**3**

votes

**0**answers

94 views

### Rarefaction Shock Wave Interaction

I am interested in explicit solutions in 1D for the interaction of a rarefaction wave with a shock wave of arbitrary strength. The book Supersonic Flow and Shock Waves by Courant and Friedricks ...

**8**

votes

**1**answer

363 views

### Dimensional regularization in odd dimensions

I am not quite sure that my question below is appropriate for this site, probably it should be addressed to the physical commutity. But I hope that some (mathematical) physicists do attend MO. I have ...

**4**

votes

**0**answers

150 views

### Quantum Drinfeld-Sokolov reduction for a module

There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the ...

**0**

votes

**0**answers

110 views

### Can we construct CHU as an internal category in a monoidal category?

I have recently read Abramsky and Heunen's paper on Operational structures and categorical physics. I have been looking at operational structures as internal categories in a monoidal category like we ...

**2**

votes

**2**answers

168 views

### Ewald's generalized theta function

Could anyone provide me some materials on the derivation of Ewald's generalized theta function (in English)? The original paper was written in German :-(
Die Berechnung optischer und ...

**3**

votes

**0**answers

154 views

### $\mathbb Z/2$-orbifolds in Virasoro representations, CFTs, VOAs

Suppose that ${\rm Vir}_c$ is a rational Virasoro algebra with central charge $c$.
Then ${\rm Vir}_c$ has finitely many irreducible modules $M_h$, parametrised by the highest weights $h$.
Furthermore ...

**7**

votes

**0**answers

69 views

### Explicit expression of WZ term for orthogonal groups

Consider the Wess Zumino term on the the space $W=I\times D$, where D is a two dimensional disk disk and $I$ is an interval, $[0,1]$, say, i.e.,
$$
\int_{I\times D} \langle(u^{-1} \, du)^3\rangle
$$
...

**2**

votes

**0**answers

90 views

### Elliptic equations with divergence-free drift terms

Given
$\
\mathbf{u}\cdot \nabla c=\Delta c-a_{1}c+\rho \text{ on }\Omega $ with a $\Omega \subset
%TCIMACRO{\U{211d} }
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}$ bounded, $div$$(\mathbf{u})=0$, ...

**0**

votes

**0**answers

138 views

### Is there a translation invariant measure on an infinite dimensional space 'without points'?

This is just a reference request. I thought I'd come across a paper demonstrating that there is a translation invariant measure on an infinite-dimensional space without 'points' whilst browsing the ...

**6**

votes

**0**answers

152 views

### Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...

**2**

votes

**0**answers

80 views

### Helmhotz decomposition and Regularity in Stokes equation

It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely
decomposed as
\begin{eqnarray*}
\
f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)}
\
\end{eqnarray*} with $f_{0}\in ...

**11**

votes

**4**answers

528 views

### Einstein field equations in perspectives from PDE and functional analysis

The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...

**3**

votes

**2**answers

482 views

### Van Vleck-Morette Determinant

There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source ...

**4**

votes

**0**answers

137 views

### semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int ...

**1**

vote

**1**answer

138 views

### pre-symplectic and foliation and its trajectories

Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?

**3**

votes

**2**answers

248 views

### Integration over special unitary group

It is known that for $SU(N)$
$$
\int \chi_{\mu_1}(UV_1)\chi_{\mu_2}(U^{-1}V_2)\, dU = \delta_{\mu_1\mu_2}\frac{\chi_{\mu_1}(V_1V_2)}{\dim(\mu_1)}
$$
where $dU$ is Haar measure on $SU(N)$ normalized ...

**8**

votes

**1**answer

251 views

### Is there a version of supersymmetry for homogeneous spaces?

The notion of "supersymmetry" that I am aware of proceeds as follows. One fixes a spacetime $\mathbb R^n$ and signature; I will write $\mathrm{SO}(n)$ for the corresponding group of orthogonal ...

**5**

votes

**2**answers

219 views

### Certain partial integrations in quantum mechanics

In classical quantum mechanics (and specifically in the introductury texts on this topic) while calculating expectation values of certain operators in the Schrödinger approach we often have to do ...

**2**

votes

**0**answers

116 views

### Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...

**4**

votes

**0**answers

94 views

### Differences of Numbers of Helicity States in 4-dimensional Strings

The question whether the states in $D=2m + 2$ dimensional string theory,
which carry a representation of $SO(2m)$, span spaces which carry
representations of $SO(2m+1)$ seems hopelessly complicated.
...

**11**

votes

**2**answers

444 views

### Book on the Three body Problem

Hi all, I am looking for a good book about the famous (infamous perhaps?) three body problem - both theoretical and numerical hardless and accomplishments.
can you help? Thanks

**1**

vote

**0**answers

293 views

### A variation of Poisson's equation in cylindrical coordinates

Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus:
$\nabla^2 V = \frac{f(V)}{R^2}$
...

**0**

votes

**0**answers

176 views

### How to understand the matrix behind a Hamiltonian?

thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be ...

**6**

votes

**1**answer

274 views

### Are Turaev--Viro invariants secretly a discretized path integral?

Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with ...

**4**

votes

**0**answers

104 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 ...

**11**

votes

**2**answers

1k 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 ...

**0**

votes

**0**answers

112 views

### Semi-Standard Young Tableaux: Do Diagrams for $O(2m)$ combine to Diagrams from $O(2m+1)$?

Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size
$K$ with Ferrers diagrams diagram $\lambda$
(i.e. the number of all fillings of $\lambda$ with natural numbers with weakly ...

**-1**

votes

**1**answer

259 views

### identifying dual of lie algebra of general linear groups

Is there any reference for the following fact? I am looking for a nice and simple proof.
Assume that $G=GL(n,C)$, the group of invertible $n\times n$ matrices with complex entries. Why can the dual ...

**23**

votes

**6**answers

4k views

### Explanations for mathematicians, about the falsifiability (or not) of string theory [closed]

Like many other mathematicians, I think string theory very attractive. This theory has wonderfully influenced many new topics in mathematics (I myself have worked on one of them), but it's not the ...

**5**

votes

**2**answers

327 views

### Generalized basis

In quantum mechanics, people introduce the notion of "continuous basis" (I actually don't know the mathematical denomination of it). It is not a Schauder basis. I would like to know what could be a ...

**18**

votes

**1**answer

1k 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 ...

**10**

votes

**1**answer

295 views

### Do circular pipes maximize flow rate?

Suppose that $U \subset \mathbb{R}^2$ is nonempty, open, connected and bounded. Consider a Poisseuille flow in the pipe $U \times \mathbb{R}$. That is: a time-independent incompressible flow of the ...

**7**

votes

**1**answer

1k 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 ...

**7**

votes

**2**answers

310 views

### Computing a cobordism group of manifolds endowed with a real vector bundle with constraints on the Stiefel-Whitney classes

I am interested in computing the cobordism group of oriented manifolds $M$ of dimension 7 endowed with real vector bundles $N$ of rank 5 with the following conditions on the Siefel-Whitney classes:
$ ...

**10**

votes

**2**answers

541 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 ...

**4**

votes

**1**answer

127 views

### Deriving Helfrich's shape equation for closed membranes

I have a bunch of papers that claim that, from the equation for shape energy:
$$ F = \frac{1}{2}k_c \int (c_1+c_2-c_0)^2 dA + \Delta p \int dV + \lambda \int dA$$
one can use "methods of variational ...

**2**

votes

**0**answers

164 views

### Similarity solutions of the imaginary time Benjamin--Ono equation

This problem arose in the course of a theoretical physics project. We seek (complex) solutions of the imaginary time Benjamin--Ono equation
$$u_t-iu u_x-iu_{H,xx}=0$$
where $u_H(x,t)$ denotes the ...

**1**

vote

**1**answer

235 views

### When does a “constant of the motion” imply a Noether current in a quantum field theory?

Assume we are given a quantum field theory described by some functional. If $J$ is a Noether current, i.e. it is associated with a symmetry of the functional and satisfies $\partial_s J^s=0$ (Noether ...

**0**

votes

**2**answers

346 views

### A question about time in Special and General Relativity. [closed]

I apologize if this question is considered too mathematically imprecise. My understanding of Special and
General Relativity comes from reading books which attempt to explain them to non-expertsin ...

**4**

votes

**1**answer

524 views

### Dijkgraaf-Witten TQFT vs. Representation Theory?

From what I had read, group characters can be "glued" together in a topological fashion and there is something to this effect in the paper by Dijkgraaf and Witten. TQFT seems to be a topological ...

**2**

votes

**1**answer

237 views

### Time has dimension $2$ with respect to the Ricci flow scaling

Terence Tao in his lecture notes on Ricci flow has written:
If we are to find a scale-invariant (and diffeomorphism-invariant) monotone quantity for Ricci flow, it had better be constant on the ...

**3**

votes

**0**answers

528 views

### On Perelman's paper

In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Grisha Perelman has written:
Fix a closed manifold $M$ with a probability measure $m$, and suppose
that our ...

**4**

votes

**1**answer

376 views

### Exact solutions to nonlinear Klein-Gordon equation

The nonlinear pde
$$
\partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0
$$
has the exact solution
$$
\phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i)
$$
...

**1**

vote

**0**answers

185 views

### An integral with Gamma functions (Part 2)

I was wondering if there is a generalization of the integral discussed here to a case like,
\begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ...

**5**

votes

**2**answers

350 views

### computing Bernoulli numbers

Is there a good way to compute the ratio ( B[n] / n! ) that occurs so often in power series coefficients? Good in the sense that you get an answer that does not overflow a double; the largest n such ...

**1**

vote

**1**answer

145 views

### A heat kernel for Schrödinger operator with low-order terms

In "Schrödinger Operator: Heat Kernel and Its Applications", Feng computes the heat kernels associated to Schrödinger operators with at most quadratic potentials.
I am trying to see how these work ...

**1**

vote

**1**answer

570 views

### What is the “fundamental theorem of invariant theory” ?

The basic question I guess can be formulated as - given two integers $N_f$ and $N_c$ what are the ways in which the fundamental and the anti-fundamental representations of $U(N_f)$ be combined to get ...

**2**

votes

**0**answers

200 views

### Analytical continuation of electrostatic potentials

I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the electrostatic ...

**1**

vote

**0**answers

340 views

### Prove that the sum of a certain infinite series is 1

Prove the (numerically-evident) proposition that
\begin{equation}
\Sigma_{i=0}^\infty f(i) = 1,
\end{equation}
where
\begin{equation}
f(i)= 2^{-4 i-6} q(i) \frac{\Gamma(3 i+\frac{5}{2}) \Gamma(5 ...

**3**

votes

**1**answer

190 views

### Quantized conserved quantities appearing from the Lie-algebra

Hi,
consider a simple situation in quantum mechanics: Your Hilbert space is $\mathcal{H}=L^2(\mathbb{R}^3)$ and you use the obvious unitary representation $\pi\colon G=O(3)\times\mathbb{R}^3\to ...