Skip to main content
59 votes
Accepted

How do we give mathematical meaning to 'physical dimensions'?

Mathematically, the concept of a physical dimension is expressed using one-dimensional vector spaces and their tensor products. For example, consider mass. You can add masses together and you know how ...
Dmitri Pavlov's user avatar
38 votes

Mathematical applications of quantum field theory

Thomae's formula is a theorem about the properties of Riemann theta functions corresponding to hyperelliptic surfaces. In a paper, Fermionic fields on ${\mathbb Z}_N$ curves by Bershadsky and Radul, ...
33 votes

Mathematical applications of quantum field theory

One of my favorite non-physical applications of topological QFTs is a proof of Mednykh's formula $$\frac{\left|\mathrm{Hom}(\pi_1(\Sigma),G)\right|}{\left|G\right|} = \frac{1}{\left|G\right|^{\chi(\...
33 votes

Why is Quantum Field Theory so topological?

As Robert Israel indicates, if you really wanted to define a QFT, it would certainly be very analytic, say, to define the path integral. So analytic, in fact, that no one can do it for even ...
Aaron Bergman's user avatar
31 votes

Mathematical applications of quantum field theory

You may be interested in reading this, by Michael Atiyah: http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf (Internet Archive) Edward Witten certainly is the master of finding ...
31 votes

Why is Quantum Field Theory so topological?

The framing of your question is a bit ambiguous and perhaps there are two different questions here depending on the context and interpretation. One could approach your question from the point of view ...
Abdelmalek Abdesselam's user avatar
25 votes

Mathematical applications of quantum field theory

@Sarah: I have some reservations about how the question is framed since it already answers itself and makes QFT seems like a subject which is completely separate from mathematics with occasional and ...
25 votes

Mathematical applications of quantum field theory

I think a lot of the trouble people are having with this question comes from the phrase 'applications of quantum field theory'. Quantum field theory is a collection of ideas under active development;...
21 votes
Accepted

QFT and mathematical rigor

As Abdelmalek Abdesselam pointed in his comment to the OP, the axiomatic approach to QFT is rather concerned with answering the question "what is a quantum field?". This is stated right at ...
Pedro Lauridsen Ribeiro's user avatar
20 votes

Quantum corrections to geometry

Physicist chiming in; "quantum corrections of a geometry" represents the vague idea that gravity should be quantum. Classically, gravity is a geometric theory: "reality" is modelled as a (Lorentzian) ...
AccidentalFourierTransform's user avatar
20 votes
Accepted

Formal mathematical definition of renormalization group flow

The renormalization group (RG) as a geometric flow (like the Ricci flow) is a very special case of the RG, namely, the one corresponding to the nonlinear sigma-model (NLSM) in two dimensions with ...
Abdelmalek Abdesselam's user avatar
19 votes

Anomaly in QFT physics v.s. determinant line bundle

The partition function should assign to each possible field configuration $\Phi$ (or field history) in your quantum field theory a number $Z(\Phi)$. That is, it should be a function on the collection ...
Severin Bunk's user avatar
19 votes
Accepted

Definition of an n-category

First of all, there are important differences between the notions of strict $n$-category, weak $n$-category, and $(\infty,n)$-category. The easiest notion is that of a strict $n$-category, and there's ...
David White - gone from MO's user avatar
18 votes

Why is Quantum Field Theory so topological?

Perhaps the main "analytic" area in Quantum Field Theory is known as Constructive Quantum Field Theory. This essentially emerged in the 1960's with the Wightman axioms. There is still work going on ...
Robert Israel's user avatar
17 votes

Why is a Topological Field Theory equivalent to a Frobenius algebra?

Summary: the equivalence relies on the mathematical formalism of TQFTs, and sends a 2d TQFT $Z$ to the state space $Z(S^1)$, which is naturally a Frobenius algebra. One potential point of confusion ...
Arun Debray's user avatar
  • 6,776
17 votes
Accepted

Where does the definition of ($\infty$-)groupoid cardinality come from?

I'll restrict to $\pi$-finite spaces (where the definition is guaranteed to make sense). Then homotopy cardinality is multiplicative in fiber sequences: if $E \to B$ is a fibration with connected base ...
Phil Tosteson's user avatar
16 votes
Accepted

Conjecture of relation between residues of Feynman integrals and mixed Tate motives

1) Counterexamples were found in the paper Brown, Francis; Schnetz, Oliver: "A $K3$ in $\phi^4$". Duke Math. J. 161 (2012), no. 10, 1817–1862. It is now the general feeling that most $\phi^4$-Feynman ...
Friedrich Knop's user avatar
16 votes

Why is a Topological Field Theory equivalent to a Frobenius algebra?

Arun Debray's answer is good, with one addendum: 2d TQFT is the same thing as a commutative Frobenius algebra. But to truly understand what's going on, one also needs some pictures. A commutative ...
John Baez's user avatar
  • 21.7k
16 votes
Accepted

Rigorous construction of fermionic field theory?

There is the construction of the C${}^*\!$-algebra of canonical anticommutation relations (CAR's), which is actually somewhat easier than the construction of free bosonic fields: given any complex pre-...
Pedro Lauridsen Ribeiro's user avatar
16 votes
Accepted

Approach to learning constructive QFT

CQFT is very much still an open research subject. I don't think it is known what the best approach is. So all I can do is share my own opinion. (And a warning: I'm just an interested observer!) ...
user1504's user avatar
  • 5,879
16 votes

Meaning of a quantum field given by an operator-valued distribution

tl;dr: The reason for operator-valued distributions is because the physically meaningful "measurements" in QFT are things that preserve locality and that can be measured at any location. In ...
Theo Johnson-Freyd's user avatar
15 votes
Accepted

Why is a Topological Field Theory equivalent to a Frobenius algebra?

As the general relation between 2d TQFT and Frobenius algebras has already been given in another answer, let me describe the Frobenius algebras occurring in the A and B-models. 1) The A-model is ...
user25309's user avatar
  • 6,820
15 votes

Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory

I keep seeing this question percolate up. I think it deserves at least one more or less correct answer. First, the 1d integrals $I(a,b) = \int_{\mathbb{R}} \exp(-\frac{1}{2}x^2 - ax - b x^4) dx$ ...
user1504's user avatar
  • 5,879
15 votes

How do we give mathematical meaning to 'physical dimensions'?

The action appears in an exponent, so it must be dimensionless. That then fixes the dimension of each term which appears in the action and "forbids you from proclaiming that $\phi$ is ...
Carlo Beenakker's user avatar
14 votes
Accepted

References for Yang-Mills Theory

If your goal is to get some understanding of the Clay Problem, you can't really go wrong with first reading the official problem statement and then reading the papers referred to in the document. On ...
Willie Wong's user avatar
14 votes

Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?

As Igor said, it's a bit late for that. If you just finished undergrad and you discovered a passion for QFT from a rigorous mathematical standpoint, what you should do, for example, is apply for the ...
Abdelmalek Abdesselam's user avatar
13 votes

Mathematical applications of quantum field theory

The study of the renormalization procedure in QFT led to the discovery of the Hopf algebra of rooted trees. Details are given in this paper.
13 votes

Does there exist a discrete gauge theory as a TQFT detecting the figure-8 knot?

Short answer: Untwisted Dijkgraaf-Witten theories with non-abelian gauge groups (e.g. $S_3$) distinguish most knots from the unknot and from each other. Longer answer: Here's my understanding of ...
Kevin Walker's user avatar
  • 12.7k
13 votes

What are the "hot" topics in mathematical QFT at the time?

I am not sure that "hot topic" is an advisable criterion for a Ph.D. research project, since this will typically mean that easy/doable questions have been done and only the hard/intractable ...
12 votes

Mathematical/Physical uses of $SO(8)$ and Spin(8) triality

Seiberg and Witten showed that the $\mathcal{N}=2$ supersymmetric SU(2) gauge theory with $N_f=4$ flavor is endowed with SO(8) flavor symmetry, and it enjoys SO(8) triality. Later, Gaiotto's ...
Satoshi  Nawata's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible