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Urs Schreiber submitted a new PF Insights post
Why Higher Category Theory in Physics?
Continue reading the Original PF Insights Post.
Why Higher Category Theory in Physics?
Continue reading the Original PF Insights Post.
mitchell porter said:Just a few days ago, for the first time, a paper by Patricia Ritter gave me an explanation that I could understand, for the relevance of n-categories to objects like branes - the categorical identities express the equivalence of different ways of doing certain integrals over a volume, e.g. where in effect you might first integrate in the x direction, then along the xy plane, then throughout the xyz volume; but you might have done all that for a different order of x,y,z... the result needs to be the same for all orderings, and that leads to the categorical formulation of higher gauge theory.
I want to emphasize, that's not exactly what she says, that's me trying to dumb it down to the simplest way of saying it. But am I even approximately correct in this interpretation?
Well, Feynman is also reputed to have said that "Math is to physics as masturbation is to sex." ;)MathematicalPhysicist said:It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?
BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.
Will it work?
Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.
BTW, @[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] do you work at the maths or physics department? :-D
Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".
Diagnosing that there is a problem(I'd say a serious one), like the mentioned about gauge fields and locality in the way they are usually displayed, and finding a good path towards its solution are two processes that not always come together.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:Regarding the former I now use the occasion of this addendum to highlight what in a more pedagogical and less personal account would have been center stage right in the introduction, namely the developments propelled by A. Schenkel and M. Benini in the last years, regarding the foundations of quantum field theory. Curiously, it had been a well kept secret for more than half a century that the mathematical formulation of Lorentzian QFT in terms of the Haag-Kastler axioms (AQFT) is incompatible with local gauge theory. At the QFT meeting in Trento 2014 I had pointed out (here) that this may be seen irrespective of details of formulation from basic principles of gauge fields, which is what in mathematics is the principle of "stacks" (higher sheaves). By a curious coincident, at the same meeting Alexander Schenkel presented (here) a detailed analysis of the AQFT construction of free QED (without matter) showing explicitly how it fails the locality axioms. As I had explained (here, see also this BA thesis for a still simple but more technical introduction ) the solution to this problem is higher homotopy/category theory, namely the local net of quantum observables has to be promoted to its homotopy version, sometimes called a co-stack or similar. Since then Beninin, Schenkel at al. have be been demonstrating this in increasing detail, I recommend to try to look at least at the introductions of these articles:
RockyMarciano said:How would you convince a physicist that at least gets to glimpse the issue (the well kept secret) that the solution lies in higher homotopy/category.
RockyMarciano said:I mean I guess going to higher homotopy allows you to obtain finer distinctions and gives you more flexibility to accommodate locality in a way that the more rigid lower categories can't, but how is this actually done and connected to the actual physics?
RockyMarciano said:what is the physical correlate of the 2-bundle?,
RockyMarciano said:how do the different physical interactions fit in all this?
RockyMarciano said:On the other hand it seems this is a movement in the direction of greater complexity,
[emoji23] [emoji23]MathematicalPhysicist said:It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?
BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.
Will it work?
Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.
BTW, @[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] do you work at the maths or physics department? :-D
Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".
Actually I had read the slides. The problem with slides separated from their oral presentation is that they are not nearly as explanatory as the actual talk. I understand you might be rationing your efforts to spread higher category among physicists though, it is a long distance race. Hope it gets somewhere, I think it is a step in the right direction.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:By explaining it, as I did. Do have a look at the slides . They are expository. They are aimed at a QFT audience. They were invited at a QFT meeting. Do have a look. It's not black magic.
The point is that homotopy in mathematics is exactly the formalization of gauge transformation in physics. If you want to be serious about describing a system with gauge symmetries, the relevant mathematics is, by necessity, homotopy theory. See the slides.
It's the correct field bundle such that the field content does contain the global instanton sectors, and not just the patchwise gauge field information. See towards the end of the slides where this is explained.
They are encoded by the Lagrangian density, as usual. The difference is this: textbooks say that the Lagrangian density is a horizontal differential form on the jet bundle of a field bundle. But this is just not true in general. For gauge theories there is no field bundle that captures the full global field content. Instead the correct field bundle is a 2-bunde. So is it's jet bundle. And the Lagrangian density which encodes the gauge QFT (interactions and all) is a differential form on that jet 2-bundle.
This is an illusion, coming from confusing unfamiliarity with complexity. Similarly, originally people said that complex numbers are overly complex, whence the name. Later they realized that, on the contrary, many a thing in real analysis becomes simpler when passing to the complex domain.
Homotopy theory is conceptually most simple. But rich in phenomena. It is just mathematics with the gauge principle natively built in.
Some would argue that it is the demand of locality that prevents from having a field that contains global instanton sectors.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:It's the correct field bundle such that the field content does contain the global instanton sectors, and not just the patchwise gauge field information.
They are encoded by the Lagrangian density, as usual. The difference is this: textbooks say that the Lagrangian density is a horizontal differential form on the jet bundle of a field bundle. But this is just not true in general. For gauge theories there is no field bundle that captures the full global field content. Instead the correct field bundle is a 2-bunde. So is it's jet bundle. And the Lagrangian density which encodes the gauge QFT (interactions and all) is a differential form on that jet 2-bundle.
RockyMarciano said:Some would argue that it is the demand of locality that prevents from having a field that contains global instanton sectors.
RockyMarciano said:Extraordinary claims require extraordinary proofs. The above are very strong statements, the physical validity of which can only be demonstrated by proving how the 2-bundle contains the global instanton sector and captures the global field content.
RockyMarciano said:First you have to prove that the usual textbook procedure can't do it(in other words that the Yang-Mills existence and mass gap problem cannot be solved within the current framework), this by itself is a highly prized enterprise, and then you have to prove that the 2-bundles of higher category do indeed give rigorous path integrals(that need to capture that global content) and solve the Yang-Mills problem.
RockyMarciano said:Without all this backing the above statements, certain scepticism from physicists is granted,
Thanks for the offer. I've found a video with the presentation of the slides, I haven't finished it yet.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:If you tell me to which point you follow the argument, and where you first feel you're thrown, I'll help you out with further comments at that point.
My background includes fiber bundles. I might be mixing up things but if the claim is that the usual QFT textbooks have a wrong mathematical framework(Yang-Mills gauge fields and gauge bundles) to handle locality, that should have some consequence in the form of rigorous proof, I would honestly like to know how is this reasoning confused?(P.S.:Maybe it's just my perception but your tone in part of your replies to my comments comes across as defensive/dismissive, not sure why. If I ask all these questions is because the approach interests me, anyway I'll keep leaving out the condescending parts).This paragraph is mixing up a few things.
RockyMarciano said:I mean if the gauge principle is about redundancy
RockyMarciano said:To be more specific,
RockyMarciano said:there is a subtle mathematical nontriviality in applying the cocycle condition to fix the above that seems to be overlooked in the slide presentation.
I can see this. By redundancy I meant that global passage to gauge equivalence at the end that you are also assuming when the dust settles. Now the intermediate steps where you are very reasonably demanding locality have to be compatible with the end global result. I was stuck trying to justify the compatibility of these two realms(local or intermediate and global or final) in a fundamental level. But I guess in doing this you are simply relying(as everyone) on the usual concept of actual infinity and the axiom of choice, right? And for this the whole apparatus of higher categories and homotopies ad infinitum fits really well so it makes sense to promote it.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:That's the thing, it is not. Only globally, when all the dust has settled, at the very end of a computation, we are intersted in passing to gauge equivalene classes. But if you insist on doing this throughout, then either locality goes out of the window or else all topological effects such as instantons go away.
Hope I managed above.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:Try to ask a concrete precise question regarding the first point in the presentation where you are not following.
Fra said:My own internal guidance is much more intuitive, and based on a vision of interacting "computer codes". But it might well be that this converges to something that is characterised by higher categories.
Thanks, I will try to get around to check that paper.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:The modern theory of computation is secretly essentially the same as category theory. This remarkable confluence has been called computational trinitarianism.
This has more recently been reinforced by the understanding that the foundations of computation is in fact a foundation for homotopy theory; this insight is now known as homotopy type theory.
I recommend this introduction:
- Mike Shulman,
Homotopy Type Theory: The Logic of Space,
in "New Spaces for Mathematics and Physics", 2018
(arXiv:1703.03007)
Fra said:For example, would the mathematical machinery of higher category theory, provide a physicist with any brilliant shortcuts to understand unification?
etc.
Fra said:You can ALWAYS inflat and theory, and create a bigger theory.
[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:Remarkably, homotopy theory is a "smaller theory": it arises from classical theory by removing axioms from classical logic. This is the fantastic insight of homotopy type theory.