec A$ automatically obeys ${\rm div}\,\vec B=0$ but this equation is violated in the presence of magnetic monopole charges because ${\rm div}\,\vec B=q_m\delta^{(3)}(\vec x)$. However, it is still possible to define $\vec A$ almost everywhere around a point-like magnetic monopole except for a semi-infinite string (line) starting at the origin (the location of the monopole), the so-called Dirac string.

This is equivalent to replacing the monopole by a long magnetic dipole connecting the original monopole with the opposite pole that is sent to infinity. Because the opposite pole is at infinity, it becomes immaterial. However, the long solenoid (magnet) connecting the two poles must be invisible as well. A necessary condition for that is that the Aharonov-Bohm effect around this solenoid produces no detectable phase shift, and this is equivalent to the Dirac quantization rule for the magnetic charge, essentially $q_m q_e\in 2\pi{\mathbb Z}$.

In mathematical terminology, the possible transformation of the phase of the wave function induced by the trip around the Dirac string is why we need to talk about "bundles": it is impossible to set $\vec A$ equal to zero everywhere so even though there is no magnetic source anywhere in the space (except for the origin), we still can't assign a natural unique phase to the wave function in the bulk of the space.

In units of the elementary magnetic monopole charge, $q_m$ – the monopole number – may be expressed either as the coefficient of $\delta^{(3)}(\vec X)$ in ${\rm div}\,\vec B$, or – which is the same by Gauss' theorem – as the surface integral $\int d\vec S\cdot \vec B$ with the right convention-dependent coefficient. Also, because all the "nontriviality" of the bundle may be concentrated to the Dirac string, the only place where $\vec A$ isn't well-defined, the magnetic flux may be reduced to the integral over a small cross section cutting the Dirac string, and therefore $q_m$ is expressed from monodromies around the Dirac string which tells you how much the fiber bundle is twisted.

All these things are the same. To see why they're the same, it's useful to realize that physicists try to "trivialize" the fiber bundle and they almost succeed, except for the Dirac string where $\vec A$ isn't well-defined. However, $\vec B$ is defined everywhere except for the origin. So anything about the nontriviality of the bundle must be encoded in the gauge-invariant functionals of $\vec A$, namely in contour integrals $\oint d\vec \ell\cdot \vec A$ and surface integrals of $\vec B$. By the usual trivial Gauss-like theorems, they give the same number for the magnetic monopole configuration.

Concerning the updated shortlist of questions,

1. the wave function is a section of a complex bundle with structure group $U(1)$ which is the gauge group
2. the gauge transformations specify the transition functions between patches; the patches should be diffeomorphic to balls but in reality, we can make the main patch as large as the whole space minus the Dirac string
3. the electromagnetic potential is always called the connection on the (gauge symmetry) bundle by the mathematicians; the inability to define $\vec A$ globally is why the bundle is non-trivial
4. the value of the connection $\vec A$ can be anything that obeys Maxwell's equations. In 3+1D, magnetic monopoles are the only localized sources that can make the bundle nontrivial so every general potential is a superposition of the well-defined $\vec A$ and the $\vec A$ from a distribution of magnetic monopoles.

See also e.g.

can one introduce magnetic monopoles without Dirac strings?