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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156 views

What are some results that assume the Connes' embedding conjecture or any of its reformulations?

As you all (may) know, the Connes embedding conjecture was disproven last year. Also, as its Wikipedia page shows, there are multiple reformulations (but it is definitely not an exhaustive list): ...
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122 views

What does the Yang-Mills flow and the Yang-Mills QFT tell about each other?

What are some known examples of what the Yang-Mills Quantum Field Theory can tell about solutions to Yang-Mills heat equation? In general, what are some known examples of what the QFT of a Lagrangian ...
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50 views

Generalized Ising Model

I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative ...
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52 views

Gradient and Hessian

As we know gradient and Hessian of a map on Banach spaces are linear transforms (Frechet derivatives). In quantum control, control objective is a map which is defined on control fields as the ...
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141 views

Feynman path integral and Wilsonian renormalization

Everything below is to be viewed in the Euclidean setting with $d$ dimensions and all measures are understood to be Borel measures. The usual problem of Quantum Field Theory is to make sense of ...
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1answer
147 views

Spin-statistic for free quantum fields

Short version of the question: Can someone explains what physicist's 'Spin-statistic theorem' says rigorously in the context of Free Quantum fields ? Contrary to general (interacting) Quantum fields, ...
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1answer
207 views

How can one recover/obtain information from the renormalization group procedure?

I know the basic idea behind the renormalization group approach as it is used in mathematical physics to study both QFT and statistical mechanics. However, I have trouble understanding how can one ...
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1answer
843 views

Definition of an n-category

What's the standard definition, if any, of an $n$-category as of 2020? The literature I can tap into is quite limited, but I'll try my best to list what I had so far. In [Lei2001], Leinster ...
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68 views

Creation and annihilation operators as operator-valued distributions

In QFT, one usually talks about operator-valued distributions. But let's take, for instance, $L^{2}(\mathbb{R}^{3})$ and its associated Fock space $\mathcal{F} = \bigoplus_{n=0}^{\infty}L^{2}(\mathbb{...
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1answer
174 views

Affine Kac-Moody algebra from quantum group exchange algebra

In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model. ...
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408 views

Axiomatic QFT, the reconstruction theorem and functional integrals

Before posting my question, let me make some remarks: [MS] Salmhofer's book on renormalization begins with a nice discussion on Feynman's path integral. At some point, the author states the following: ...
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160 views

Derived geometry and theoretical physics

Is there any link between derived geometry and theoretical physics? for example with particle physics or quantum mechanics? Specifically something that included the obstruction bundle. If possible I ...
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1answer
122 views

Supersymmetry charge $Q$ as anti-linear and anti-unitary operator

We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$: $$ (-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0 $$ which defines the anti-...
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Representations of 2-groups and quantum double constructions

Let $G$ be a finite group. The category of its representations (complex linear, finite dimensional, throughout this whole question) is equivalent to $\mathbb{C}[G]$-modules. V. Drinfeld constructed a ...
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107 views

Intuition for conformal nets

I was planning on reading the work of Arthur Bartels, Christopher L. Douglas and André Henriques on the 3-category of conformal nets as discussed in these papers: Coordinate-free nets, Conformal ...
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Wightman reconstruction theorem-details of the proof

First of all forgive me if this question is not well suited for this forum: it is motivated by physics however after all my concerns are mathematical so I hope it would be appropriate to post it here. ...
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515 views

Is $C^{*}$-algebra the most modern way to study QFT?

I am not an expert on either QFT or $C^{*}$-algebras, but I'm trying to learn the basics of QFT. In all books/papers and other materials that I know, QFT is studied mainly using a lot of functional ...
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440 views

Creation and annihilation operators in QFT

As I said before, I'm not a QFT expert but I'm trying to understand the basics of its rigorous formulation. Let's take Dimock's book, where the foundation of QM and QFT is discussed. If we consider, ...
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179 views

Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$ The $\Spin(1,3)$ is the ...
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170 views

Multidimensional series: an application of quantum field theory

While computing the quantum vacuum energy of a real scalar field defined on $\mathbb{R}\times \mathbb{T}^3$, I encountered the following sum: $$ \sum_{n_1^2+n_2^2+n_3^2\geq 1}^{\infty} \frac{1}{(n_1^2+...
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1answer
418 views

How do you know that you have succeeded-Constructive Quantum Field Theory and Lagrangian

Quantum Field Theory is a branch of mathematical physics which is begging for a better understanding. In fact there are no rigorous constructions of interacting QFT in four dimensions. By a rigorous ...
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627 views

What are your opinions on Zeidler's QFT books? [closed]

I am interested in mathematically rigorous treatment of quantum field theory, constructive QFT in particular. I have read 'QFT, A Tourist Guide for Mathematicians' and am going to read "Quantum ...
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1answer
297 views

A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
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188 views

Is there a general theory for Wilsonian renormalization?

I know that Wilson's renormalization group is not a theory per se and that there are many ways to implement it in a given system. Also, renormalization group techniques are applied in a large number ...
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1answer
74 views

Representation of an arbitrary element on a fermionic Fock Space

Let $\mathcal{H}$ be a Hilbert space with orthonormal basis $\{\varphi_{k}\}_{k\in I}$. Take $\mathcal{H}^{\otimes n} := \overbrace{\mathcal{H}\otimes\cdots\otimes \mathcal{H}}^{\mbox{$n$ times}}$. An ...
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Research Problem combining Algebraic Geometry and QFT

A student who specialises in algebraic geometry has contacted me to ask if they could collaborate with me on some problem which relates mathematical physics and algebraic geometry. I think his idea ...
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1answer
562 views

Is there a physical reason that fields in QFT are globally defined?

I have been trying to read a physics textbook on Quantum Field theory. There seems to me to be a bit of a disconnect in most texts I have looked at between quantum mechanics and quantum field theory, ...
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1answer
266 views

Reformulation - Construction of thermodynamic limit for GFF

I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...
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1answer
277 views

Derive how the level quantization for 3d quantum Chern-Simons theory path integrals?

Let us consider abelian and non-abelian 3d quantum Chern-Simons theory path integrals: abelian Chern-Simons theory on non-spin manifolds --- $$ \int [DA]\exp(i \frac{k}{2\pi} \int_X (A \wedge dA )) ...
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1answer
140 views

Anti-symmetric operators for the Dirac or Majorana spinors

In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) ...
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2answers
400 views

Path integral derivation of extended TQFT

I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of ...
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1answer
691 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 ...
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1answer
186 views

Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?

My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \...
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1answer
661 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 ...
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2answers
475 views

QFT and its notations

I know hardly anything about quantum field theory (QFT) but I'm giving a try to understand some ideas of it. As far as I understand, in QFT one is interested in studying measures such as: \begin{...
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2answers
193 views

Relativistic scattering theory vs non-relativistic one

In relativistic scattering theory (e.g. in quantum electrodynamics) the existence of the $S$-matrix as well as of Moller operators is postulated as far as I understand (although at some stage it has ...
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125 views

What is the meaning of the coefficients of the Alekseev-Torossian associator

Drinfeld associators became a central object in mathematics and mathematical physics. They appear in deformation quantization, quantum groups, in the proof of the formality of the little disks operad, ...
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1answer
852 views

Vafa-Witten invariants for mathematicians

As Richard Thomas has written (we paraphrase just slightly), mathematical physicists Vafa and Witten introduced new "invariants" of four-dimensional spaces in a paper: A Strong Coupling Test of S-...
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1answer
133 views

Integration of a particular quartic form

I would like to solve the following integral: \begin{equation} \int \prod_i d x_i e^{a x_i^2 + b x_i^4 + c x_i^2 x^2_{i+1}} \end{equation} This integral can be for sure lead back to a common ...
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1answer
163 views

Donaldson Invariants in 2 dimensions

I am trying to understand the correspondence between Donaldson invariants and different correlation functions in certain topological quantum field theories. To be exact, among others I am reading ...
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3answers
321 views

Meaning of divergent integrals

In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...
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48 views

Expression for the (1+1)-dimensional retarded Dirac propagator in position space

Where an expression for the (1+1)-dimensional retarded Dirac propagator in position space can be found, especially including the generalized funcion supported on the light-cone? In particular, is it ...
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What is the value of the partition function of CFT on a compact conformal manifold?

Is the value of the partition function of a 2d CFT on a compact conformal manifold well defined? Or is there some kind of "anomaly" that makes it dependent on a bulk or some other kind of further ...
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76 views

Uniqueness of Witten-Dijkgraaf 2D TQFT at 0th dimension

If I understand correctly, Dijkgraaf-Witten TQFT in dimension 2 is the following. Fix any finite group $G$, we define a field over a closed 2-manifold to be a principle $G$ bundle (it's automatically ...
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805 views

A toy model in 0-d QFT

Questions For any positive integer $r$, compute $$(\frac{d}{dY})^r e^{(Y^2)}| _{Y=0}.$$ The answer should directly relates to a counting problem about Feynman diagrams. Is there a tutorial for how ...
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108 views

Inverse semigroups and partial symmetries

I recently ran across the idea of inverse semi-groups in the context of partial symmetries, where the symmetry only acts on part of the system and not the entire system (e.g., in quasi-crystals). My ...
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Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory

(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$: $$ \int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i} $$ We can generalize this ...
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2answers
2k views

Formal mathematical definition of renormalization group flow

I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow. I have tried reading up the exact definition of what ...
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143 views

Bridgeland stability for restricted Kahler moduli?

Let $X$ be a simply-connected, smooth, projective Calabi-Yau threefold. To my understanding, Bridgeland introduced stability conditions on triangulated categories to give a proper mathematical ...
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239 views

Are vertex operator algebras ever conspiratorial?

I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...

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