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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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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" ...
Isaac's user avatar
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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 ...
Ben Sprott's user avatar
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2 votes
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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 ...
Azam's user avatar
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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 ...
Azam's user avatar
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6 votes
0 answers
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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 ...
CBBAM's user avatar
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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 ...
Isaac's user avatar
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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 ...
Mr. Proof's user avatar
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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. ...
CBBAM's user avatar
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4 answers
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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,...
MathMath's user avatar
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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 ...
0x11111's user avatar
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4 votes
2 answers
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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 ...
MathMath's user avatar
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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 ...
Gabriel Palau's user avatar
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]...
Matthew Niemiro's user avatar
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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}^...
Malkoun's user avatar
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1 answer
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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 ...
Isaac's user avatar
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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, ...
zeta's user avatar
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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 ...
zeta's user avatar
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1 answer
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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 ...
Malkoun's user avatar
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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 ...
Student's user avatar
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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 ...
JustWannaKnow's user avatar
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 ...
Student's user avatar
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0 answers
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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 $\Lambda$...
MathMath's user avatar
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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? ...
curiouser's user avatar
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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 ...
E. KOW's user avatar
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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 ...
Josh Lackman's user avatar
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5 votes
1 answer
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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 ...
Mehmet Coen's user avatar
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 ...
Ken's user avatar
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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^...
Connor Mooney's user avatar
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 ...
CBBAM's user avatar
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1 answer
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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 ...
Ezzeddine El Sai's user avatar
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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\...
Lili Si's user avatar
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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 ...
Nicholas James's user avatar
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 ...
Gabriel Palau's user avatar
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 ...
Estwald's user avatar
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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 ...
Isaac's user avatar
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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 ...
Isaac's user avatar
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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 ...
Guillermo García Sáez's user avatar
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 $\...
Estwald's user avatar
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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 ...
Jannik Pitt's user avatar
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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 ...
CBBAM's user avatar
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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 ...
MathMath's user avatar
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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 ...
Isaac's user avatar
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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_{\...
Isaac's user avatar
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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 ...
JustWannaKnow's user avatar
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 ...
Hollis Williams's user avatar
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 ...
Kimyeong Lee's user avatar
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 ...
Andi Bauer's user avatar
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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 ...
truebaran's user avatar
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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 ...
Confused Physicist's user avatar
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. ...
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