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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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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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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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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The Lebesgue covering dimension of the Cosmic String interval topology
Take the spacetime $(M,g)$ that satisfies Einstein's Field Equations exactly where $g$ is locally:
$$g= - c^2 dt^2 + d \rho^2 + (\kappa^2 \rho^2 - a^2) d \phi^2 - 2 ac d\phi dt + dz^2 \ $$
in the ...
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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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Trouble understanding Lax method for KDV equation for inverse scattering method
I am trying to learn the Lax pair condition on my own so that I can eventually learn the inverse scattering method. I am following a paper by Tuncay Aktosun ("Inverse scattering transform and the ...
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Size of Hilbert space in geometric quantization from index theorem
In these notes on geometric quantization by Nair, on page 24, the Bohr-Sommerfeld rule in quantum mechanics is interpreted in terms of the Atiyah-Singer index theorem.
To be precise, the polarization ...
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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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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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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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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 ...
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Can I relate the L1 norm of a function to its Fourier expansion?
I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
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Stable graphs: Feynman diagrams and Deligne-Mumford space
I do not know very much about quantum field theory, but I have seen, in my reading, that stable graphs can appear in QFT in the form of, I think, Feynman diagrams. By stable graph I mean a "graph with ...
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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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Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach
Let $P\rightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $\alpha: G\times F\rightarrow F$ a smooth left action.
There is an associated fibre bundle $E\rightarrow ...
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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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Rigorous Euler-Lagrange equations for fields
I'm looking for rigorous discussions on the derivation of the Euler-Lagrange equation for field as it is usually discussed in classical field theory books. More precisely, if the action is given by:
$$...
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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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An inequality for a "generalised random energy model"
Let, for all $i, j$, $Z_{i,j}$ be a standard normal, chosen iid. For each $n\geq 1, k\geq2$, define the Hamiltonian $H_{n,k}: [k]^n \to \mathbb{R}$ by
$$(j_1,j_2,\ldots,j_n) \mapsto \sum_{i=1}^n Z_{i, ...
CommunityBot
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On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
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Perturbation method for time-periodic singular system of ODEs
I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form:
$$
\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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What kind of Lagrangians can we have?
In any physics book I've read the Lagrangian is introuced as as a functional whose critical points govern the dynamics of the system. It is then usually shown that a finite collection of non-...
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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 $2\pi$" ...
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Anomaly in QFT physics v.s. determinant line bundle
In a quantum field theory (QFT) lecture, a math-physics professor explains the anomaly in physics, say the non-invariance of the partition function of an anomalous theory under background field ...
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Is the Gödel universe Wick rotatable?
Take Wick rotatability being as the way defined in the following article by Helleland and Hervik:
Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
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Definition of second quantization
The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write:
Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the ...
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7-sphere x 4-sphere manifold and its physical significance [closed]
I am looking for sources about this manifold 7-sphere*4-sphere and its relations to theoretical physics.
Where to go to read about 7-sphereX4-sphere manifold and its physical significance?
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What is the dual of a hyperbolic configuration of points?
Let $C_n$ denote the configuration space of $n$ distinct points in hyperbolic $3$-space $\mathcal{H}$. If $\mathbf{x} := (\mathbf{x}_1, \dots, \mathbf{x}_n) \in C_n$, where $\mathbf{x}_i \in \mathcal{...
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Why computing $n$-point correlations?
I am trying to find a sufficiently convincing answer to this question, but it has been taking so much of my time and I can't get anywhere. It also follows my previous question on PSE.
In axiomatic QFT,...
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Mathematical explanation of the failure to quantize gravity naively
One often hears in popular explanations of the failure to find a "Grand Unified Theory" that "Gravity goes off to infinity, but cutting off the edges gives us wrong answers", and other similar ...
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Deriving Sommerfeld radiation condition from limiting absorption principle
For the Helmholtz equation
$$
-(\Delta + k ^2) u = f, \label{1}\tag{1}
$$
imposing the Sommerfeld radiation condition
$$
\lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0
$$
on $u$ ...
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Reference for rigorous interacting many-body quantum mechanics
Are there good references for (both zero and finite time) interacting systems of quantum many-body theory? More precisely, I would be interested in references discussing the following topics:
Second ...
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Mixing for a gas of hard spheres
The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. ...
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Flat Maurer-Cartan connection iff flat Berry connection
I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$.
The first is the canonical or $H$-...
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What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$
FYI: I asked this question here couple of days ago but got no answer yet.
$n$ is an integer
We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are ...
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What are "branes", and why do they form a category?
I've been trying to read Kapustin–Witten - Electric–Magnetic Duality And The Geometric Langlands Program recently, as someone whose mathematical interests are in the Langlands program. I have some ...
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Is a Wick rotatable spacetime necessarily strongly causal?
There are a few viable ways to formulate Wick rotatability that preserve distinct features.
One is mentioned in the post:
Obtain Lorentzian manifolds from Riemannian ones by Wick rotation
There's also ...
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Green's function for a linear PDE initial value problem
For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem
$$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...
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Is every strongly causal spacetime purely electric?
Take a time 4-dimensional orinted Lorentzian manifold $(M,g)$.
A spacetime is called Strongly Causal at point $p$ if and only if for every neighbourhood $U$ of the point $p$ there exists a ...
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Prove that the following function is positive
Consider the following function:
$$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right)
$$
This is Mehler's kernel, and can be ...
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Group cohomology and TFTs
Ok, so I am still trying to make my way through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Actions", and I have a question (what's new). The authors start by showing ...
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Mathematical proof of Regge symmetry
In the representation theory of the group $SU_2$ a big role is played by so-called $6j-$symbols. Let me sketch its definition (some other interpretations could be found here).
Denote a representation ...
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How causal is a strongly causal purely electric spacetime?
Take a generic Lorentzian spacetime $(M, g)$ where $M$ is a time-oriented 4d manifold and $g$ is the Lorentzian metric that is strongly causal and purely electric.
According to this answer:
Is every ...
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Bergman kernel via Riemann Theta function
Let $\Sigma$ be a Riemann surface of genus $g$. A Bergman kernel $B(p,q)$ is a bilinear meromorphic form on $\Sigma \times \Sigma$ with poles of order $2$ along the diagonal $p = q$ and holomorphic ...
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Is there any case of remormalization in which we have to solve it by ways in two different systems? [closed]
In renormalization of physics, $$\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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