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 ...

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, ...

Community wiki

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 ...

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 ...

Community wiki

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 ...

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 ...

Community wiki

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 ...

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) ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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-...

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!)
...

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 ...

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 ...

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 $I(a,b) = \int_{\mathbb{R}} \exp(-\frac{1}{2}x^2 - ax - b x^4) dx$ ...

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 ...

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 ...

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 ...

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.

Community wiki

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 ...

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 ...

Community wiki

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 ...

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

#### Related Tags

quantum-field-theory × 376mp.mathematical-physics × 212

reference-request × 45

conformal-field-theory × 39

dg.differential-geometry × 35

quantum-mechanics × 35

topological-quantum-field-theory × 33

fa.functional-analysis × 31

string-theory × 27

at.algebraic-topology × 25

qa.quantum-algebra × 22

ag.algebraic-geometry × 19

statistical-physics × 18

rt.representation-theory × 17

pr.probability × 15

schwartz-distributions × 15

gauge-theory × 15

renormalization × 15

physics × 13

chern-simons-theory × 13

ct.category-theory × 11

gt.geometric-topology × 11

lie-groups × 11

oa.operator-algebras × 11

differential-topology × 11