**2**

votes

**0**answers

104 views

### Infinite total variation of complex measure in Feynman path integral [closed]

I am trying to understand this: If one tries to define a Feynman path integral as a Wiener integral, then the complex measure could be of infinite total variation. What exactly does this mean? How ...

**50**

votes

**2**answers

3k views

### Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black
holes one billion light-years away. One of the black holes had 36
times the mass of the sun, and the other 29. After the merging the
...

**1**

vote

**0**answers

23 views

### Construction of point process having same pair correlations as GUE

The distribution of the pair correlations of the eigenvalues of the GUE satisfies (in the limit, when being normalized appropriately)
$$
g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + ...

**3**

votes

**3**answers

121 views

### Symplectic manifolds with dense group of periods

Let $ (M, \omega) $ be a symplectic manifold. The de Rham class of $\omega$ induces a homomorphism $[\omega]: H_2(M) \to \mathbb{R}$, whose image $\Gamma_{\omega} \subseteq \mathbb{R}$ is called the ...

**5**

votes

**1**answer

217 views

### Is there an alternate name for the symplectic convolution?

Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution
$$[F\star G](x,p) = \int \!dy\,dk\, ...

**2**

votes

**0**answers

37 views

### TAP expression for entropy [closed]

This paper by Barton and Cocco: http://www.phys.ens.fr/~cocco/Art/articlejstat.pdf
claims on page 17 (Formula (30)) an expression for the "high-temperature" entropy of an Ising model, given its ...

**10**

votes

**2**answers

418 views

### How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $ A_i=
\biggl(\begin{matrix}
C_i+E_i & B_i \\
B_i^T & D_i-F_i
\end{matrix} \biggr)
$, where $B_i$ is an arbitrary $n\times n$ ...

**1**

vote

**1**answer

91 views

### A Gaussian integral over complex variables by a defined Green's function for a Gaussian ensemble of random matrix

We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...

**3**

votes

**0**answers

106 views

### Spectrum of the Laplace-Beltrami operator on a domain of finite volume in the hyperbolic space $H^n$

What is known about the ($L^2$) spectrum of the minus Laplace-Beltrami operator ($- \Delta$) with zero boundary conditions on $B =H^n/\Gamma$, where $H^n$ is $n$-dimensional hyperbolic space ...

**1**

vote

**1**answer

119 views

### Transferring connection information to associated bundles and back

This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try.
At the risk of repeating well known stuff I tried ...

**7**

votes

**0**answers

179 views

### Flat connection from gauged WZW model

$\newcommand{\g}{\mathfrak g}$
$\newcommand{\h}{\mathfrak h}$
In short my question is :
Has someone worked out the flat connection that one should get from the gauged WZW model in genus 0 ?
Some ...

**7**

votes

**2**answers

419 views

### States in C*-algebras and their origin in physics?

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$
Now, of course there is also in classical physics and quantum ...

**1**

vote

**0**answers

66 views

### Non-trivial but simple concrete examples for some categories related to Tensor/Fusion categories

I'm writing a note on Tensor and Fusion categories, the readers of which are physicists rather than mathematicians. So instead of giving abstract definitions I have to give examples to inspire each ...

**2**

votes

**1**answer

121 views

### How to compute $t_0$ and $r^0$ in Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations?

I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations.
In the book a guide to quantum groups, on page 83, there is an example of solutions of the ...

**3**

votes

**1**answer

252 views

### Why curvature is equivariant as a moment map?

In Atiyah and Bott's famous paper "The Yang-Mills Equations over Riemann Surfaces", they treated curvature as a moment map of the gauge group acts on the connection space of a principal bundle $P$ ...

**5**

votes

**1**answer

102 views

### Does a classical wave detect compact dimensions?

Please excuse if the question is too easy; I'm just not familiar enough with PDEs.
I'd like to understand a little bit classical implications of "adding compact dimensions" in Physics, that is, what ...

**-3**

votes

**1**answer

159 views

### How to solve this system of Partial Differential Equations for reflection from a surface

The problem is to find the surface $z(x,y)$ determined by the following system of partial differential equations:
$${\partial z \over \partial x}f(x,y,z)+{\partial z \over \partial ...

**0**

votes

**0**answers

90 views

### Metric calculation from tetrad gives wrong answer

I'm reading the following article by Kinnersley
http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958
and cannot reproduce one (rather trivial) result.
On page 5 of the paper, in ...

**0**

votes

**0**answers

84 views

### Proper time and asymptotic flatness

I have asked this question at physics stackexchange but got no response. I thought I could try my luck here:
I'm trying to understand the concept of asymptotic flatness in general relativity, and ...

**0**

votes

**0**answers

63 views

### Looking for A. J. Tolland's exposition of non-renormalizability [duplicate]

In the MathOverFlow thread "Mathematical explanation of the failure to quantize gravity naively" there are references to "A.J. Tolland's very nice exposition of nonrenormalizability". However, ...

**1**

vote

**1**answer

156 views

### Intuition behind the “Lapse Function”

I came across the following definite of the Lapse Function:
$N=\sqrt{\frac{1}{2}g(L,\overline{L})}$
where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this ...

**5**

votes

**3**answers

367 views

### Path integral methods

Are there detailed expositions of the path integral methods in (mathematical) physics other than Feynman-Hibbs and Glimm-Jaffe?

**1**

vote

**0**answers

45 views

### Smoothness of the twistor space of a lorentzian manifold, or “convexity wrt null geodesics”

Null lines in Minkowski space form a 5-dimensional manifold, represented as a (real) quadric $\mathbf{PN}\subset\mathbb{C}\mathbf{P}^3$. This is a well-known fact, on which R. Penrose’s twistor ...

**6**

votes

**1**answer

162 views

### Is this function Schwartz?

I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it.
Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that
$$
...

**11**

votes

**0**answers

198 views

### Squeezing physics out of formal deformation quantizations

I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...

**9**

votes

**1**answer

466 views

### What is the spin connection in 9 dimensions as opposed to 5 dimensions?

From Spin Connection in 5 dimensions I can define a massless fermion's covariant derivative on a curved manifold as
$$
\nabla_\mu \psi = (\partial_\mu - {i \over 4} \omega_\mu^{ab} \sigma_{ab}) \psi
...

**1**

vote

**1**answer

101 views

### An analytic family of in fact non-existent improper Riemann integrals

Question:
Are there any useful interpretations or "applications" of the formula
$$\intop_0^\infty e^{-ax}\frac{\sin x}{x}\,dx=\frac{\pi}{2}-\arctan(a)\qquad \forall\;a\in \mathbb{R},
$$
in which the ...

**25**

votes

**4**answers

1k views

### Does quantum mechanics ever really quantize classical mechanics?

I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at the answer to this question, but it's not what I'm looking for.
Basically, classical mechanics ...

**0**

votes

**1**answer

98 views

### gradient descent in space of functions

Differential equations of the form
$$\frac{d}{dt}\vec{x} = - \nabla E(\vec{x})$$
can be analyzed using phase portrait method. In particular, if the function $E$ (we call it energy) has local minimums, ...

**7**

votes

**1**answer

176 views

### Brauer-Picard for a fusion category coming from a quantum group

In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A ...

**4**

votes

**1**answer

188 views

### How are the real-space RG transformations defined?

I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...

**1**

vote

**0**answers

95 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 ...

**22**

votes

**6**answers

2k 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 ...

**4**

votes

**1**answer

177 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 ...

**1**

vote

**0**answers

106 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

74 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 $\mathbb{R}^n$ ...

**5**

votes

**1**answer

119 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 ...

**5**

votes

**1**answer

128 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

88 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

391 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

114 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| ...

**4**

votes

**3**answers

262 views

### An extremal type problem on segments

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

**2**

votes

**0**answers

102 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 ...

**7**

votes

**3**answers

443 views

### In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$?

Let $X$ be a Calabi-Yau threefold and let us fix a homology class $\beta\in H_2(X,\mathbb Z)$, just for simplicity. The generating series of Gromov-Witten invariants of $X$ in class $\beta$, $$\mathsf ...

**4**

votes

**1**answer

162 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\\
...

**6**

votes

**1**answer

429 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 ...

**4**

votes

**1**answer

221 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 ...

**1**

vote

**1**answer

90 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

207 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 ...

**3**

votes

**0**answers

70 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 ...