**1**

vote

**0**answers

24 views

### How can I express the following integral as special function?

How can I express the following integral as special function? where the integral is from 0 to infinity
$ f_{int}(x)=\int^{\infty}_{0}dy\frac{e^{(-x+y)}}{(1+e^{(-x+y)^{2}})\sqrt{y}} $

**4**

votes

**1**answer

132 views

### Survey paper on isoperimetry

I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...

**1**

vote

**0**answers

48 views

### Floquet-Bloch solutions of the quasiperiodic Schrödinger equation

I am concerned about the Schrödinger equation
$-x''(t)+q(t)x(t)=Ex(t).$
Here, the potential $q$ is real and quasiperiodic with frequency vector $\omega$. That is, we let $T^d$ be the $d$-dimensional ...

**0**

votes

**0**answers

41 views

### Does an arbitrary product of $f$ and $f^\dagger$ belong to a universal enveloping algebra of the Heisenberg algebra? [migrated]

The Heisenberg algebra is essentially the canonical commutation relations (CCR) for bosons $[f,f^\dagger]=1$. $f$ is called an annihilation operator in physics ($f^\dagger$ creation operator). ...

**0**

votes

**1**answer

84 views

### References: Solutions of the Bethe Ansatz Equations [closed]

Could someone show me good references to find solutions of the Bethe Ansatz Equations, for simple cases (using algebraic geometry or others interfaces with mathematics)?

**6**

votes

**1**answer

166 views

### Domains of raising and lowering operators in QM

I should add that what is known in physics as SUSY QM, is often called the Crum-Darboux method in Mathematics, so don't be confused about this.
Let $H : \operatorname{dom}(H) \subset L^2(\Omega) ...

**2**

votes

**1**answer

112 views

### Well-posedness of heat equation with distributional right hand side

The question is about well-posedness of heat equation
$$
\frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T],
$$
subjected to ...

**4**

votes

**1**answer

183 views

### Uhlenbeck's theorem novelty

This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this ...

**6**

votes

**1**answer

99 views

### Is there a maximum principle for stress in continuum mechanics?

I'm working with the equilibrium equations in linear elasticity, which I have not worked with in the past. My engineering colleagues seem to "know" that the maximum Von Mises stress occurs on the ...

**2**

votes

**0**answers

70 views

### Do position and momentum measurements determine a wave function?

Suppose we have a function $f\in L^2(\mathbb R^n)$ and we know the functions $x\mapsto|f(x)|$ and $p\mapsto|\hat f(p)|$, where $\hat f$ is the Fourier transform of $f$.
Can we reconstruct the function ...

**2**

votes

**2**answers

267 views

### Geometrical interpretation of a Schrödinger operator

Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function
$$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$
such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some ...

**2**

votes

**1**answer

97 views

### Quaternion Wishart matrices of half-integer dimension?

For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution
...

**1**

vote

**0**answers

121 views

### Electrodynamics modelled by U(1) gauge theory [closed]

As the article 'Electrodynamics in general spacetime' greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ...

**2**

votes

**0**answers

72 views

### “Semiclassical approximation” in random matrix theory

I am reading Planar Diagrams by Brezin, Parisi, Itzykson and Zuber. If you specialize the discussion in section 5 there we seem to derive the eigenvalue distribution of $N \times N$ random Hermitian ...

**1**

vote

**0**answers

76 views

### translation invariance of the Laughlin wave function

This is a translation into math of the following question, posted on PhysicsOverflow.
Let $H:=L^2(\mathbb C)$.
For every $N$, let $\psi_N\in\Lambda^N H\cong (L^2(\mathbb C^N))^{S_N}$ be the function
...

**0**

votes

**0**answers

30 views

### Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...

**6**

votes

**0**answers

183 views

### Can one classify irreducible unitary representations of the Weyl algebra?

I saw in this MO post: Is there a machinery describing all the irreducible representations ? that classifying irreducible representations of the Weyl algebra is essentially intractable. My question is ...

**8**

votes

**2**answers

274 views

### What is the BRST-anti-BRST formalism?

What is the BRST-anti-BRST formalism?
Is the Sp(2) doublet the ghost, antighost pair?
Introductory accounts of this subject seem to be hard to find. I would appreciate a reference for someone who ...

**5**

votes

**1**answer

373 views

### Harmonic Crystal using Random Walk

Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.
Basically, it is a ...

**16**

votes

**3**answers

902 views

### How mirror of quintic was originally found?

In the 90-91 pager
"A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY",
Candelas, De La Ossal, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically ...

**1**

vote

**0**answers

53 views

### Bogomol’nyi’s Formula for the Critical Action

I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is
...

**3**

votes

**0**answers

78 views

### A technical question in Feix's construction of hyperkahler metric on cotangent bundles

I am now reading Feix's paper Hyperkahler metrics on cotangent bundles
and I have a technical question to ask.
In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its ...

**2**

votes

**0**answers

70 views

### Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom

Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation
$$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$
can ...

**4**

votes

**0**answers

81 views

### Ground State Degeneracy of 2+1D U(1) Chern Simons Theory?

I am a physics graduate student trying to understand more mathematical aspects of gauge theories.
How can I understand ground state degeneracy of a simple Chern Simons Theory: 2+1D U(1) $S= \int_M ...

**1**

vote

**2**answers

120 views

### Deformation quantization of a closed Riemann surface with genus >1

Quantization of of an elliptic curve can be done in different ways.
In C^*-algebraic version,
one can start with the C^*-algebra ...

**1**

vote

**0**answers

74 views

### The normalization axiom of a quantization

Guillemin, Ginzburg and Karshon explain a quantization in their book [Chap 6,MR1929136] as follows.
The quantization is a process which associates to a symplectic manifold $M$ a Hilbert space ...

**10**

votes

**0**answers

189 views

### “extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...

**7**

votes

**0**answers

133 views

### What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be:
Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...

**0**

votes

**0**answers

35 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 ...

**0**

votes

**0**answers

64 views

### Equivalence of ensembles and $\sum_{N=0}^{\infty} \int_{-Bn}^{\infty} 1_{\text{{N>K or u>A}}}(u)e^{-\beta |\lambda_m|(\frac{u}{2}+n)}du $

I am thinking about a step in the proof of the equivalence of ensembles in StatMech on page 25. step from 3.19 to 3.20.
It seems to be argued there that for $A$ and $K$ large enough, the term
...

**0**

votes

**2**answers

108 views

### Can any antilinear involution be trivialized by a change of basis?

Consider an antilinear involution, that is an antilinear map on a complex vector space, whose matrix $M$ obeys $MM^*=1$ where the star denotes complex conjugation. Can we find a change of basis whose ...

**3**

votes

**3**answers

182 views

### Reference request for a treatment of Schwinger–Dyson equations

Is there a treatment of Schwinger–Dyson equations with no mention of Green's functions? Is there perhaps a purely algebraic analog?

**1**

vote

**0**answers

143 views

### A simple question in Hitchin's paper “The Geometry of Three-forms in Six Dimensions”

I am reading Hitchin's beautiful paper "The Geometry of Three-forms in Six Dimensions".
Everything goes smooth up to now except for a tiny problem in Section 6.2, which can be formulated as follows. ...

**5**

votes

**0**answers

178 views

### Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...

**1**

vote

**0**answers

110 views

### Dislocations,Disclinations Latices, Displacement fields and scaling

I am looking up Frank, and Burger vectors and associated material on dislocation/disclination. It seems straightforward describing a lattice and what dislocation means. It is even possible to restrict ...

**-3**

votes

**1**answer

134 views

### A problem that involves matrix and Lorentz Transformation [closed]

To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes.
$1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G ...

**0**

votes

**0**answers

130 views

### path integral and index theorem

I actually have an integral which is used to prove Atiyah-Singer index theorem for spin complex in a path integral fashion. The integral I need to evaluate is following (in simplified form)
$\int ...

**1**

vote

**0**answers

142 views

### Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...

**6**

votes

**1**answer

180 views

### Generalizing “variation of parameters”

I'm stuck on generalizing an ODE formula and could use your help!
One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here ...

**1**

vote

**1**answer

95 views

### GOE convergence

As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see this question for more background). The question is: ...

**2**

votes

**0**answers

80 views

### Jackiw-Pi identity

In their paper http://journals.aps.org/prd/abstract/10.1103/PhysRevD.42.3500 (Classical and quantal nonrelativistic Chern-Simons theory) Jackiw and Pi introduced an unusual identity involving ...

**-2**

votes

**1**answer

88 views

### What are the formula of representation of quasicrystals and the law or mechanism of the formation [closed]

I vaguely recall that formula of representation of quasicrystals is relevant to tiling plane,and tiling plane without period is relevant to recursiveness, and do not know the mechanism or physics ...

**1**

vote

**1**answer

213 views

### When does a moduli space admit a spin structure?

This is a very vague question.
Is there any example of spin structures on a moduli space? References are requested.
I have vaguely heard that Witten discussed when a sigma model is spin. Somehow I ...

**6**

votes

**3**answers

340 views

### Anderson localization - an embarassment of riches

I am looking for a good, not too technical discussion of Anderson Localization, and some explanation of why it exists. Googling "Anderson Localization" produces an infinite number of possibilities, ...

**2**

votes

**2**answers

237 views

### Gauge-theoretic formulation of Maxwell equations [duplicate]

Does any one know how to write the Maxwell equations as an equation on a principal $U(1)$-bundle?
In Freed & Uhlenbeck's Instantons and Four manifolds, the authors claim that the Maxwell ...

**2**

votes

**0**answers

87 views

### First Variation of Dyson Series/Magnus Expansion

Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...

**1**

vote

**1**answer

93 views

### 2D semilinear elliptic PDE

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. ...

**1**

vote

**0**answers

230 views

### Feynman integrals in algebraic geometry [closed]

In quantum field theory, multi-loop Feynman integrals are basic ingredients of calculating high order corrections. Recently, I have come across the paper A Feynman integral via higher normal ...

**5**

votes

**0**answers

95 views

### Stationary point processes with arbitrarily slow decorrelation

A point process $P$ (a probability measure on simple, locally finite point configurations $\mathcal{C}$ on $\mathbb{R}$ - I'm restricting to the one-dimensional setting) is stationary when ...

**0**

votes

**1**answer

229 views

### Largest eigenvalue of the sum of hermitian matricies [closed]

Is there an expression for the largest eigenvalue of the sum of two hermitian matricies in terms of the spectrum of the same matricies?