**9**

votes

**2**answers

365 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 ...

**4**

votes

**0**answers

48 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). ...

**-3**

votes

**0**answers

53 views

### I would like to study Industrial Mathematics but needs to know it importance for project managers and the the development of third world countries [on hold]

Key Importance of Industrial Mathematics in the development of third world countries?
Why industrial Mathematics important for Project Managers?

**10**

votes

**0**answers

362 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

104 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

56 views

### Petrov type D spacetime

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

151 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 ...

**1**

vote

**1**answer

77 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

565 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:
...

**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

505 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

130 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

119 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

51 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

103 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

54 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

711 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

57 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

251 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

181 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

60 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

266 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

249 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

339 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

323 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

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

173 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

148 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 ...

**5**

votes

**2**answers

267 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 ...

**6**

votes

**2**answers

361 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 ...

**0**

votes

**0**answers

91 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 ...

**0**

votes

**1**answer

289 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

115 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

208 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 ...

**12**

votes

**2**answers

295 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 ...

**3**

votes

**0**answers

125 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 ...

**1**

vote

**1**answer

129 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

78 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?

**10**

votes

**1**answer

833 views

### Vanishing of $\hat{A}$ genus and positive scalar curvature

Classicly, for a spin Riemannian manifold $M$, the $\hat{A}(M)$ genus will be $0$, if the scalar curvature is positive.
The proof is to use the Lichnerowicz formula. we have the index of the Dirac ...

**3**

votes

**1**answer

164 views

### higher order Noether identities

Noether's second variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities, which are relations among the Euler–Lagrange equations.
How about relations ...

**7**

votes

**0**answers

145 views

### Deformation of Noether's first theorem

Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...

**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

86 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 ...

**27**

votes

**2**answers

668 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} ...

**3**

votes

**0**answers

223 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 ...