**78**

votes

**12**answers

16k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**14**

votes

**3**answers

835 views

### Number theory and physics

I was following some lectures by Edward Frenkel about Langlands correspondence. He was describing some analogies between number theory and theoretical physics (Mirror symmetry). At some point ( my ...

**26**

votes

**6**answers

5k views

### Mathematician trying to learn string theory

I'm a mathematician. I want to be able to read recent ArXiv postings on high energy physics theory (String theory) (and perhaps be able to do research). I want to understand compactifications, ...

**1**

vote

**0**answers

58 views

### Understanding the Segal-Sugawara construction

I am trying to understand the Segal-Sugawara construction from the book "vertex algebras and algebraic curves" by Frenkel, Ben-Zvi in 3.4.8. As an absolute layman in the area of mathematical physics ...

**42**

votes

**7**answers

7k 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 ...

**5**

votes

**2**answers

1k views

### Space with 720° / not 2$\pi$ rotational symmetry?

Is there a space with a 720°, but no 360° rotational symmetry? Possibly one that can be mapped onto something more conventional like R(3) or R(3,1)?
The reason I am asking is because in quantum ...

**0**

votes

**1**answer

93 views

### Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality.
As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. ...

**2**

votes

**1**answer

161 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

**1**answer

154 views

### Equation of motion for the Lagrangian $\mathcal{L} = \text{Tr}(\partial^\mu G \partial_\mu G^{-1})$, $G$ is unitary $N \times N$ matrix? [closed]

What is the equation of motion for the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G \partial_\mu G^{-1})$$where $G$ is an $N \times N$ unitary matrix? Could anyone supply a reference to its ...

**4**

votes

**1**answer

71 views

### Compact Quantum Groups and FRT-Algebras

As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $*$-sub-algebra is an FRT-algebra, which ...

**1**

vote

**0**answers

85 views

### Line bundles with vanishing cohomology on Calabi-Yau manifold

Suppose we have some line bundle $L(D)$ on Calabi-Yau threefold. Let's call this line bundle "rigid" if $H^0(X,L(D)) \simeq \mathbb{C}$ and $H^i(X,L(D))=0$ for $i=1,2,3$.
Is anything known about such ...

**1**

vote

**0**answers

66 views

### One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...

**0**

votes

**2**answers

204 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 ...

**4**

votes

**1**answer

112 views

### Generating Functional for the Dirac Field, equivalence of expressions

As with the Klein-Gordon field, we can alternatively derive the Feynman rules with the free Dirac theory by means of a generating functional. In analogy with the scalar field theory where $Z[J]$ is ...

**3**

votes

**0**answers

72 views

### Branches of 3j symbols

Question
Is there a quick way to identify the branches in a 3J symbol?
Context
I need to compute Wigner 3J symbols/Clebsch–Gordan coefficients,
$$
\begin{pmatrix}
\ell_1 &\ell_2 &\ell_3\\
...

**4**

votes

**1**answer

151 views

### Sign of 3j symbol (in view of interpolation)

Question
Is there a closed formula for the sign of a 3j symbol?
Context
I need to compute Wigner 3J symbols/Clebsch–Gordan coefficients,
$$
\begin{pmatrix}
\ell_1 &\ell_2 &\ell_3\\
...

**0**

votes

**1**answer

73 views

### Self-adjointness of the components of the magnetic derivative

On $L^{2}(\mathbb{R}^{n})$ define the operator $\Pi_{j} u := (-i\partial/\partial x_{j} - A_{j})u$, where $A_{j} \in L^{2}_{loc}(\mathbb{R}^{n})$ represents the $j$-th component of the magnetic ...

**5**

votes

**0**answers

100 views

### Relativistic correction to the Hydrogen atom spectrum, reference? [closed]

Can anyone provide me a reference to the relativistic correction to the Hydrogen atom spectrum worked out? I've tried googling, but I haven't found anything (perhaps I'm just bad at googling). Thanks!
...

**4**

votes

**2**answers

190 views

### An extremal type problem on segments

I am interested in the following extremal-type problem.
Let us define $\Psi$ as
$$\Psi(x)=\max_{f\in L^2[0,x] \,\,with\,\,\|f\|_2=1}\Bigg|\int_0^x\int_0^xf(t)f(s)\ln|t-s|dsdt\Bigg|$$
on ...

**4**

votes

**1**answer

329 views

### What are the ramifications of the Hodge conjecture to mathematical physics?

I read a little bit the survey:
http://www.claymath.org/sites/default/files/hodge.pdf
Well I understand its a conjecture in Algebraic Geometry, so it seems there should be some consequences of ...

**2**

votes

**1**answer

103 views

### Expectation equation, harmonic functions, do not understand why equation is true

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| ...

**31**

votes

**1**answer

2k 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. ...

**2**

votes

**0**answers

83 views

### Complex scalar field, computation of propagators, four point function [closed]

This is a followup to my previous question here.
In quantum field theory, the Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^*\phi.$$Can ...

**4**

votes

**0**answers

99 views

### Time decay for Hartree equation with Coulomb potential

Are there any time-decay results for the solution of the Hartree equation
\begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in ...

**1**

vote

**0**answers

35 views

### Can the cavity method be used to analyze graphs with loops which are short?

In statistical physics, the cavity method can be regarded as a generalization of the Bethe-Peierls iterative method in tree-like graphs to the case of graphs with loops that are not too short. I would ...

**28**

votes

**7**answers

4k 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, ...

**4**

votes

**1**answer

179 views

### Conformal compactification of Kerr spacetime

I'm looking for a book/paper where the conformal compactification of Kerr spacetime is calculated. I've seen plenty of reference for the Minkowski, but none (explicitly calculated) for Kerr.
Thank ...

**6**

votes

**1**answer

169 views

### Complex scalar field, generating functional?

The Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^* \phi.$$Can anyone work out or provide me a reference to the computation of the ...

**7**

votes

**1**answer

106 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

**1**answer

55 views

### Discrete summation of Gaussian functions. Decay time problem

I am facing the following problem. I have a function which is defined through a discrete sum of Gaussians
$$F_M(t) = 2\sum\limits_{n=1}^{M}e^{-t^2 \sigma^2 n^2}\times \sum\limits_{k=n}^{M}p_k p_{k-n} ...

**6**

votes

**1**answer

177 views

### Gauge field quantization, electromagnetism

Classical electromagnetism (with no sources) follows from the actions$$S = \int d^4x\left(-{1\over4}F_{\mu\nu}F^{\mu\nu}\right),\text{ where }F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$The ...

**4**

votes

**0**answers

30 views

### Second-order term of the Fedosov quantised product

In Fedosov's version of quantisation of functions on a symplectic manifold, the product is given in terms of a symplectic connection. I have looked through Fedosov's book in deformation quantisation, ...

**1**

vote

**1**answer

33 views

### $P_{x}(T_{A}<\infty)<P_{x}(T_{B}<\infty)$ imply $Cap_{N}(A)<Cap_{N}(B)$, where $Cap_{N}$ is Newtonian capacity

We start a Brownian motion at $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a large enough ball that contains compact sets $A$ and $B$. In other words, the B.M. starts on the exterior of $A$ and $B$.
Then ...

**6**

votes

**4**answers

2k views

### Does the derivative of log have a Dirac delta term?

Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics":
$\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...

**80**

votes

**0**answers

6k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**3**

votes

**0**answers

62 views

### Proof of asymptotic non-flatness

in a few papers I came across a statement that the Kerr-NUT metric
$g_{uu}=\rho\overline{\rho}(r^{2}-2mr-l^{2}+a^{2}\cos^{2}x)$
$g_{ur}=1$
$g_{uy}=-2\rho\overline{\rho}l\cos ...

**6**

votes

**1**answer

154 views

### The proof that a vertex algebra can lead to a Wightman QFT

On p. 13 of "Vertex Algebras for Beginners", 2nd edition, Kac writes:
"Under certain assumptions and with certain additional data one may reconstruct the whole QFT from these chiral algebras, but we ...

**32**

votes

**9**answers

4k views

### Dimensional Analysis in Mathematics

Is there a sensible and useful definition of units in mathematics? In other words, is there a theory of dimensional analysis for mathematics?
In physics, an extremely useful tool is the Buckingham Pi ...

**29**

votes

**5**answers

6k views

### Poincaré Conjecture and the Shape of the Universe

Has the solution of the Poincaré Conjecture helped science to figure out the shape of the universe?

**1**

vote

**0**answers

201 views

### Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...

**1**

vote

**0**answers

256 views

### quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement.
Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...

**6**

votes

**0**answers

198 views

### Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary

It is well-known that, in globally hyperbolic spacetimes, the time separation function $\tau$ (aka Lorentzian distance function) enjoys the following property: fix a point $p$ and a point $q \in ...

**2**

votes

**0**answers

212 views

### Multivalued solution of PDE ${v_{xx}v_{yy}-v_{xy}^{2}}={(1+v_{x}^{2}+v_{y}^{2})^2}$

Let's start with a definition:
Definition: A scalar k-th order differential equation on a smooth manifold $M$,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for ...

**4**

votes

**0**answers

245 views

### Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...

**2**

votes

**1**answer

88 views

### How can I understand the braiding terms introduced in the plaquette operator of the Walker-Wang TQFT

Walker-Wang models as introduced in (3+1)-TQFTS and Topological Insulators are an example of 3+1D lattice TQFTS. In analogy with the Levin-Wen models in 2+1D the authors define an exactly solvable ...

**39**

votes

**9**answers

3k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**6**

votes

**1**answer

228 views

### Foliations of Lorentzian manifolds by Spacelike Hypersurfaces

Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...

**4**

votes

**2**answers

147 views

### Obtaining Killing fields from the tetrad

I'm reading the following article by Newman
http://scitation.aip.org/content/aip/journal/jmp/4/7/10.1063/1.1704018
about the generalization of the Schwarzschild metric. My question is the following: ...

**2**

votes

**1**answer

179 views

### Quaternion Wishart matrices of half-integer dimension?

For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution
...

**25**

votes

**0**answers

1k views

### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...