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3
votes
1answer
73 views

Rigorous definition of superselection sector/quasiparticle type in anyon systems

The systems I have in mind are for example Kitaev's toric code model (arXiv:quant-ph/9707021) and Kitaev's honeycomb model (arXiv:cond-mat/0506438). What I'm looking for is a mathematically rigorous ...
0
votes
0answers
55 views

Does model theory in mathematics have any usefulness in the modeling of physical systems?

The term Model Theory in mathematics seems to have a somewhat precise definition here. Reading through that reference you'll largely see discussions strictly relating to mathematical concepts, but one ...
0
votes
1answer
74 views

Dilemma: Fusion space from a direct sum of anyons or NOT

In Preskill's note, 9.1.2 in page 44, concerning the fusion space, it states that: The fusion rules of the model specify the possible values of the total charge $c$ when the constituents have ...
1
vote
0answers
49 views

What insights does category theory offer in terms of grand unified theories?

What insights does category theory offer in terms of grand unified theories? Any references to books or papers that give categorical descriptions of any of the common grand unified theories would be ...
1
vote
0answers
107 views

Topological theta term as a topological quantum field theory?

It is well known that the theta term $\int d^4x\frac{\theta}{4\pi}Tr[F\wedge F]=\int d^4x\frac{\theta}{4\pi}\epsilon_{\mu\nu\sigma\lambda}Tr[F^{\mu\nu}F^{\sigma\lambda}]$ is a topological term, ...
3
votes
0answers
90 views

What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
1
vote
0answers
43 views

About category theory and physics [duplicate]

Could the ideas of category theory be applied to Physics, maybe simplifying how algebraic topology and sheaf theory and other hard-to-explain subjects are used in physics?
0
votes
0answers
38 views

What are the “name” and “coname” of operators?

So, I've been reading articles about categorical quantum mechanics, and I keep coming across definitions of "name" and "coname" of an operator. Googling these basically only turn up the papers I've ...
6
votes
0answers
99 views

target category of extended field theory

For a topological field theory to be a true “extension” of an Atiyah-Segal theory, the top two levels of its target (ie its $(n-1)^{\text{st}}$ loop space) must look like $\text{Vect}$. What other ...
5
votes
3answers
265 views

TQFT associates a category to a manifold

Any 3d TQFT (topological-quantum-field-theory) associates a number to a closed oriented 3-manifold, a vector space to a Riemann surface, a category to a circle, and a 2-category to a point. This ...
14
votes
3answers
592 views

about the Atiyah-Segal axioms on topological quantum field theory

Trying to go through the page on Topological quantum field theory - The original Atiyah-Segal axioms - "Let $\Lambda$ be a commutative ring with 1, Atiyah originally proposed the axioms of a ...
7
votes
1answer
331 views

Are identity types interpreted physically in an infinity-topos formulation of equations of motion?

In reference to Urs Schreibers paper/book on foundations of field theory Differential cohomology in a cohesive infinity-topos I wonder: are identity types there used "only" for the computations, or ...
3
votes
1answer
252 views

Etale bundles and sheaves

Before answering, please see our policy on resource recommendation questions. Please try to give substantial answers that detail the style, content, and prerequisites of the book or paper (or ...
9
votes
1answer
870 views

How algebraic geometry and motives appears in physics?

First, I'm not a physicist so I have just a little background in physics. I have been reading some noncommutative geometry books and papers (Connes, Rosenberg, Kontsevich etc) and a lot of high ...
6
votes
0answers
156 views

Quantization as a functor [duplicate]

Can anyone give an mathematical elaboration of the following statement: Quantization is a functor carrying the category of Hilbert space and linear maps to that of Symplectic manifolds satisfying ...
1
vote
2answers
236 views

Studying the logical structure of physics as a mathematical object per se? [closed]

I was wondering is there a branch of mathematical physics which studies the underlying logical structure of physics as a mathematical object per se? Let me explain what I mean by that. I'm ...
1
vote
0answers
40 views

Ontic state space from operational structures

Operationalism eschews a notion of system state in favour of the empirical behaviours of preparation and measurement. Abramsky has formalized operational structures as Chu spaces. It would be nice ...
5
votes
2answers
489 views

Does the mathematics of physics require impure set theory?

Suppose for the sake of this question that all mathematics is ultimately reducible to set theory in such a way that the only mathematical objects there really are, are sets. Now, there is a common ...
0
votes
0answers
127 views

Can quantum field theory be seen as an epistemic restriction on (quantum) causal structure

Suppose we take Vicary's quantum harmonic oscilator as a kind of "toy quantum field theory". Next, take the category of internal comonoids to not represent the background causal structure. We ...
3
votes
1answer
148 views

The (co)algebra for the (co)monad of a light switch

If we take a light switch to embody an entire category, we could take the light switch to be a set with two elements and the morphisms are all endofunctions. Let's say, for fun, that we define the ...
7
votes
2answers
219 views

Could motives aid in the study of the Navier-Stokes equations?

Recently, mathematicians and theoretical physicists have been studying Quantum Field Theory (and renormalization in particular) by means of abstract geometrical objects called motives. Amongst these ...
2
votes
1answer
107 views

Causality and operationalism: from sets and functions to monads

When working in a laboratory, the most basic behaviour is to turn a knob or dial and then see a transformation of some data output. An example is increasing a magnetic field and seeing Zeeman ...
5
votes
1answer
107 views

Quantum causal structure

We take causal structure to be some relation defined over elements which are understood to be morphisms of some category. An example of such a relation is a domain, another is a directed acyclic ...
11
votes
5answers
2k views

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

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 ...
8
votes
3answers
261 views

What categorical mathematical structure(s) best describe the space of “localized events” in “relational quantum mechanics”?

In a recent (and to me, very enlightening) paper, entitled "Relational EPR", Smerlak and Rovelli present a way of thinking about EPR which relies upon Rovelli's previously published work on relational ...