# 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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### Non-computable numbers in constructive mathematics

Edited in order to take into account feedback from comments:
If we have an uncountable formal language L with well-defined semantics, can every set in the set-theoretic universe V be explicitly ...

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### Asymptotic behavior of an integral on a d-dimensional torus

I am trying to evaluate the asymptotic behavior of the following integral as $t \to \infty$:
$$
I(t; \mathbf{v}) = \int_{[-\pi, \pi]^d} \frac{\sin(t f(\mathbf{k}))}{\sin(f(\mathbf{k}))} e^{i t \mathbf{...

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### Equivariant KR-theory of representation sphere

I would like to say my question first.
Let $G$ be a compact Lie group acting on a good space $X$ in a good way. Let $V$ be a $G$-representation whose real dimension may be less than 8, and let $S^V$ ...

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### Prof. Alan Huckleberry's fascinating lectures [closed]

Does anyone know what textbook Professor Alan Huckleberry was using in these lectures, especially the mathematical physics ones? Some of the lectures are missing, and I want to follow his lectures and ...

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1
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### Proving non-existence of non-frictional CVTs?

This is a bit of a weird question because the problem is more about how you could even go about formalizing a hypothesis more than how to prove it — but it seemed like a fun idea and I figured someone ...

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### Can a maximal rank subgroup of a simply connected Lie group have simply connected factors?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\rank{rank}$Take a simply connected Lie group $G$ such as $\SU(N)$ and a maximal rank subgroup $H$, i.e. $\rank(G) = \rank(H)$. Assume that $H$ takes ...

58
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### Possible new series for $\pi$

In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \...

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### Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators

Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation:
$$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...

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### Possible obstructions to global Wick-rotation in distinguishing spacetimes

Take the time-orientable $3+1$ dimensional spacetime $(M,g)$ that is locally Wick-rotatable at any arbitrary point $p \in M$ to a Riemannian manifold $(N,h)$.
Local Wick-rotatability of $(M,g)$ ...

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### On the $\phi^4$-model on infinite lattice

It is mentioned in this answer Is there a program to solve The Yang–Mills Existence and Mass Gap problem similar to the Hamilton's program to solve Poincaré Conjecture?
that it is an open ...

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### Are causally isomorphic spacetimes Wick-related?

Take the time-orientable spacetimes $(M_1,g_1)$ and $(M_2,g_2)$ that are locally(to be clarified below) Wick-related and both are globally Wick-rotatable(to be clarified below) to the same Riemannian ...

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### About Friedrichs historical contribution to QFT cited in Reed and Simon

In the Reed and Simon book, Appendix X.7, they mention that Friedrichs provided the first examples of inequivalent representations of the canonical commutation relations via the Weyl relations in the ...

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### Does there exist an electromagnetic analogue of Einstein's field equations?

This will look like a physics question but it doesn't have anything to do with reality so its a vague math question if anything.
I recently learned about gravitoelectromagnetism which describes an ...

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### Is the topological dimension of spacetime fixed for causally isomorphic spacetimes?

Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are not homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
The Lorentzian metrics $g_1$ and $...

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### Status of generalization of timelike tube theorem to algebras of causal completions

The timelike tube theorem states that the additive algebra $A_{\text{add}}(U)$ of operators in a spacetime region $U$ is equal to the additive algebra $A_{\text{add}}(E(U))$ of the timelike envelope $...

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### Convergence of Liouville correlation functions

A key object in Liouville conformal field theory is the random Liouville measure $M$ defined heuristically as $M(d^2x) = :e^{2bX(x)}: d^2x$, where $X$ is a Gaussian free field and $:e^{2bX}:$ denotes ...

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### Hartle-Hawking state as a universal maximum entropy weight on the observer algebra

$\newcommand{\HH}{\mathrm{HH}}$Consider a general spacetime containing an observer, and let $\mathcal{A}_{\mathrm{obs}}$
denote the algebra of observables available to the observer. It has been ...

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### Equality between operators, on dense subspace, from a quadratic form point of view

Let $L \ge 1$ and consider a finite box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$. The set of functions:
$$\psi_{p}(x) = \frac{1}{L^{d/2}}e^{i\langle p,x\rangle} \quad p\in \frac{2\pi}{L}\mathbb{Z}^{...

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1
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227
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### Inequality with Hermite polynomials

Consider the (physicist's) Hermite polynomials $H_n(x)$ which are divided by
$$\sqrt{\sqrt{\pi} 2^n n!}$$
for the purpose of normalization.
These are orthogonal with respect to the weight function $e^{...

2
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### Topology of the energy space for the gauged Ginzburg-Landau

Given a scalar field $\phi \in H^1_{\mathrm{loc}} (\mathbb R^2 \to \mathbb C)$ and connection $1$-form $A \in H^1_{\mathrm{loc}} (\mathbb R^2 \to \mathbb R^2)$ on the plane, define the gauged Ginzburg-...

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### Who says understanding physics helps mathematicians? (A reference request) [Take the word "who" literally.]

If I wanted to make a somewhat bold and rather vague claim in print that it is widely acknowledged among mathematicians that knowledge of mechanics (in the sense in which physicists understand that ...

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### Order isomorphism + manifold homeomorphism => path topology homeomorphism?

Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
Let's call this map $\phi: (M_1, \...

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### Renormalization from cohomology point of view

In order to construct a Euclidean quantum filed theory one usually needs to take care of the renormalization problem. Let us consider a simple model like $\phi^4$ in dimension two. In this case just ...

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### Possible research topics for a beginner in Topological QFT?

I am highly interested in Topological Quantum Field Theory (TQFT) and am currently planning on doing a project on this topic this year. Some relevant background: Algebra (Groups, Rings, Fields, basics ...

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354
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### Can the Causal Structure recover the manifold topology for non-chronological spacetimes?

Given a time-oriented spacetime $(M,g)$, a binary relation $\ll$ can be defined on this spacetime where $p \ll q$ for $p, q \in M$ if and only if there exists a time-like path connecting $p$ and $q$.
...

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### Possible relation between causal-net condensation and algebraic K theory

Causal-net condensation is a natural construction which takes a symmetric monoidal category or permutative category $\mathcal{S}$ as input date and produces a functor $\mathcal{L}_\mathcal{S}: \mathbf{...

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### Topology and local isometry, spinning cosmic string

Suppose one is given the spacetime $(M,g)$ where $M$ is a fixed differentiable manifold and $g$ is a Lorentzian metric whose local expression is:
$$g= -(dt + a \, d \phi)^2 + d\rho^2 + \kappa^2 \rho^2 ...

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### What is the "intrinsic reason" for the failure of Schwarzschild coordinates in general relativity?

It is well known that the Schwarzschild metric fails at r = 2M (in units where c = G = 1) and this is the result of choosing "bad" coordinates. I find this surprising because the coordinates ...

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### Wick ordering, probability vs physics

Consider a collection of creation $a^\dagger$and annihilation operators $a$. In physics one defines Wick ordering (also known as normal ordering) as a prescription to place all creation operators ...

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### Is there a canonical smooth structure on tame Fréchet orbit type stratifications?

In finite dimension orbit type stratifications, it is known that the orbit space $M/G$ resulting from an action of a proper Lie Group $G$ on a smooth manifold $M$, satisfying a set of certain ...

3
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### Generalizing the Narasimhan–Seshadri theorem

There is a theorem that (stable, topologically trivial) holomorphic $G$-bundles are in one-to-one correspondence to flat $K$-bundles (with the appropriate corresponding condition), where $K$ is the ...

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1
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### Wick product of free fields and wave front sets in the sense of Lars Hörmander

Let $\phi$ be the neutral, massive and free scalar field in $\mathbb{R}^4$. That is, $\phi$ is a tempered distribution whose values are unbounded operators on the Bosonic Fock space.
Note that the ...

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### Can an ellipse roll down a tilted sine curve without jumping?

Background
Assume that we have a solid ellipse with uniform density, and that it rolls along a curve.
In the following MO question, I asked along what curve an ellipse rolls down fastest. It was ...

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1
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### Recursive relation to represent the last element of a matrix using determinant [closed]

$J$ is a $N\times N$ matrix, each element of $J$ is sampled from a Gaussian distribution with zero mean and variance $N^{-1}$. The resolvent matrix is defined as $R^{(N)} = [\mathcal{E} \mathbb{I} - J]...

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### Basis vectors using anti-commuting operators?

Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy:
$$A_{i}A_{j}+A_{j}A_{i} = A_{i}...

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### Systems of (hyperbolic) 2nd order PDEs with lower order constraints

Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\...

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### I'm looking for the NLab page on particle species

This is just a reference request.
I came across an NLab page on particle species described as certain vector bundles. But I can't seem to find it again when I searched recently.
If someone can point ...

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### Representation of anti-commuting matrices in $\mathbb{C}^{2}$

This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...

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### Stability of rigid bodies spinning around $z$-axis under gravity

Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...

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### Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...

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### Predicting the peak "amplitude" of a damped sine wave in the frequency spectrum with FFT

In one line: Given an exponentially decaying sine wave $x(t)$, how can we predict the amplitude of the resulting peak in frequency spectrum using discrete Fourier transform.
In nuclear magnetic ...

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2
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### Algebra/Algebraic geometry in statistical mechanics

This is a soft question. I am currently studying statistical mechanics and I found this one by chance: Algebraic statistical mechanics
And I also found some workshops on interactions between ...

14
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1
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### Hilbert's sixth problem and QFT description

The Wikipedia entry on Hilbert's sixth problem about QFT description is “Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, modern quantum field theory can also be considered ...

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### Symplectic structure of Higgs branch

I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the ...

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### On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...

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### Perturbation method for time-periodic singular system of ODEs

I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form:
$$
\begin{cases}
\dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...

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### Why does deformation quantisation have anything to do with $\mathbf{E}_2$/little disks?

Kontsevich proved that any Poisson manifold $M$ has a quantisation $\mathcal{O}_\hbar(M)$: an associative algebra recovering the $\mathcal{O}(M)$ with its Poisson bracket by taking $\hbar=0$. Later he ...

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### Recommendation to understand mean field theorem

I am studying Rodnianski and Schlein - Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Everything was clear for me and I reproved everything before inequality (3.22) (except ...

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1
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### What is the Lebesgue covering dimension of this topological space?

Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal.
Take the induced topology defined by the Lorentzian metric called Alexandrov topology.
This topology matches ...

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1
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### What are double groups mathematically?

In physics and chemistry, there is the concept of double groups. These are double covers of the usual point groups, obtained by "adding an element $R$, which represents rotation by $2\pi$" ...