[Eternal Student]

Hi.

Can you figure out why cutting a Mobius strip in half by cutting once (i.e. rotating the scissors through  one complete circle) results in a larger, but single strip, whereas cutting the strip by rotating the scissors by two complete circles results in two linked but unequal ribbons?

    That one I can just about get my head around.

   Here's a video that can help to visualise how and why things are happening.   

[Approx. 4 minutes, no spoken words, just some text and animations,  it all becomes very easy to understand]
You ( @varsigma ) may not need it but anyone else trying to follow might appreciate some stuff being shown and explained in the good old fashioned 3-D world.

Best Wishes.

[varsigma (OP)]

That one I can just about get my head around.

The image (I haven't watched the video yet) illustrates what happens if you cut down the middle of the strip.
One reason you get a longer thinner strip is, you cut down the centre of rotation for the twist (recall to make a Mobius strip you tape the ends together after you flip one of them over).

I've made at home about 6 twisted strips and cut them different ways. I have found a way to generate a ribbon graph with 5 twists in it. It's a complete K₂ with a twisted loop on each vertex. One loop has 1 the other has 4 twists.  The loops are linked. Weird or what?

It's because I didn't complete the cutting procedure. I'm investigating how many ways there are to cut a strip with one twist so it does or doesn't come apart. Of course there are then all the ways I can cut open a strip with more than one twist in it.

Graph theory, don't get me started.

[varsigma (OP)]

Another question.

The image posted by Eternal Student shows a Mobius strip and a cut strip. Can you see that the one on the right is the one on the left when it's glued back together, but as presented has more twists in it? How many are there and how does it affect what happens when you cut this one down the middle?

[Eternal Student]

Hi.

I haven't watched the video yet

I've made at home about 6 twisted strips and cut them different ways.

    That's fine, you're under no obligation to watch a video.   Many of us don't watch videos that get posted on a forum thread.
    It's mainly there so that if anyone else was trying to follow they will have something tangible.  It's a 4 minute video,  I would recommend it and it's a lot faster than trying to make your own strip and cutting it in real life.

The image .... illustrates what happens if you cut down the middle...

   Warning, a thumbnail image of a You Tube video may be different for different people.  I think it depends on whether they have ever started to watch any of it -  there may be some temporary files already in their browser cache.

Best Wishes.

[varsigma (OP)]

That's fine, you're under no obligation to watch a video.   Many of us don't watch videos that get posted on a forum thread.
    It's mainly there so that if anyone else was trying to follow they will have something tangible.  It's a 4 minute video,  I would recommend it and it's a lot faster than trying to make your own strip and cutting it in real life.

But, in real life you can choose to do things with strips of paper and scissors, some tape etc, that a video doesn't.
I did watch the video, it reminded me of something I did recently, but more often and in different ways.

To me, it's remarkable that you can easily make something which immediately gets you thinking (it worked for me) about paths along the strip or its edge.
It has one edge, although locally it has two. It has one side but locally, etc.
It's a thing that generally, I find is more interesting if I draw lines and don't cut along them.

Moreover a strip of paper can be an edge in a ribbon graph.

[varsigma (OP)]

Ok here's the skinny:

A path in or through a graph visits both vertices and edges. This is already a way to partition different kinds of paths in graphs.

Graphs are free objects because they don't have to have directions (one way signs) along edges. If they do, then paths are restricted. The order you move along edges or visit vertices is now more important.
Ribbon graphs encode an order for path walks between sets of vertices. When you upgrade an edge from a one dimensional line to a ribbon, by "thickening" it, it has two edges, if you orient it by drawing a directed edge around it or a circular arrow, that's a restriction which imposes an order for the edges, pathwise.

I'm plowing through a long paper from arxiv which is mostly equations, but there are some diagrams. The message that gives me is there can be an awful lot of equations in a graph.

Well, ok, the Rubik's puzzles are all graphs as well as polytopes. There are books, full of notation, that people have written, and probably a lot that haven't about what these things are, mathematically or algebraically.
Give me an algebraic object and I'll show you a graph . . .

[varsigma (OP)]

News from the front.

I've realised that making Mobius strips or twisted loops of paper in general gets you into knot theory.
But, you need a sort of algebra of cutting along lines which are embedded circles.
You need a rotation algebra in that many dimensions--three is the minimum needed to make a knot.

So if I wanted to explain what it's about to a young person, I'd say there's a 4-dimensional algebra of rotations, hiding inside a Rubik's cube. It's hard to see it because it needs to fix a 4th dimension and rotate everything around it.

Unfortunately this universe has a time dimension and you can't fix it and rotate around it, because it keeps vanishing at a point. You probably can't do that in a universe with more dimensions, because motion depends on that last detail about time disappearing, all the time. But with more spatial dimensions and time, you can have motion of objects through other objects or themselves; you can turn a sphere inside out.

[varsigma (OP)]

I've been aware for some time that there are many ways to define the complex plane.

I've been walked through a proof of the real and the complex plane being isomorphi√√c. Both are vector spaces, but how does that tell you anything about "imaginary" numbers like i?

I've seen a method that gives you a geometric view of imaginary roots of a polynomial. For example a parabola is either concave up or down. In the first case, if it has a positive y intercept it won't have any real roots. The method reflects the parabola through the y intercept, and rotates it into another dimension out of the plane (where the roots are).

Likewise, the square root of -1 can be said to "live" halfway between 1 and -1, in a rotated plane. And I think the easiest way to see the isomorphism is because of the way you can represent a complex number as a 2x2 matrix. This matrix is the sum of two 2x2 matrices, the real and the imaginary part. The real part is a fixed plane (times I, the identity rotation), the imaginary part is a rotated plane (times a positive rotation of 90 deg.)

That's what I see when I note that the imaginary part is a rotation matrix.

[varsigma (OP)]

In physics you have gauge fields.

My best heuristic for this idea is the Aharonov-Bohm effect.
What this is is a measurement of a shift in the interference peak--the central peak shifts or splits in two because of a phase change in the fermion matter field, The phase is of the amplitudes for each fermion's probability of being you know, a this or a that, or both or neither. It doesn't matter, in terms of paths for particles which are fermions.

The matter field's amplitude is given at any point by this wavefunction of probability (not actually the "physical" thingy!).

So long story short, the magnetic potential, shielded from the electron beams going around it in a coherent split matter-wave, gauges the phase of the field, of . . . amplitudes.

The shift in phase, from the gauge field, breaks a symmetry. This symmetry-breaking is a measurement, they are one and the same thing.

In the Hopf gauge, the algebra breaks a symmetry at vertices, where the order of the edges is gauged by a ciliation.
I'm looking at algebras which are about braids and knots, they don't need ribbon graphs but I have seen Kauffman do this for knot polynomials using his bracket notation.

This stuff is all active, on the drawing board right now kind of stuff. Knots, ye gods.

[Origin]

the central peak shifts or splits in two because of a phase change in the fermion matter field

What is a "matter field"?
What is the phase change you are referring to?

So long story short, the magnetic potential, shielded from the electron beams

What electron beam?  Where did that come from?  How do you shield a magnetic potential?

[Kryptid]

Is this a new theory?

[varsigma (OP)]

Is this a new theory?

Nope.
I recommend you look up the Aharonov-Bohm experiment.  A beam of particles with a definite velocity is a matter-field. So is a lump of matter at rest.

A diffraction grating with two slits is a matter-field, a moving field of fermions interacts with (the geometry of) a fixed field of matter. Bingo, you measure a pattern of interference by breaking some symmetry. This symmetry can be the global phase of the fermion field. It can be that thing because you can't ever measure or detect it.

If that sounds a bit weird, yes it does.

[Origin]

If that sounds a bit weird, yes it does.

Could you answer my questions from reply #29? 
Thanks.

[varsigma (OP)]

I'm not a performing seal, mate.

About removing circles from a Mobius strip, which you can do with an untwisted one, in which case it's a cylinder.

Where does it go? To the same place you can get it back from, i.e. anywhere you can glue back the edges that weren't there before you used scissors to cut out "a circle worth" of twists.
 More to this, is the fact that a circle is a 1-torus, so pedantically you remove a twisted strip with the scissors, but it's embedded (immersed) in a 1-torus--approximately since paper has a small thickness. When you glue paper together along edges, it fails to correspond to the mathematical version.

So this cutting and gluing with a strip that is closed and connected, ok, maybe it needs to be compact if you want there to be bijective maps, but yada yada, it speaks to what a circle is, or what anything mathematical like the number of twists is.

If you can increase this number by cutting the strip, then it follows that removing a circle acts on this number. What is a number? You can partition a number, you can categorify a number.

A number is not just a number.

[Origin]

I'm not a performing seal, mate.

I realize that. You made some statements that I didn't understand so I asked for some clarity.  I thought that was the point of a discussion forum.  If you don't want to answer that's fine with me.

[Eternal Student]

Hi.

You made some statements that I didn't understand so I asked for some clarity.

   I didn't originate the posts but this is what I think was meant:

What is a "matter field"?

     In QFT (Quantum Field Theory) it is any field for which oscillations in that field or energy in that field could be considered as a particle of matter.   Usually we're thinking of fermions - typical matter particles like an electron, a quark or a composite particle like a proton.   

    I'm not entirely sure that @varsigma really needed to consider QFT for what was being discussed.    The Aharonov-Bohm effect is explainable or can be described with just a non-relativistic, plain vanilla version of quantum mechanics.

   

What is the phase change you are referring to?

     In the standard experiment, it would be a shift in what is called the complex phase of the wave function.   Basic idea:   Wave functions have a value that is a complex number,  they have real and imaginary parts.    So  a wave function might be something that looks like this:



     If you look at the place where the wave function is being measured by the... disc shaped speedometer looking thing with the big arrow spinning around....  you'll see the magnitude of this wave function never changes.  However, the portion of it that is in the imaginary axis  (say running in the up-down direction of the page) compared to the portion of it that is the real axis (say running in the direction that looks like it's coming out of the page or going into the page) does change.  Sometimes all of the magnitude of the value is in the imaginary axis, sometimes all of it is alog the real axis.  Overall the complex valued wave function seems to spin or rotate when you view the bare complex value of the wave function.   The modulus, or overall size of the wave function is what is usually important,   the modulus squared is proportional to the probability of finding the particle at this place,  so in this diagram that never changes and the particle is as likely to be found at place X now as it will be likely to be found there in 20 seconds or tomorrow.    The fact that the complex value of the wave function is rotating in the complex plane doesn't matter in the slightest.   Indeed for almost all purposes, the phase or exact way in which the magnitude of the wave function is spread between the real and imaginary axis doesn't affect any physical observation you could make.   All we usually care about is the overall magnitude of the value and the complex phase of the wave function was, historically, thought to be essentially irrelevant and something that was only there as an artifiact of the mathematics.

     However, the individual oscillations in the real and imaginary axis WILL matter if you brought two quantum mechanical wave functions together,  i.e. made one wave packet approach another wave packet and merge.   On merging, you just add the wave functions much as you would for any other sort of wave mechanics.   However, it is not the modulus or magnitude of the two wave packets that you would add, you add the direct or bare complex value of the two wave functions.   If the two wave packets had their rotations in the complex plane synchronised or "in phase",  then the real part adds to the real part,  imaginary part will add to the imaginary part,   all of it gets made bigger and you end up with one really big valued wave function that still spins or rotates around the Complex plane as before (it's just got a bigger magnitude or longer spinning arrow on the speedometer). 
     On the other hand if the two wave packets that merge had their rotations in the complex plane precisely 180 degrees,  or one half a cycle, out of phase,  then when you add them you end up with the oscillation tending to cancel out:  Whenever one was going big in the positive direction along the real axis, the other is going big in the negative direction of the real axis  etc.   When you add the two wave packets together, you get only destructive interference and 0 oscillation all together.   Hence 0 probability of finding the particle(s) in this place where the wave packets merged but were so badly out of phase.

   

What electron beam?  Where did that come from?

    What the Aharanov-Bohm experiment did was to take a beam of electrons and split it into two.  Pass one of those groups of electrons around the left side of a barrier and the other electrons around the right side of the barrier.  While travelling, the complex phase of any beam doesn't matter at all.   Now re-combine the two beams after they get around the barrier and let the beam hit a screen that will glow on contact with an electron.
     If the two split beams remained in phase,  then you get a certain pattern on the screen.   If the two beams went out of phase then you'll get a different pattern on the screen due to all the destructive interference as discussed just above.

    Here's a diagram (the magnetic field will be explained next):


Aharanov Bohm.jpg (64.35 kB . 978x663 - viewed 981 times)

How do you shield a magnetic potential?

    Well you don't.     What you can shield is an ordinary magnetic field (the thing that should exert force on a moving charge).    The potential, it seems, you can do nothing about.    What was done in the Aharabov-Bohm experiment was to use a special sort of barrier that they sent electrons around.   The simplest one to use is just a very long coil.   As you may know, inside a coil of wire carrying an electric current, there is a very strong magnetic field.   Outside of the coil the magnetic field is really small and effectively negligble at most distances.    The magnetic field wasn't really "shielded", although that could have been done, but it was just easier to use a coil and assume the magnetic field exists only inside of it and not around the outside of it.
     The thinking of the time was that since there wasn't any magnetic field outside the coil and in the space where the electrons were being passed,  the behaviour of the split beams should be much the same regardless of which side of the coil they were passed around.    In particular, the complex phase of their wave functions were in phase before they were split and they ought to just stay in phase.  So, on being recombined at the other side of the coil,  you'll get only the usual pattern on the screen.
    As you may have guessed... you don't.... the complex phase of the beams does seem to be different depending on which side of the coil they were passed around and you end up with a pattern on the screen that suggests destructive interference has occurred when the two beams were recombined.   

      For quite a while this was quite a serious issue:   There was no magnetic field where the electrons were travelling, what on earth was affecting one beam differently to the other?   Turn off the coil and everything is back as it should be... good old pattern 1 on the screen when you recombine the electrons.   Somehow the magnetic field that existed only inside the coil seemed to be influencing the electrons that were only ever outside of the coil.    Sure, if some charged particle was moving and it went through a magentic field you'd expect to see some effect but you need to realise that the magnetic field is precisely 0 valued outside of the coil, there is no magnetic force of any sort to be felt where the electrons were travelling.

    Let's use gravity as an example because that's usually easier for us to appreciate.   Let's assume I can put two equal sized planets next to planet earth,  one on the left and one on the right,  equal distance from the earth,  everything nice and symmetric.   You would have a new gravitational force pulling to the left but also a new one pulling right, so overall no net force.   You would carry on with your life as usual and expect to notice no change in anything whether these pair of planets were there or not.    What the Aharanov-Bohm experiment suggests is that... actually there may be a difference.   Although there is no net gravitational force acting on you, you have certainly changed your gravitational potential.   There are loads of things for which we still think "potential" is fairly arbitrary and all we need is some arbitrary reference point we ca call 0 potential.    In electronics, the earth line is reference 0 volts and everything works provided the +5 Volt line is at a potential 5 Volts higher.   We don't try to guess what the actual potential of the earth line was and we assume it doesn't matter.   If we took the box of electronics to another planet,  drove an earth pylon into the ground there and used their "earth" as our electronic earth, we would still expect the box of electronics to work.   

     The Aharanov-Bohm experiment suggests that for magnetic potential (and possibly other potentials) things are not as we once thought.   It made a difference when the coil was turned on because the magnetic potential of the space the electrons were in had changed, even if there was actually 0 magnetic field (no magnetic force if you prefer) in the space.   It's maybe not that interesting these days where we flippantly add another field whenever required - but back in the day it was important to imagine that magnetism (and possibly every other force) may demand that two fields existed throughout all of space.   There is a magnetic field, which can exert a force on moving particles but there is also a magnetic potential field.   More importantly, the potential field may actually have physical effects rather than just being a mathematical tool.   This alone is important, it means that maybe knowing that the +5V line in your box of electronics is 5 volts higher than the earth line is NOT enough,  it may actually matter what the absolute potential at the earth line really is.

    Unlike the electric or gravitational fields,   the magnetic potential field cannot be described just as a scalar field,  we require a vector valued field for magnetic potential (i.e. it has both a magnitude and also a direction).   This was another reason why the notion of a magnetic potential was contested or considered to be just a mathematical artifact.   An explanation for the Aharanhov-Bohm experiment is not attainable if the magnetic potential is just a scalar field.   There is a perfect symmetry around the coil in the experiment ,  an electron passing on an equivalent path on either side of the coil should have been put into all the same potentials for all the same amounts of time,  so even if it did affect the complex phase of their wave functions, they should be affected equally.   On re-combining the beams at the other side of the coil, you'd still expect to have both beams remain synchronised or in phase - and get a perfectly standard pattern for constructive superposition being shown on the screen.
   If we allow the magnetic potential to be vector valued instead of just a scalar, then we can break this symmetry.   The potential field has direction in addition to numerical value and there can be a net circulation of the vector field around the coil:   We imagine that electrons passing around one side can be travelling against the direction of the magnetic vector field,  the ones travelling around the other side would be travelling in the same direction as the magentic potential vector.

    That's probably all I know about the Aharanov-Bohm effect,  I hope it's of some use to you.

 Best Wishes.

[varsigma (OP)]

Here's how i think about the original A-B experiment.

It was an engineering problem setting up the experiment. An electron beam in a SEM can be focused on a small wire which is the beam-splitter. This split beam has a "coherence factor", it doesn't stay coherent for very long. the beam is recombined by a second wire. the two wires have opposite charges, to split the beam the first must have a -ve electric potential, to recombine the two partial beams the second must be +ve.

The paths of these two split beams aren't affected, the phase of the matter-wave is. Electrons see this geometry of a magnetic potential acting on their phase--it's a parallel transport of a phase vector along a surface, which is flat in our classical experimental frame, but 'rolled up' into a truncated cone in the phase-space. That is a description of the effect in terms of a geometry--is there really a cone-shaped potential erm, quantum waveguide? Yes, the electrons say. We just can't ever see the thing directly.

And my conjecture is that a measurement--the change in the pattern--is a symmetry-breaking "operation". There is an action on a symmetry group. Which one? Well it's likely living in SU(2) x U(1) somewhere.

[varsigma (OP)]

I think I just got a beam again about the unit being a "sum over orbits".

This is the unit for the module you have when you take a set of permutable objects and embed it in a field of numbers, generally you want them to have some other symmetry, a shape or a color map maybe. Geometry need not be a consideration here because your module is abstractly a string of points along a boundary. An ambient space.

The Rubik's cube and 3-d puzzles in that class or permutable sets, comes with an identity permutation--easy to define as just " any face has one and only one color on (identically!) all of its squares"

--hang on the missus needs a cuppa, back in a ticky boo.

Ok so the deal is, I can choose any permutable set so I can choose one of these, made for that purpose, mathematical toys

Screenshot from 2024-03-07 11-28-37.png (299.65 kB . 479x488 - viewed 567 times)

Each is in a permutation of the set, this is the sum of orbits of all the pieces such that they leave the set unchanged. There are a lot of orbits to sum over, but you don't have to do that. Each puzzle in the posted image is in this state and you can prove that there are permutations that change the positions as you do them, but leave the puzzle unchanged when you finish. The sum includes the group action on positions, there are only eight for the 2 x 2 x 2 puzzle in the photo. So each piece has the same size orbit, which is 8.

 You need to say that this number is "one orbit", and we are done (at least, we are done when you write down this mathematical proof).

But you can just say it without the proof, if you're a professor of mathematics and know that fact about orbits and stabilizers, because you know you're talking about one of the symmetric groups S_n. So she might mention you have to include the S_n-equivariance "over orbits".

You really are saying that the number which is given by orbit sums, is "one number" it's a categorification of N₀. which is the 0-category of natural number objects.

Whew

[Origin]

That's probably all I know about the Aharanov-Bohm effect,  I hope it's of some use to you.

Outstanding explanation, thank you!

[paul cotter]

I second Origin, on his appraisal of Eternal Student's excellent contribution. I will add a few comments about magnetic potential: there are two magnetic potentials, the vector magnetic potential which was alluded to and the scalar magnetic potential. The vector magnetic potential is defined by B=∇xA, ie the magnetic flux equals the curl of A(magnetic vector potential). The magnetic scalar potential is defined by H=∇V, ie the magnetic field strength equals the gradient of V(the scalar magnetic potential). With regards to the reality of the magnetic vector potential as opposed to a mathematical artefact consider the following : a very large toroid of extremely high relative permeability energised by a winding with a sinusoidal current sufficient to produce a very high flux with negligible leakage. Through the centre of said toroid we add a secondary of loops well spaced from the energised winding and we find an emf induced on this secondary winding. Why? the secondary although looped around the toroid is spatially separated from the flux which is constrained within the toroid. We know that dB/dt gives us the value of the emf but what actually moves the electrons in the loops of secondary, where B does not reach? I say it is the vector magnetic potential, A which is responsible for the behaviour of transformers.