Do Category Theory and/or Quantum Logic add value in physics?

Question

I know they have their adherents, but do more or less esoteric branches of mathematics such as Category Theory and/or Quantum Logic provide powerful tools for new theory development or are they just occasionally-useful presentational frameworks?

More particularly, is it worth investing the time to study these formalisms?

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Nigel Seel

Comments

@RonMaimon: Linkbox & Likebox ...I'll read it if I find the time. But now I wonder, does this programming language, describing a process (intended for the purpose of biology, although I think that doesn't really matter), have any ontology behind it? Like set theory does by describing things and their structur, and categrory theory does too in a way, I think.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user NiftyKitty95

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@NickKidman: Nothing computational has "ontology", it's describing data structure. I don't see categories as particularly deep--- they are just handy ways of categorizing common simple arguments (of course, I just realized I said "they are just...(something completely different)... above", this is why they actually are deep even though they are not).

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Ron Maimon

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@RonMaimon: I'm interested in the general mathematical framework as well as notations at the moment, because I'm planing to write a script which translates formal mathematical language (e.g. using the Mizar library) into a easy to adjust representation/notation/language. I want to play around and look out for some general structure, maybe some notation in which certain concepts are easier to grasp. I remember talking to you in several threads (e.g. the tensor diagrams, and your rant about feynman diagrams and their complexity). If you have any comments/suggestions, let me know in a chat.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user NiftyKitty95

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@NickKidman: That is a fantastic project, very close to my interests. I'm likebox at g mail. This is something I spent a lot of time thinking about ~10 yrs ago, but the formalisms I ended up using for proteins aren't optimal for general purpose theorem proving. I don't have an answer. Are you getting paid to do this?

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Ron Maimon

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@RonMaimon: Hehe, I wish.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user NiftyKitty95

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Is it ok to restric this to categories? Quantum logic should be a separate question. I view categories as just a way of doing homology computations, where you have a lot of different abelian groups to keep track of.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Ron Maimon

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@RonMaimon: "...and there is no substitute at present (although I always think there should be, and I always struggle to find it)." What would be your approach, idea or suggestion?

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user NiftyKitty95

Answer 1

Recently, it is realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states.

The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order in higher dimensions may need higher category to describe them.

One can also show that the quantum phases with short-range entanglements and symmetry $G$ in any dimensions can be "classified" by Borel group cohomology theory of the symmetry group. (Those phases are called symmetry protected trivial (SPT) phases.)

The quantum phases with short-range entanglements that break the symmetry are the familar Landau symmetry breaking states, which can be described by group theory.

So, to understand the symmetry breaking states, physicists have already been forced to learn group theory. It looks like to understand patterns of many-body entanglements that correspond to topological order and SPT order, we will be forced to learn tensor category theory and group cohomology theory. In modern quantum many-body physics and in modern condensed matter physics, tensor category theory and group cohomology theory will be as useful as group theory. The days when physics students need to learn tensor category theory and group cohomology theory are coming, may be soon.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Xiao-Gang Wen

Answer 2

Category theory has some potential for physics. Quantum logic I am less sanguine about. It has always struck me as a way of expressing something we understand in set theoretic language. It has always struck me as a formalistic study that brings little additional content.

Philip Goyal demonstrated how the above summation over intermediate points is equivalent in a discrete setting to the concatenation of measurements, where the summation is over all possible outcomes [1]. The complex amplitudes are products of complex numbers, and in a discrete setting this is a multiplication rule which requires complex numbers. The result is that quantum mechanics is reduced to a simple system of associative and commutative mathematics of complex numbers with no reference to classical mechanics, or any notion of space or spacetime. However, the one requirement is that the points intermediate to the initial and final points be intermediate in time.

The Goyal logic is a summation of Stern-Gerlach experiments. The intermediate point corresponds to some intermediate measurement between the source of particles and the final Stern-Gerlach (SG) apparatus. If the outcome of the intermediate SG apparatus is ignored, or no measurement is performed of their outcomes, the split beams recombine as a discrete summation. So the intermediate SG apparatus represents a sum over elementary quantum events. This summation in the complex algebra corresponding to this logic recovers the quantum superposition.

These summations over SG experiments occur in a sequence. There is no ambiguity in the ordering of these events. Further, the process appears well defined in a discrete setting. The Goyal approach for discrete quantum mechanics, even if the number of elements is enormous, but not infinite, indicates some sort of quantization of time, and a discrete spacetime.

This has elements of Zariski topology. Consider the affine space $A_n$ as the n-dimensional space over a closed field $F$. The topology is constructed from closed sets defined by the polynomial set $S~\in~F$ by $$ V (S)~=~\{x~\in~A_n| f(x)~=~0;~\forall f~ \in~S\} $$ For two polynomials in the set S we have the following rules: $$ V (p_1)\cup V (p_2)~=~V (p_1\times p_2),~ V (p_1)\cap V (p_2)~=~ V (p_1~+~p_2) $$ which serve as the representation map between the logic of outcomes and the algebra of quantum operations demonstrated by Goyal. This closed set topology defines the Zariski topology on the affine set $A_n$. So a connection to quantum mechanics exists within this system with respect to Zariski topology. This is the topology of {\’E}tale and Grothendieck, or topos theory. An overview of topos theory in physics is in [2] by Isham.

[1] P. Goyal, K. H. Knuth, J. Skilling, "Origin of Complex Quantum Amplitudes and Feynman's Rules," {\it Phys. Rev.} {\bf A 81}, 022109 (2010) http://arxiv.org/abs/0907.0909

[2] C. J. Isham, "Topos Methods in the Foundations of Physics," {\it"Deep Beauty,} ed. Hans Halvorson, Cambridge University Press (2010) http://arxiv.org/abs/1004.3564

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Lawrence B. Crowell

Comments

A discursive answer! I think you have demonstrated connections but they seem mostly rephrasings of existing results. Has CT or QL really propelled any fundamentally new insights, I wonder? I may post another query about the possibilities of discrete spacetime.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Nigel Seel

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As yet there is no categorical theory of physics that I know about. There are people who are trying to advance topos theory, but as yet nothing definative has come of it. The discrete QM work of Goyal is maybe a link to a categorical or topos in physics based on known physics.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Lawrence B. Crowell

Answer 3

My short answer is No, they're not too useful, but let me discuss some details, including positive ones.

Categories, especially derived categories, have been appropriate to describe D-brane charges - and not only charges - beyond the level accessible by homology and K-theory. See e.g.

http://arxiv.org/abs/hep-th/0104200

However, I feel it is correct to say that the string theorists who approached D-branes in this way did so because they first learned lots of category theory - in mathematics courses - and then they tried to apply their knowledge.

I am not sure that a physicist would "naturally" discover the categories - or even formulated them in the very framework how they're usually defined and studied in mathematics. And on the contrary, I guess that the important qualitative as well as quantitative insights about the D-branes - including the complicated situations where category theory has been relevant - could have been obtained without any category theory, too.

But of course, people have different reactions to these issues and these reactions reflect their background. And I - a non-expert in category theory - could very well be missing something important that the category theory experts appreciate while others don't.

Most famously, Joe Polchinski - the very father of the D-branes - reacted wittily to the notion that the D-branes should have been rephrased in terms of category theory. In a talk, he spoke about an analogy with a dog named Ginger. We tell Ginger not to do many things and do others, Ginger. What Ginger hears is "blah blah blah blah Ginger blah blah blah".

In a similar way, Polchinski reprinted "what mathematicians say". It was a complicated paragraph about derived categories and their advanced methodology applied to D-branes. What Joe hears is "blah blah blah blah D-branes blah blah blah blah T-duality blah blah D-branes blah."

Some physicists also try to generalize gauge theory to some "higher gauge theory" using category theory but I don't think that there are any consistent and important theories of this kind. What they're doing is similar to the theories with $p$-forms and extended objects except that they don't do it right.

As always, category theory may offer one a rigorous language to talk about analogies etc. - but I don't think that physicists need anything beyond the common-sense understanding how the method of analogies works. So if you learn category theory - which is pretty tough - I think you should have better reasons than a hope that the theory could be useful for physics.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Luboš Motl

Comments

Thanks, I had a brief exposure to category theory in the context of formalising datatype composition and transformation in software engineering. However, you have to have a lot of disparate areas of maths under your belt to ascend to the level of abstracting common attributes. My feeling was that it was rather remote from physics, motivating the question.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Nigel Seel

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I think that your intuition was very correct, @Nigel Seel.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Luboš Motl

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+1, surprisingly, I agree. But you need to become intuitively familiar with categories anyway, to follow mathematics, and do algebraic topology (which is the reason it became dominant, and also the reason it appears in physics). But I always sense that the formalism of categories is suboptimal, that there are big missing pieces, because the arrows don't describe the whole structure.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Ron Maimon

Answer 4

Nigel,

A few comments. Firstly I think we should separate the questions for Category Theory and Quantum Logic here, as I think they are rather different in a sense I will explain below.

Category Theory: My view is that most physicists look for insights and theories from a Geometric viewpoint. This somewhat unites Relativity Theorists and String Theorists. Also the Fibre Bundle model of particle physics (Yang-Mills) is largely a geometric structure too. So mainstream physicists are likely to continue to look to geometric models for answers and insights. This doesnt have to be "narrowly geometric" however, as there is some interesting work on Topological insights into physics: solitons, etc. However the most interesting material to me has been the topological discoveries which have a geometric form. Even Statistical Mechanics has some topological theorems of this sort.

Having said this there is a "school" of fundamental "pre-geometry" physicists (which can be found at another site that some of the Stack physics group use called the "Institute for Fundamental Questions"). This link is to an author there I have briefly interacted with:

http://www.fqxi.org/community/forum/topic/482

They may not use Category Theory itself however, but some are looking for a way to derive spacetime geometry and quantum theory (thereby providing a framework for Quantum Gravity) from something else: but what can that "something else" mathematically be? Chris Isham quoted in another Answer is motivated by this too.

Futhermore although Category Theory is billed as a "unifying" theory, physicists tend to get excited by more specific unifications between specific models: the String Theorists have shown us that too, although such unifications also (seem to) happen outside of both String Theory and Category Theory too.

EDIT: There is an online Category Theory, Philosophy and Physics Wiki: a bit technical in detail but some sections outline why the researchers believe that Category Theory is useful in their area e.g in Path Integral Modelling.

The unifying attraction of Category Theory in a less Geometric world, like Abstract Algebra or Software Engineering Logics is higher - it might have value there.

Quantum Logic: Unlike mathematical theories or Software Logics this arose as a result of a theorem from Von Neumann about what Quantum Mechanics was supposed to be telling us about physical reality. However this theorem has been disputed, see http://write4science.com/Assets/pdfs/Global%20logic.pdf (around page 61 of 74), and although controversy still exists quantum logic per se seems to dead. It has left us with related lattice structures though, and again these might prove useful to those interested in a novel "Quantum Gravity Foundation".

Roy.

Extra Note: Chris Isham has also managed to keep QL going with his Temporal Quantum Logic http://en.wikipedia.org/wiki/HPO_formalism.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Roy Simpson

Comments

I scanned through Dr Goldfarb's paper (the FQXi link) but it's a bit too "new-age" for my tastes :-).

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Nigel Seel

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I was somewhat critical too, the criticisms are on the site somewhere, but I just gave this link as a flavour of this kind of alternative. The site is a little difficult to navigate, but other papers are there too. And some people get grants from FQI!

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Roy Simpson

Answer 5

Sorry for necromancing this post:

I have spent a few years trying to use Category Theory to develop new ideas for physics.  I think the problem that the field generally faces is that of translation.  Most people translate familiar structures into category theory believing this will give some clues as to its abstraction.  For instance, suppose we have a quantum field theory that defines operator valued fields at every point in a Manifold.  An evolving spacetime is a manifold map, and the "dynamics" of this system generally means that the quantum field evolves according to the background evolution.  In categorical terms, we define a category of manifolds and a category of Hilbert spaces and a (T)QFT is then a functor from Manifolds to Hilbert spaces.

All this seeks to do is make the subject obscured in some strange new language.

The real way to use category theory is to completely dispense with familiar structures.  The point of category theory is that it comes with its own basic structure : morphism, object, composition, associativity of composition, functor, monad etc.  Doing real physics in categorical terms means using these structures alone as a heuristic to talk about reality.  For instance, there have been attempts to codify probability theory in terms of a functor from Polish spaces to Polish spaces .  The real key ingredient in this construction is the monad (see how it comes up in the general notion of epistemics).  I contend that a Monad on the category of an apparatus (which can be any category), is an update to your understanding of  the "system" under investigation.

Don't translate existing structures into precisely defined categories.  Instead, use intuition to tell you how your particular science translates into the heuristic of 'category structure' (as described above).

Finally, there is a large number of quantum theorists who understand a serious problem in the interpretation of the quantum mechanical wave-function (either as purely informational or an aspect of reality like an electric field).  I contend that the only way to solve this problem, is by dispensing with the whole notion of state and dynamics.  To do this, we use category theory (dispensing with Sets) and focus on uniting  the evolution of a "universe", and the way in which information is defined.  We focus on 'transformation', and causal structure.  Spekkens has written about this and also on how to use categories as a heuristic for Bayesian statistics.

Comments

Thanks for this interesting new point of view :-)

It is not true (as often claimed on other physics Q&A sites) that bringing older (good) posts on the front page again has to be a nuisance, on the contrary there might be (new) users who have not yet seen it when originally posted, and looking at a post a second time later does not hurt...

A new answer that adds something valuable is always worthwile, so you do not have to apologize at all for answering an older question ;-)

Cheers

Answer 6

Look at this and see if it helps http://arxiv.org/pdf/0905.3010v2

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user iii

Comments

I saw that some time ago; at least it was simple enough I could understand the first few pages. I'd love to have a category theorist around on the Exchange in order to answer questions.

This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Carl Brannen