**13**

votes

**1**answer

2k views

### The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results:
- Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980).
- A Galois correspondence for depth 2 irreducible subfactors (Izumi-Longo-...

**31**

votes

**1**answer

2k views

### Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...

**9**

votes

**1**answer

2k views

### Kuenneth-formula for group cohomology with nontrivial action on the coefficient

For a trivial action on the coefficient, we have the following Kuenneth formula
for group cohomology:
$$
H^n(G_1 \times G_2; M) \cong
[\oplus_{i= 0}^n H^i(G_1;M) \otimes_M H^{n-i}(G_2;M)]
\oplus [\...

**91**

votes

**12**answers

18k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**85**

votes

**16**answers

7k views

### Does Physics need non-analytic smooth functions?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is ...

**42**

votes

**9**answers

4k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**27**

votes

**7**answers

9k views

### Why does bosonic string theory require 26 spacetime dimensions?

I do not think it is possible really believe or experimentally check (now), but all modern physical doctrines suggest that out world is NOT 4-dimensional, but higher.
The least sophisticated ...

**16**

votes

**3**answers

2k views

### What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...

**7**

votes

**3**answers

1k views

### Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces?

I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.
Background
A simple consequence of the singular value decomposition is that any vector $...

**4**

votes

**0**answers

118 views

### Is there a maximal finite depth infinite index irreducible subfactor?

A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $.
It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
It's cyclic if its lattice of ...

**37**

votes

**6**answers

9k 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 ...

**44**

votes

**7**answers

8k views

### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....

**31**

votes

**6**answers

6k views

### Book on mathematical “rigorous” String Theory?

I've been looking high and low for a mathematical Book on String Theory. The only Book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and Scarlatti. I only ...

**33**

votes

**10**answers

4k views

### Is there a mathematical axiomatization of time (other than, perhaps, entropy)?

Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements ...

**18**

votes

**5**answers

5k 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

**6**answers

5k views

### A remark of Connes

In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that
I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point ...

**9**

votes

**1**answer

1k views

### Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact?
in which book we have this fact, the number of ...

**9**

votes

**1**answer

341 views

### Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...

**7**

votes

**2**answers

703 views

### Building a Physical Model to Solve Sudoku

Before asking my questions, allow me to begin with a separate example to help clarify what I'm driving at. For terms that are not defined formally, please interpret them as you feel would be most ...

**89**

votes

**26**answers

20k views

### A soft introduction to physics for mathematicians who don't know the first thing about physics

There have been similar questions on mathoverflow, but the answers always gave some advanced introduction to the mathematics of quantum field theory, or string theory and so forth. While those may be ...

**50**

votes

**11**answers

6k views

### What is Quantization ?

I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...

**45**

votes

**13**answers

12k views

### A reading list for topological quantum field theory?

Can you suggest a reading list, or at least a few papers that you think would be useful, for a beginner in topological quantum field theory? I know what the curvature of a connection is, know basic ...

**58**

votes

**1**answer

5k views

### The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...

**59**

votes

**3**answers

4k views

### What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...

**83**

votes

**3**answers

6k views

### Has the Lie group E8 really been detected experimentally?

A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum ...

**24**

votes

**5**answers

3k views

### Is symplectic reduction interesting from a physical point of view?

Do you think that symplectic reduction (Marsden Weinstein reduction) is interesting from a physical point of view? If so, why? Does it give you some new physical insights?
There are some possible ...

**34**

votes

**9**answers

4k views

### Dimensional Analysis in Mathematics

Is there a sensible and useful definition of units in mathematics? In other words, is there a theory of dimensional analysis for mathematics?
In physics, an extremely useful tool is the Buckingham Pi ...

**35**

votes

**3**answers

4k views

### Quantum mechanics formalism and C*-algebras

Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many ...

**25**

votes

**3**answers

3k views

### Classical geometric interpretation of spinors

A lot of notions in differential geometry have direct meaning in Physics. For example:
A Riemannian metric is a way to encode distances on a manifold and in Physics it is the gravitational field. ...

**27**

votes

**3**answers

2k views

### What are D-branes, really?

In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...

**25**

votes

**4**answers

3k views

### When is a symplectic manifold equivalent to a cotangent bundle?

Let $X$ be a differentiable manifold. Its cotangent bundle $T^*X$ carries a canonical 1-form $
\alpha$ whose exterior differential $\omega = d\alpha$ endows $T^*X$ with the structure of a symplectic ...

**17**

votes

**3**answers

1k views

### What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-...

**15**

votes

**11**answers

4k views

### Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences? [closed]

Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular ...

**12**

votes

**2**answers

3k views

### Classical Limit of Quantum Mechanics

There is a well-known principle that one can recover classical mechanics from quantum mechanics in the limit as $\hbar$ goes to zero. I am looking for the strongest statement one can make concerning ...

**24**

votes

**7**answers

2k views

### Does every ODE comes from something in physics?

Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself.
Say I have a nasty ODE, nonlinear, maybe extremely ...

**14**

votes

**5**answers

2k views

### Can the equation of motion with friction be written as Euler-Lagrange equation, and does it have a quantum version?

My (non-expert) impression is that many physically important equations of motion can be obtained as Euler-Lagrange equations. For example in quantum fields theories and in quantum mechanics quantum ...

**13**

votes

**2**answers

941 views

### What do correlation functions compute in CFT?

I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power ...

**5**

votes

**3**answers

2k views

### Bertrand theorem - central forces

Here is a version of Bertrand theorem. Let us consider a force $F(r)$ which depends only on the distance to a given point. If all trajectories which remain bounded are closed, then either $F(r)=ar$ ...

**13**

votes

**2**answers

1k views

### Hodge decomposition in Minkowski space

This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section.
Let us recall Maxwell’s equations: Given a semi-...

**11**

votes

**1**answer

1k views

### decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$
1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus \wedge^{6}V^{\star}\...

**10**

votes

**3**answers

510 views

### What happens to Newtonian systems as the mass vanishes?

This question is closely related to another one I asked recently, and may be thought of as a warm-up to that one.
Consider $\mathbb R^n$ with its usual metric, and pick a one-form $b$ and a function $...

**7**

votes

**3**answers

2k views

### randomness in nature [closed]

What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...

**7**

votes

**2**answers

432 views

### States in C*-algebras and their origin in physics?

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$
Now, of course there is also in classical physics and quantum ...

**29**

votes

**2**answers

876 views

### Volume of the unitary group

I saw a very remarkable asymptotic formula (or a conjecture?) for the volume of of the unitary group $ U(n)$ which is the following:
$$\log[\mathrm{Volume}(U(n))] \sim_{n\rightarrow \infty} \frac{n^...

**5**

votes

**1**answer

758 views

### basic questions on quantum integrable systems

I have been learning about (classical) integrable systems lately, e.g. in the examples of a lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions:
What ...

**14**

votes

**6**answers

1k views

### Can the “physical argument” for the existence of a solution to Dirichlet's problem be made into an actual proof?

Caveat: I don't really know anything about PDEs, so this question might not make sense.
In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace ...

**12**

votes

**2**answers

1k views

### Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$.
I found this in claim a ...

**6**

votes

**3**answers

871 views

### Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...

**3**

votes

**2**answers

1k views

### Are Fukaya categories Calabi-Yau categories?

Let X be a compact symplectic manifold. There is an idea, I think probably originally due to Kontsevich, that we should be able to get Gromov-Witten invariants of X out of the Fukaya category of X. ...

**8**

votes

**1**answer

724 views

### Coordinate-free derivation of the Einstein's field equation from the Hilbert action.

It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here):
$$S = \int_M R \mu_g,$$
is given by ...