**5**

votes

**1**answer

104 views

### Foliations of Lorentzian manifolds by Spacelike Hypersurfaces

Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...

**18**

votes

**2**answers

582 views

### Reflection of light from function graph

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ ...

**2**

votes

**0**answers

121 views

### How to write BRST-BV for dg-Lie?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?

**4**

votes

**0**answers

87 views

### PDE with distributional coefficient [migrated]

I'm a undergraduate major in physics and when learning mathematical aspects of quantum physics I run into for example this problem: for a 1-dim system with delta potential, $-\frac{d^2}{dx^2}\phi + ...

**2**

votes

**2**answers

336 views

### space at the Planck scale [closed]

All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings ...

**1**

vote

**0**answers

101 views

### Spectral sequences and Batalin-Vilkovisky formalism

I have been studying the BRST quantization in quantum field theory recently and noticed that the subject is very much related to algebraic topology and cohomology. A quick google search led me to the ...

**1**

vote

**0**answers

120 views

### Dirac operator in Generalized Geometry

I am wondering how the Dirac operator can be built in the context of Hichin's generalized geometry.
In particular, I have the following questions:
On a spin manifold, is the conventional spin ...

**1**

vote

**0**answers

65 views

### Dixon's diagram for BRS cohomology

The article by J. A. Dixon titled Calculation of BRS cohomology with spectral sequences (Comm. Math. Phys. Volume 139, Number 3 (1991), pages 495-526) describes in words a diagram that is not printed. ...

**1**

vote

**0**answers

31 views

### modern exposition of exact ground state of classical XY model or Ising model

What is the state of art technique in solving exact ground state of Heisenberg model, meaning minimization of the H terms (hamiltonian) provided infinite spin space?
...

**1**

vote

**1**answer

211 views

### How to construct (another) Landau-Ginzburg model for a compete intersection Calabi-Yau?

For Calabi-Yau variety $X$ which is a complete intersection
$$
f_1=f_2=\ldots=f_r=0
$$
in ${\mathbb P }^n$ (hence $\mathrm{dim}\,X=n-r$) it is possible to construct a Landau-Ginsburg model in the ...

**5**

votes

**2**answers

845 views

### Physicist trying to understand modern mathematics

I'm a physicist trying to gain a deep understanding of mathematics that is required for my work.I intend to specialize in string theory which is a very math intensive branch of theoretical physics ...

**0**

votes

**0**answers

97 views

### Perturbation of Laplacian via Kato-Rellich theorem

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian
$$-\Delta+V(x)$$
is self-adjoint on $H^2(\mathbb{R}^3)$.
My idea is to use Kato-Rellich theorem; ...

**3**

votes

**1**answer

225 views

### book about string theory a la Von Neumann [duplicate]

Can we summarize string theory (in its actual state) in some principles and fundamental equations like electromagnetism, general relativity, quantum mechanics and classical mechanics ?
I am looking ...

**4**

votes

**1**answer

139 views

### reference for higher spin - not gravitational nor stringy

Other than the papers of Berends, Burgers and van Dam, are there any papers that study the general case of deforming a free field theory with higher spin fields to be interactive?

**0**

votes

**0**answers

80 views

### What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex
$$
V ...

**7**

votes

**2**answers

186 views

### How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?

Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original ...

**2**

votes

**1**answer

98 views

### Proper domain for operators

in this paper on arxiv in equation 27, two operators
$$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$
and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + ...

**1**

vote

**1**answer

98 views

### Commutators of Schur polynomials of Lie algebra elements

Question:
Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...

**1**

vote

**2**answers

252 views

### Witten index non-trivial in the context of Quantum Mechanics?

Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^*$, such that we have $H = D^*D$.
I will now consider the one-dimensional case on a compact set:
So ...

**14**

votes

**2**answers

606 views

### Exact Definition of Dirac Operator

Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ...

**12**

votes

**3**answers

1k views

### On $e^{\pi\sqrt{4\cdot163}}$ and unusual connections

We are familiar with the expansion of the j-function,
$$j(\tau) = \tfrac{1}{q}+744+ 196884{q} + 21493760{q}^2 + \dots\tag1$$
and maybe with the approximation,
$$e^{\pi\sqrt{652}} = ...

**0**

votes

**1**answer

173 views

### Some quantities which definitions are (somehow) similar to the classical Divergence

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...

**31**

votes

**7**answers

3k views

### Companion to theoretical physics for working mathematicians

In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...

**1**

vote

**0**answers

83 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}} $

**5**

votes

**1**answer

169 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

74 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 ...

**6**

votes

**1**answer

193 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

119 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

215 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

**2**answers

139 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

82 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

279 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

107 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

138 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

80 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

92 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

32 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

202 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

310 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

398 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

921 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

54 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

83 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

81 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

88 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

131 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

83 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

234 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

142 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

42 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 ...