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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Reference request for $\phi^{4}_{d}$ theory - where to begin?

When I started studying the basics of $\phi^{4}_{d}$, I looked for papers or lecture notes which would give me some general ideas about the topic and which would construct and/or prove the basic ...
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2 votes
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Fourier transform of the hyperboloid

Equip $\mathbb{R}^{d+1}$ with the Lorentzian form $\langle x, y\rangle=-x^0y^0+{\bf x}\cdot{\bf y}$ where $x=(x^0,{\bf x})$ and $\cdot$ is the usual Euclidean dot product. We define the hyperboloid $\...
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4 votes
1 answer
113 views

Cluster expansion, Mayer expansion and perturbative renormalization group

This is a second part of my previous question, which I decided to split into two parts not to mix up different topics at one giant question. Again, according to V. Rivasseau (section 1.5 of ...
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5 votes
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What is a large field problem?

I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify. On page 2, Rivasseau talks about the large field problem and, if I understood it ...
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Cutoff and regularization

Some variant of this question has probably been asked before on this site but my idea is to work with an explicit example. Suppose we discretize the momentum space, so we work with: $$\Lambda^{*} := \...
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6 votes
1 answer
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The role of estimates in field theories

I have been taking a look at some papers in constructive quantum field theory and I got the impression that there is a systematic of estimating things like e.g the effective action or the free energy ...
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2 votes
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57 views

Pullbacks of LCS-valued distributions

Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\...
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Arithmetic analogues in Liouville quantum gravity

I recently discovered about Minhyong Kim's work on what can be coined "Arithmetic Gauge Theory/Arithmetic Chern Simmons Theory". Since Liouville quantum gravity is fully understood, I was ...
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3 votes
0 answers
177 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 ...
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7 votes
1 answer
232 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 \...
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1 answer
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Can Fock spaces be replaced by arbitrary Hilbert spaces under some hypothesis to justify path integrals?

I was reading this post from PSE and it reminded me an old question of mine, in which the use of creation and annihilation operators were discussed. Both questions got answers which agreed on the fact ...
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Borel vs genuine equivariant cohomology in quantum field theory

A lot of important work in quantum field theory involves Borel equivariant cohomology of certain geometric objects, usually with the goal of computing integrals over some complicated moduli stack. In ...
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2 votes
0 answers
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Localization for generalized Borel cohomology

For both equivariant de Rham cohomology and equivariant K-theory (in the "naive" or Borel sense), we have localization formulae which allow us to compute this cohomology in terms of the ...
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2 votes
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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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3 votes
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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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102 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 ...
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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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3 votes
0 answers
141 views

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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1 vote
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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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2 votes
0 answers
87 views

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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4 votes
1 answer
148 views

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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0 answers
187 views

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
143 views

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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3 votes
1 answer
109 views

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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18 votes
4 answers
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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 ...
2 votes
0 answers
102 views

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
166 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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10 votes
1 answer
340 views

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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6 votes
0 answers
350 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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4 votes
1 answer
103 views

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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2 votes
0 answers
106 views

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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1 vote
0 answers
41 views

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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3 votes
1 answer
117 views

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
3k views

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
0 answers
183 views

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
167 views

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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3 votes
0 answers
86 views

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
2k views

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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0 votes
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
344 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
617 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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3 votes
0 answers
117 views

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
0 answers
188 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
121 views

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
90 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
290 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
116 views

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
576 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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