**1**

vote

**0**answers

95 views

### Proof of Arnold-Liouville theorem in classical mechanics [on hold]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point.
Especially, I am talking about the part that ...

**0**

votes

**0**answers

42 views

### When is the discrete logarithmic energy not approximable by its ostensibly more general counterparts?

In my answer at Maximum of the Vandermonde determinant / minimum of the logarithmic energy it is shown that
that for each large enough natural $n$ there is some $a=(a_0,\dots,a_{n-1})$ with ...

**2**

votes

**0**answers

94 views

### Gauge freedom in the tetrad

I'm reading the following paper about Petrov type D space times called "Type D vacuum metrics":
http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958
by Kinnersley. I have a ...

**3**

votes

**1**answer

105 views

### Stationary distribution of last passage percolation

Consider last passage percolation model on $\mathbb{Z}^2$. I am interested to know if there is any known result for the stationary distribution of passage times, given some distribution for the ...

**1**

vote

**1**answer

62 views

### Conformally covariant distributions

In Conformal Field Theory (in $D$ dimensions) one considers (in particular) correlation functions of the form
$$
\langle O(x)O(y)\rangle,
$$
where $O$ is a scalar primary field. Scale covariance ...

**0**

votes

**0**answers

45 views

### Extracting information from a differential equation if the zero eigenvalue eigen-function is known

Given the second order linear homogeneous differential equation
$$
-\dfrac{d^2}{dx^2}\psi_m(x) + V(x)\psi_m(x)=E_m\psi_m(x)
$$
with eigen-functions $\psi_m(x)$ and eigenvalues $E_m$, what information ...

**4**

votes

**1**answer

85 views

### Gauss-Milgram formula for fermionic topological order?

For Bosonic topological order, a very useful formula was proved to be true:
$\sum_a d_a^2 \theta_a=\mathcal{D} \exp(\frac{c_-}{8}2\pi i) $
(for more detail: $d_a$ is the quantum dimension of anyon ...

**6**

votes

**3**answers

679 views

### Ambidexterity and Quantization

Here the nlab says about Hopkins-Lurie's ambidexterity paper:
"The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in ...

**0**

votes

**0**answers

54 views

### Kerr metric affine parameter

I'm going through the chapter about Kerr space-time of Chandrasekhar's "Mathematical theory of black holes", and have a question about the following transformation:
the idea is, that one wants to ...

**4**

votes

**0**answers

54 views

### Infinitesimal variation of spectrum of Schrödinger operator with changing domain

Suppose we have a Schrödinger operator
$$-\frac{d^2}{dx^2}+V(x)$$
defined on $[a,b]$ with Dirichlet boundary conditions. I am interested in whether there are any results for the variation of the ...

**1**

vote

**0**answers

259 views

### Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here
B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and
noncompact Calabi-Yau manifolds. Nuclear Physics ...

**4**

votes

**3**answers

329 views

### $A \wedge A \wedge A$ in Chern-Simons

I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form
$$A \wedge dA + \frac{2}{3}A \wedge A ...

**7**

votes

**3**answers

262 views

### Do non-normal states exist in the Solovay model?

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ...

**0**

votes

**0**answers

71 views

### How does one express a Lagrangian via differential forms? [duplicate]

I asked this question here on Physics.SE; and I accepted an answer, which thinking about it later I was dis-satisfied with; to save clicking on the link I'm reproducing the question below:
In ...

**0**

votes

**1**answer

76 views

### Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...

**4**

votes

**2**answers

170 views

### Find the expansion of the exact solution (beyond Taylor)

In a paper by Kitagawa & Ueda Squeezed spin states they give an argument that the minimum variance in one-axis twisting Hamiltonian scales like $V_{min} \propto S^{-2/3}$. I will shortly describe ...

**5**

votes

**1**answer

144 views

### What's the relation between spin model for subfactors theory and physics?

In the sense of subfactor theory, a spin model is a commuting square of the form
$$\begin{matrix}
\Delta &\subset & M_n(\mathbb{C})\cr
\cup &\ &\cup\cr
\mathbb{C} &\subset ...

**6**

votes

**2**answers

349 views

### Is there a nice “synthetic” way for doing differential geometry on infinite dimensional vector spaces?

If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...

**5**

votes

**2**answers

262 views

### Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side:
$$
\Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial ...

**0**

votes

**1**answer

273 views

### Physics that needs “new” math [closed]

Just curious: I can't think of a single example that a physicist did not had his mouth open in amazement when he learnt that all (OK, lets say the foundations) the math he needs for his brand-new ...

**1**

vote

**0**answers

114 views

### General procedure to find the determinant of an operator?

I want to learn to find the determinant of an operator.
I am given an operator like
$\Sigma _{\alpha\beta}=-k^2g_{\alpha\beta}+i\theta\epsilon_{\alpha\beta\gamma} k^\gamma$
$k^2=k^μk_μ$, $g^{αβ}$ ...

**3**

votes

**1**answer

207 views

### An integral equation

I have a Fredholm integral equation of second kind
$$\frac{-1}{2\pi \omega'^2}+\int_{-\infty}^{\infty}\frac{1}{\pi ...

**0**

votes

**0**answers

87 views

### Tetrad transformation

I have been reading an article about Type D gravitational fields, and came across the tetrad transformation that I cannot understand. The author evidently introduced a new coordinate, but just a ...

**1**

vote

**1**answer

111 views

### Null geodesic congruence

I came across a statement in Chandrasekhar's "Mathematical Theory of Black Holes" that I don't understand (rather say disagree):
Assume we have a Newman Penrose tetrad $\lbrace l, ...

**1**

vote

**0**answers

74 views

### AKSZ sigma models for higher spin

The AKSZ framework constructs 2D sigma models in the BV formalism. Is there a generalization of the AKSZ approach to higher spin?

**3**

votes

**0**answers

120 views

### On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation
$$\triangle \varphi = 4 \pi \rho \qquad (1)$$
defined on 3-dimensional oriented Riemannian manifold $(M,g)$,
where $g$ is metric and ...

**12**

votes

**2**answers

285 views

### Is there a relationship between Fourier transformations and cotangent spaces?

Maybe a trivial question but I can't seem to find it treated anywhere.
Consider a smooth manifold $Q$ (configuration space) and its cotangent bundle $T^*Q$ (phase space). Any function ...

**1**

vote

**0**answers

70 views

### Energy inequalities for Sobolev spaces of negative integer

I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| ...

**1**

vote

**1**answer

83 views

### Derivative of a time evolution operator w.r.t. a parameter

Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function.
For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U_y(b,a)$ be the time evolution ...

**3**

votes

**0**answers

218 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 ...

**27**

votes

**2**answers

658 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} ...

**5**

votes

**0**answers

129 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 ...

**3**

votes

**2**answers

169 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 ...

**8**

votes

**0**answers

84 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 ...

**1**

vote

**1**answer

62 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

110 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

43 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

129 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

123 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 ...

**25**

votes

**2**answers

783 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 ...

**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 ...

**3**

votes

**1**answer

169 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

170 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 ...

**12**

votes

**4**answers

415 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) ...

**4**

votes

**1**answer

91 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 ...

**1**

vote

**2**answers

469 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 ...

**3**

votes

**1**answer

89 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 ...

**10**

votes

**1**answer

562 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

**0**answers

101 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

**1**answer

184 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:
...