**0**

votes

**0**answers

28 views

### 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**

votes

**0**answers

68 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 ...

**3**

votes

**1**answer

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}...

**8**

votes

**2**answers

307 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 ...

**7**

votes

**1**answer

152 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 ...

**2**

votes

**1**answer

139 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. $H=\...

**7**

votes

**1**answer

236 views

### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if almost ...

**4**

votes

**1**answer

316 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

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}}...

**3**

votes

**1**answer

179 views

### Computational trick used in QFT and the Jones Polynomial

Given a compact, simple Lie group $G$, and a compact, oriented three manifold $M$, we can consider the following smooth homotopy groups:
$(C^\infty(M;G)/{\sim},\cdot )$: Here, $\sim$ is smooth ...

**0**

votes

**0**answers

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=...

**5**

votes

**1**answer

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?
...

**3**

votes

**1**answer

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, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.
...

**5**

votes

**1**answer

275 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**

votes

**1**answer

196 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 ...

**2**

votes

**2**answers

189 views

### Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities

This question is an expansion of another question that I asked over at Math Stack Exchange.
In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ [...

**30**

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, ...

**91**

votes

**12**answers

18k 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" ...

**6**

votes

**1**answer

322 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{...

**2**

votes

**1**answer

176 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 $x^2+y^2+z^2=R^2$...

**10**

votes

**0**answers

299 views

### What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...

**1**

vote

**0**answers

59 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 ...

**7**

votes

**1**answer

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 ...

**1**

vote

**2**answers

4k views

### Calculating moment of inertia in 2d planar polygon

I've derived equations for 2d polygon's moment of inertia using Green's Theorem (constant density \rho)
$$I_y = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) ( x_i y_{i+1} - ...

**6**

votes

**1**answer

67 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, ...

**13**

votes

**1**answer

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 ...

**7**

votes

**0**answers

165 views

### Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? :
$V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). $...

**5**

votes

**0**answers

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}}...

**0**

votes

**0**answers

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 ...

**12**

votes

**1**answer

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 ...

**5**

votes

**0**answers

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 ...

**22**

votes

**4**answers

1k views

### How is tropicalization like taking the classical limit?

There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and ...

**1**

vote

**0**answers

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 ...

**4**

votes

**1**answer

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 ...

**2**

votes

**1**answer

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.

**4**

votes

**0**answers

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
...

**10**

votes

**3**answers

596 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 ...

**4**

votes

**1**answer

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\...

**7**

votes

**4**answers

4k views

### Applications of Euler-Cauchy ODEs

The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$
x^2 y'' + a x y' + b y = 0
$$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but ...

**7**

votes

**4**answers

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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**0**answers

104 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 ...

**38**

votes

**8**answers

4k views

### 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 ...

**1**

vote

**1**answer

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,...

**3**

votes

**2**answers

181 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 ...

**4**

votes

**0**answers

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 ...

**6**

votes

**0**answers

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 ...

**6**

votes

**1**answer

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, ...

**1**

vote

**0**answers

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.
...

**9**

votes

**0**answers

329 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 ...