Questions tagged [quantum-field-theory]

Use this tag for questions about quantum field theory in theoretical/mathematical physics. Quantum Field Theory is the theoretical framework describing the quantization of classical fields allowing a Lorentz-invariant formulation of quantum mechanics. Associate with [tag:mathematical-physics] if necessary.

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How are topological invariants obtained from TQFTs used in practice?

Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places: Atiyah, Topological quantum field theory Lurie, Topological Quantum Field Theory ...
Melissa's user avatar
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26 votes
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Was von Neumann's 1954 ICM address "Unsolved Problems in Mathematics" outdated?

I recently tried to "explain" the generalized probability theory aspect of quantum theory (as one common part of both quantum field theory and quantum mechanics), in the sense of motivations for the ...
Thomas Klimpel's user avatar
14 votes
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458 views

What are D-branes (in a topological field theory)?

In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
Dan Kneezel's user avatar
10 votes
0 answers
369 views

Multivariable Integral, How to compute it?

Q How to evaluate a multivariate integral with a Gaussian weight function? $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm f}\left(x_{1},x_{2},\...
Sijo Joseph's user avatar
9 votes
0 answers
229 views

Motivation for quantum cohomology rings

I can't seem to find a good source for the motivation for defining the big quantum cohomology ring with its quantum product. Collecting the Gromov-Witten invariants in a generating function seems like ...
failedentertainment's user avatar
9 votes
0 answers
422 views

What are the mathematical foundations of the renormalisation group?

Briefly, RG refers to mathematical tools that allows systematic investigation of the changes of a physical system as viewed at different distance scales. These methods are very important in quantum ...
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9 votes
1 answer
696 views

In what sense is quantum field theory mathematically incomplete?

Is the Yang-Mills existence and mass gap (Millenium Prize problem) essentially what is required? Or are there more problems in putting QFT on strong mathematical foundations? For example, the ...
user68441's user avatar
7 votes
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181 views

Cauchy representation and branch point order

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. The main application is for dispersion ...
MarkWayne's user avatar
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7 votes
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Reference for Dimensional Regularization

I am studying a bit of theoretical physics (QFT and string theory), and I obviously stumbled upon dimensional regularization. I have been told that this technique has in fact a solid mathematical ...
Daniel Robert-Nicoud's user avatar
6 votes
0 answers
426 views

Can anyone help to perform this heinous integral? (Peskin & Schroeder's Quantum field theory (4.76)-(4.78)) (Including my own trial )

I am reading the Peskin & Schroeder's An introduction to Quantum field theory, p.105~p.106 (Construction of a cross-section from the invariant matrix element (p.104) ) and stuck at understanding ...
Plantation's user avatar
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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. ...
Qi Tianluo's user avatar
6 votes
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Is there a relation between Super Riemannian manifolds and Kähler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
Trimok's user avatar
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5 votes
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Geometric algebra and quantum field theory

What does the reformulation of QFT with GA look like? I read that GA can be applied to almost every kind of physics, but QFT is rarely mentioned. Is there a lot of research going on in this ...
JonnyPython's user avatar
4 votes
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126 views

How to set up Chern-Simons theory, compute its "topological gauge shift" term, and compute its quantum path integrals with Wilson lines?

Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold $X$ for a Lie group (the "structure/gauge group") $G$. Is it possible to do so ...
I.A.S. Tambe's user avatar
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In what situations are Path Integrals well-defined?

In physics I have come across contexts where apparently path integrals are well-defined, and others where they are not. However I have no clear understanding of when and why they succeed or fail to be ...
leob's user avatar
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139 views

Calculation involving hypergeometric functions

I am trying to derivethe $N$ defined via the following integral: $$\displaystyle \frac{4\pi^2}{\omega N^2}= \int_0^L \frac{dz}{z} \left(1-\frac{z^2}{L^2}\right)^{-1} |V_k(z)|^2 $$ where $$\...
Nirmalya Kajuri's user avatar
4 votes
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What is $H_{3}Spin(3)$, and how is this related with the twist of framing on a 3-manifold?

From the question, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action $$I(g)=\frac{1}{4\pi}\int_{M}\mathrm{Tr}(\omega\wedge d\omega+\frac{2}{3}...
Valac's user avatar
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4-dimensional integral representation of modified Bessel function

In 4-dimensional Euclidean QFT, the propagator can be expressed in terms of a modified Bessel function of the second kind: \begin{align} \Delta_E (x_1-x_2)&\equiv\int\frac{d^4p}{(2\pi)^4}\frac{e^{...
user143410's user avatar
4 votes
0 answers
284 views

Instanton Moduli Space on ALE Spaces?

I've recently been learning about moduli spaces of instantons on $\mathbb{C}^{2}=\mathbb{R}^{4}$. From what I can gather, one can consider the framed moduli space of torsion-free sheaves on $\mathbb{...
Benighted's user avatar
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4 votes
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Non-negative integral implies non-negative integrand (up to total derivatives)?

Take a functional $S[\varphi]$ on a $d$-dimensional space-time of the form $S[\varphi]=\int d^d x\, L(x,\varphi,\partial_\mu \varphi,\partial_\mu \partial_\nu \varphi, \dots)$ where $\varphi(x)$ is ...
vvega's user avatar
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Is the physicist's notion of the generalized Dynkin index of a representation of a Lie-Algebra well defined?

Let there be a Lie-Algebra $\mathfrak{g}$ (I am most interested in the cases, where $\mathfrak{g}$ is simple or semisimple) and a irreducible representation R. Let $t^a$ be a base of the Lie-Algebra ...
warpfel's user avatar
  • 523
4 votes
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A reference containing computational examples for Quantum information

I need a reference, book, lecture note, webpage, or whatever containing a lot of computed examples. What I need is not just something having definitions, statement, proof. I don't need any statement-...
AmirHosein Sadeghimanesh's user avatar
4 votes
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523 views

Problems 2.3 in Peskin's book

It's about how to evaluate the integral below, $$ D(x-y)=\int \frac{d^3 p}{(2\pi)^3}\frac{1}{2\sqrt{p^2+m^2}}e^{-i\vec p\cdot (\vec x-\vec y)} $$ it describes the amplitude for a scalar particle in ...
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4 votes
0 answers
1k views

Contour Integration - Quantum field theory

I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk \frac{ke^{ik\...
user35952's user avatar
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4 votes
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267 views

(Topological quantum field theory) identifying objects of cobordism category

I am beginning the study of Topological quantum field theory(TQFT) and I am confused with the basic notions. Before writing down the question, to check if I understood the definition correctly, I ...
Tesuma's user avatar
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4 votes
0 answers
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4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
user avatar
4 votes
0 answers
253 views

Steps toward making the Feynman integral rigorous

I would like to know if there has been any progress recently toward giving the Feynman integral a rigorous definition. I heard that Richard Borcherds is working on giving quantum field theory a ...
Leonard Huang's user avatar
3 votes
0 answers
53 views

Does the order of operators reverse when taking the Hermitian adjoint of a product of many operators?

${}^\zeta$ Defining the vacuum state $\lvert 0 \rangle$ as that where $\hat a(\vec p)\lvert 0 \rangle=0$ and normalizing it as $\langle 0\lvert 0 \rangle = 1$, then we again build the Hilbert space as ...
FutureCop's user avatar
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Properties of reducible representations

I have the following doubt. Let's assume we have two mixed states $\rho_1 = \Sigma_i a_i \omega_i^{1}$ and $\rho_2 = \Sigma_i b_i \omega_i^{2}$ on the same algebra, where the states $\omega$ are all ...
MBlrd's user avatar
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3 votes
0 answers
138 views

Propagators and PDEs

In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. Mathematically, these derivations are somewhat magical (typically one ...
Bettina's user avatar
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3 votes
0 answers
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Derivative after applying residues theorem, that removes original poles

I have a question related to a typical integration in particle physics. Suppose one has a function $f(t) = \dfrac{i}{t^2-\omega^2 }$ with $t, \omega \in \mathbb{R}$ and $\omega>0$. I am interested ...
ilBri's user avatar
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3 votes
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Integrals of functions with poles on the real axis (Sokhotski–Plemelj theorem) in the sense of Distributions/Generalized Functions.

The Sokhotski–Plemelj theorem [Wikipedia link] states that $$ \lim_{\epsilon \to 0 } \int_{\mathbb R} dz \; \frac{f(z)}{z \pm i \epsilon} = \mp i \pi f(0) + \mathcal{P} \int dz \; \frac{f(z)}{z} \;, ...
duality's user avatar
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3 votes
0 answers
72 views

How to solve this integral for three-point correlation function?

I need to solve the following integral as part of the computation of a three-point correlation function in a charged massless quantum field theory, where I can also assume conformal invariance. $$ \...
mathematica_beginner's user avatar
3 votes
0 answers
75 views

Why does it seem possible from Physics to say something about orientability locally?

In Quantum Field Theory, if I understand correctly, Physicists showed that certain process are not symmetric under time reversal. This should be a local thing, mathematically, and would show that ...
user242318's user avatar
3 votes
0 answers
1k views

Prerequisites for Quantum Fields and Strings: A Course for Mathematicians

I'm interested in learning more about the mathematical structure underneath of quantum field theory and string theory. I've taken a few courses on quantum field theory before, so am getting more ...
leob's user avatar
  • 351
3 votes
0 answers
331 views

n-dimensional Gaussian integral with absolute values

Given two matrix $A$ and $D$ and a column vector $x$, what is the value of the following integral? $\int d^nx \; \; e^{x^T A x + \mid x \mid^T D \mid x \mid + B x}$ where $\mid x \mid_i = \mid x_i \...
Ninja Warrior's user avatar
3 votes
0 answers
90 views

Quantum Groups for Generic q and 3d-TQFT. What breaks?

I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the ...
Campbell's user avatar
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3 votes
0 answers
110 views

Integrate $ I=\int_0^1 dx\ln\bigg(\frac{A}{A-x(1-x)B}\bigg) $

I am currently working on a mathematical problem in QFT when I came across the integral: $$ I=\int_0^1 dx\ln\bigg(\frac{A}{A-x(1-x)B}\bigg) $$ I have no idea how to do such an integral, or where to ...
Gradient137's user avatar
3 votes
0 answers
242 views

What is the difference between a massless pinor and a spinor?

I was reading the paper "The Pin Groups in Physics: C, P, and T" by M. Berg, C. Morette-DeWitt et al. in which they analyze the (double) covering groups of (Lorentzian) orthogonal groups $\...
NDewolf's user avatar
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434 views

Explicitly proving that the Hamiltonian is Lorentz covariant

I want to show explicitly that the Hamiltonian $$ H = -\Omega V + \int d\tilde{\textbf{k}}\ \omega (a^\dagger(\textbf{k}) a(\textbf{k}) + b(\textbf{k}) b^\dagger(\textbf{k}) ) $$ is Lorentz ...
soap's user avatar
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3 votes
0 answers
86 views

What is the relation between BRST quantization and gauge fixing quantization

To quantize gauge field, one usually use gauge-fixing procedure and then plus ghost field, my question is what the relation between BRST quantization and gauge fixing quantization is? Because the ...
abc's user avatar
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3 votes
0 answers
95 views

Is it possible to give one general definition of the normal ordering symbol?

In Quantum Field Theory one usually defines the Normal Ordering Symbol by means of examples and a description of its action: the normal ordering $N$ applied to one expression will be the expression ...
Gold's user avatar
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3 votes
0 answers
109 views

calculating Feynman amplitude of a graph

I'm trying to understand Feynman's theorem mentioned in this paper, Chapter 0.0.2. In this paper, the Feynman amplitude of a graph $G$ is a number obtained as a result of the following process: (...
cjackal's user avatar
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3 votes
0 answers
90 views

Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...
yan's user avatar
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3 votes
0 answers
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I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
Teddy Baker's user avatar
3 votes
1 answer
294 views

Poisson summation formula for the Casimir effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...
user47224's user avatar
3 votes
0 answers
144 views

Wilson lines, boundary condisions, surface defects of TQFTs

I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
user avatar
3 votes
0 answers
287 views

Could Motives aid in the study of the Navier-Stokes equations?

Recently, mathematicians and theoretical physicists have been studying Quantum Field Theory (and renormalization in particular) by means of abstract geometrical objects called motives. Amongst these ...
Max Muller's user avatar
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2 votes
0 answers
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Square of the Dirac Delta Shenanigans

The following is a purely mathematical problem, but it often arises when dealing with transition probabilities in Quantum Field Theory. In QED the transition probability amplitudes often contain a ...
Noumeno's user avatar
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2 votes
0 answers
63 views

Proof explanation for $\left[\hat a(\vec p)^\dagger\hat a(\vec p), \hat a(\vec q)^\dagger\hat a(\vec q)\right]=0$

We have the usual commutation relations for creation/annihilation operators in QFT, $$\left[\hat a(\vec p),\hat a(\vec q)\right]=\left[\hat a(\vec p)^\dagger,\hat a(\vec q)^\dagger\right]=0\tag{1}$$ $$...
Sirius Black's user avatar