**-2**

votes

**0**answers

29 views

### Green`s function [on hold]

Find the Green's function $G(x,y, x',y')$ for Laplace's equation in
$0<x'<a$, $0<y'<b$; with $G=0$, $x'=0$, $G_{x'}=0$, $x'=a$, $G_{y'}=0$, $y'=0$, $G=0$, $y'=b$,
and $0<y'<b$, ...

**23**

votes

**2**answers

512 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} ...

**1**

vote

**0**answers

114 views

### Unusual generalization of the law of large numbers

I have seen in physical literature
an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...

**21**

votes

**1**answer

2k views

### What is about nonassociative geometry?

At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below:
Audience: You showed ...

**0**

votes

**0**answers

76 views

### Can someone help me find a Mathematical documentary that aired on British television within the past 10 years about Leibniz? [on hold]

I have been searching for a documentary that aired on British television between around 2006 and 2012 which was centred around the German Mathematician, Gottfried Leibniz. All that I can remember ...

**1**

vote

**0**answers

162 views

### unfolding as resolution

Has anyone described 'unfolding' as used in mathematical physics (e.g. on-shell AND off-shell) as analogous to a resolution in algebra - higher derivatives are unfolded in terms of new variables?

**33**

votes

**3**answers

3k 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 ...

**4**

votes

**0**answers

101 views

### TQFT characterization of braiding statistics

In the TQFT language, quasiparticles correspond to Wilson loop operators. It is well-known that quasiparticles can have non-trivial braiding statistics.
Take $2+1$ dimensional Abelian Chern-Simons ...

**2**

votes

**2**answers

124 views

### Momentum a cotangent vector

Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.
Furthermore, many sources ...

**4**

votes

**1**answer

547 views

### Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello!
Let's say I have
$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$
with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$).
Now separate the Amplitude and Phase of the solution:
...

**1**

vote

**1**answer

173 views

### Is the structure constant additive on connected components?

This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would ...

**7**

votes

**0**answers

66 views

### What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$.
When exactly are two unitary matrices related in this ...

**20**

votes

**1**answer

947 views

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

I'm not 100% sure that this question is appropriate for this site. If it's not, please tell me and I'll delete it.
Recently I encountered in a class the fact that there is a generating function of ...

**1**

vote

**1**answer

45 views

### Petrov classification/Weyl scalars

There is one calculation in Chandrasekhar's "Mathematical Theory of Black Holes" that I cannot understand. Here is the setup:
We want to show that Petrov type D (i.e. two principal null directions) ...

**3**

votes

**1**answer

97 views

### Reference of $\hbar$-differential operator from symplectic geometry perspective

I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...

**0**

votes

**0**answers

33 views

### Orthogonality relation for associated Legendre functions

The associated Legendre Polynomials $P_l^m(x)$ obey orthogonality relations for fixed $m$ and fixed $l$:
$$
\begin{align}
\int_0^\pi P_k^m(\cos\theta)P_l^m(\cos\theta)\sin\theta d\theta ...

**0**

votes

**1**answer

103 views

### Positivity of logarithmic energy of certain measures

Let $\Gamma$ be a smooth closed curve in the complex plane (for all practical purposes). Assume $f$ is a real-valued continuous function defined on $\Gamma$ and let $d\mu=fdm$, where $dm$ is the ...

**1**

vote

**1**answer

107 views

### String theory target spaces

In basic string theory Lagrangians (e.g. the Polyakov or the Nambu-Goto), the variables include a function $x:X\rightarrow M$ embedding a world-sheet $X$ into some target space $M$, which can be ...

**1**

vote

**2**answers

153 views

### Mathematical statistical qm book-recommendation

I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum ...

**13**

votes

**2**answers

2k views

### What does Yang-Mills and mass gap problem has to do with mathematics?

I'm not very experienced in this topic, but I read a short description of the Yang-Mills existence and mass gap problem, and as long as I understood it has mainly physical consequences and ...

**13**

votes

**3**answers

1k views

### Applications of set theory in physics

In the introduction of the paper "Links between physics and set theory", the following quote of Eris Chric is stated:
"Set theory perhaps is too important to be left just to ...

**24**

votes

**2**answers

723 views

### Interplay between Loop Quantum Gravity and Mathematics

It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...

**1**

vote

**3**answers

255 views

### What are some good sanity checks for simulating BNLS?

After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states:
$$
i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} ...

**12**

votes

**4**answers

394 views

### Obtain any 3-manifold from repeating surgeries on knots in $S^3$

In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...

**3**

votes

**1**answer

151 views

### Two equivalent descriptions of a physical system yielding a non-trivial mathematical formula

First I would like to admit that this question may not be entirely appropriate for this site, but I will give it a go none the less.
One often hears stories about how string dualities lead to highly ...

**1**

vote

**2**answers

309 views

### Certain inverse problem related to moments

Suppose $D\subset \mathbb C$ is a smoothly bounded domain and it contains the origin. Let $ds$ denote the arc length measure on $\partial D.$ I am interested in the following two inverse problems ...

**4**

votes

**1**answer

87 views

### Isomorphism of various gauge groups under homotopy

Given a compact, simple Lie group $G$, and a compact, oriented three manifold $M$, we can consider the following smooth homotopy groups:
$(C^\infty(M;G)/{\sim},\cdot )$: Here, $\sim$ is smooth ...

**3**

votes

**1**answer

84 views

### Blow-Up for Semi-Linear Wave Equations

I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...

**79**

votes

**25**answers

15k 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 ...

**10**

votes

**1**answer

525 views

### Associativity of Kontsevich's star product up to second order

In Deformation quantization of Poisson manifolds, Kontsevich gives the quantization formula
$$f \star g = \sum_{n=0}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma,\alpha}(f,g).$$
He gives ...

**4**

votes

**1**answer

550 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 ...

**4**

votes

**0**answers

92 views

### power laws emerging from the sandpile model

Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?

**1**

vote

**0**answers

205 views

### Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$
I am looking for a proof for ...

**2**

votes

**0**answers

218 views

### Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...

**1**

vote

**1**answer

169 views

### Yang-Mills Functional and Energy

I have a question about the meaning of Yang-Mills Functional.
It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is:
...

**13**

votes

**2**answers

993 views

### Symplectic formulation of statistical physics

Does there exists a symplectic formulation of statistical physics?
I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated ...

**5**

votes

**1**answer

254 views

### Domains of raising and lowering operators in QM

Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and ...

**62**

votes

**9**answers

6k views

### Theoretical physics: Why not just R^4?

You and I are having a conversation:
"Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."
"Something like that"
"...And ...

**1**

vote

**0**answers

40 views

### modern exposition of exact ground state of classical XY model or Ising model

What is the state of art technique in solving exact ground state of Heisenberg model, meaning minimization of the H terms (hamiltonian) provided infinite spin space?
...

**0**

votes

**1**answer

191 views

### How would I apply Wick's theorem to the time-ordered product of three fields?

I think I know how to apply Wick's theorem in order to expand the time-ordered product of quantum fields, but I just want to verify my understanding. Could someone perform it for the arbitrary ...

**0**

votes

**1**answer

183 views

### Some quantities which definitions are (somehow) similar to the classical Divergence

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...

**18**

votes

**2**answers

627 views

### Reflection of light from function graph

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ ...

**5**

votes

**1**answer

137 views

### Foliations of Lorentzian manifolds by Spacelike Hypersurfaces

Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...

**8**

votes

**3**answers

578 views

### Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In the paper "Exotic ...

**1**

vote

**1**answer

148 views

### 2D semilinear elliptic PDE

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. ...

**57**

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 ...

**2**

votes

**0**answers

133 views

### How to write BRST-BV for dg-Lie?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?

**2**

votes

**2**answers

359 views

### space at the Planck scale [closed]

All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings ...

**1**

vote

**0**answers

112 views

### Spectral sequences and Batalin-Vilkovisky formalism

I have been studying the BRST quantization in quantum field theory recently and noticed that the subject is very much related to algebraic topology and cohomology. A quick google search led me to the ...

**1**

vote

**0**answers

132 views

### Dirac operator in Generalized Geometry

I am wondering how the Dirac operator can be built in the context of Hichin's generalized geometry.
In particular, I have the following questions:
On a spin manifold, is the conventional spin ...