Mathematical methods in quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

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Vector application [on hold]

A rope is hung at both ends from a horizontal beam, and a weight m is suspended from it. The left part of the rope exerts a force G at P, while the right part of the rope exerts a force H. Find the ...
2
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2answers
54 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 ...
4
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1answer
106 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
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0answers
29 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 ...
9
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211 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 $n\times n$ ...
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1answer
57 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
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84 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 ...
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1answer
104 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 ...
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150 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
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2answers
369 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 ...
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50 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 ...
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45 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 ...
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1answer
123 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 $P$ ...
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38 views

Complex tetrad vs Real metric

I have a question on the relationship between the complex tetrad in general relativity and the metric. All the papers I've sen so far just usually state the metric and the (null) tetrad without ...
5
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1answer
98 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 ...
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1answer
154 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 ...
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78 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 ...
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75 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 ...
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59 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, ...
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1answer
123 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 ...
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110 views

Path integral methods

Are there detailed expositions of the path integral methods in (mathematical) physics other than Feynman-Hibbs and Glimm-Jaffe?
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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
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1answer
153 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 $$ ...
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159 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 ...
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1answer
450 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 ...
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69 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 ...
23
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4answers
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 ...
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1answer
92 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, ...
6
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1answer
153 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 ...
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72 views

On Different Ways of Proving Isoperimetric Inequalities [closed]

Update: Thanks to Douglas Zare's comment, My previous questions in this thread turned out to be equivalent to the Isoperimetric problem. Thus I edited my question to make it a bit different. ...
4
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1answer
174 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 ...
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81 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 ...
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5answers
1k 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 ...
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1answer
173 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 ...
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97 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 ...
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71 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 $\mathbb{R}^n$ ...
4
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1answer
94 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 ...
4
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1answer
118 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
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81 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\\ ...
5
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0answers
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
1answer
359 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
1answer
110 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
2answers
197 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 ...
2
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0answers
90 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
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1answer
154 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
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1answer
233 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
1answer
190 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 ...
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1answer
69 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
1answer
193 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 ...
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69 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 ...