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.
373
questions
72
votes
2
answers
9k
views
The amplituhedron minus the physics
Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it?
All the descriptions I've so far encountered assume ...
68
votes
5
answers
17k
views
Mathematics of path integral: state of the art
I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called ...
67
votes
4
answers
9k
views
Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?
The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
63
votes
10
answers
7k
views
Mathematical applications of quantum field theory
I understand that quantum field theories are interesting as physics; however, there is also a large community of mathematicians who are interested in them. For someone who is not at all interested in ...
53
votes
4
answers
8k
views
Why is Quantum Field Theory so topological?
I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always ...
52
votes
6
answers
13k
views
Mathematical explanation of the failure to quantize gravity naively
One often hears in popular explanations of the failure to find a "Grand Unified Theory" that "Gravity goes off to infinity, but cutting off the edges gives us wrong answers", and other similar ...
51
votes
9
answers
9k
views
The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch?
Starting from 80-ies the ideas either coming from physics, or by physicists themselves (e.g. Witten) are shaping many directions in mathematics. It is tempting to paraphrase E. Wigner, saying about "...
49
votes
4
answers
7k
views
How undecidable is the spectral gap?
Nature just published a paper by Cubitt, Perez-Garcia and Wolf titled Undecidability of the Spectral Gap, there is an extended version on arxiv which is 146 pages long. Here is from the abstract:"Many ...
44
votes
4
answers
15k
views
What is Chern-Simons theory?
What is Chern-Simons theory? I have read the wikipedia entry, but it's pretty physics-y and I wasn't really able to get any sense for what Chern-Simons theory really is in terms of mathematics.
Chern-...
37
votes
6
answers
8k
views
Doing geometry using Feynman Path Integral?
I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space.
Coming from a background of studying Quantum Field Theory from the books like ...
36
votes
6
answers
4k
views
Examples of applications of the Borel-Weil-Bott theorem?
In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes:
A representation Ri of a group G should be seen as a quantum object. This ...
34
votes
5
answers
11k
views
What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
29
votes
4
answers
7k
views
Mathematical foundations of Quantum Field Theory
Is there any reasonable approach, essentially different from Wightman's axioms and Algebraic Quantum Field Theory, aimed at obtaining rigorous models for realistic Quantum Field Theories? (such as ...
28
votes
2
answers
4k
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 ...
28
votes
2
answers
4k
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 ...
26
votes
2
answers
4k
views
BRST cohomology
I am reading some work on Mirror Symmetry from Physics perspective,the physicists seem to use some aspects of BRST quantization and BRST cohomology. What is BRST Quantization and BRST cohomology, in ...
25
votes
1
answer
4k
views
What are Gromov-Witten invariants in terms of physics?
What do Gromov-Witten invariants (of say a Calabi-Yau 3-fold) represent, or what are they supposed to represent, in terms of string theory? When I compute GW invariants, am I actually computing some ...
25
votes
1
answer
5k
views
What is the Batalin-Vilkovisky formalism, and what are its uses in mathematics?
I checked Wikipedia, I know it is a powerful quantization in physics, but I am wondering what is its relation in mathematics (like mirror symmetry as in wikipedia). A related thing is quantum master ...
25
votes
1
answer
2k
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 ...
25
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. ...
24
votes
6
answers
3k
views
Quantum fields and infinite tensor products
As I understand it, a naive interpretation of the state space of a quantum field theory is an infinite tensor product
$$\otimes_{x\in M} H_x,$$
where $x$ runs over the points of space. This ...
24
votes
0
answers
1k
views
p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)
I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be, it seems to me, a close relation between p-adic string theory and the refinement of ...
22
votes
2
answers
8k
views
How to learn QFT from mathematical perspective?
I want to learn QFT, because I have heard of its applications in mathematics, I am not interested in scattering cross sections and such. Where can I start to learn? Only books I found are either way ...
22
votes
2
answers
4k
views
2d Ising model in conformal fields theory and statistical mechanics
I am not completely sure that this question is appropriate for this mathematical site. But since in the past I did get on MO couple of times nice answers to rather physical questions, I will try. ...
22
votes
1
answer
2k
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 ...
22
votes
1
answer
1k
views
Classical and Quantum Chern-Simons Theory
Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway.
Let $\Sigma$ be a two-manifold and $M$ a ...
21
votes
3
answers
22k
views
How to count symmetry factors of Feynman diagrams?
I have enough fears that this question might get struck down. Still let me try.
I shall restrict myself to $\frac{\lambda \phi^4}{4!}$ perturbed real scalar quantum field theory and call as "...
21
votes
5
answers
5k
views
What do mathematicians currently do in conformal field theory (or more general field theory)
I am wondering what currently our mathematicians do related to conformal field theory, (I know currently it is a central topic, but I have only a vague idea what mathematicians do in there), or more ...
20
votes
1
answer
2k
views
Gauss linking integral and quadratic reciprocity
In the setting of Mazur's "primes and knots" analogy, prime ideals $\mathfrak p\subset\mathcal O_K$ correspond to "knots" $\operatorname{Spec}\mathcal O_K/\mathfrak p$ inside a "3-manifold" $\...
19
votes
3
answers
2k
views
4D TQFT from a modular tensor category
I know the construction of a 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 ...
19
votes
1
answer
1k
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 ...
18
votes
8
answers
4k
views
Path integrals outside QFT
The main application of Feynman path integrals (and the primary motivation behind them) is in Quantum Field Theory - currently this is something standard for physicists, if even the mathematical ...
18
votes
4
answers
3k
views
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 ...
18
votes
1
answer
1k
views
Do all $\mathcal{N}=2$ Gauge Theories "Descend" from String Theory?
I asked this on PhysicsSE, but I think it also fits here as it's related to algebro-geometric connections to string and gauge theory.
I'm thinking about the beautiful story of "geometrical ...
18
votes
2
answers
931
views
Infinite dimensional 2-Hilbert spaces
Is there a definition of an infinite dimensional 2-Hilbert space?
Finite dimensional 2-Hilbert spaces have been discussed by Baez in
http://arxiv.org/abs/q-alg/9609018
In the more recent paper by ...
18
votes
0
answers
529
views
Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians
Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
17
votes
2
answers
2k
views
Quantum corrections to geometry
In this video Alain Connes made a comment about the ,,quantum corrections'' of the geometry. I would like to understand this notion in some details since I haven't found anything about this in the ...
17
votes
3
answers
4k
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 ...
17
votes
2
answers
1k
views
What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?
Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$
superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $...
17
votes
4
answers
1k
views
Braided Hopf algebras and Quantum Field Theories
It is well-known, that there are a lot of applications of classical Hopf algebras in QFT, e.g. Connes-Kreimer renormalization, Birkhoff decomposition, Zimmermann formula, properties of Rota-Baxter ...
17
votes
0
answers
993
views
"Next steps" after TQFT?
(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.)
Recently, I've been ...
16
votes
3
answers
2k
views
Why is a Topological Field Theory equivalent to a Frobenius algebra?
How can a physicist understand a 2-dimensional topological field theory as a Frobenius algebra? Are there some explicit examples in order to understand this relation?
The definition (e.g. on ...
16
votes
1
answer
3k
views
Donaldson-Thomas Invariants in Physics
First of all, I am sorry for there are a bunch of questions (though all related)and may not be well framed.
What are the DT invariants in physics. When one is computing DT invariants for a Calabi-Yau ...
16
votes
2
answers
2k
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 ...
16
votes
2
answers
2k
views
Mathematical/Physical uses of $SO(8)$ and Spin(8) triality
Triality is a relationship among three vector spaces. It describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation ...
16
votes
0
answers
1k
views
"extended TQFT" versus "TQFT with defects"
There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...
15
votes
2
answers
2k
views
4d Constructive Quantum Field Theory
As a follow up to my previous question (How does Constructive Quantum Field Theory work?), I was wondering what difficulties physicists have had constructing 4d axiomatic qfts. Why has CQFT's success ...
15
votes
5
answers
2k
views
How should I think about B-fields?
So, physicists like to attach a mysterious extra cohomology class in H^2(X;C^*) to a Kahler (or hyperkahler) manifold called a "B-field." The only concrete thing I've seen this B-field do is change ...
15
votes
1
answer
674
views
Practical consequences of the geometric cobordism hypothesis
As far as I understand, the cobordism hypothesis provides a construction of all (appropriately defined) fully-extended TQFTs. In particular, given a fully-dualizable object in a certain category, one ...
15
votes
1
answer
520
views
Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert spaces on other n-manifolds
Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both $n$-torus and $n$-sphere but higher dimensional Hilbert spaces to some other $n$-manifolds? Here I am assuming ...