If I've managed to finally understand what the gist of Bernstein and Phillips article is about, here's what I think it is. The geometry of the gauge field (read: field of phase changes) concerned looks like a truncated cone capped with a spherical dome. That means, there is a hemisphere which has a line of latitude extended to a surface with constant curvature. So the bundle of fibers from that circle of latitude and lying over the conical surface (a section of the total space), has planes 'stacked' along each fiber at a constant angle, each electron sees a 'wide' ramp it transports the direction of its phase angle around, in the connection. The curvature of the connection corresponds (topologically) to the solenoid field in the A-B experiment, and the shift in phase corresponds to the angular excess of a geodesic around the truncated cone (this unrolls to a straight line in the plane, and all the phase vectors are parallel there). So what that actually says about what a gauge field is in terms of electrons interacting with a field potential, I still can't say in so many words. But the electrons encounter a spacetime which is curved by the presence of a potential, the electromagnet is shielded and there is no magnetic field (?). Another clue there is that the spherical part of the curved space corresponds to that part of the spacetime that does contain a magnetic field (i.e. is inside the shield).

One other thing the article mentions is that the base space in the A-B experiment (indeed, in any quantum experiment) is three-dimensional. The total space is the set of fibers over each point in the base, and each fiber is a circle just as in the bundle of directions over a surface. It's much easier to take a plane slice through the A-B experiment, perpendicular to the solenoid, and to then consider the three dimensional bundle of phases over this slice (of ordinary three dimensional space). In the A-B setup, a beam of electrons is split by a charged wire (the dimensions of the wires and the solenoid are quite small, the solenoid itself is less than 1/7 the diameter of a human hair, and the apparatus is installed in an electron microscope). The charged wire, of course, "emits" an electric field, an electric field shifts the phase of the entire electron matter-field (wave). The wire divides the beam in half, the outgoing twin beams experience a global change in phase, hence when they reach the shielded solenoid their phase vectors are parallel. If the beams then encountered a magnetic field they would see a spherically curved surface, and the phase angles would not change, it's only because the partial beams see a conically curved surface that there is a relative change in phase. This change in relative phase is otherwise known as a gauge transformation. As the authors also explain, the reason a magnetic vector potential is a gauge field is that it acts on charged particles by changing their direction without changing their energy. The frequency of the matter-waves is constant, but the spatial arrangement changes, as if the wavelength changes from point to point.

On a two-dimensional surface which is locally Euclidean, the bundle of directions over every point is a three-dimensional space. If the surface is also globally Euclidean then there is zero curvature and all the fibers are just topologically equivalent to circles (of directions). If say, the surface is a hemisphere then there is a gradient to account for, and the fiber bundle needs more structure. The gradient is defined by this set of inclined planes in the bundle, such that moving from a pole towards the equator means you will encounter steeper and steeper planes, inclined from left to right (or vice-versa), so now the total space has a gradient defined on it, and clearly moving around the equator will lift the path along the inclined planes. (You'll have to just imagine this abstract space). A gradient is a case of a covariant quantity, whereas a direction is a case of a contravariant quantity. When transporting a fixed direction (vector) along a gradient, if you change direction, the fixed vector contra-varies (because you effectively change to a different coordinate system), and your gradient co-varies. In the curved surface that the phase of electron matter-waves sees, as transport around the surface occurs, the gradient is constant for a constant magnetic potential (it's geometrically equivalent to the surface of a truncated cone), the parallel transport is of twin beams (separated by a one-dimensional charged wire with its own electric field) around the truncated cone. Unrolling the cone onto a plane surface, reveals that all the directions are parallel in the flat space (which is tangent to a circle of latitude on a sphere where the sphere is the geometry of the 'classical' magnetic field). This is similar to a geodesic for a free-falling (Newtonian) particle in a gravitational field. After accounting for the curvature in the time dimension, the two-dimensional spacetime slice rolls up into a cone, without the curvature you have a Newtonian cylinder. But this rolling up of a slice of spacetime assumes that you can identify opposite ends of an interval of time and make them equivalent (i.e. time-symmetric, or something) . . .

Found it. Please Register or Log in to view the hidden image! In the left hand Newtonian picture, the apple has a force, gravity, acting on it. In the right hand Einsteinian picture the apple follows a straight path through spacetime after detaching from the tree branch. The initial green section of each geodesic is where the apple is stationary and has the force of the (molecules in the) branch it's hanging from acting on it, this is a straight line in Newton's picture because time isn't curved. Now identifying opposite ends of both spacetime slices is two isometric embeddings in a third dimension (distances are preserved). Which looks like this: Please Register or Log in to view the hidden image! Now we can place the cylinder inside the truncated cone so the geodesic curves are congruent, and see that the force of gravity (the downward arrows) isn't 'really' there, instead it's just the curvature of time . . . Note that light travels much further than the apple during the free-fall (but, so what? well, it accounts for the initial curvature in the apple's path, when it isn't falling). In the case of a divided beam of electrons "free-falling" through a magnetic potential, the outcome (what can be measured) is a difference in phase when the beams are recombined. This is a quantum phase-space, gravity doesn't seem to have anything to do with the phase of a matter-wave, so the similarity seems to be a complete coincidence. Nonetheless gravity is a gauge theory, it has a gauge particle--the graviton--mediating the 'force' of gravity. Really a gauge particle is in some sense a convenient way to describe mathematically the degrees of freedom of a physical vector field: a scalar field can be described by a spin-0 particle, three dimensional vector fields by a spin-1 particle, etc.

Another major difference between the A-B experiment and a free-falling apple "experiment", is the shape of the connections on the respective topological bundles. There is one apple (not two, nor two "beams" of apples, and apples don't have relative phases), it follows a single path which winds around the isometric embedding (think of this as just a mathematical trick that aligns the Newtonian and Einsteinian pictures). In the A-B experiment, it's about interference (phase shifting), there are two paths for each electron (if you think about doing an A-B experiment one electron at a time, it actually looks very similar to a double-slit experiment with a shielded magnet between, and after, the two slits). Also, the geometry of the gauge field around the magnet is inclined at an acute angle, relative to the incoming beams. This is made explicit in a diagram in Bernstein and Phillips' article, but not really explained. The two-dimensional slice perpendicular to the electromagnet is not at any angle, so what gives? Bernstein and Phillips explain that for a cone with sufficiently small internal angle, and for sufficiently large paths on its surface, the difference (an angular excess) between the flat surface of the cone and the flat plane becomes apparent. Typically these aren't further elaborated, the article is "math-free". But I found something that suggests what these sufficient conditions are related to: (which is here) Please Register or Log in to view the hidden image! On the left is the "unrolled cone". The cone itself covers 1/3 of a circular disc (there are three equal areas in the disc itself, one of which is the actual cone); the three geodesics wrap around the rolled-up cone on the right. I'm interested in the green + blue "path". Looking again at the left picture, imagine the points at the end of both the green and blue lines are exactly halfway around the cone, and that this point is where the two electron beams encounter the potential field. That kind of lines things up with the Bernstein and Phillips article. More on the geometry of gauge fields here (lots of diagrams, some math, it appears to be aimed at students who are familiar with the SchrÃ¶dinger eqn. etc) . . .