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

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Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories. More concretely, I want to obtain a SDE of type as ...
2
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0answers
69 views

relative quantization on fibration

Let $\pi:X\to B$ be a holomorphic submerssion of two Kaehler varieties $X,$ $B$ and $(B,\omega)$ be quantizable and fibres $X_s$ also are quantizable, then $X$ is quantizable?. I want to define ...
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1answer
60 views

Gap-opening perturbations of the periodic Schrödinger operator

I am trying to understand this short paper and I am getting stuck right at the end. Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$). We are considering the operator $$A=-\dfrac{d^2}...
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1answer
153 views

Geometric Construct for Integrating Symmetric Tensors?

I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds. The motivation comes ...
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36 views

Fisher metric for shift-invariant probabilities

I'm just discovering what seems to be the tremendous heuristic value of the (century-old, more or less) canonical Riemannian metric (Fisher metric) on the $n$-dimensional simplex $\Sigma_n:=\{(p_i)_{i=...
3
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1answer
99 views

Classification of $2k$-vectors modulo orthogonal transformations

Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where $$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}...
5
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1answer
153 views

Is there a way to embed Clifford algebras into the corresponding tensor algebra?

$\newcommand{\talg}{\mathcal{T}(V)}$$\newcommand{\clalg}{\mathcal{Cl}_q(V)}$$\newcommand{\qalg}{\mathcal{I}_q(V)}$Is there a way to embed Clifford algebras into the corresponding tensor algebra? ...
5
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1answer
276 views

Confusion regarding statement of mirror symmetry for elliptic curves

I am a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
3
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1answer
142 views

Boundary conditions for Klein-Gordon equation

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$. ...
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1answer
177 views

Spin structure for varieties, especially finite field

I wonder about the notion of a spin structure for varieties over any field and results in this direction. For example, I wonder if there is something like a spin-bundle for the sphere $x^2+y^2+z^2=R^2$...
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323 views

Why does the Bogolyubov transformation work? - In language of Clifford Algebras?

Letting the standard Clifford algebra of dimension $2k$ be denoted by $Cl_{2k}$, let's denote the corresponding complex Clifford algebra via $$\mathbb{C}l_{2k}\equiv Cl_{2k}\otimes_{\mathbb{R}}\mathbb{...
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60 views

Reason of the scaling factor $n^{2}$ in Hydrodynamic limits

In some books about hydrodynamic limits, example De Masi and Pressuti, when taking about the transition from micro to macro to get the hydrodynamic limit of some process it is mentioned that in order ...
5
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69 views

Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?

Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$. Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral $k$-currents in $M$ and write ${\cal D}^{\mathit{int}}...
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24 views

diffusion and potentials in several dimensions

In a Lagrangian field theory there are conserved quantities, like total energy. This allows geometric arguments to be made about the behaviour of a system with known potential. (E.g. on asymptotic ...
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274 views

Digital physics and “Gandy-like” machines

Various physicists, famously John Wheeler, have asserted that physical information is the central object of study in physics, in the sense that an object or concept is "physically meaningful" if it ...
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130 views

Deformation quantization of Poisson bracket without star-product

Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$, $$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...
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40 views

The ground state energy of an atom as a function of an external electric field

I think this question belongs to mathematical physics. The Hamiltonian of an N-electron atom in a homogeneous electric field is $$ H =\left( \sum_{i=1}^N \frac{p_i^2}{2m } - \frac{Z e^2}{r_i} - E_z ...
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1answer
108 views

Calculus of variations when functional involves inverse of the function

Typically the Euler-Lagrange equations are defined for the functional $$ J[u] = \int_a^b L(x,u,u') dx. $$ However, I was wondering if anyone knows if they can be solved when the expression involves ...
4
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74 views

Periodicity of KdV equation in relation to zero-curvature equation

In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation $$ \partial_t U - \partial_x V + [U,V] = 0 $$ which gives rise to the monodromy matrix ...
2
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1answer
108 views

How to obtain a classical r-matrix from a quantum R-matrix?

Let $R$ be a quantum R-matrix. Is there a procedure to dequantize $R$ and obtain a classical r-matrix? Thank you very much.
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98 views

Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...
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117 views

The Yamabe problem and $\phi^4$ scalar field theory?

The other day I happened to be browsing this page on wikipedia: https://en.wikipedia.org/wiki/Mass_gap In the middle of the page is the equation $$\square\phi+\lambda\phi^3=0$$ where $\square$ is the ...
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4answers
493 views

General Relativity and Differential Geometry intuitions of Second Bianchi Identity

In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity- $R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$ It is said that ...
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8answers
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Physical meaning of the Lebesgue measure

Question (informal) Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the ...
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119 views

Why do we care about simplicity of the spectrum in Oseledets' theorem?

Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana). The simplicity of the spectrum has been studied ...
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104 views

Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2. I was wondering if there exists any reference concerning the ...
7
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1answer
288 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
6
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1answer
175 views

Time-Energy Uncertainty Relation in relativistic Quantum Mechanics

There is an old intriguing result in non-relativistic QM, stating (roughly) that there is an Heisenberg Time-Energy Uncertainty Relation. Unfortunately, in QM time is not an operator like space, ...
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59 views

Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well. ...
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2answers
183 views

Two point function of a free scalar field in Euclidean space-time

This question was previously asked here http://physics.stackexchange.com/questions/251927/two-point-function-of-a-free-massless-scalar-field-in-euclidean-space-time though I did not get there an ...
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330 views

Why do we study symplectic geometry? [closed]

What is the motivation behind studying smooth manifolds with a non-degenerate closed two-form? The subject certainly originated from physics, but is there a deeper reason for why it is still an ...
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3answers
597 views

References for Yang-Mills Theory

We are looking to run a working seminar about the Yang-Mills story. We hope that our seminars is of interest to analysts (working with curvatures and Ricci flows on Riemannian manifolds), the ...
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84 views

Is $n+\frac {1}{2}$ in Kendall-Mann numbers and quantum harmonic oscillator related?

It is known that quantum harmonic oscillator is connected to the symmetric group of infinite order which is isomorphic to the permutation group. According to Cayley's theorem any finite group is ...
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1answer
82 views

Invariance of spin coefficients

I have a question about how spin coefficients (Newman Penrose formalism) transform. I know that if we perform a tetrad rotation, say of Class III: $(l,n,m,\overline{m})\mapsto \left(\frac{1}{A}l, An,...
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148 views

intuitive connection between The KdV equations and the Virasoro bott group

I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
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0answers
95 views

A mathematical biology reference request

Is there any mathematical articles that describe the differential equation modelling of locomotion of amoeba using pseduopodia? I am looking for physics based models of pressure difference modeling of ...
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247 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal? (functional integrals in probability theory) Clearly my question looks at the same time fairly ...
2
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1answer
161 views

Choosing a coordinate transformation

I was reading the following paper http://scitation.aip.org/docserver/fulltext/aip/journal/jmp/4/7/1.1704018.pdf?expires=1460721373&id=id&accname=2112043&checksum=...
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2answers
907 views

What is the relation between sphere spectrum and supersymmetry?

In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab). Are there some ...
2
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1answer
115 views

Hessians on Kahler Manifolds

This is primarily a linear algebra question, but for motivation I want to state this question in its natural, global context. Whenever we have a non-relativistic quantum field theory (renormalized, ...
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1answer
159 views

find solution of complex number recurrence equation

I have the following recurrence equation: $$(\mu\ n + \nu) f_{n} + J\Phi^{*} \sqrt{n+1}f_{n+1} + J\Phi\ \sqrt{n}f_{n-1} = 0$$ for complex numbers $f_{n}$ where $n = 0,1,2,3,...,\infty$ and complex $\...
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1answer
66 views

Importance of a Hamiltonian integrable system be a bi-Hamiltonian system?

Once I know a complete integrable system $(f_1=H,\ldots,f_m):M^{2m}\to\mathbb{R}$, $H$ being the Hamiltonian of the system. What is the importance to know about a second Hamiltonian representation for ...
13
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1answer
288 views

Summation of series involving $\sinh$ of a square root

Consider the following series: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})} $$ From the physical ...
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Reference for supergroupoids in supersymmetry?

I would like to know some references on supergroupoids in supersymmetry. A supersymmetry is invariance under a supergroup action (nLab, Supersymmetry). It is know that groupoids provide a local ...
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105 views

Christoffel-Darboux type identity

The classical Christoffel-Darboux identity for Hermite polynomials reads $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y}.$$ I am ...
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1answer
261 views

Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains $$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$ The standard ...
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195 views

Quantum Mechanics derivation of Wallis' Formula?

Recently there was a proof of the Wallis Product using quantum mechanics on the arXiv. However, there are many proofs of the result, Wikipedia has 4. Fine Print the first proof has on Wikipedia, the ...
5
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1answer
181 views

When two vertex (operator) algebras can be patched-up to a full CFT on a genus 0 surface?

Theorem 3 of the nLab article "Full field algebra" states that Theorem 3. Two vertex operator algebras $V$ may appear as the left and right chiral halfs of a full conformal field theory precisely ...
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28 views

Vector fields for volumetric-deviatoric decomposition

The strain tensor $\epsilon(u) = \frac12 (\nabla u + (\nabla u)^T)$ in linear elasticity can be decomposed additively into volumetric and deviatoric strains \begin{gather*} \epsilon_D(u) &= \...
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1answer
56 views

underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: ...