41

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 12/03/2024 22:56:11 »

I began thinking recently about what this might have to say about another kind of physics where the toys are states of matter, condensates and superconducting solids.

Well, I think you need to turn the diagram monoids into a switching fabric. How you define a switch then depends on what you need to do in the classical sense, vs what you would like to do in the quantum-exchange sense.

Switching a classical switch is affine if you don't care about whether it's off or on--there is no start.
Hence a string of 0 characters with 1 here and there (i.e. sprinkled in the 0s) will switch a switch off or on, it's the Boolean XOR product of two inputs, which inputs? One has to be monitoring--read only--the state of the switch. I surmise this is a necessarily continuous measurement. The other input monitors a string of 0s, waiting for a 1.

In the quantum domain the monitoring is quite different to the classical version. This could be exemplified by the braid groups that are being used in theories. A braid on two strands is different to a switch that cross two strands, because the switch only needs left over right, or right over left, not both.

The switch 'factors' the braid group on 2 strands, and so it's a permutation switch. You would have to know which way the switches were crossed to have any chance of constructing a braid, and then all you could do with it is unbraid it.

That could say something about braid groups and why they're popular with the physicists.

42

New Theories / Re: How does Hamdani explain the twins paradox?

« on: 12/03/2024 20:56:30 »

In case of symmetrical motions, average relative velocity is zero.

?? I don't think you can just claim that separate motions are symmetrical. What do you mean symmetrical? What symmetry?

43

New Theories / Re: How does Hamdani explain the twins paradox?

« on: 12/03/2024 20:52:46 »

I gather that "canoe" can have umpteen meanings, dependent on context. Very economical!

That's my understanding of waka, where in Hawaiian it's vaka. Not surprising because Hawaii and NZ were the last places on the map, sort of.

So a waka or whaka  means a canoe. But canoes carry people and people carry stories around. So a waka is a thing that moves people and stories--legends, myths such as a god, Maui, who fishes land out of the sea--from place to place.

Te reo uses pa for a place, repeat that and papa means places, but also movement. Hence one's ancestry is whakapapa;;  carrying stories and people around.

p.s. an algebraicist might say that pasting these phonemes together is tensoring them to modify the meme. There is a linguistic order (grammar) imposed. Te Reo is a Maori treasure, all words are gifts you are supposed to "look after", because you give them back. That's a nice way to close things, Closure comes at the end, the terminals locate the meme.

44

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 11/03/2024 23:28:19 »

Ok, now that I've bootstrapped my concerns about how important it all is--to whom, and in what context--I can take that Hillary step and forget about the step I just made.

Abandon physics for a while, stop thinking about geometry. This is advice Lenny Kauffman gives in his notes about knots and links. Stop thinking about how complex a knot looks, and check out the locally defined symmetries and abstract those.

As to a solution for the icosahedrally-sliced puzzle:-  I have recognized that I can choose an initial color map. This would be where a 4-color map fails to be 4 disjoint regions, you only get 3 and the 4th is two regions. I have managed to get the puzzle there, but then the numbers can be pointing anywhere (one of three directions), although it's possible to arrange these, the puzzle is tedious to use. But mathematics gets you there without that kind of fuss..

If I choose this particular permutation of the set of colored tiles, then remove the red piece which is located by the color mapping, so it lies at a centre of the three colored regions which are disjoint but arranged in lots of 5 tiles, then it has a permutation on its edges. There is exactly one such permutation on the icosahedron, all the other tiles are surrounded by a combination which is not a permutation. Lovely isn't she?

oops again there are in fact three more permutations around the red triangles in the disjoint red tiling, there because of the 3-cycle (i would guess). Ok so the inital state has 4 and no more than 4 permutations, the same as the number of regions which are disjointly colored, So now I need to be able to count up the operations (compositions of tile swaps around the hole) so that I return to the ah, mountain.

I have to embrace the tIger first (that's tai chi stuff, btw).

45

New Theories / Re: How does Hamdani explain the twins paradox?

« on: 11/03/2024 23:07:40 »

I live in a country that has two languages. One could be called the Maori tongue, which they call Te Reo.

In Te Reo, the sound at the end of a word is where you 'locate' the word. Words imply action or movement and aren't static descriptive things as in English. The meaning is fluid, and this isn't surprising given their ancestors were ocean explorers.

So in my country, a university is a Wananga, or Waananga. You take this apart, so the last sound (phonemic syllable), is 'nga', which translates variously as 'here', 'this way', etc. another word 'tenei' translates as 'this is', so you also get 'here is'--fluidity of meaning.

The rest of wananga is I believe a contraction of 'waha', meaning voice, or 'from the mouth', etc. and the 'a' connecting the two makes 'nga' into 'anga' meaning 'facing' (a direction), 'open', or 'unadorned, naked'.

So the whole word 'wananga' translates as 'voices facing (each other) here', i.e. a public forum. Polynesian languages are simpler than the one we use.

'The Maori language' -- 'Te Reo Maori', translates as 'the normal language'. Ironic just a little.

46

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 11/03/2024 22:17:25 »

I was once told, along with the rest of the class, that some people have difficulty with abstraction.

Well, maybe. I like to think that isn't really true, it's possibly more the case that people have difficulty thinking about a thing that requires a bit of focus. Some people pull engines and other bits of car apart, and put them back together.

Some people write software and have to stay focussed on the problem and finding solutions. Some people keep computers and software working in a useful manner. Some people design communications networks. I like to think those are skills I might have already, if I just look for them hard enough. I like to think it's about letting go of how important you or someone else thinks it is, and just do it because maybe you can. Maybe you should stop listening to people who tell you it's hard to do.

47

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 11/03/2024 12:57:05 »

I've been thinking on my spherical toy, the group action and where the vector space I need to write down the right stuff all is.

But I've thought about free and restricted group actions before, We probably all have.
So this thing has design and engineering restrictions, The puzzle is a solution, but it restricts the action because to have any in the permutation space, it has to be designed with a removable piece.

This frees up three pieces around the hole, and they can exchange places with it. You don't seem to have cycles and the inverse operation acts as the identity. So you get to add or remove one permutation from the set, but it's a big set.

So it walks through the graph, and it's a monoid in an affine algebra: no path starts or ends anywhere. There is no initial or final permutation and no home base for this hole in the surface.

Perhaps I should pay more attention.
oops I forgot about the cycle on orientations, or the direction the number is pointing, on a face when you repeat the exchange. a 3-cycle. I should have said you can't compose any cyclic permutations on the positions, because the hole is the cab, the triangles are passengers.

48

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 10/03/2024 19:00:13 »

Yes, complexity. We enjoy being able to solve complicated puzzles. There are many videos on the internet of many people doing this, explaining how to simplify the problem by doing in in parts. You partition the solution.

That is, whatever internal algebra one uses to apply whatever algorithms you cook up, you solve part of such a puzzle, then figure out how to solve another part, so that parts add together, A partition of algebras.

You want a free one and a fixed one. You realize the fixed one is one you will need to be able to partition again, then "re-fix". Solving a Rubik's cube means essentially that if you solve one layer, you need to be able to unsolve and resolve, freely, this layer. You take a measure (the solved bit) and distribute it so it will go back together, under some algebraic operation.

49

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 10/03/2024 18:13:17 »

So here I am dithering somewhat, about various possibly related subjects, ribbon graphs and the Hopf gauge, algebraic graphs as free or restricted monoids, and 3d permutation puzzles.

At my age I should be playing cards with people my age. But here I am playing with toys. Well maybe as we get older we learn how to play with toys the right way. I like to see If I can take something apart and put it back together.

Another little beauty of a problem I was meant to find some kind of computational solution--a mock up if you will--for a switching problem. This is, if you have n inputs and to start with, the same number of outputs how do you connect a request for a connection to an output, so it doesn't block the network? In a thing we were handed a diagram of, there is a set of switches that do the routing of such a request from an input to a given output, called a Benes network.

But I know now that it's a graph of a construction, of fixed and free elements from B₂, the braid group on two letters. You compose these vertically and horizontally, in an ambient space, this is also in graph theory.

To fix braid elements, you restrict their orbits. A way to think about what fixing a braid group element means, is take a switch apart that switches between the two, which is how many there are in B₂. The identity isn't fixed if you can switch it to a transposition. To fix it you take the two strands in the switch you took apart, and say it's a direct sum of the single strand in B₁, and glue it back into the graph. You do the same with the fixed crossing but just say it's one crossing. Now your fixed set has a pair of elements with 0 and 1 crossing. The switches permute, the fixed elements don't.
The action on the switching elements is free, and Abelian because you can change the switch with addition modulo 2, addition is Abelian, and, for a pair or a bank of two switches, the group action is isomorphic to the Klein 4-group, There's the basis for your free module.

But all this is so you can have a lot of inputs and outputs and a way to connect them. I was told our efforts should look at the difference between routing one, and routing many in parallel,  parallel processing was the name of the course, at the time. There is a communications protocol you need to have, in a communications network.

One approach is to locate the graph of your switching network in a Hotel, that can connect any room to any other room. It's for the guests, this one. So the Hotel Benes, has a desk clerk who routes room A to room B, he knows which number room is which, and which guest is staying, which rooms are empty. He's the desk clerk.

So he might have hired a phone switchboard operator, who knows which switches to flip to get the guest in room A connected to room B. In the parallel version there are several operators, so they compete for access to the computational resource, the switches and their settings.

It gets complicated.

50

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 08/03/2024 17:35:47 »

What it needs is less freedom.

That's ironic because the design of the puzzle means only one of the triangular faces can move at a time. The movement of a face or a tile into the hole exchanges the positions of the hole and the tile. But the tile has an oriented figure, a number, on it so the orientation rotates on the tile, whereas the hole doesn't have a number or a color.

The color map on the edges of the hole is a three-character combination of 4 x colors, so it's a word in a monoid and it's a free one; only some words are permutations.  Perhaps that's why it lacks a fixed arrangement and you have to decide what to do to "solve" it. I showed that it's a four-color map and you can arrange at most three sets of five tiles or triangular faces, so the 'pentad' shares a single vertex. The remaining fourth color can't be mapped like that so it's distributed over two separate regions, at most a tetrad plus a single tile.
 
Nonetheless, the thing is still a good pedagogical model that demonstrates group actions on sets, combinations have a subset of permutations, directions can be permuted. It's nice in a mathematical way but a bit useless as a puzzle.

Or you could drink lots of coffee say.  Yeah, nah.

p.s. watching Monica Vazirani again and thinking over her advice to her grad students, about categorification and how decategorifation gets you into the Hom sets (category of homomorphisms), it's about choosing a single number and partitioning it.

The prime numbers are like single points along a line, with composite numbers in between them. A prime number has a set of divisors with two elements, 1 and the prime are the only divisors of a number which is prime.

So exclusion: 1 is the trivial case--every number is divisible by 1. So using this, we have the single divisor remaining. Vazirani shows how Young's lattice of partition shapes S, can be used to derive useful stuff. You have have to watch, listen, and hopefully understand what's being presented. But what you want to do with a prime p is glue it to a boundary, maybe the real line, and this is just 1 ⊕ p, two points on the boundary, the  1 is the distance between p and the number 1 less than p. So easy it makes your eyes water. Or is that onions, again . . .

(I have abused the notation above, usually you see "otimes" sometimes "boxtimes" for the operation that adds another point to the boundary. I used oplus which is usually taken to mean a direct sum.)

But you can put every prime into a single cell of a Young's diagram and take the direct sum, using the usual ordering of numbers to get that shape, by moving up and down the lattice of shapes. It's cool stuff, but it's only rock 'n roll.

So you want a row of primes and a partition of numbers (compositions of cells with singletons--prime numbers--in them) and you can partition everything in as many dimensions as you need.

I suggest if you are interested, you google the nice lady's name on Youtoob. There is a series of lectures on Combinatorics and group representation theory. She is a rock star of advanced algebra.

51

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 08/03/2024 08:40:44 »

I suspected something about the "thickened" K₂ graph I made by cutting a twisted, connected strip of paper.
The recipe, or the algorithm to make this bad boy, is to start near an edge (a local one!) and cut in a narrow-ish spiral, so you go through the centre and towards the "other" edge. Edges are local when you cut near them with scissors, but you have a global view too. The edges in the resulting looped thingus, are also where you cut near, if that's what you do.

I can keep cutting along either loop, separating it from the nice flat edge. Eventually paper and scissors won't support this. But mathematics will. If I have a topological strip I can keep cutting and make increasingly smaller loops. The paper gets skinnier and the loops will get more like little circles, eventually the twists will vanish when the strips are one-dimensional. But they will stay linked together.
And they will be really small, like smaller than atoms small.

p.s. I think I've learned how to play my little plastic toy into being a reduced algebra, with icosahedral symmetry and it seems to have the chops. It totally shreds S₅ because there are 4 copies on the thing. Because of the face-coloring, if you're strict about the map, it must be a missing red one. This is the reduced copy because it isn't in the set of remaining red triangles. Reducing a permutable set in mathematics is a matter of constructing a partition of the number, in this case 5, and using Young diagrams, you look for a parabolic subgroup. Yawn. This puzzle does it for you

It's a nice example of a polytope, a thing which is several things in the same place, or with the same status. A graph which is permutable, has orientable elements, etc. You orient the faces of the graph. You color the edges of a hole that wanders around the icosahedrally-sliced 2-sphere, mapping 3 of 4 colors (a restriction on the set and the color map) to its edges. Total Picasso, a rasguero of modules on a free monoidal category.

It needs something, though

52

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 06/03/2024 22:23:33 »

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 218 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

53

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 06/03/2024 21:51:22 »

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.

54

Physics, Astronomy & Cosmology / Re: why do a lot of people confuse between interference and diffraction?

« on: 06/03/2024 12:09:15 »

a single source of wavefronts generates waves that will diffract when they encounter surfaces.

Pedant hat firmly in place, I would say "edges" , not "surfaces". Surfaces reflect and/or absorb, but em radiation doesn't "bend" round a surface!

I've been pondering on the photon model of radio waves. We generate them by a continuous process - essentially, sinusoidal alternating current in a wire - with no essential discontinuity of frequency or duration. The energy received from a radio transmitter depends only on how long you listen to it  - it doesn't arrive in discrete packets. Why assume that it is quantised?

Ok sorry, I might have still had my topologist's hat on, where a surface can have less than two dimensions "effectively".

You can treat a barrier like a 1-dimensional 'surface' if it acts as a waveguide, more or less, e.g. a breakwater, but along the surface, not below it. That comes with the idea that it's  some kind of aerial if we are talking radio waves. That 'could' be a gotcha that isn't. A radio broadcast antenna is a 1-dimensional waveguide.

55

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 06/03/2024 09:39:58 »

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.

56

Physics, Astronomy & Cosmology / Re: why do a lot of people confuse between interference and diffraction?

« on: 05/03/2024 21:08:47 »

The solution to diffraction is to fly higher. The solution to interference is to request one transmitter to shut down. It helps to know which problem you are dealing with.

Right. I would say the essential difference is that a single source of wavefronts generates waves that will diffract when they encounter surfaces. Otherwise interference is measured or observed when there are two or more sources.
And being a part-time physicist, I note that radio waves are photons with a long wavelength, so a classical result would be expected and I can ignore the fact that photons are quanta.

Actually that's what a lot of physics does. Perhaps because a classical approximation does work quite well. Nasa doesn't design rockets that have to cope with quantum gravity even though we're pretty sure gravity is quantized.

57

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 05/03/2024 21:01:07 »

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.

58

Physics, Astronomy & Cosmology / Re: why do a lot of people confuse between interference and diffraction?

« on: 05/03/2024 12:22:57 »

'sigh'

I thought there was a reasonably big clue about measuring an effect. It means you break a symmetry.

So what is the measured difference between interference and diffraction, what are the measurements in either effect so you can say what their difference is? Would that answer the question about the confusion between?

Another problem with light doing this, is that all waves interfere and diffract around barriers. Ocean waves do it, radio waves do it. All waves do it.

59

Physics, Astronomy & Cosmology / Re: why do a lot of people confuse between interference and diffraction?

« on: 05/03/2024 08:28:54 »

Can't you distinguish between interference and diffraction without invoking quantum physics?

No. At least most modern day physicists will tell you it can't be done. The classical theory of diffraction obviously is a good enough result, to first order, ('chortle').

About Don Quixote and Picasso. Sometimes learning mathematics is like reading one or trying to understand the other.

Then you realize, you are reading one and trying to understand the other. The explanations are straight out of that wonderfully paradoxical story, and the diagrams and equations are Picasso on bloody steroids I tell ye.

60

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 05/03/2024 08:21:29 »

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.