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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Are analytic solutions for the Navier-Stokes equations sufficient?
Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space.
However, I am wondering, whether it is possible to consider just analytic ...
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Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure
Recently I have been learning about Rozansky-Witten invariants, mainly through Hitchin-Sawon's paper "curvature and characteristic numbers of hyperkahler manifolds" and through Justin Sawon'...
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1D topological defects in $d>3$ spatial dimensions
I am trying to construct a 1D topological defect solution in 4 spatial dimensions, i.e., a solution to some PDE (likely the equations of motion of some Lagrangian) on $\mathbb{R}^{4}$ which is ...
- 147
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Definition of average $\langle \langle \cdot \rangle \rangle$
I started reading the paper Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model and I would like to clarify the notation $\langle \langle \cdot \rangle\rangle$ the authors use in ...
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Is there any case of remormalization in which we have to solve it by ways in two different systems? [closed]
In renormalization of physics, $$\sum_{j=1}^{\infty}j=-\frac{1}{12}$$ We may obtain the result in two ways: first we may redifine the sum so we have used two system of math with different definition ...
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Anderson localization for time-dependent noises
Anderson localization concerns the localization properties of the Schrödinger operator with a Hamiltonian of the form
$$H=-\Delta+V(x),$$
where $V$ is a highly oscillatory random potential. A simple ...
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Distinguishable under manifold topology but indistinguishable under the Alexandrov topology
Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.
In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
- 308
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Fokker Planck equation in the Stratonovich approach
I'm a physics master student and I have difficulties understanding how to derive the Fokker Planck equation from the Stratonovich SDE.
With the Ito SDE it is simple since the noise is independent of $...
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Definition of this formula for the $2p$ functions
I am reading this paper about constructive renormalization for fermions and I got a really basic question about it. There, the effective Lagrangian (with UV cutoff $\Lambda_{0}$ and IR cutoff $\Lambda$...
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Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by $\phi_{f_{\theta(t)}(x)}^{\tau'}$?
Given a control family $F:=\{f_1,\dotsc,f_n\}$, and $\phi_f^\tau(x)$ is the flowmap of the dynamical system
$$
\begin{cases}
z'(t)=f(z),\\
z(0)=x,
\end{cases}
$$ at end time point $\tau$.
Suppose $a_i&...
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Structure of all Wightman QFTs
I have two related questions related to constructive/axiomatic QFT.
Is there a structure on the collection of all QFTs, as defined by the Wightman axioms? Do they form some type of category?
...
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Classification of real Clifford algebras
$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, ...
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Temporal evolution of a globally hyperbolic spacetime
Any globally hyperbolic spacetime can be assigned a global function of time as Hawking has demonstrated for stably causal spacetime. (Any globally hyperbolic spacetime is also stably causal).
For ...
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How to compute this path integral?
Let $\mathbb{R}^2$ be phase space with coordinates $(p,q)$ and let $\epsilon>0\,.$ Then given any path $\gamma:[0,1]\to \mathbb{R}^2$ and any large enough $N>0\,,$ we can approximate $\gamma$ by ...
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On Dirac/ Clifford matrices
Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$.
The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition
$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$
where $\{A,B\}=...
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Question about a step in the proof of the min-max principle
I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
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What is a particle in the context of QFT with interactions?
I'm a bit of a novice, so bear with me.
My understanding of the story is as follows.
From Lagrangians to Irreducible Representations
The story of the types of possible particles begins with the ...
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Properties of the stress energy tensor in Wightman formulation of CFT
In various papers that I have been reading about applying the Wightman axioms to conformal field theory, the authors write things like the following about the stress-energy tensor:
$$\int \mathrm{d}x^...
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Is this set a manifold?
Take a general spacetime that is not strongly causal.
Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric that satisfies the Einstein's Field ...
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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 ...
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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 ...
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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.
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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 ...
- 308
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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 ...
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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{...
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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 ...
- 421
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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 ...
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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 ...
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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 ...
- 1,054
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What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)=...
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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
$$\rho(M)=\frac{n \sum_i \sigma_i^4}{\...
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Spectrum asymptotics for a product of $k$ random matrices?
How does the spectrum of a product of $k$ random matrices behave around 0?
In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=...
- 2,380
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Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support
In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the
Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V_n)_n$ ...
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Nancy Cartwright's dichotomy
Nancy Cartwright introduced an interesting distinction with regard to modeling of physical phenomena. According to Cartwright, a mathematical theory is not applied directly to such phenomena. Rather, ...
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Biot-Savart-like integral for a toroidal helix
The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon.
Let us consider a toroidal helix parametrized as follows:
$$
x=(R+r\cos(n\phi))\cos(\phi)...
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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 ...
- 3,151
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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 ...
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A variant of quantum harmonic oscillators
We have the following variant of harmonic oscillators.
$$
\left\{
\begin{array}{**lr**}
T = a + a^\dagger\\
a | n \rangle = \sqrt{[n]} |n-1 \rangle \\
a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
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Mathematical justification for the use of an energy shell in the microcanonical ensemble
I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics.
Consider $\Lambda$ to be ...
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Splitting of the conformal group into $PSL(2,\mathbb{R})$ and other factorizations
In 1+1 dimensions of Minkowski spacetime, the conformal group can be split into two copies of $PSL(2,\mathbb{R})$ acting on null lines. I'm curious to know if a similar split exists for the conformal ...
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How many Coulomb branches do we (conjecturally) know?
Physics preamble: Attached to any $3$ or $4$ dimensional SCFT it is expected to be a Poisson variety $\mathcal{M}_C$ called the Coulomb branch. It should admit a symplectic resolution.
Moreover, ...
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Proving the Replica Trick works
The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit
$$
\log(Z) = \lim_{n\to 0}\...
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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,...
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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 ...
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Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
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