[varsigma (OP)]

I made a mistake earlier about the geometry and topology of a gauge field. Both are involved in the two experiments Bernstein and Phillips discuss. I like the article because of what it explains, if not necessarily the way they go about it.

They actually state that neutron interferometry demonstrates the topology of the gauge field, and the A-B experiment demonstrates the geometry. The gauge field is the magnetic vector potential. This was considered a mathematical nicety, required to complete Maxwell's system of equations that describe the E and B fields.

You can call it a redundant degree of freedom. Bernstein and Phillips say its geometry is a truncated cone, with a hemisphere glued back. The topology (in spin precession in a real magnetic field) is a Mobius strip; the spin vector is transported around the strip so takes two complete rotations over the base space, a circle, to be parallel again to its initial direction. It's really a space of phase shifts in the spin vector. Moreover, the global twist in a Mobius strip is indeterminate; it's there, but not in any localised sense.

This is why the global phase of the field--the neutron matter field--is indeterminate as well.

The A-B experiment takes the notion of parallel transport in a bundle of directions (the tangent vector spaces at each point) to the notion of a bundle of phases, in such a field. Instead of the geometric version of parallel transport, you have the topological version, a path-lifting rule.

One big difference between the two experiments is the mass of the quantized field elements, neutrons vs electrons. The fiber bundles are classical, the equipment is classical.
The gauge field is there because electrons see a phase shift, they interact with the magnetic field (not detectable) determined by the vector potential.

[alancalverd]

nobody has ever insulted me with the title of philosopher!

I had assumed you were a PhD.

But a gentleman wouldn't draw attention to it, surely?

[Bored chemist]

nobody has ever insulted me with the title of philosopher!

I had assumed you were a PhD.

But a gentleman wouldn't draw attention to it, surely?

I wasn't aware that PhDs were handed out by gentlemen.

[varsigma (OP)]

Some last few details to unpack about what Bernstein and Phillips are saying.

They say that the base space of the experiment, in which the beam and the coil are three dimensional, is also three dimensional and the total space is four dimensional. The extra dimension is the "circle of phases" for quantum fields over each point in the base space.

So consider a two dimensional base space in which the electrons move, and the total space is three dimensional. This is otherwise called a pullback. You can push forward from this to a different four dimensional space, maybe take some laws of physics along.

[varsigma (OP)]

We all have to deal with philosophy. Physics, "by itself", doesn't really try to cover philosophical questions or answers.

A good example I can think of is when, during an electronics tutorial a student asked why is time mathematically a real number, like in all the formulas? The lecturer more or less said, "because it works that way". Or, doing it another way doesn't work, so that's why.

Wikipedia's article on the A-B effect and its implications, or the physical nature of the EM field as it were, goes over some of the problems philosophy has with it.

The Aharonov?Bohm effect is important conceptually because it bears on three issues apparent in the recasting of (Maxwell's) classical electromagnetic theory as a gauge theory, which before the advent of quantum mechanics could be argued to be a mathematical reformulation with no physical consequences. The Aharonov?Bohm thought experiments and their experimental realization imply that the issues were not just philosophical.

The three issues are:

    1. whether potentials are "physical" or just a convenient tool for calculating force fields;
    2. whether action principles are fundamental;
    3. the principle of locality.

--https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#Magnetic_solenoid_effect

So there's that word "physical"; also 3. implies that measurement is local, or at least localised.

I think because of the way measurements change (or break) a symmetry, there has to be a gauge field for measurement to make physical sense, and not just in the A-B effect.
p.s. on second thought, that might be a bit controversial. It might not be true for classical measurement.

[alancalverd]

We all have to deal with philosophy. Physics, "by itself", doesn't really try to cover philosophical questions or answers.

Best way to deal with philosophy is to hold  your nose and walk away. Philosophical questions do not have answers because if they did, philosophers would be out of work.

Why is time a real number? Because it is the dimension that separates sequential events, and real numbers are those that we invent to delineate sequence and quantity.

[varsigma (OP)]

The neutron diffraction/interference experiment was another one that had to wait for technology to catch up.

If you read the SciAm article the authors go over the details but, they don't go into why a silicon 'waveguide' does the job, or why it needs to be a certain shape. One requirement of the experiment is that, after splitting the neutron beam, one half has to go through a real magnetic field, so its path is longer than the other half.

The topology of spin precession in fermionic matter (fields), as the article explains, is the Mobius strip. I did a bit of graph theory so I know you can embed a Mobius strip in a solid torus; actually you just have to embed the edge on the surface so you have a torus knot, with a single twist. If the base space is a circle you have a fiber with two points--two angles 180? apart--over each point in the base.

Bernstein and Phillips use the phrase "path lifting rule", but it seems this is nowadays just "the lift" and there are two components--the horizontal lift and the vertical lift. Since the base space and the total space are 1-dimensional there is no vertical lift in spin precession in fermions. This is a nice heuristic.

p.s. it could also qualify as a cool thing to say at parties.
p.p.s. if you have to explain it, you can say it's because the geometry of the magnetic vector potential is topologically a hemisphere, and all fermion geodesics stay at the same latitude, if they start that way so they are perpendicular to the B field lines. The spin of all fermion fields has no vertical lift in general, whether the phase of their matter fields does and the phase of the field, not the spin, can be lifted vertically.
By now whoever you thought was interested probably isn't.

[varsigma (OP)]

So it's complicated. Field theories are complex and as that article explains, you have geometry and topology to explain the effects. You also have Hamiltonians and Lagrangians and lots of equations.

The simplest way to see what it all means is in terms of information, and how to encode quantum effects:=phase shifts, with classical information, and how to process it. So clearly you can't encode the global phase of a matter-wave or its spin vector, you have to have a difference so you need at least two distinct waves (!).

Also, an external magnetic field is a way to encode a phase difference, and, clearly it's not the field itself but the potentials that are "effective". But the external field strength is the control factor.

[alancalverd]

....none of which is known to the neutrons. Just goes to show how a simple experimental fact can require a very complex mathematical model to interpret it. OK, I'm somewhat abusing the word "simple" because neutron engineering isn't trivial, but I think the point is made.

[varsigma (OP)]

A physics joke.

A proton and a neutron go into a magnetic field bar. The neutron heads straight towards the bartender, but the proton tries to and ends up smacking into a wall.

The bartender says to the neutron, "Your mate ok?", The neutron says, "I think he's a bit charged up about something".

[varsigma (OP)]

Topology joke.

A pair of topological fiber bundles go into a bar. After finding some ambient space, one of them sees a sign on the wall saying "Edges only Night".

He says to the other bundle, "We might as well come back another time, or we could have a completely pointless evening".

[varsigma (OP)]

I have a question about ChatGPT. This is the standard online version so I think it's still probably 3.0

Has anyone explained anything to it, so it says "I see", then tells you why?
It's been explaining the development of a surface to me,  I've been explaining why Bernstein and Phillips use a truncated cone for the, ah, gauge field geometry seen by charged fermions.

'meh'

[Origin]

A physics joke.

A proton and a neutron go into a magnetic field bar. The neutron heads straight towards the bartender, but the proton tries to and ends up smacking into a wall.

The bartender says to the neutron, "Your mate ok?", The neutron says, "I think he's a bit charged up about something".

I believe the neutron would also be affected by a magnetic field due to the neutrons magnetic moment.  Of course it would be much less than the proton.

[alancalverd]

And the neutron would deflect in the opposite direction.

[varsigma (OP)]

I believe the neutron would also be affected by a magnetic field due to the neutrons magnetic moment.  Of course it would be much less than the proton.

The neutron scattering experiment was finely tuned. Actually both experiments that Bernstein and Phillips explain are highly engineered to very close tolerances.

The velocities of particles in incoming beams is a known. So the time neutrons spend in a magnetic field is also known, and the field strength is known.

The total rotation angle induced along the path through the magnetic field is equal to the Larmor precession frequency multiplied by the time the neutrons spend in the field. The angle can therefore be calculated from measurements of the velocity of the beam, the intensity of the field and the distance across the field. In the version of the experiment done by Rauch, Bonse and their colleagues the neutrons travel through a magnetic field 1.5 centimeters wide at a speed of 2,170 meters per second, so that each neutron spends a little less than seven microseconds in the field.

When the electromagnet is operating at maximum current, the strength of the field is 400 gauss, which corresponds to a Larmor frequency of 433 million degrees per second. At this rate, in seven microseconds the spin vector of each neutron rotates about eight full turns. If each 360-degree rotation of the spin vector restored a neutron to its original state, one would expect to observe eight cycles of maximum and minimum counts. The actual result is significantly different. As the magnetic field increases from zero to its maximum the number of neutrons detected at the counter passes through only four cycles.

I would first (shut up and) calculate the deBroglie wavelength and energy of the neutrons. Neutrons have no electric charge (U(1) charge), their magnetic moment is explained by, I think weak hypercharge and that explains why free thermal neutrons decay spontaneously. It's a weak SU(2) sector interaction between field energies. Why is the B field maximum 400 gauss , that has to be an electromagnet that can pack a few field lines into  R^3

p.s. I think I managed to get ChatGPT to agree that magnetic field lines are an heuristic device, which in essence "decorate" a magnetic field with closed loops and lead to the notion of magnetic flux. The flux density through a surface is heuristically the density of the field lines through the surface. Also because of the closed loop it must be conserved.
Inside a solenoid the (electric) current is in the same direction as the magnetic flux along the field lines, outside it's inverted and opposes the current.

[alancalverd]

Inside a solenoid the (electric) current is in the same direction as the magnetic flux along the field lines, outside it's inverted and opposes the current.

Not sure we are looking at the same picture here! The field of a solenoid  is merely an extension of the field of a single loop. The current direction is  circular, with the field lines axial to the solenoid.

Confusion arises because for most practical purposes we make solenoids by winding a wire over a cylinder, and talk about magnetic field in ampere-turns or ampere-turns per meter, so you may be misled into thinking that the current passes from one end of the solenoid to the other. Fact is that "turns" is actually irrelevant! If you had a flat sheet of perfect conductor, passed a current from one edge to its opposite, and rolled it into a cylinder with its axis parallel to those edges, you would get the same field for the same number of amps per meter, with the current obviously circling round the cylinder and not along it. The "turns" business is just a matter of engineering practicality: it's much easier to make a long homogeneous field by driving 1 amp through 100 turns of wire than by driving 100 amps around 1 turn of rolled sheet - though superconductors do allow very large currents to circulate through very few turns.

[varsigma (OP)]

About the experiment that requires a working neutron interferometer.

Bernstein and Phillips use a diagram (of course) to illustrate some of the finer details. Imagine an incoming beam of spin-polarized neutrons so all spin up. This is represented by a sequence of alternating red and grey stripes above a line. So for an unpolarized beam, the stripes would span this line and change colour across it. The stripes represent the probability amplitude of spin up or down as a regularly spaced pattern.

So a rotation of this spin amplitude in an external magnetic field rotates the pattern across the line; a 180? rotation pushes a red stripe over to a grey stripe, and this is meant to be interpreted as a flip from spin up to spin down. The phase of the amplitudes doesn't shift.

With a flat surface of pure silicon, neutrons aren't reflected but for a thick enough potential barrier, neutrons reflect internally such that constructive interference generates a diffracted beam at right angles to the transmitted beam. This depends on how flat the external surfaces are and on the amount of silicon between them. The same effect means a second barrier will add 1/2 the partial beams together and a third barrier will generate a single output beam, with 1/2 the intensity of the input beam.

All that sounds a bit like what happens when you have three polarizing filters and some light.
. 

[varsigma (OP)]

The "turns" business is just a matter of engineering practicality: it's much easier to make a long homogeneous field by driving 1 amp through 100 turns of wire than by driving 100 amps around 1 turn of rolled sheet - though superconductors do allow very large currents to circulate through very few turns.

Yes, I would say that it's because of the convenience of having the same current density in a metal coil as in a metal cylinder.
In both cases you increase the density, but you should bear in mind that electrons behave a bit differently when their motion is restricted by the dimensions of a metal conductor.

As the authors do near the end of that SciAm article, consider a flat ring of metal with magnetic flux through the centre. A current around the ring can go in two directions, and according to B & P, the parallel transport of the phase vector is that of a geodesic on a cone; slice the cone open and it's flat--the geodesic is a straight line and the phase vector is pointing in the same direction everywhere along it.

This geometric cone isn't "really there", unless you use electron phase to determine that it is. It might only look like a truncated cone, this vector potential field, or classical gauge field, if you're a fermion with U(1) charge.

[varsigma (OP)]

Something I read recently about physical theories means I should revise my view of simplification.

The argument is that simplification of complexity in the natural world leads to useful theories, because we can formulate simple relations and operations, then use them to fabricate more complex 'structures'. The close relation between mathematics and physics is fairly obviously a requirement or necessary condition for the fabrication of theories and real devices (telescopes, microscopes, interferometers, . . .) which illustrate or support the ideas we have about "the universe".

So this paper refers to sophistication instead; in what sense is the sophistication of ideas (theories) a better path to understanding than simplification is? Or in other words, how can simplification and sophistication, work together?

An example of what this paper says is a sophistication, is the construction of a topological space, equivalent to a geometry. So wrapping the circle around itself is a sophistication; simply stated any number of windings is a map to the integers (#crossings in the graph) and from the circle to itself.
Now add the condition or restriction that none of the circles can be contracted because real 3-space has a line removed and all the wrappings go around it. So the 3-space, with a line removed is a circle, topologically. Circles or loops that don't go around the removed line are contractible.

[alancalverd]

how can simplification and sophistication, work together?

Said it before, will say it again. Rocket science is just two equations. Rocket engineering is a lot more complicated. The trick is to keep adding bits of science until your model is good enough for practical purposes.