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Daniel Geisler (Apr 06 2020 at 20:11):Does renormalization and self-organization mean the same thing? Both relate to the completion of one level of physics and the ability to begin a new level.
John Baez (Apr 06 2020 at 20:23):Renormalization has a specific meaning, very different from "self-organization" (which is vague).
Daniel Geisler (Apr 06 2020 at 20:55):period.gif
The picture is of the periods of the Mandelbrot exponential map, often referred to as tetration. Now consider the tangent point where the red meets the green. The red is period one and the green period three. Moving through the tangent point results in period three phenomena simplifying to period one. Lets say this matches some physical phenomena. I have long considered this a case of self-organization, but recently it occurred to me that it could also be a case of renormalization. I hope this clears the murkiness of my motivation.
Sam Tenka (Apr 06 2020 at 20:58):Daniel Geisler said:
period.gif
The picture is of the periods of the Mandelbrot exponential map, often referred to as tetration. Now consider the tangent point where the red meets the green. The red is period one and the green period three. Moving through the tangent point results in period three phenomena simplifying to period one. Lets say this matches some physical phenomena. I have long considered this a case of self-organization, but recently it occurred to me that it could also be a case of renormalization. I hope this clears the murkiness of my motivation.
Ooh are you interested in dynamics on complex manifolds? I'm wondering how you define tetration (I looked up Mandelbrot exponential and didn't find what looked like a match with your usage) --- specifically, if tetration is repeated exponentiation, then how does one resolve multivalued-ness of exponentiation over the complex numbers?
Daniel Geisler (Apr 06 2020 at 21:07):My definition of tetration originate at fixed points, this deals with the multivalued-ness. I have also been able to compute a neighboring fixed point starting at another fixed point.
Nikolaj Kuntner (Jul 11 2020 at 18:01):Is there any proper text discussing Galilean group, as a Lie group as well as proper mathematical physics and representations.
Of course there's several books on the Poincare group. Sometimes group theoretical aspects of invariance of the Schrödinger equation are touched upon in texts and the Wikipedia page on it makes some notes on central extensions <https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group>. But the only one I found systematically approaching it was a 70 page paper by Jean-Marc Lévy-Leblond from the 70's.
John Baez (Jul 12 2020 at 23:42):Interesting question! The Wikipedia article cites Lévy-Leblond.
John Baez (Jul 12 2020 at 23:43):I also see this older one by E. Inönü & E. P. Wigner:
https://link.springer.com/article/10.1007/BF02782239
John Baez (Jul 12 2020 at 23:44):Also Ryder has an older paper on "Physical and nonphysical representations of the Galilei group":
https://link.springer.com/article/10.1007%2FBF02738849
John Baez (Jul 12 2020 at 23:45):I really enjoyed the discussion of mass as a 2-cocycle for representations of the Galilei group in Guillemin and Sternberg's Symplectic Techniques in Physics.
John Baez (Jul 12 2020 at 23:47):I haven't read Lévy-Leblond. I think any really systematic treatment of reps of the Galilei group should also discuss reps of the Poincare group, how the Galilei group is an Inönü-Wigner contraction of the Poincare group, what happens to reps of the Poincare group as , and how mass becomes a cocycle.
Arthur Parzygnat (Jul 13 2020 at 05:47):I used to be very curious about this when I was a graduate student and was never satisfied with just taking limits inside of a matrix. @John Baez , are you saying that Guillemin and Sternberg discuss all of this more rigorously?
John Baez (Jul 13 2020 at 21:04):What's "all of this"? And: more rigorously than what?
Guillemin and Sternberg are mathematicians; they precisely state mathematical facts and often prove them. In section 53 of Symplectic Techniques in Mathematical Physics they explain how to calculate the 2nd cohomology of the Galilei group and how this is connected to the Poincare group. In section 54 they discuss "Galilean and Poincare elementary particles" using symplectic homogeneous spaces of these groups. In section 55 they discuss Coppersmith's theory of the limit.
I should really learn all this stuff!
Arthur Parzygnat (Jul 14 2020 at 05:43):I know about the authors and the existence of the book you're referring to. What I had not known was that their book discussed this topic. As a physics student, I was only exposed to the relationship between the two groups by taking the limit inside of a matrix and that's the end of the story. In particular, no discussion about what that means at the level of group representations. Thanks for the breakdown.
ADITTYA CHAUDHURI (Jul 17 2020 at 06:25):In the n-cat lab article https://ncatlab.org/nlab/show/topological+order#mathematical_foundation about Topological Order it is mentioned that "The mathematical frame work of topological order involves tensor category, or more precisely n-category, for topological orders in n+1 dimensions." I tried searching for literatures which describe Topological order using Tensor categories and n-categories but unfortunately could not find any. I want to understand "Topological Order" from the perspective of a mathematician (especially from the point of view of category theory). I would be very grateful if someone suggests any literature in this direction.
Thank you.
Arthur Parzygnat (Jul 17 2020 at 08:00):Based on my discussions with Emil Prodan, there's a non-categorical take on these things in his book "A Computational Non-commutative Geometry Program for Disordered Topological Insulators", but I have not had time to look at his book (at the time he told me, I was only beginning to learn about C*-algebras).
Arthur Parzygnat (Jul 17 2020 at 08:01):You might find the conference at the Simons Center helpful http://scgp.stonybrook.edu/archives/18438 (this is not the one mentioned on the nlab) at the very least to get an intuition for the subject (I particularly liked Mathai's talk, who is a mathematician) and the workshop "Mathematics of topological phases of matter" http://scgp.stonybrook.edu/archives/17921, particularly the talks of Nick Bonesteel (not a mathematician, but his talks are much more understandable). I think I mentioned at some point to learn the Levin--Wen model (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.71.045110). If you're serious about the subject, make this a priority so that you have a good working example (I only browsed this article, but it looks like another reasonable start https://arxiv.org/abs/0904.2771).
Arthur Parzygnat (Jul 17 2020 at 08:01):However, I have tried on numerous occasions to understand what people call the "mathematically precise" papers in this subject and have failed miserably. Many say they are described by n-categories, but I have never seen anyone explain to me exactly how in both a mathematically and physically comprehensible fashion. The only paper I have read come close to understanding is the one by Dan Freed and Greg Moore on twisted equivariant matter, though that doesn't address these concerns completely. I would go so far as to say a mathematically precise definition of topological order together with a wide range of physically reasonable examples as special cases of said definition is not yet known. However, I would be absolutely delighted if someone showed me a reference that achieves these goals.
(Here are some recent potentially promising papers I found while browsing, but I have not checked: https://link.springer.com/article/10.1007/s00220-019-03338-2, https://quantum-journal.org/papers/q-2020-06-04-277/pdf/)
ADITTYA CHAUDHURI (Jul 17 2020 at 09:33):@Arthur Parzygnat Thank you very much for the references and thanks a lot for your insightful comments.
Simon Burton (Jul 17 2020 at 10:01):It's a big subject with many perspectives. Here is my attempt to understand "anyons" from the point of view of category theory: https://arxiv.org/abs/1610.05384
ADITTYA CHAUDHURI (Jul 17 2020 at 10:23):@Simon Burton Thank you very much for the link of your paper!!
Arthur Parzygnat (Jul 17 2020 at 10:50):@Simon Burton , you have some pretty cool and interesting looking papers! I also like that you shared a translation of Fadeev's paper on your website. Have you been working on entropy lately?
Simon Burton (Jul 17 2020 at 11:56):@Arthur Parzygnat Thanks! Yes, last year I submitted a paper to ACT about Huffman coding and entropy. The idea is that you can establish a bijective relationship between entropy and Huffman trees, so this is a kind of categorification of entropy. This connection is well known in CS, but i discovered a way to compose these trees that seemed to be novel. The paper was not accepted to ACT2019, but i did present a poster about it. I was originally inspired by this blog post from Tom Leinster: https://golem.ph.utexas.edu/category/2019/03/how_much_work_can_it_be_to_add.html
John Baez (Jul 17 2020 at 18:44):ADITTYA CHAUDHURI said:
In the n-cat lab article https://ncatlab.org/nlab/show/topological+order#mathematical_foundation about Topological Order it is mentioned that "The mathematical frame work of topological order involves tensor category, or more precisely n-category, for topological orders in n+1 dimensions." I tried searching for literatures which describe Topological order using Tensor categories and n-categories but unfortunately could not find any.
You may do better looking under "topological phase" rather than "topological order". This thesis looks interesting, and has lots of interesting references:
I'm also really fond of the work of Liang Kong:
His talk slides here lead up to describing a tricategory for a 2+1-dimensional condensed matter system, which is exactly right because the categorical diagrams actually look like the stuff that's going on in the matter!
John Baez (Jul 17 2020 at 18:46):Arthur wrote:
However, I have tried on numerous occasions to understand what people call the "mathematically precise" papers in this subject and have failed miserably.
If you have a specific question about a specific paper I'd enjoy hearing it. I'm not an expert on this stuff, so I'm not claiming I'll be able to answer it! But it might be fun for me, or us, to try to understand some particular aspect of some particular paper.
Arthur Parzygnat (Jul 17 2020 at 19:15):@John Baez , did Liang Kong ever write a paper on this subject? Upon searching his name and "Levin-Wen" I just found this https://arxiv.org/pdf/1307.8244.pdf, so maybe he did!
As for questions, I had many 5 years ago but have since moved on and forgot most of them. If I lose steam on my current research front, I'll perhaps revisit this stuff because it is really fascinating and brings together a lot of my interests.
I do vaguely remember two questions, which were even at a very elementary stage.
One was how one would go from a finite lattice with points separated by a fixed distance and a Hamiltonian that commutes with a unitary operator describing the symmetry to a continuum model on with a discrete symmetry (such as from an infinite crystal). The symmetry in the infinite case would give rise to a Brillouin zone modeled by (by taking the spectrum of your unitary operator viewed as acting on ). In the finite lattice case, for each , you get a Hamiltonian, which provides you with a "bundle" over your Brillouin zone, which has natural sub-bundles coming from the different energy eigenvalues (provided there is no degeneracy splitting/crossing). Then what happens to this geometric picture as you go to the limit? What are the precise details for this limit? Of course, one would also like to explore this in higher dimensions (since any complex bundles over are trivial---though discrete symmetries also play another role, which I will ignore for now).
Another question (which I was hoping would grow into a research project) was how you could do this more realistically with an actual crystal by modeling your physical space as a finite lattice but with a groupoid symmetry (without imposing periodic boundary conditions or an infinite-size lattice) and what the analogous geometric structure (coming from the bundle perspective mentioned above) looks like. How are the morphisms in the groupoid implemented as operators on your quantum system? Is there a meaning to the spectrum of those morphisms? If so, what does the analogous "Brillouin zone" look like? What is the associated bundle, and what is the analogue of the sub-bundles coming from the different energy levels of the Hamiltonian?
ADITTYA CHAUDHURI (Jul 17 2020 at 19:19):@John Baez Thank you Sir.
Arthur Parzygnat (Jul 17 2020 at 19:28):I should also mention that the only reason I thought the second question would lead to a research project is because I could not find this idea described anywhere. Of course, if it is described somewhere, I'd like to know (as far as I am aware, the work of Bellissard, Schulz-Baldes, and Prodan does not address this, but I could have easily missed something).
John Baez (Jul 17 2020 at 19:52):Arthur Parzygnat said:
I do vaguely remember two questions, which were even at a very elementary stage.
One was how one would go from a finite lattice with points separated by a fixed distance and a Hamiltonian that commutes with a unitary operator describing the symmetry to a continuum model on with a discrete symmetry (such as from an infinite crystal). The symmetry in the infinite case would give rise to a Brillouin zone modeled by (by taking the spectrum of your unitary operator viewed as acting on ). In the finite lattice case, for each , you get a Hamiltonian, which provides you with a "bundle" over your Brillouin zone, which has natural sub-bundles coming from the different energy eigenvalues (provided there is no degeneracy splitting/crossing). Then what happens to this geometric picture as you go to the limit? What are the precise details for this limit?
Oh. I was asking about questions you had about papers, not difficult fundamental questions in statistical mechanics. But maybe those were the questions you had!
Depending on how rigorous you want to be, and also depending on whether the Hamiltonian you're talking about is "exactly solvable" or not, taking the limit you mention could be extremely hard. Mathematical physicists who like lots of analysis work on questions like this, typically with lots of epsilons and deltas.
Ben Sprott (Jul 18 2020 at 19:02):I am working on, what I consider, a branch of science and physics, which posits that experiments can be a central aspect to science. We model experiments as monads and so data arises naturally in the form of containers like multisets. This is then attached to probability theory naturally as there is a natural transformation from the multiset functor to a measures functor, the finite distribution monad. The "data science" has been completely worked out by Jacobs in "Structured Probabilistic Reasoning". I have attached this to physics in this paper and this paper. I know there are mistakes in these papers and they are not very well written. Regardless, perhaps we can talk about this work here.
Ben Sprott (Jul 18 2020 at 19:38):Naturalness in Category Theory is a difficult notion to pin down. Here is some discussion about naturalness:
https://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html
I try to study and generate applications of Category Theory to the natural world. I believe that naturalness has been a guiding principle in my work over the years. Another principle that has guided me is Occams's razor. I get the feeling they may be related, but I have never seen this written about. Does anyone have any thoughts on this?
I have been developing ideas about science couched in the language of monads. They seemed rather natural to me. I am now discovering sketches, but have a very limited understanding of both of these. They both seem like a way to have natural presentations of theories or analysis of ideas. Is one more natural than the other? I feel that monads are more natural, but that might just be because I don't understand sketches very well. I think monads are more natural because there is this effect that, when you are looking for a given monad, it may only exist on a particular category. It forces you to accept the particular site where everything that ought to be true, is true. Cleverness, or brittle constructions are not as useful.
I suppose the counter argument could be made that sketches work by choosing the kinds of limits you are going to use and then you can achieve various presentations of theories given that kind of limits you chose. So, you work in the universe or logic of particular limits and all things that ought to be true with those limits are true.
It has been put forth that monads and sketches are not related, in that they don't present the same kinds of things (or theories). Taking a look at Barr and Well's CT for computing science text, the first example they give is the "free monoid construction". How is that any different from the list monad, who's category of algebras is the category of monoids? Also, we know that you can generate monads from sketches. They are clearly related, and I would argue, presentations of the same things.
Here we see a publication where the authors assert the following
"Lawvere theories and monads have been the two main category theoretic formulations of universal algebra"
I believe Lawvere Theories are just sketches restricted to finite products.
The fact that sketches tend to be more like complicated constructions, would suggest to me that they are less natural than monads.
Why is this all important? Look at this quote in Spivak's latest paper
"Previous work has shown how to allow collections of machines to reconfig-
ure their wiring diagram dynamically, based on their collective state. This notion was
called “mode dependence”, and while the framework was compositional (forming
an operad of re-wiring diagrams and algebra of mode-dependent dynamical systems
on it), the formulation itself was more “creative” than it was natural."
Naturalness mattered to the tool he choses in his paper, namely the category of polynomial endofunctors on set and their monads
If you read this book by Hosseenfelder, you see that what physics may be in need of now is a new guiding principle, other than the standard understanding of beauty. I am proposing that the new principle which we should be looking for is the categorical notion of naturalness.