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.
269
questions
10
votes
1answer
308 views
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 ...
9
votes
1answer
203 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 ...
5
votes
0answers
185 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):
...
0
votes
0answers
133 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 ...
0
votes
0answers
54 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 ...
0
votes
0answers
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 ...
5
votes
0answers
148 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 ...
2
votes
1answer
152 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, ...
6
votes
1answer
211 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 ...
21
votes
1answer
860 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 ...
2
votes
0answers
69 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{...
6
votes
1answer
178 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.
...
11
votes
2answers
426 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:
...
3
votes
0answers
165 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 ...
3
votes
1answer
129 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-...
5
votes
0answers
77 views
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 ...
3
votes
0answers
111 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 ...
5
votes
0answers
140 views
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. ...
7
votes
2answers
529 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 ...
6
votes
2answers
454 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, ...
4
votes
0answers
184 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 ...
1
vote
0answers
171 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+...
11
votes
2answers
520 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
...
3
votes
2answers
634 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 ...
5
votes
1answer
314 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 ...
10
votes
0answers
193 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 ...
2
votes
1answer
75 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 ...
5
votes
0answers
162 views
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 ...
12
votes
1answer
579 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, ...
4
votes
1answer
271 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 ...
3
votes
1answer
285 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 ))
...
0
votes
1answer
144 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) ...
8
votes
2answers
411 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 ...
17
votes
1answer
707 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 ...
7
votes
1answer
195 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] = \...
17
votes
1answer
678 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 ...
4
votes
2answers
485 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{...
6
votes
2answers
201 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 ...
10
votes
0answers
126 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, ...
9
votes
1answer
879 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-...
1
vote
1answer
135 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 ...
5
votes
1answer
165 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 ...
4
votes
3answers
331 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. ...
5
votes
0answers
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 ...
4
votes
0answers
149 views
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 ...
5
votes
0answers
77 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 ...
12
votes
2answers
808 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 ...
4
votes
0answers
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 ...
14
votes
2answers
1k views
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 ...
21
votes
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 ...
