# 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.

376
questions

0
votes

1
answer

61
views

### Extracting each field operator as Wightman fields from a set of time-ordered products satisfying Eckmann-Epstein axioms

The paper by Eckmann-Epstein proves that Schwinger functions at "coinciding points" uniquely defines "time-ordered products".
In physics, these "time-ordered products" ...

-2
votes

0
answers

83
views

### The succinct, abstract derivation of the holographic principle

I am looking for a five line derivation of the holographic principle that involves only mathematics. I figured it should be something like a quantum field on a curved spacetime, so it would just be ...

2
votes

0
answers

144
views

### Renormalization from cohomology point of view

In order to construct a Euclidean quantum filed theory one usually needs to take care of the renormalization problem. Let us consider a simple model like $\phi^4$ in dimension two. In this case just ...

5
votes

1
answer

194
views

### The equivalence of stochastic quantization and path integral quantization

I am looking for a reference in which the equivalence of stochastic quantization and
path integral quantization has been shown. It would be great if I can see such a relation for a Euclidean quantum ...

6
votes

0
answers

102
views

### Wick ordering, probability vs physics

Consider a collection of creation $a^\dagger$and annihilation operators $a$. In physics one defines Wick ordering (also known as normal ordering) as a prescription to place all creation operators ...

1
vote

1
answer

240
views

### Wick product of free fields and wave front sets in the sense of Lars Hörmander

Let $\phi$ be the neutral, massive and free scalar field in $\mathbb{R}^4$. That is, $\phi$ is a tempered distribution whose values are unbounded operators on the Bosonic Fock space.
Note that the ...

1
vote

0
answers

146
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 ...

4
votes

1
answer

178
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. ...

4
votes

4
answers

433
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,...

0
votes

0
answers

140
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 ...

4
votes

2
answers

210
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 ...

1
vote

0
answers

155
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 ...

10
votes

1
answer

418
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

50
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

128
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

2
answers

355
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

132
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 ...

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 ...

2
votes

0
answers

121
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 ...

6
votes

2
answers

622
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 ...

1
vote

2
answers

228
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

275
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?
...

5
votes

0
answers

124
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 ...

1
vote

0
answers

152
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 ...

5
votes

1
answer

582
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 ...

9
votes

2
answers

402
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 ...

3
votes

0
answers

166
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^...

5
votes

0
answers

200
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 ...

7
votes

1
answer

544
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 ...

0
votes

0
answers

101
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\...

17
votes

0
answers

1k
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

38
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

95
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

327
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

407
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 ...

2
votes

0
answers

344
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

179
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 $\...

15
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 ...

11
votes

1
answer

900
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 ...

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 ...

3
votes

1
answer

265
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 ...

4
votes

2
answers

293
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_{\...

2
votes

1
answer

163
views

### The ultraviolet limit as a limiting case of the renormalization group flow?

In his paper Constructive Renormalization Theory, V. Rivasseau describes the idea of Wilson's approach of solving path integrals step by step. In section 1.4, page 5, however, there is a statement ...

9
votes

1
answer

264
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 ...

9
votes

0
answers

292
views

### Dual Coxeter numbers, Langlands dual groups, black holes and twisted compactification of 6d (2,0) A D E theories on a circle

A 6-dimensional (2,0) superconformal quantum field theory comes in Lie algebra A, D, E types. These theories do not have classical Lagrangian and are purely quantum.These theories on a torus ...

4
votes

0
answers

282
views

### CFT as an axiomatic field theory

I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every ...

2
votes

1
answer

192
views

### Another formula for the Schwinger term — problems with a calculation

$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...

15
votes

1
answer

710
views

### Practical consequences of the geometric cobordism hypothesis

As far as I understand, the cobordism hypothesis provides a construction of all (appropriately defined) fully-extended TQFTs. In particular, given a fully-dualizable object in a certain category, one ...

9
votes

1
answer

388
views

### Propagators and PDEs

I have already asked this at MSE but did not get an answer.
In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. ...