# Tagged Questions

**7**

votes

**1**answer

245 views

### Applications of set theory in physics

In the introduction of the paper "Links between physics and set theory", the following quote of Eris Chric is stated:
...

**2**

votes

**1**answer

100 views

### Two equivalent descriptions of a physical system yielding a non-trivial mathematical formula

First I would like to admit that this question may not be entirely appropriate for this site, but I will give it a go none the less.
One often hears stories about how string dualities lead to highly ...

**0**

votes

**1**answer

62 views

### Mathematical statistical qm book-recommendation

I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum ...

**1**

vote

**2**answers

233 views

+50

### Certain inverse problem related to moments

Suppose $D\subset \mathbb C$ is a smoothly bounded domain and it contains the origin. Let $ds$ denote the arc length measure on $\partial D.$ I am interested in the following two inverse problems ...

**9**

votes

**4**answers

288 views

### Obtain any 3-manifold from repeating surgeries on knots in $S^3$

In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...

**4**

votes

**1**answer

85 views

### Isomorphism of various gauge groups under homotopy

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 ...

**3**

votes

**1**answer

66 views

### Blow-Up for Semi-Linear Wave Equations

I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...

**79**

votes

**25**answers

15k views

### A soft introduction to physics for mathematicians who don't know the first thing about physics

There have been similar questions on mathoverflow, but the answers always gave some advanced introduction to the mathematics of quantum field theory, or string theory and so forth. While those may be ...

**10**

votes

**1**answer

502 views

### Associativity of Kontsevich's star product up to second order

In Deformation quantization of Poisson manifolds, Kontsevich gives the quantization formula
$$f \star g = \sum_{n=0}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma,\alpha}(f,g).$$
He gives ...

**4**

votes

**1**answer

534 views

### Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello!
Let's say I have
$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$
with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$).
Now separate the Amplitude and Phase of the solution:
...

**4**

votes

**1**answer

538 views

### basic questions on quantum integrable systems

I have been learning about (classical) integrable systems lately, e.g. in the examples of a lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions:
What ...

**4**

votes

**0**answers

76 views

### power laws emerging from the sandpile model

Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?

**1**

vote

**0**answers

203 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 ...

**3**

votes

**0**answers

218 views

### Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...

**1**

vote

**1**answer

146 views

### Yang-Mills Functional and Energy

I have a question about the meaning of Yang-Mills Functional.
It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is:
...

**13**

votes

**2**answers

964 views

### Symplectic formulation of statistical physics

Does there exists a symplectic formulation of statistical physics?
I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated ...

**5**

votes

**1**answer

249 views

### Domains of raising and lowering operators in QM

Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and ...

**62**

votes

**9**answers

6k views

### Theoretical physics: Why not just R^4?

You and I are having a conversation:
"Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."
"Something like that"
"...And ...

**1**

vote

**3**answers

233 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} ...

**1**

vote

**0**answers

37 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?
...

**0**

votes

**1**answer

186 views

### How would I apply Wick's theorem to the time-ordered product of three fields?

I think I know how to apply Wick's theorem in order to expand the time-ordered product of quantum fields, but I just want to verify my understanding. Could someone perform it for the arbitrary ...

**0**

votes

**1**answer

180 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 ...

**18**

votes

**2**answers

614 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 $ ...

**5**

votes

**1**answer

129 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 ...

**8**

votes

**3**answers

568 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 the paper "Exotic ...

**1**

vote

**1**answer

146 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. ...

**58**

votes

**3**answers

4k views

### What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...

**2**

votes

**0**answers

129 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?

**2**

votes

**2**answers

350 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

108 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

128 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

70 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. ...

**4**

votes

**1**answer

143 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?

**5**

votes

**2**answers

882 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 ...

**1**

vote

**1**answer

220 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 ...

**1**

vote

**0**answers

106 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

234 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 ...

**23**

votes

**5**answers

3k 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, ...

**3**

votes

**2**answers

499 views

### An interesting equation of some practical interest.

I have encountered a problem in my elec. eng research that I find rather challenging, being a bear of very little brain. The question has been considered under a slightly different aspect here: ...

**0**

votes

**0**answers

84 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

208 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

103 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}} + ...

**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

**1**answer

102 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

256 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 ...

**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}} = ...

**74**

votes

**0**answers

5k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**14**

votes

**2**answers

637 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 ...

**67**

votes

**29**answers

9k views

### Where does a math person go to learn statistical mechanics?

The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've ...

**6**

votes

**2**answers

146 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 ...