# Tagged Questions

**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

242 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

215 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

114 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

86 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

432 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

281 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

175 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

272 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

103 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

111 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

253 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

324 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

284 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

308 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:
$ ...

**9**

votes

**2**answers

519 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

124 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

163 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

229 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

345 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

508 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

235 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

524 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

358 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

183 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

345 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

144 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

556 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

197 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

332 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

185 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 ...

**3**

votes

**2**answers

291 views

### Probability distribution for two-state system that depends on residence time

I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...

**7**

votes

**1**answer

289 views

### BRST cohomology definition

Is there written anywhere a full definition of BRST cohomology? All I have found so far is BRST cohomology in _______.
As far as I can see, BRST cohomology is the ...

**32**

votes

**2**answers

1k views

### Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...

**10**

votes

**4**answers

760 views

### Quantum algorithms for dummies

I want to try my hand at designing quantum algorithms to solve certain problems. I feel like I understand (for example) how Grover's algorithm and Shor's algorithm work, and I'm excited to apply the ...

**5**

votes

**0**answers

290 views

### unitary equivalence

Let $U$ be the bilateral shift operator in $l^2(Z)$, and let $V$ stand for a rotation on an irrational angle $\alpha$ in $L^2(T, \mu)$, where $T$ is a circle with a rotation-invariant Lebesgue measure ...

**3**

votes

**1**answer

160 views

### Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms ...

**3**

votes

**1**answer

187 views

### Exact sampling from 2D Ising model where coupling is constant?

What progress has been made towards sampling from the 2D lattice Ising model with the following Hamiltonian:
$H=-J\sum_{\langle i,j \rangle}S_iS_j - \sum_i b_iS_i$
Where the first sum runs over all ...

**19**

votes

**0**answers

273 views

### Infinitely many planets on a line, with Newtonian gravity

(I previously asked essentially this on physics.stackexchange, but was actually
hoping for answers with something closer to a proof than what I got there.)
Suppose we have a unit mass planet at each ...

**3**

votes

**1**answer

199 views

### What are the invariant definitions of spinorial quantities from mathematical physics?

When physicists write expressions involving spinors $\psi \in S \otimes V$, where $S=S_+ \oplus S_-$ is a complex spinor representation of a spin group $Spin(2d)$ and $V$ is a complex representation ...

**8**

votes

**6**answers

1k views

### Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with Intuitionistic logic. It is ...

**1**

vote

**0**answers

118 views

### Global maximum of non-concave function

Hello!
I have a set of function, such as
$f_1(x_1,x_2,x_3,x_4,x_5,x_6)=\frac{x_1}{1+x_1-x_6} + \frac{x_2}{1+x_2-x_5} + \frac{x_3}{1+x_3-x_4} + \frac{x_4}{1+x_4-x_3} + \frac{x_5}{1+x_5-x_2} + ...

**3**

votes

**2**answers

499 views

### Energy functional

During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works ...

**4**

votes

**0**answers

266 views

### Which orbits of a separable representation of the infinite unitary group are closed?

Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following:
Is it true that all ...

**2**

votes

**1**answer

250 views

### Explicit 2-Cocycles of G=Z2×Z2xZ2 over U(1)

We know that group cohomology $H^2(G,U(1))$ consists of 2-cocycles $\beta(A,B)\in U(1)$ corresponding to elements in the group $H^2(G,U(1))$, where $A\in G,B \in G$. Note that $\beta(A,B)$ satisfies ...