**2**

votes

**0**answers

61 views

### Implicit/Explicit Time Dependence for Melnikov Functions

My question concerns an article by Koiller and Carvalho found here: http://link.springer.com/article/10.1007/BF01260390
On page 645, they parameterize the time variable $t$ in terms of one of the ...

**2**

votes

**0**answers

80 views

### Complex structure on the set of prequantization line bundles

For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the ...

**2**

votes

**0**answers

142 views

### The Cauchy Problem in General Relativity: Existence of a Hausdorff Development

This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...

**2**

votes

**1**answer

56 views

### Finite differencing scheme for Hamilton's equation with planar linkages

I am trying to simulate the movement of a planar linkage in the plane whose position and momentum obey Hamilton's equations, which is to say that $${{dq}\over{dt}} = {{dH}\over{dp}}$$ and ...

**2**

votes

**0**answers

123 views

### Error in trace of operator

Imagine you have a Schroedinger operator $H:=-\frac{d^2}{dx^2}+V$ with $V \in C[a,b]$ on some compact interval $[a,b]$. The boundary conditions are supposed to be taken in such a way that this ...

**0**

votes

**0**answers

66 views

### Do principally polarized abelian varieties enjoy a genus expansion?

This is a vague question from an interested outsider:
It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...

**9**

votes

**1**answer

429 views

### Vector bundles, Higgs bundles and the Langlands program

This question is somewhat vaguely structured. But, I hope someone can make it more precise (or) it is indeed possible to answer it in the form that I am stating it.
Background : I recently chanced ...

**6**

votes

**3**answers

612 views

### Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...

**1**

vote

**2**answers

212 views

### How to evaluate the wiener measure of sets?

I would like to understand how the Wiener measure of some simple sets can be evaluated.
I will sketch the construction of Wiener measure I have in mind:
We denote the one point compactification of ...

**5**

votes

**1**answer

181 views

### Self-adjoint extensions and delta potentials

Is there a self-adjoint extension of an operator that corresponds to a particle in a box $[a,b] \times [c,d] \subset \mathbb{R}^2$ with a delta potential, i.e., $-\Delta + \lambda \delta_y $ on ...

**4**

votes

**1**answer

171 views

### Asymptotic behavior of Schrödinger operators

I am currently dealing with $1$ or at most $2$-dimensional Schrödinger operators on compact domains. A classical result of spectral theory is the Weyl approximations for this operator
$H = -\Delta ...

**7**

votes

**2**answers

414 views

### Free Boson Correlator $ \langle X(z)X(w) \rangle =- \ln |z - w| $

In physics papers, the massless free boson has a definition involving an action:
$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$
The random functions $X(z)$ are ...

**4**

votes

**3**answers

165 views

### What to read for many-body problems in 3D Schrodinger equation

I am a graduate student just started learning dispersive PDE in MSRI's summer program. I roughly finished reading the paper by Klainerman and Machedon "ON THE UNIQUENESS OF SOLUTIONS TO THE
...

**1**

vote

**1**answer

220 views

### Comparison of Different Types of QFT

As far as I can tell, there are a number of major types of quantum field theory. For example,
Constructive QFT, which has two major branches (Algebraic/Axiomatic QFT and Functorial QFT).
Topological ...

**6**

votes

**1**answer

154 views

### Are there any explicit probability conserving solvers for Pauli equation?

I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one.
But when I tried to add spin into account in this scheme, it ...

**1**

vote

**1**answer

68 views

### Smallest subalgebra of $\mathfrak{su}(4)$ arrising from a control problem on $SU(4)$

What is the smallest subalgebra of $\mathfrak{su}(4)$ containing the span of the set $A = \{A, B_1, B_2\}$ where:
$A = i (J^x \sigma_x \otimes \sigma_x + J^y \sigma_y \otimes \sigma_y + J^z \sigma_z ...

**9**

votes

**2**answers

1k views

### Review of Tim Maudlin's New Foundations for Physical Geometry [closed]

Tim Maudlin, a philosopher of science at NYU, has a book out called:
New Foundations for Physical Geometry: The Theory of Linear Structures.
The section on about the book says the following:
...

**1**

vote

**0**answers

130 views

### The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$:
...

**2**

votes

**1**answer

108 views

### Heat transfer: boundary conditions with fluid velocity

The following equation is considered:
$$
\frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f.
$$
I have difficulties in formulating boundary conditions for this equation.
If ...

**3**

votes

**1**answer

119 views

### Equivalence of the construction of the Lagrangian in a book of Sternberg to the “usual” construction

I have a question regarding the following cited text from [1]:
Let $F$ be a representation of the structure group $G$ of the principal bundle $P_G\to M$ (a (semi-)Riemannian manifold), and let ...

**2**

votes

**0**answers

115 views

### Extension of a bounded operator on manifold

I have a problem, which is quite urgent, as I have only today discovered an error in a proof i had in a thesis which is to be handed in tomorrow.
The problem, if stated in as full generality as ...

**6**

votes

**2**answers

522 views

### References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...

**1**

vote

**1**answer

167 views

### System of quadratic complex equations

I want to solve this system of N non-linear equations without using a numerical method:
$x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$
With
$\left| ...

**2**

votes

**0**answers

138 views

### The mathematics in understand anyons [closed]

I've been about particles called anyons which exist within a two dimensional framework. I've also found out that these particles can have an angular momentum equal to any real number. Normally, in ...

**3**

votes

**1**answer

174 views

### quantum states and observables for the non-commutative torus

The noncommutative torus $A_\theta$ is a $C^*$-algebra corresponding to an irrational foliation of the torus $\mathbb{T}^2$ by lines of slope $\theta \notin \mathbb{Q}$.
As far as I am reading it is ...

**2**

votes

**1**answer

218 views

### How to calculate the first and second homotopy groups of the following space constructed from $U(4)$

In solving a physics problem, I came across a weird topological space constructed from $U(4)$, the group of $4\times4$ unitary matrices. I want to know the first two homotopy groups of it. Here is how ...

**1**

vote

**0**answers

176 views

### Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$
I am looking for a proof for ...

**6**

votes

**1**answer

355 views

### 4d Constructive Quantum Field Theory

As a follow up to my previous question (How does Constructive Quantum Field Theory work?), I was wondering what difficulties physicists have had constructing 4d axiomatic qfts. Why has CQFT's success ...

**2**

votes

**1**answer

286 views

### How does Constructive Quantum Field Theory work?

Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via ...

**1**

vote

**1**answer

306 views

### A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding
$$\phi: (M,\omega)\to (\mathbb CP^N, ...

**5**

votes

**1**answer

306 views

### Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle ...

**3**

votes

**0**answers

303 views

### Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?
Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...

**3**

votes

**0**answers

246 views

### The dogma of the natural numbers in physics

As is well known "God made the natural numbers; all else is the work of man" (Leopold Kronecker). However, "what would correspond more to the spirit of physics would be a mathematical theory of the ...

**1**

vote

**1**answer

128 views

### Is the structure constant additive on connected components?

Let $M$ be a Riemann surface and $\mu$ a metric on it, which could be non-compact. Moreover let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its ...

**8**

votes

**2**answers

611 views

### Mathematical equivalent to ladder operators?

A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...

**1**

vote

**3**answers

176 views

### What are some good sanity checks for simulating BNLS?

After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states:
$$
i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} ...

**0**

votes

**1**answer

196 views

### A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$
is integral, for some almost complex structure ...

**3**

votes

**1**answer

136 views

### All symplectic manifolds have $Mp^c$-structures?

Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times_{\mathbb Z_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a ...

**6**

votes

**0**answers

150 views

### Characterize spin cobordism invariants in dimer models

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...

**8**

votes

**0**answers

226 views

### State of the art of BPS and Donaldson-Thomas invariants for toric Calabi-Yau threefolds

I am trying to understand what has been done with regards to computing BPS invariants and Donaldson-Thomas type invariants of Calabi-Yau threefolds. To make the question more focused, let's say that I ...

**2**

votes

**0**answers

82 views

### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...

**1**

vote

**2**answers

369 views

### $\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$

Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type.
What are the Lipschitz functions with zero integral with respect to the measure $\mu?$
Clearly any ...

**7**

votes

**1**answer

200 views

### What is the general form of the duality transform for the Fock space?

I am interested in properties of the symmetric Fock space, looked at via the associated Wiener space. It is well known that for a Hilbert space $k$, the symmetric Fock space ...

**3**

votes

**1**answer

77 views

### Moments of the position operator and wavepacket spreading

I've noticed that when papers in mathematical physics concern themselves with the rate at which a wavepacket spreads, they almost always try to bound the moments of the position operator (the operator ...

**1**

vote

**0**answers

99 views

### Combinatorial Pin structure

David Cimasoni and Nicolai Reshetikhin have a paper on the combinatorial description of spin structure http://arxiv.org/abs/math-ph/0608070, where it shows the equivalence of spin structure to the ...

**0**

votes

**0**answers

70 views

### Measure generated by Semigroup $\exp[-t|p|]$

I am studying Ingrid Daubechies' paper 'An Uncertainty Principle for Fermions with Generalized
Kinetic Energy'.
On page 514, the measure $\mu_{x,y;t}$ is introduced as generated by the semigroup ...

**0**

votes

**0**answers

38 views

### How can one use stability analaysis of finite differences methods in linear Schrodinger to the NLS?

Specifically, I've seen a lot of analysis of grid stability for solving Linear Schrodinger with Forward Euler, Backward Euler and Crank-Nicolson. However, most of the usages I've seen for the same ...

**0**

votes

**0**answers

59 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 ...

**6**

votes

**2**answers

396 views

### Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...

**3**

votes

**2**answers

221 views

### What are the modular transformation properties of q-Pochhammer symbols?

Do q-Pochhammer symbols, defined as
$(a;q) = \prod_{k=0}^{\infty} (1- a q^k)$
have known modular transformation properties? That is, if we write $q = q[z] = e^{2\pi i z}$, is there any reasonably ...