Questions tagged [mp.mathematical-physics]

Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

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5 votes
2 answers
695 views

What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?

In the same spirit of this question: How much of mathematical General Relativity depends on the Axiom of Choice? I want to go radically further ahead and ask for what remains of mathematical general ...
9 votes
2 answers
532 views

Physical intuition behind Kontsevich's deformation quantization formula

Kontsevich gives a construction that produces deformation quantization of $C^\infty(M)$ for general Poisson manifolds $M$. The resulting formula (on $\mathbb{R}^n$) is $$ f\star g = \sum_{n=0}^\infty \...
158 votes
14 answers
39k 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 "...
13 votes
0 answers
847 views

What is a BPS state and why is it the cohomology of a moduli space?

The notion of a BPS state has existed in physics for a long time: I do not understand it completely, but I get the impression they are the supersymmetric analogue of ground states, and then physicists ...
8 votes
1 answer
212 views

How does the Tannaka duality work for weak Hopf algebras and fusion categories?

I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works ...
0 votes
1 answer
484 views

Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds

$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
2 votes
1 answer
66 views

Spectral threshold effect: examples

I know that the effect of homogenization can be treated as a spectral threshold effect. I want to know more examples of spectral threshold effects in mathematical physics.
5 votes
1 answer
177 views

In the rep theory of Quantum Double, why does the fusion of 2 "pure fluxes" yield a "pure charge"?

Physicist here, so my notation may be different from standard math notation. For the quantum double $D(G)$ of a group $G$, we may write representations of $D(G)$ in the following way: Consider a ...
5 votes
0 answers
196 views

Difficulty of solving $Ax=b$ in terms of limiting spectral density of $A$?

Suppose $A$ is a random real-valued $n\times n$ matrix and we want to know the difficulty of solving $Ax=b$ when entries of $b$ are sampled IID from Normal$(0,1)$. Can we say anything about the ...
40 votes
3 answers
5k views

How much of mathematical General Relativity depends on the Axiom of Choice?

One of the cornerstones of the mathematical formulation of General Relativity (GR) is the result (due to Choquet-Bruhat and others) that the initial value problem for the Einstein field equations is ...
1 vote
1 answer
163 views

Is there a notion of a higher Kullback-Leibler divergence?

For discrete probability distributions $P$ and $Q$ defined on the same sample space, $\mathcal{X}$, the Kullback-Leibler divergence is defined as $$ D_{\mathrm{KL}}(P \parallel Q)=\sum_{x \in \mathcal{...
5 votes
4 answers
813 views

Limit of an integral vs limit of the integrand

I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral $$ I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
22 votes
1 answer
2k views

Current status of axiomatic quantum field theory research

Axiomatic quantum field theory (e.g. the wightman formalism and constructive quantum field theory) is an important subject. When I look into textbooks and papers, I mostly find that the basic ...
5 votes
0 answers
192 views

Is there any overlap between the geometric and analysis oriented approaches to mathematical QFT?

The impression I have is that the mathematical approach to quantum field theory can broadly be categorized into one that is more geometrical/topological, for example in gauge theories, and another ...
4 votes
1 answer
537 views

How to classify continuous/differentiable maps from $T^2$ to $U(N)$?

I read a physics paper, of which the main idea is based on the topological classification of maps from 3-torus to the space of $N\times N$ unitary matrices. To quote their equation (4), which gives a ...
55 votes
6 answers
15k views

(How) is category theory actually useful in actual physics?

An answer to a recent question motivated the following question: (how) is category theory actually useful in actual physics? By "actual physics" I mean to refer to areas where the underlying ...
19 votes
0 answers
3k views

What does a product of many Gaussian matrices converge to?

Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$. Is ...
5 votes
0 answers
72 views

Braided monoidal category of (generalized) operator algebras

In the paper "Quantum Collections" Kornell (2016) proved that the category of $W^*$ (von Neumann) (and iirc also $C^*$) algebras, equipped with a suitable choice of tensor product, forms a ...
39 votes
9 answers
4k views

Interpretation of the action in classical mechanics

In classical mechanics the dynamics on a manifold $M$ are characterised by the minimisation of a functional $$ \min_{q \in C^\infty(\mathbb{R},M)} \int_{\mathbb{R}}L(q(t),\dot{q}(t))dt, $$ where $L:TM\...
0 votes
0 answers
100 views

A variant of quantum harmonic oscillators

We have the following variant of harmonic oscillators. $$ \left\{ \begin{array}{**lr**} T = a + a^\dagger\\ a | n \rangle = \sqrt{[n]} |n-1 \rangle \\ a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
1 vote
0 answers
126 views

Concentration of a combinatorial sum

Let $X=(x_1,\ldots,x_p)$ be an $p \times n$ random matrix with iid entries from $\{\pm 1\}$, distributed so that $\mathbb P(x_{ij} = 1) \equiv 1/2$, where $x_i=(x_{i1},\ldots,x_{in})$. Let $y$ be a ...
0 votes
0 answers
61 views

How to derive the formulas of the spin-weighted spheroidal eigenvalues (2.16a)-(2.16g) in arXiv:gr-qc/0511111?

I am reading the article "Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions", which is on https://arxiv.org/abs/gr-qc/0511111. I want to ...
0 votes
0 answers
43 views

Solving the Global Relationship of the Dampened String $q_{tt} + \frac{\rho}{2} q_{t} - q_{xx}=0$ with the Method of Fokas

I was interested in learning more about the Fokas method for solving partial differential equations after hearing the impressive claim that it could resolve any linear partial differential equation. I ...
6 votes
1 answer
389 views

How many Coulomb branches do we (conjecturally) know?

Physics preamble: Attached to any $3$ or $4$ dimensional SCFT it is expected to be a Poisson variety $\mathcal{M}_C$ called the Coulomb branch. It should admit a symplectic resolution. Moreover, ...
7 votes
1 answer
509 views

Is Segal's notion of conformal field theory a quantum field theory in the sense of Wightman axioms?

For context, I heard that the Atiyah-Segal formalism of quantum field theory is equivalent to the traditional Wightman approach (at least within the scope of certain theories). However, I could not ...
1 vote
1 answer
224 views

Spin connection vs. Cartan connection

I am studying the tetradic Palatini formalism of general relativity. In this formalism, one usually considers a manifold $M$, which is either non-compact or compact with Euler-characteristic $\chi(M)=...
0 votes
0 answers
92 views

Additivity of purity of random matrix products

Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as $$\rho(M)=\frac{n \sum_i \sigma_i^4}{\...
6 votes
1 answer
311 views

Rotations, harmonic oscillators, Gaussians, ladders

I am trying to understand better the quantization of the harmonic oscillator. Here are three ways of thinking about the harmonic oscillator. Eigenfunctions of the differential operator: $H = -\frac{...
6 votes
1 answer
233 views

Spectrum asymptotics for a product of $k$ random matrices?

How does the spectrum of a product of $k$ random matrices behave around 0? In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=...
1 vote
1 answer
152 views

Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support

In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V_n)_n$ ...
21 votes
2 answers
2k views

Modular Tensor Categories: Reasoning behind the axioms

(Sorry for the length of the question, I'm trying to communicate what is bothering me as thoroughly as possible) In the construction of modular tensor categories (MTC) from ground zero, we put ...
0 votes
1 answer
283 views

Nancy Cartwright's dichotomy

Nancy Cartwright introduced an interesting distinction with regard to modeling of physical phenomena. According to Cartwright, a mathematical theory is not applied directly to such phenomena. Rather, ...
1 vote
0 answers
67 views

Biot-Savart-like integral for a toroidal helix

The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon. Let us consider a toroidal helix parametrized as follows: $$ x=(R+r\cos(n\phi))\cos(\phi)...
0 votes
0 answers
275 views

Spectral theorem for commuting operators

Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation ...
6 votes
1 answer
178 views

Degenerations of $OG_{+}(5,10)$ over the Kapustin-Witten $P^1$ and tetragonal genus 7 curves

If $X_{i,j}$ is a $2 \times 4$ matrix and $Y_{j,k}$ is a $4 \times 2$ matrix, then there is a $P^1$ family of ideals defined by the four equations $XY = 0$ and the six equations $s \cdot minors(X) + t ...
0 votes
0 answers
59 views

A recurrence relation with two variables

How to solve the following recurrence relation? $$f(i,j) = 2 f(i,j-1) + (\alpha^j+\beta^j) f(i-1,j), 0<\alpha,\beta < 1$$ With the boundary condition $$ f(0,0) = f(1,0) = f(0,1) = 1 $$ A special ...
1 vote
0 answers
112 views

Mathematical justification for the use of an energy shell in the microcanonical ensemble

I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics. Consider $\Lambda$ to be ...
4 votes
1 answer
286 views

Sharpest version of semiclassical Calderon-Vaillancourt theorem

Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
1 vote
0 answers
37 views

Splitting of the conformal group into $PSL(2,\mathbb{R})$ and other factorizations

In 1+1 dimensions of Minkowski spacetime, the conformal group can be split into two copies of $PSL(2,\mathbb{R})$ acting on null lines. I'm curious to know if a similar split exists for the conformal ...
9 votes
1 answer
1k views

Proving the Replica Trick works

The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit $$ \log(Z) = \lim_{n\to 0}\...
2 votes
1 answer
306 views

Does there exist a Python package that samples random special unitary matrices such that the matrices are parameterized

For reference, the linked paper is Composite parameterization and Haar measure for all unitary and special unitary groups by Christoph Spengler, Marcus Huber and Beatrix C. Hiesmayr (J. Math. Phys. 53,...
7 votes
2 answers
372 views

Non-associative deformation quantization

Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review An introduction to nonassociative ...
5 votes
1 answer
157 views

What structure is preserved by pseudo-homeomorphisms of pseudo-Euclidean spaces?

Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb ...
3 votes
1 answer
200 views

Interesting question about the Thomson problem for arbitrary number of electrons

This question is crossposted from here I believe this is a pretty hard question and so I decided to repost the question in the Math Overflow forum. If there is something wrong with doing this, I am ...
4 votes
1 answer
1k views

What is Quantum Geometry supposed to be about?

There are meaningful questions we can ask about Euclidean geometry which could not have been posed in the time of Riemann or even of Hilbert, and which would have made no sense at all to Euclid. For ...
2 votes
0 answers
127 views

Naked curvature singularity vs Cauchy horizon in stably causal space-time

There is a result that says (theorem 2.11) that any stably causal space-time $M$ is either a product $\Sigma\times \mathbb{R}$ or the time-like gradient $\nabla f$ of a time function $f:M\rightarrow \...
3 votes
1 answer
127 views

Applications of maximal surfaces in Lorentz spaces

I have been working on minimal surfaces, only recently learnt about maximal surfaces and "maxfaces" in Lorentz spaces. I can clearly see the mathematical motivations. But I wonder if zero-...
0 votes
0 answers
118 views

Equilibrium position of $ n $ free charges as polynomials roots

I asked the same question on here but received no answer. The classic problem of the electrostatic equilibrium positions of a linear system of $ n $ free unit charges between two fixed charges is well ...
3 votes
0 answers
179 views

On the spectrum of Fokker–Planck with linear drift

The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
1 vote
0 answers
88 views

"Classifying" causally closed sets in Minkowski space

Let $M = \mathbb R^{D+1}$ be Minkowski space. Recall that the causal complement of a set $A \subseteq M$ is the set $A^\perp \subseteq M$ where $p \in A^\perp$ there is no timelike path between $p$ ...

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