17 votes
Accepted

Where does the definition of ($\infty$-)groupoid cardinality come from?

I'll restrict to $\pi$-finite spaces (where the definition is guaranteed to make sense). Then homotopy cardinality is multiplicative in fiber sequences: if $E \to B$ is a fibration with connected base ...
Phil Tosteson's user avatar
11 votes
Accepted

What is a particle in the context of QFT with interactions?

In quantum field theory (QFT) the notion of a field is fundamental, not the notion of a particle. You can search for local excitations of the fields and identify these with particles, but this can ...
Carlo Beenakker's user avatar
10 votes
Accepted

Is Segal's notion of conformal field theory a quantum field theory in the sense of Wightman axioms?

My understanding is that Segal invented his formalism (which was then adapted by Atiyah) by thinking about the same thing Wightman was thinking about: formalising the theory of local operators. In ...
Theo Johnson-Freyd's user avatar
9 votes
Accepted

Is there, mathematically speaking, a QFT with the following properties?

First of all, strictly speaking you are talking about a finite-dimensional (in fact, 2-dimensional) caricature of QFT. More precisely, the only way in which your model can be thought of as a QFT is if ...
Pedro Lauridsen Ribeiro's user avatar
7 votes

Why computing $n$-point correlations?

Quite generally, three-point (and higher order) correlators are used to reveal the non-Gaussian (read: nonclassical) character of the fields, see for example Experimental characterization of a quantum ...
Carlo Beenakker's user avatar
7 votes
Accepted

How do these definitions of factorization algebra compare?

I believe that Definition 2 and Definition 3 are equivalent. This involves that Definition 2 implies that F is multiplicative ("for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, ...
Daniel Bruegmann's user avatar
7 votes

Defining the multiplication of distributions in the context of QFT : Colombeau algebra vs Regularity structure?

For free fields Colombeau algebra is equivalent to normal ordering (Wick ordering) of products of creation and annilation operators. The latter is more easily described, which may explain the reduced ...
Carlo Beenakker's user avatar
6 votes

Explicit form of this unitary transformation

Denote by $b_x^*$ and $b_x$ the right-hand sides of (1) and (2). They satisfy the anticommutation relations, and therefore there is an isomorphism from the $C^*$-algebra generated by the $a_x$'s onto ...
Mikael de la Salle's user avatar
5 votes
Accepted

Explicit form of this unitary transformation

Let me define (for $x\in\mathbb{Z}$) the Hermitian operators $$\gamma_{2x-1}=a_x+a_x^\dagger,\;\;\gamma_{2x}=i(a_x-a_x^\dagger).$$ The $\gamma_n$'s are known in physics as Majorana operators. Each ...
Carlo Beenakker's user avatar
5 votes

Reference for rigorous interacting many-body quantum mechanics

A textbook that covers much ground in a mathematically rigorous way is Mathematical Methods of Many-Body Quantum Field Theory by Detlef Lehmann (2004). This book offers a comprehensive, ...
Carlo Beenakker's user avatar
5 votes

Why computing $n$-point correlations?

Now, suppose I am not interested in QFT (in the sense that I don't want to quantize a classical field) but, instead, I want to study many body quantum mechanics. Tough luck! :-) These are ...
Igor Khavkine's user avatar
4 votes

Why computing $n$-point correlations?

In physics, you certainly want to do more than just find the energy eigenvalues that you get from the 2-point functions. You also want to evaluate matrix elements of operators in the corresponding ...
Michael Engelhardt's user avatar
4 votes
Accepted

How do we give a rigorous mathematical meaning to expressions like $\delta^4(0)$ or $\lim\limits_{x \to y} \delta^4(x-y)$?

Q1 refers to the identity $$(2\pi)^{-4}\int_{\mathbb{R}^4}e^{i\mathbf{k\cdot x}}d^4 x=\delta^4(\mathbf{k}).$$ So "$\delta^4(0)=(2\pi)^{-4}$ times the volume of $\mathbb{R}^4$". My answer to ...
Carlo Beenakker's user avatar
4 votes
Accepted

Structure of all Wightman QFTs

AFAIK, people have not spent much time formalizing Wightman-style axioms for QFT in a category framework. On the other hand, categories and functors have been essential elements in formulating ...
Igor Khavkine's user avatar
3 votes

How do these definitions of factorization algebra compare?

OK, here is the full story, which confirms what is written in Daniel Bruegmann's answer. In what follows, I will work with a fixed prefactorization algebra $F$ and an $n$-manifold $M$. I will make use ...
Ken's user avatar
  • 1,749
3 votes

Physical intuition behind Kontsevich's deformation quantization formula

I don't know how much this answers your questions, but it might be helpful to observe that the formula you mention does reproduce the deformation of classical observables in quantum mechanics, non-...
Josh Lackman's user avatar
  • 1,188
3 votes

Precise mathematical relation between chirality (or $\gamma_5$) and (spatial) orientation in $1+3$ Minkowski spacetime

Chirality is related to orientation via spin-momentum locking. The projector $P_\pm=\tfrac{1}{2}(1\pm\gamma_5)$ projects a spinor onto two subspaces that are decoupled in the evolution equation of a ...
Carlo Beenakker's user avatar
3 votes

"canonical" framing of 3-manifolds

The answer is a paper aptly titled Canonical framings for 3-manifolds by Rob Kirby and Paul Melvin. Here is a 2012 email correspondence between me and Kirby concerning that paper (glad I didn't walk ...
Chris Gerig's user avatar
  • 17.1k
3 votes

"canonical" framing of 3-manifolds

This is maybe not the answer you're looking for, but it's certainly too long for a comment. First, there's no contradiction here, and nothing too strange. A 2-framing on $M$ is a framing of $TM\oplus ...
Marco Golla's user avatar
  • 10.4k
3 votes

Reference for rigorous interacting many-body quantum mechanics

I believe all of these topics and more are also covered in Dereziński, Jan; Gérard, Christian, Mathematics of quantization and quantum fields, Cambridge Monographs on Mathematical Physics. Cambridge: ...
Igor Khavkine's user avatar
3 votes
Accepted

Link invariants from Hecke relations of higher order

Since you've referenced the paper by Delaney, Rowell and Wang (where the authors talk about braided fusion categories), I'll add a comment (in response to your question about taking more interesting ...
Sachin Valera's user avatar
2 votes
Accepted

Reference request: Gaussian measures on duals of nuclear spaces

There is a choice to be made here: working with Banach spaces or with spaces of distributions like $\mathscr{S}'(\mathbb{R}^d)$. There are pros and cons for each of these two settings. Let me stick to ...
Abdelmalek Abdesselam's user avatar
2 votes

Why computing $n$-point correlations?

The $n$-point functions for $n\ge 3$ are required if you are interested in how your system responds to an external probe $\phi$ (e.g. an electromagnetic field). You have to couple this to your system ...
gmvh's user avatar
  • 2,758
2 votes

Link invariants from Hecke relations of higher order

For the Kauffman polynomial, the Birman-Murakami-Wenzl algebra (BMW algebra) https://en.wikipedia.org/wiki/Birman%E2%80%93Wenzl_algebra plays a similar role as Hecke algebra. As an algebra, the BMW ...
Tetsuya Ito's user avatar

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