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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On the spectrum of fokker plank 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 ...
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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 ...
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"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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K-finiteness of unitary representations of Poincaré-like groups?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\ISO{ISO}$I'd like to know if there are any papers that study the following problems:
Determine when decomposing the unitary irreps of $\ISO(d,1)$ into ...
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How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]
Crossposted at SciComp SE
I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method.
I find ...
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finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
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Solve the recurrence relation with 2 variables
We have the following recurrence relation:
\begin{equation}
f(n,m) = f(n-1,m) g_{\alpha, \gamma}(n,m) + f(n,m-1) g_{\beta, \gamma}(n,m) \\
g_{\alpha, \gamma}(n,m)= \sum^{n}_{i = 0} \sum^{m}_{j = 0} \...
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Finding solutions to nonlinear wave equations with time-periodic boundary conditions (closed timelike curves)
Is it known how to solve nonlinear dispersive wave equations, such as the Klein-Gordon equation with a $\phi^4$ interaction with time-periodic boundary conditions? The motivation behind time-periodic ...
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How to solve recurrence relation with 2 variables？
I have the following recurrence relation and boundary condition?
$$
f(n,m) = \frac{\alpha n}{n+m} f(n-1,m) + \frac{\beta m}{n+m} f(n,m-1) + 1
$$
$$
f(n,0) = \frac{1-\alpha^{n+1}}{1-\alpha}, f(0,m) = \...
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Probabilistic interpretation of von Neumann's approach to quantum mechanics
One of the basic postulates in the mathematical formalism of quantum mechanics is that the probability of a measurement of an observable $A$ in the state $\psi \in \mathscr{H}$ to return a value in a ...
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Mach's principle, Newton's law and Hilbert sphere?
(This question has originally been posted on reddit, but I thought, that the question raised in the post above, might fit as well here on MO.)
I wanted to share with you something I stumbled upon ...
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What structure is preserved by pseudo-homeomorphisms of pseudo-Euclidean spaces?
Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb ...
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symmetrization and invariants
Let $V$ be a supervector space over $\mathbb C$ and let $T^n(V):=V \otimes V \otimes \cdots \otimes V$ and let $S^n(V)$ be the super vectorspace of symmetric tensors. Then we have a cannonical ...
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Langevin dynamics or stochastic gradient flow for grand canonical ensemble
We know that for a measure exp(-U(X)) (canonical ensemble), we can use the dynamic dX=-DU(X)+ noise to sample the measure as t goes to infinity.
Is there any dynamic corresponding to the grand ...
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Physical relevancy of two curious PDE's
My research has brought me to the following linear parabolic second order PDE:
$$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) $$
for $c(t,x)=-\frac{t}{x}$ and $...
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Large deviation principle for product of iid bounded symmetric random variables
Let $n$ and $k$ be positive integers. Let $X$ be the empirical mean of $n$ iid Rademacher random variables. Note that the distribution of $X$ is symmetric about 0, and also $|X| \le 1$ w.p 1. Let $X_1,...
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Categorifying the definition of a principal $G$ bundle
For a Lie group $G$, we can define a principal $G$ bundle as a submersion of manifolds $\pi:P \to X$ equipped with a free right $G$-action on $P$ that is transitive on the fibres over $X$.
What goes ...
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Number of divergence free symmetric two tensor in dimension 4 [duplicate]
In a $4$ dimensional (semi)-Riemannian manifold $(M^{4}, g)$, both Einstein tensor $G= \operatorname{Ric}(g)- \frac{R(g)}{2}g$ and stress-energy tensor $T$ symmetric and divergence-free. Is there any ...
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Reference Request for a particular approach of (rigorous) statistical mechanics
I was reading Mathematical Aspects of Quantum Field Theory by. E. de Faria and W. de Melo, and the following caught my attention.
In (Hamiltonian) mechanics, the states of a system are described by ...
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Asymptotic analysis for a double integral related to Airy functions
Let $Ai(x,y)$ be the Airy kernel which is given by
\begin{equation}\label{equ2.12}
Ai(x,y)=
\begin{cases}
\dfrac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, & x\ne y, \\
Ai'(x)^2-xAi(x)^2 & x=y. \\
\end{...
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About the proof of Lebesgue decomposition theorem for Hilbert spaces
Let $\mu$ be a Borel measure on $\mathbb{R}$. By the Lebesgue decomposition theorem, there exists measures $\mu_\text{pp}$, $\mu_\text{ac}$ and $\mu_\text{sing}$ such that $\mu = \mu_\text{pp}+\mu_\...
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Definitions fundamental forms and their geometric Intuition
Let $(M^{n+1}, g)$ be a Lorentzian manifold (spacetime) that contains a Riemannian/spacelike hypersurface $(\Sigma ^{n},h).$ Then we can define the second fundamental form of the hypersurface in many ...
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Uniqueness of Borel functional calculus for unbounded self-adjoint operators
I was reading these short notes on the Borel functional calculus where the author discusses the uniqueness property of this calculus for both bounded and unbounded self-adjoint operators.
When it ...
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Simple examples of Gromov-Witten invariants not being enumerative
I understand why Gromov-Witten invariants in general are not enumerative, so it's not necessary to explain this. However to test something I'm working on, I'm looking for examples of concrete ...
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inclusion of von Neumann algebras implies reversing inequality of its modular operators
I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999)
Since $\mathcal N \subseteq \mathcal M$, it follows by standard arguments that $\Delta_{\mathcal ...
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Heat kernel coefficients for Laplacian in instanton background
The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is ...
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Characterization of the Hamiltonian's spectrum in quantum mechanics
This is basically the same question I made on physics stack exchange The spectrum of the Hamiltonian in quantum mechanics, but I got no answers so far and decided to move it to mathoverflow with some ...
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Segal's axioms for CFT
In Segal's papers about Conformal Field theory, https://www2.math.upenn.edu/~blockj/scfts/segal.pdf, in section $1$, he describes the evolution of a system (a string moving about in a manifold $M$) by ...
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Schrödinger equation approximation – continuity of eigenvalues with respect to potential
The question has been crossposted from Stackexchange after receiving no answers.
Setup: the time-independent Schrödinger equation (eigenvalue problem):
$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$
(On ...
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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 ...
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how to derive this elliptic integral?
I am reading the article arXiv: 2207.09961, there are some interesting elliptic integrals, i.e. the formula (3.7) and (3.8). You can also see this image
where $p_0(z)=\sqrt{-Q_0(z)}$ and $Q_0(z)=-\...
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What is the status of the cobordism hypothesis?
Let $\mathscr{C}$ be a symmetric monoidal (weak) $n$-category. A framed extended TQFT of dimension $n$ with values in $\mathscr{C}$ is a symmetric monoidal functor from the framed bordism $n$-category ...
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Applications of super-mathematics to non-super mathematics
Supergeometry and more broadly supermathematics has been around for few decades. Since its introduction by physicists, there has been an some mathematical interest in them.
Although interesting in its ...
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Meaning of a quantum field given by an operator-valued distribution
I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me.
Let $\mathcal{H}$ be a Hilbert space in which ...
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Solving a Differential equation from intersection theory via series expansion
I have the following differential equation $\nabla_\omega \psi=\varphi$ where $\nabla_\omega(\psi)=d(\psi)+\omega(z)\wedge\psi$.
With the local coordinates of $y=z-x_i$ the series expansions is
$$
\...
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Solution of an ODE upon singular perturbation
The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.
The original system involves $N$ massless electric charges at position $\...
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When is a stationary measure of a Markov chain "exponentially localized"?
Here exponentially localized can be thought in a non-rigorous manner as a measure that is mostly supported on a sparse number of nodes.
Some intuition can gained by thinking about a diffusion process, ...
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Hamiltonian-ization of a dynamic system
On affine space, a sufficiently smooth continuous-time Hamiltonian dynamic system $\dot p = \nabla_q H, \dot q = -\nabla_p H$ conserves $H$, preserves the volume form (e.g. if we are looking for ...
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Random variables with density distributions given by squared Hermite polynomials
I was wondering whether anything is known on the following: Let
$h_k (x)= (-1)^k e^{x^2/2} \frac {d^k}{dx^k} \, e^{-x^2/2}$, $k \geq 0$, be the classical
Hermite polynomials ($h_0(x) = 1$, $h_1(x) = x$...
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Approach to learning constructive QFT
First I would like to apologize if this post breaks any rule regarding career advice or opinion-based questions. Given that construct QFT (CQFT) is a rather small community, I found this is the only ...
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Interpretation of spectral measures in quantum mechanics
Let us define a pure vector state of a quantum system as a vector $\psi$ in a Hilbert space $\mathscr{H}$ with norm $\|\psi\| = 1$. Let $\mathscr{B}(\mathscr{H})$ be the Banach space of bounded linear ...
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GNS Representation — A theorem from Thirring’s book
After the GNS representation for $C^{*}$-algebras is presented in Thirring's book Quantum mathematical physics, the author states the following theorem.
The Spectral Theorem: For any given Hermitian (...
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Evolution equation in renormalization group for infinitely-many variables
Let $\varepsilon > 0$, $L \gg 1$ and define the torus $\mathbb{T} = \varepsilon \mathbb{Z}^{d}/L\mathbb{Z}^{d}$. Let $K$ be a smooth, strictly decreasing function. To make things easier, consider ...
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Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform?
I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$.
I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^...
- 1,803
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Express $Q_0 u + Q_1 \Delta u + Q_2 \Delta^2 u + Q_3 \Delta^3 u=0$ as a conservation law for $u(\vec x, t) : \mathbb R \times \mathbb R \to \mathbb R$
In the study of certain PDEs, it is beneficial to write them as a conservation law so that the energy of the system may be defined. More facts such as causality can be proven by considering surface ...
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Understanding the boundary condition of spherical waves in the flat spacetime
I am trying to understand one of the two boundary conditions one has to impose to find the solutions of the wave equation in the flat space-time inside a collapsing null shell. For the spherical wave, ...
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Fluctuation-dissipation theorem for Markov processes
In the context of particle systems of non-gradient types (see e.g. here, Step 2 on page 633), I recently encountered the concept of fluctuation-dissipation theorem (FDT). Since it is a major result in ...
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Physical intuition for curvature on higher order frame bundles?
$\DeclareMathOperator\SO{SO}$A priori: I apologize if this isn't up to Mathoverflow standards, I've had very little luck getting questions on this subject answered elsewhere.
I'm looking for a physics ...
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Relationship between heat kernel and Maxwell-Boltzmann distribution
There appears to be a connection between the heat kernel and Maxwell-Boltzmann distribution, but I have not seen this in the literature before. I'd appreciate any kind comments or corrections/...
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Roadmap to geometric Langlands for a mathematical physics student
I am a student of both mathematics and physics, who has recently been studying string theory. My mathematics background is mostly differential geometry (principal bundles, Lie groups, etc.), although ...
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