**1**

vote

**1**answer

66 views

### Minkowski spacetime in Newman Penrose formalism

I have a rather basic question for which (surprisingly!) I cannot find a short and clear answer anywhere:
I'm currently looking at the Newman Penrose (NP) formalism (I use primarily Chandrasekhar's ...

**1**

vote

**0**answers

45 views

### Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...

**11**

votes

**2**answers

276 views

### Quantum Hamiltonian for an Inverse Cube Force Law

If you have a nonrelativistic quantum particle in $\mathbb{R}^3$ in an attractive inverse cube force, its Hamiltonian is
$$ H = -\nabla^2 - \frac{c}{r^2} $$
where I'm keeping things simple by ...

**4**

votes

**1**answer

300 views

### Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau

Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$:
$$
\int_M
e^{n \mathbf{e}}
...

**0**

votes

**0**answers

57 views

### Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular:
$$
\int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M},
$$
in which ...

**14**

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 ...

**9**

votes

**2**answers

640 views

### How should a mathematician approach the physics literature concerning percolation?

I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...

**2**

votes

**1**answer

98 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?

**15**

votes

**6**answers

2k views

### Can I derive the Boltzmann distribution by an invariance argument?

In statistical mechanics, the Boltzmann distribution gives the probability of a system being in state $i$ as
$$\displaystyle \frac{e^{- \beta E_i}}{\sum_i e^{-\beta E_i}}$$
where $E_i$ is the energy ...

**24**

votes

**3**answers

2k views

### Shape of snowflakes

Is there a mathematical theory that explains the shape of a snowflake? Why is it not round?
Update Tree-like metric spaces appear often as limits of sequences of metric spaces (say, asymptotic cones ...

**0**

votes

**1**answer

76 views

### Existence and uniqueness for two-dimensional time-dependent Schrödinger equation

I currently have to deal with time-dependent Schrödinger equations in two variables on bounded domains and wanted to find out about uniqueness and existence of solutions.
Unfortunately, I am a ...

**3**

votes

**1**answer

175 views

### Action of G_2 on certain 7x7 skew-symmetric matrices

I have been working with something related to Goldman bracket for $G_2$ gauge group. There I have something like "$\text{Tr}(M_{\gamma}O_i)$", where $M_{\gamma}$ is a monodromy which takes value in ...

**1**

vote

**1**answer

117 views

### A proof of the Ibragimov et al. commutation relation

Let $u_a(x),\,a=1,2,\ldots n$ be a $n$-component field in Minkowski spacetime
$x^\mu,\,\mu=0,1,2,3$ and let $u_{a,\,\mu}=\frac{du_a}{dx^\mu}$. Let us introduce two operators (we use Einstein ...

**5**

votes

**1**answer

577 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:
...

**4**

votes

**3**answers

2k views

### Ruelle Perron Frobenius Operator

Hello everybody! Recently in my research, I came across the Perron-Frobenius operator .. I would like to intuitive interpretation of this operator, ie, physical interpretations are possible, articles ...

**34**

votes

**7**answers

6k 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 ...

**1**

vote

**0**answers

41 views

### anomaly polynomial of generalized Hitchin system

I would like to ask about mathematical background of this object. So, I am trying to puzzle out with 4d $\mathcal{N}=2$ SQFT. As far as I can gather this theroy can be described in terms of ...

**17**

votes

**5**answers

4k 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?

**7**

votes

**3**answers

2k views

### Total energy of the Universe

In popular science books and articles, I keep running into the claim that the total energy of the Universe is zero, "because the positive energy of matter is cancelled out by the negative energy of ...

**5**

votes

**0**answers

110 views

### Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a “Wick rotation”?

We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like
...

**10**

votes

**2**answers

484 views

### Witten's proof of Morse inequalities, question on eigenvalues?

See here. I present Theorem 6 and Corollary 7 as follows.
Theorem 6. For large $s$, $\Delta_s$ and $H$ have the same number of eigenvalues in the interval $[0, s^{-2/5}]$.
Corollary 7. $\dim ...

**5**

votes

**0**answers

74 views

### Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? :
$V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). ...

**10**

votes

**0**answers

398 views

### Which journals publish applied mathematics with mostly pure mathematics content?

In the spirit of Which journals publish expository work? please advise:
What consistently high quality journals$^1$ today publish results that would otherwise go to a pure mathematics journal if ...

**2**

votes

**1**answer

108 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 ...

**-1**

votes

**0**answers

72 views

### Petrov type D spacetime [on hold]

I have a rather general question about the Petrov type D spacetime.
The standard definition (also used in the official Wikipedia article) is that the Petrov classification is according to which Weyl ...

**6**

votes

**2**answers

161 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 ...

**13**

votes

**5**answers

1k views

### Is there a relation between 4-dimensional general relativity and exotic smooth structures on $\mathbb{R}^4$?

Let's say General Relativity is the study of the Einstein equation on smooth Lorentzian manifolds, i.e. pseudo-Riemannian manifolds of signature $(n-1,1)$.
I've heard more than once people say that ...

**14**

votes

**5**answers

1k 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 ...

**27**

votes

**6**answers

3k views

### Good references for Rigged Hilbert spaces?

Every now and then I attempt to understand better quantum mechanics and quantum field theory, but for a variety of possible reasons, I find it very difficult to read any kind of physicist account, ...

**1**

vote

**2**answers

509 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 ...

**1**

vote

**0**answers

147 views

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

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 ...

**3**

votes

**1**answer

120 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 ...

**0**

votes

**0**answers

53 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

110 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 ...

**0**

votes

**0**answers

59 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 ...

**6**

votes

**3**answers

740 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

58 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 ...

**2**

votes

**1**answer

254 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 ...

**2**

votes

**0**answers

182 views

### The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$:
...

**4**

votes

**0**answers

61 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 ...

**7**

votes

**3**answers

277 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 ...

**24**

votes

**1**answer

1k 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. ...

**4**

votes

**1**answer

252 views

### Subset of causal spacetime+Imprisonment Condition+Compact Closure -> Stably Causal spacetime?

My question arose after studying the article "John K. Beem: Conformal Changes and Geodesic Completeness". (http://projecteuclid.org/euclid.cmp/1103899983) One of the results there is:
Let $(M,g)$ ...

**4**

votes

**3**answers

349 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 ...

**9**

votes

**5**answers

1k views

### Spectral theorem for self-adjoint differential operator on Hilbert space

I need a reference concerning a theorem that shows the following result, stated very roughly:
Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert ...

**3**

votes

**3**answers

324 views

### Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...

**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

77 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

176 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

151 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 ...