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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Gauge invariance of a QFT path integral

If we consider the usual formal construction of a path integral over fields with gauge symmetries e.g as in Weinbergs "The Quantum Theory of Fields - Volume 2" the notion of gauge invariance ...
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Representations of minimal model primary fields in the Coulomb-gas Formalism

This question is in some sense a follow-up to [1]: is it known how to construct the primary field operators of the unitary minimal models $\mathcal{M}(m+1,m)$ in the Coulomb gas formalism? (This would ...
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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 ...
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Slice in momentum space?

This is probably a very basic question but I tried physics stack exchange already and I got no answers, so I'm asking the same question here. I was reading this article and the author considers the ...
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Topological analogs of Galois representations and Selmer groups

There is an interesting analogy between primes in number fields and knots in 3-manifolds. This is can be explained by the analogy between Artin-Verdier duality theorem for number rings and the ...
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Class of spectral zeta functions whose analytic extension takes a particular form

In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
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Why do quantum observables form an associative algebra in some contexts?

In elementary quantum mechanics, we learn that quantum observables are self-adjoint operators that act on the Hilbert space of states. However, in more advanced context, we talk of local operators, ...
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Motivation for the axioms in Wick product

Here is a link for the definition of Wick product https://encyclopediaofmath.org/wiki/Wick_product, which defines the Wick product recursively. My question is where do these two equations come from? I ...
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Do Chern-Simons terms qualitatively alter the behavior of the Yang-Mills gradient flow?

I'm reading about the Yang-Mills heat flow, and I'm curious how adding a Chern-Simons term alters its solutions. This is probably elementary or folklore, but I don't know well enough to say. ...
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4 votes
1 answer
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Divisibility by 2 of invariants forms on reductive Lie algebras and anomaly cancellation for gauge theories

Let $G$ be a connected reductive group over $\mathbb C$ and let $\rho:G\to \operatorname{Sp}(2n,\mathbb C)$ be a homomorphism. You can think about $\rho$ as a linear symplectic representation of $G$ ...
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The exact domain on which the Euclidean Dirac operator is self-adjoint

I use the convention of the Weinberg QFT textbooks, that is, $(-,+,+,+)$. According to Weinberg QFT vol 2 p. 369, he says the Euclidean Dirac operator \begin{equation} {D}:=[i\partial_i +t_\alpha A_{i ...
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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 ...
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Is the timelike free boson CFT a valid CFT as per Segal's functorial CFT prescription?

Is the timelike free boson CFT a valid CFT as per Segal's functorial CFT prescription? I am aware that the Euclidean free boson theory is a well-defined CFT, but I was wondering whether one might run ...
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8 votes
1 answer
156 views

From the conceptual idea of the RG to its actual implementation

Everytime I want to understand a little more about the ideas behind Renormalization Group techniques, I get troubled by a gap between the general picture one usually presents (e.g. in books or ...
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Wightman QFTs corresponding to minimal models

Is it known (rigorously) whether or not there exist (1+1)D Wightman QFTs which can (in some reasonable sense) be said to correspond to physicists' unitary minimal models $\mathcal{M}(m+1,m)$, $m\in\...
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327 views

Yang–Mills existence and mass gap official statement on Euclidean $\mathbb{R}^4$, why not Minkowski $ \mathbb{R}^{3,1}$?

Yang–Mills existence and mass gap problem is officially stated by Clay Mathematics Institute: Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple gauge group G, a non-trivial ...
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An introductory reference for tensor networks

I found a good reference on Tensor Networks: https://arxiv.org/abs/1912.10049. But I need an introductory reference with detailed proofs on Tensor Networks. Do you know another reference?
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A spectral sequence in Khovanov Homology

Szabo equipped the mod $2$ Khovanov complex with a family of differentials $\{d_{i} \}_{i=1}^{\infty}$ such that each $d_{i}$ has bigrading $(i,2i-2)$ where $d_1$ is the mod $2$ Khovanov differential ...
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Is there an analogous notion of 'free quantum field of arbitrary spin' on a $4-$dimension finite lattice?

It is well-known that on the Minkowski spacetime $\mathbb{R}^4$, there exist a free quantum field of arbitrary spin. In the book "QFT : A Tourist Guide For Mathematicians" by Folland, a ...
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1 answer
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Supersymmetric SYK Model in 3D?

In a 2017 article More on supersymmetric and 2d analogs of the SYK model by Murugan, Stanford and Witten, the authors take a model called the SYK model (named after Sachdev, Ye and Kitaev) and study ...
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26 votes
2 answers
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How do we give mathematical meaning to 'physical dimensions'?

In so-called 'natural unit', it is said that physical quantities are measured in the dimension of 'mass'. For example, $\text{[length]=[mass]}^{-1}$ and so on. In quantum field theory, the dimension ...
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8 votes
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Donaldson invariants for piecewise-linear $4$-manifolds

It is well known that in dimension $4$, the notion of piecewise linear manifolds and the notion of smooth manifolds are the same [1][2]. On the other hand, the computations of Donaldson invariants ...
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6 votes
0 answers
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Two questions about Fock spaces

Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection ...
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2 votes
1 answer
135 views

Vacuum state generating functional

In Theorem 1 of this paper Segal stablish a relation between states and generating functionals. He assert that in order to $\mu$ be a generating functional must satisfy $$ \sum_{j,k\in F} \mu (z_j-...
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Regularization of fermionic field theory

My journey into fermionic field theory led me to this very nice paper by M. Salmhofer, which gives an overview of such theories with applications to condensed matter theory. The path integral approach ...
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24 votes
1 answer
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Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?

Edition On July 15 2021, the description of the question has been considerably modified to meet the requirement of making this question more OP-independent and thus more useful for general readers. ...
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1 answer
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Propagation of Klein-Gordon solutions in extra dimensions

In his paper "Von Neumann Algebras of Local Observables for Free Scalar Field" Araki used the solutions of the equation $$\frac{\partial ^{2}h}{\partial x^2}-\frac{\partial ^{2}h}{\partial t^...
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8 votes
2 answers
311 views

Formula for the anomalies of spin Chern-Simons theories?

$\newcommand{\SH}{\mathit{SH}}\newcommand{\Z}{\mathbb Z}$Let $G$ be a compact Lie group and $\lambda\in H^4(BG;\Z)$. The data $(G, \lambda)$ determine a 3d topological field theory called Chern-Simons ...
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11 votes
1 answer
575 views

State of rigorous effective quantum field theories

It's well-known that there are no rigorously constructed and physically relevant QFTs. There is, however, a lot of mathematical work on effective field theories and renormalization, such as the books ...
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Seeking a precedent – two-stage Gaussian integration?

Sometimes, by iteration, linear algebra can be used to solve non-linear equations. For example, consider the system $$Ax=a \qquad B(x)y=b(x), $$ where $a$ is a vector with scalar entries, $A$ is a ...
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2 votes
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181 views

Frontiers of QM and QFT

This is somehow a more mature version of an old question of mine. I'd like to have a more clear picture of the difference between QFT and QM from a mathematical point of view. Okay, so we begin with a ...
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5 votes
0 answers
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Associating noncommutative geometries to 2D conformal field theories

I have recently been reading a bit about noncommutative geometry and string theory and it looked to be an open question (or at least this was open two decades ago) whether there are constructions ...
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3 votes
0 answers
87 views

Construction of Dirac field theory

In what follows, I'm following Folland's book and Reed & Simon. Notation: Points in $\mathbb{R}^{4}$ are denoted by $p =(p_{0},p_{1},p_{2},p_{3})$. Also, I'm using Reed & Simon's notation for ...
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6 votes
1 answer
274 views

Fermions, their path integrals and effective actions

I just read the nice exposition Fermionic Path Integral on nLab and began to wonder about some details to which references appear to be lacking. Suppose we live on Euclidean space as in the ...
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1 vote
1 answer
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Invariance of Lorentz measure

Let $m > 0$ be fixed. If $x=(x_{0},x_{1},x_{2},x_{3})$ and $y = (y_{0},y_{1},y_{2},y_{3})$ are elements of $\mathbb{R}^{4}$, we denote the Lorentz inner product by: $$ x\cdot \tilde{y} := x_{0}y_{0}...
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15 votes
3 answers
2k views

QFT and mathematical rigor

One way to approach QFT in mathematical terms is provided by the so-called Gårding-Wightman axioms, which defines in rigorous mathematical terms what a quantum field theory is supposed to be. If I'm ...
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7 votes
1 answer
561 views

Rigorous construction of fermionic field theory?

In section X.7 of Reed & Simon's book there is a nice rigorous construction of the free scalar field theory which applies to the Klein-Gordon field. Question: Are there references which discuss, ...
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6 votes
1 answer
230 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 ...
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5 votes
1 answer
329 views

Initial conditions in the Klein-Gordon equation

I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$) \begin{equation}\label{kg} \left\lbrace \begin{array}{ll} (\square+m^2)F(x)=0\\ ...
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2 votes
0 answers
55 views

Reflection positivity on weighted $L^2$-spaces

Denote by $(t, x_{1}, \ldots, x_{d-1})$ the coordinates of $x \in \mathbb{R}^{d}$ and set $$\mathbb{R}^{d}_{+}=\left\{t, x_{1}, \ldots, x_{d-1} \in \mathbb{R}^{d}|t > 0\right\}. $$ Write $\theta$ ...
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2 votes
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Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry

Question: How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...
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4 votes
1 answer
388 views

Representations of the Lorentz group

The first few lines of this post is based on this lecture notes, but similar expositions can be found in other physics books such as Peskin & Schroeder's book. On chapter 8 of the linked notes, ...
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50 views

Antisymmetric tensor coordinates and tensorial spaces

I am currently working on some geometric aspects of higher-spin models for which there appear antisymmetric tensor coordinates $X^{\mu\nu}=-X^{\nu\mu}$, with $\mu,\nu=1,...,N$, which have been ...
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2 votes
1 answer
225 views

Free field rigorous quantization - possibly a misunderstanding?

I'm sorry if this is not the right place to ask this question but I've been struggling with this for days now (and I think this is too technical/specific for math stack). Notation: A conjugation $C$ ...
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1 vote
0 answers
70 views

Ordering in Cobordism Category

Let $Cob^{3}$ denote the cobordism category of $1$ dimensional manifolds i.e the objects are finite disjoint union of circles and morphisms are represented by surfaces. Is it possible to treat the ...
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4 votes
1 answer
297 views

Bochner-Minlos for moment-generating functions?

It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions? I have ...
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10 votes
1 answer
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Alternative approaches to topological QFTs

A while ago I read the paper 'Quantum Field Theory and the Jones Polynomial' by Edward Witten. This article uses a lot of concepts from physics like BRST symmetry and the Chern-Simons action which ...
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11 votes
1 answer
367 views

Mathematical construction of $\phi^4$ Euclidean field theory

One possible approach to constructive field theory is to define it on a lattice and take the scaling limit, and there are famous results stating that in $d\geq4$ this cannot lead to a non-trivial ...
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6 votes
0 answers
279 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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2 votes
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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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