Questions tagged [quantum-field-theory]

For questions about mathematical problems arising from quantum field theory, the branch of physics which describes subatomic particles and their interactions in terms of perturbations of the corresponding scalar, vector or tensor fields.

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9 votes
3 answers
1k views

Stable graphs: Feynman diagrams and Deligne-Mumford space

I do not know very much about quantum field theory, but I have seen, in my reading, that stable graphs can appear in QFT in the form of, I think, Feynman diagrams. By stable graph I mean a "graph with ...
4 votes
1 answer
165 views

Reference request: Gaussian measures on duals of nuclear spaces

I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. ...
1 vote
0 answers
136 views

Recommendation to understand mean field theorem

I am studying Rodnianski and Schlein - Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Everything was clear for me and I reproved everything before inequality (3.22) (except ...
19 votes
1 answer
1k views

Anomaly in QFT physics v.s. determinant line bundle

In a quantum field theory (QFT) lecture, a math-physics professor explains the anomaly in physics, say the non-invariance of the partition function of an anomalous theory under background field ...
4 votes
4 answers
407 views

Why computing $n$-point correlations?

I am trying to find a sufficiently convincing answer to this question, but it has been taking so much of my time and I can't get anywhere. It also follows my previous question on PSE. In axiomatic QFT,...
52 votes
6 answers
13k views

Mathematical explanation of the failure to quantize gravity naively

One often hears in popular explanations of the failure to find a "Grand Unified Theory" that "Gravity goes off to infinity, but cutting off the edges gives us wrong answers", and other similar ...
4 votes
2 answers
194 views

Reference for rigorous interacting many-body quantum mechanics

Are there good references for (both zero and finite time) interacting systems of quantum many-body theory? More precisely, I would be interested in references discussing the following topics: Second ...
0 votes
0 answers
132 views

Dependence of functional integral on the function space

In physics, the following functional integral is considered \begin{gather} Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf )) \end{gather} It is usually said that the integration is performed ...
1 vote
0 answers
152 views

AQFT from a Lagrangian

In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of ...
6 votes
2 answers
598 views

Explicit form of this unitary transformation

Disclaimer: This question has its motivation from physics. It is probably not entirely rigorous at the moment. I just want to clarify some steps and try to make the arguments rigorous afterwards, if ...
10 votes
1 answer
409 views

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

The cardinality of a finite set $X$ is the number of its elements. Once you know that, you would define the groupoid cardinality of a $\infty$-groupoid $X$ as the quantity $$\lvert X\rvert := \sum_{[x]...
1 vote
0 answers
47 views

How to check if reflection positivity holds for the Atiyah n-point functions?

In the Atiyah problem on configurations of points, one defines smooth complex-valued functions $D(\mathbf{x}_1, \ldots, \mathbf{x}_n)$ on the configuration space of $n$ distinct points in $\mathbb{R}^...
1 vote
1 answer
109 views

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

This is a bit of a qualitative question, but I have great difficulty finding a reference that clarifies the point I have been confused about. So, I guess I need to ask here.. Let us restrict atttetion ...
6 votes
1 answer
1k views

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

I am still learning QFT, on my own. I am using A. Zee's nice book called quantum field theory in a nutshell. When I got to Wick's theorem, I couldn't help but notice an analogy between a formula I ...
6 votes
2 answers
343 views

"canonical" framing of 3-manifolds

In Witten's 1989 QFT and Jones polynomial paper, he said Although the tangent bundle of a three manifold can be trivialized, there is no canonical way to do this. So if I understand correctly, ...
1 vote
0 answers
126 views

Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character

In Witten's 1989 QFT and Jones polynomial paper, he wrote in eq.2.22 that Atiyah Patodi Singer theorem says that the combination: $$ \frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi} $$ is a ...
2 votes
0 answers
115 views

Rigorous QFT from integration over subspace

Many perturbative QFTs suffer from the lack of a rigorous definition of a "good enough" measure over the space of paths (or fields) $P$, $$\mathcal{Z} = \int_{{x \in P}} e^{iS(x)} Dx$$ There ...
1 vote
2 answers
222 views

Link invariants from Hecke relations of higher order

Alexander theorem says oriented links in $\mathbb{R}^3$ can be represented by closures of braids. Markov theorem says that braids related by Markov moves produce isotopic braid closures, and vice ...
1 vote
0 answers
77 views

Definition of this formula for the $2p$ functions

I am reading this paper about constructive renormalization for fermions and I got a really basic question about it. There, the effective Lagrangian (with UV cutoff $\Lambda_{0}$ and IR cutoff $\Lambda$...
4 votes
1 answer
269 views

Structure of all Wightman QFTs

I have two related questions related to constructive/axiomatic QFT. Is there a structure on the collection of all QFTs, as defined by the Wightman axioms? Do they form some type of category? ...
4 votes
0 answers
119 views

Tensor product - Vertex / Chiral algebras

Two questions regarding tensor product of modules over vertex / chiral algebras: First question: For (rational?) vertex operator algebras there is a notion of fusion product of modules inducing a ...
9 votes
2 answers
394 views

How do these definitions of factorization algebra compare?

Question Several sources define (homotopy) factorization algebras in a seemingly different manner (I am looking at [CG], [Gi], and [CFM].) I wish to know how they compare with each other. I apologize ...
1 vote
0 answers
145 views

How to compute this path integral?

Let $\mathbb{R}^2$ be phase space with coordinates $(p,q)$ and let $\epsilon>0\,.$ Then given any path $\gamma:[0,1]\to \mathbb{R}^2$ and any large enough $N>0\,,$ we can approximate $\gamma$ by ...
4 votes
1 answer
547 views

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

I'm a bit of a novice, so bear with me. My understanding of the story is as follows. From Lagrangians to Irreducible Representations The story of the types of possible particles begins with the ...
3 votes
0 answers
163 views

Properties of the stress energy tensor in Wightman formulation of CFT

In various papers that I have been reading about applying the Wightman axioms to conformal field theory, the authors write things like the following about the stress-energy tensor: $$\int \mathrm{d}x^...
9 votes
2 answers
532 views

Physical intuition behind Kontsevich's deformation quantization formula

Kontsevich gives a construction that produces deformation quantization of $C^\infty(M)$ for general Poisson manifolds $M$. The resulting formula (on $\mathbb{R}^n$) is $$ f\star g = \sum_{n=0}^\infty \...
22 votes
1 answer
2k views

Current status of axiomatic quantum field theory research

Axiomatic quantum field theory (e.g. the wightman formalism and constructive quantum field theory) is an important subject. When I look into textbooks and papers, I mostly find that the basic ...
5 votes
0 answers
192 views

Is there any overlap between the geometric and analysis oriented approaches to mathematical QFT?

The impression I have is that the mathematical approach to quantum field theory can broadly be categorized into one that is more geometrical/topological, for example in gauge theories, and another ...
8 votes
1 answer
349 views

Euclidean and Minkowski Majorana spinors - inconsistency with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf Somehow I find some inconsistency with his Tables of Euclidean and ...
0 votes
0 answers
100 views

A variant of quantum harmonic oscillators

We have the following variant of harmonic oscillators. $$ \left\{ \begin{array}{**lr**} T = a + a^\dagger\\ a | n \rangle = \sqrt{[n]} |n-1 \rangle \\ a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
7 votes
1 answer
509 views

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

For context, I heard that the Atiyah-Segal formalism of quantum field theory is equivalent to the traditional Wightman approach (at least within the scope of certain theories). However, I could not ...
17 votes
0 answers
993 views

"Next steps" after TQFT?

(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.) Recently, I've been ...
1 vote
0 answers
37 views

Splitting of the conformal group into $PSL(2,\mathbb{R})$ and other factorizations

In 1+1 dimensions of Minkowski spacetime, the conformal group can be split into two copies of $PSL(2,\mathbb{R})$ acting on null lines. I'm curious to know if a similar split exists for the conformal ...
0 votes
0 answers
85 views

Some version of non-commutative Wick formula

Let $V$ be a vertex algebra. The traditional non-commutative Wick formula is a tool to calculate term like $[a_\lambda:bc:]$. However, I need to calculate terms of the form $[:ab:_\lambda c]$. I found ...
4 votes
1 answer
324 views

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

The question is as in the title. In QFT literature, $\delta^4(0)$ is said to stand for the volume of entire $\mathbb{R}^4$, where $\delta^4(x)$ is the $4-$dimensional delta function. Or when defining ...
10 votes
1 answer
402 views

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

This is a bit of a qualitative question. A rigorous treatment of QFT comes down to making sense of multiplication of distributions, as far as I understand. This is in the aim of constructing and ...
11 votes
1 answer
829 views

Approach to learning constructive QFT

First I would like to apologize if this post breaks any rule regarding career advice or opinion-based questions. Given that construct QFT (CQFT) is a rather small community, I found this is the only ...
14 votes
4 answers
2k views

Meaning of a quantum field given by an operator-valued distribution

I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me. Let $\mathcal{H}$ be a Hilbert space in which ...
2 votes
0 answers
287 views

Segal's axioms for CFT

In Segal's papers about Conformal Field theory, https://www2.math.upenn.edu/~blockj/scfts/segal.pdf, in section $1$, he describes the evolution of a system (a string moving about in a manifold $M$) by ...
6 votes
0 answers
177 views

Infinite-dimensional BRST reduction

Fix a base field $k$. First let me loosely describe the BRST reduction in the finite-dimensional setting. For a finite-dimensional Lie algebra $\mathfrak{n}$, we can form the Clifford algebra $\...
18 votes
4 answers
3k views

What are the "hot" topics in mathematical QFT at the time?

I am currently finishing my Master's studies in mathematical physics. One topic which always interested me a lot were modern mathematical approaches to Quantum Field Theory (QFT) as well as the ...
2 votes
0 answers
80 views

Evolution equation in renormalization group for infinitely-many variables

Let $\varepsilon > 0$, $L \gg 1$ and define the torus $\mathbb{T} = \varepsilon \mathbb{Z}^{d}/L\mathbb{Z}^{d}$. Let $K$ be a smooth, strictly decreasing function. To make things easier, consider ...
7 votes
1 answer
605 views

Is there a program to solve The Yang–Mills Existence and Mass Gap problem similar to the Hamilton's program to solve Poincaré Conjecture?

According to Wikipedia: "Hamilton's program was started in his 1982 paper in which he introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré ...
34 votes
5 answers
11k views

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
3 votes
1 answer
237 views

How should I understand rigorously the definition of normal ordering of free fields

Let $\phi(x)$ be a free Hermitian scalar field in $4D$ Minkowski spacetime with the metric $(1,-1,-1,-1)$. Then, though I wrote it as $\phi(x)$, it is in fact an operator-valued tempered distribution ...
10 votes
1 answer
2k views

Quantum Field Theory: completing the "A Bridge between Mathematicians and Physicists" series

I decided to read the series "A Bridge between Mathematicians and Physicists" written by Eberhard Zeidler. But when I read the preface of the first book I realized that at first this series ...
4 votes
2 answers
289 views

Making sense of $1+1$ massless bosonic free field as a "distribution" rather than tempered

The question has been motivated by the fact that the $1+1$ massless bosonic free field suffers the infrared problem as a "tempered distribution". The reason is essentially that $\int_{\...
9 votes
1 answer
259 views

Physics application of Wilson surface observables

There is some work which generalises the usual Wilson loop in QFT to higher dimensions and constructs non-abelian Wilson surface functionals in the context of non-abelian gerbes. It seems to me that ...
4 votes
1 answer
213 views

Understanding the Osterwalder-Schrader conditions as formulated by Glimm and Jaffe

$\newcommand{\real}{\mathrm{real}}$I am having trouble with understanding the axiom (OS3) in this book by Glimm and Jaffe. It defines \begin{equation} \mathcal{A} = \left \{ A(\phi) = \sum_{j = 1}^N ...
3 votes
0 answers
74 views

Convergence in perturbative renormalization

Consider the following: $$G(\phi,W) = -\log \int d\mu_{C}(\psi)e^{-W(\phi+\psi)} \tag{1}\label{1}$$ which is very common in QFT. Here $d\mu_{C}$ is a Gaussian measure with covariance $C$. I want to ...

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