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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How should I understand rigorously the definition of normal ordering of free fields
Let $\phi(x)$ be a free Hermitian scalar field in $4D$ Minkowski spacetime with the metric $(1,-1,-1,-1)$.
Then, though I wrote it as $\phi(x)$, it is in fact an operator-valued tempered distribution ...
2
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Uncertainty principles and Anderson localization principles
The motivation of the question comes from the paper "Some harmonic analysis questions suggested by Anderson-Bernoulli models. Geom. Funct. Anal. 8 (1998), no. 5, 932–964" by Shubin, Vakilian ...
8
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What structure does Rep(vertex algebra) have?
Let $V$ be a vertex algebra. If $V$ is particularly nice, it is known that its category $\text{Rep} V$ of modules is a modular tensor category, see e.g. [1] [2].
However, this has always seemed to me ...
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D'Alembert's Principle: rigorous formulation using notions from modern differential geometry
Is there a rigorous definition of D'Alembert's principle of virtual dynamic work in the language of differential geometry? Some questions I'm hoping to answer are:
How to view the configuration space ...
14
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Dual polyhedra and electric circuits
Good morning,
I hope this question is not too far out of the scope of the forum. I am posting it here because this doesn't seem to be a very standard problem.
Yesterday we were calculating the ...
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0
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Euler-Lagrange equation for a section of complex line bundle
Suppose $M$ be any smooth manifold, and $E$ be a hermitian line bundle over $M$ with hermitian metric $h$. The action functional $S(\phi, \psi)$ for two section variabled $\phi$ and $\psi$ is given by
...
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Can two eigenfunctions be almost linearly dependent in a region?
Consider the Schrödinger operator $H=-\Delta+|x|^a$ on $\mathbb{R}$, where $a>0$. Since the potential is growing at $\infty$, we have compact resolvent thus the eigenvalues are discrete and tend to ...
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How do we solve this rather simple ODE (Loewner equation with driving function $\sqrt t$)?
Remember the following result for the Loewner equation:
If $\lambda:[0,\infty)\to\mathbb R$ is continuous, then for all $z\in\mathbb C\setminus\{\lambda(0)\}$ there is a uniqe $\zeta(z)\in(0,\infty]$ ...
2
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Heat conduction type equation in 4D
[I asked a similar question, Linear PDE, analytic continuation, Green's function and boundary conditions, and was told that a follow-up question should be a separate post.]
I'm interested in a ...
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Generalizing Kato-Seiler-Simon-type inequalities to diamagnetic operators
I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the ...
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Linear PDE, analytic continuation, Green's function and boundary conditions
I'm looking at the linear PDE in 3+1 dimensions,
$$
\left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1}
$$
Where $\xi$ is generally a ...
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Is there an explicit solution to the reaction diffusion system in the following special form?
Suppose $\Omega \subset R^N$ is a smooth bounded domain. Is there an explicit solution to the reaction diffusion equations (RDE) in the following special form?
\begin{equation}
\left\{\...
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Convergence in perturbative renormalization
Consider the following:
$$G(\phi,W) = -\log \int d\mu_{C}(\psi)e^{-W(\phi+\psi)} \tag{1}\label{1}$$
which is very common in QFT. Here $d\mu_{C}$ is a Gaussian measure with covariance $C$. I want to ...
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0
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Second derivative of the logarithm of the modified Bessel function of the first kind
This question makes sense entirely without the probabilistic perspective, but let us quickly describe how it arises in our setting. Let $X,X’$ denote two i.i.d. random variables having the ...
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Linearization stability condition
The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs.
Theorem.
Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
2
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1
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The ultraviolet limit as a limiting case of the renormalization group flow?
In his paper Constructive Renormalization Theory, V. Rivasseau describes the idea of Wilson's approach of solving path integrals step by step. In section 1.4, page 5, however, there is a statement ...
3
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References for applications of Young diagrams/tableaux to Quantum Mechanics
I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book:
Wybourne, B.G.; "...
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Hyperbolic random geometric graphs with less clustering
The hyperbolic random geometric graph $G_{\mathbb{H}}$ consists of a $N$ uniformly random points (within a disk of radius $R$ centred at the origin) of the hyperbolic plane $\mathbb{H}$, connected ...
4
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Motivation for Heisenberg's modeling of observables
What's the motivation for observables to be modeled by self-adjoint operators? I can't seem to find any place where this is laid out clearly. Maybe von Neumann's book is decent, but it's not ...
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Physics application of Wilson surface observables
There is some work which generalises the usual Wilson loop in QFT to higher dimensions and constructs non-abelian Wilson surface functionals in the context of non-abelian gerbes.
It seems to me that ...
2
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1
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Power series expansion of the order parameter in the Kuramoto model
In this review of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$,
$$
r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta
$$
...
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0
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769
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Mathematical physics roadmap
I am looking for a roadmap to be able to understand more of the physical theory behind mathematical physics (at the level where I can read physics oriented textbooks/monographs)
I have a background in ...
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Approximating e-bits by CHSH black boxes
In the CHSH game two parties Alice and Bob independently get a bit $x$ and $y$ which is $0$ or $1$ with probability $\frac{1}{2}$.
Without communicating each of them has to send a referee a bit $a$ ...
2
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1
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On a theorem of Carlson on the necessary and sufficient condition for a matrix to have $m$ real eigenvalues
Background: In the physics of open quantum systems the Lindbladian $\mathcal{L}$ governs the evolution of quantum states through the Lindblad master equation.
The Lindblad operator usually has ...
4
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0
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CFT as an axiomatic field theory
I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every ...
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1
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216
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Spectral perturbation theory of discrete spectra in presence of continuous spectrum
This is a 2 part question:
1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete ...
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0
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Approximating spectra of (finite rank pertubations of) Laurent operators by spectra of (pertubations of) periodic finite operators
A tridiagonal matrix is a matrix which only has elements on three diagonals.
So for $\alpha, \beta, \gamma \in \mathbb{C}$ consider the bi-infinite tridiagonal Laurent operator $T$ with $\beta $ on ...
1
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1
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Lax pair and cubic nonlinear Schrödinger equation
Motivation: I'm trying to understand the Section 4 in the this section 4 in paper" Low regularity conservation laws for integrable PDE by Killip-Visan-Zhang ö
It reads as follows: many completely ...
2
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1
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Equivalence of Hilbert space norm associated to the harmonic oscillator and a sum of Sobolev and weighted $L^2$ norms
I have seen an equivalence claimed in a few places, but I do not know of a reference that actually proves it with details and it has been a while since I took graduate courses on all this. Apologies ...
15
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Practical consequences of the geometric cobordism hypothesis
As far as I understand, the cobordism hypothesis provides a construction of all (appropriately defined) fully-extended TQFTs. In particular, given a fully-dualizable object in a certain category, one ...
0
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1
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Mathematical characterization of gravitational geons as reference request, and their properties as main question
I've edited (ten days ago) a question on Physics Stack Exchange, this Mathematical characterization of gravitational geons, post with identifier 726281 the users of the site were kind adding in the ...
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2
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Infinite clusters for loopless percolation
I feel like this is maybe an incredibly trivial problem, and I'm just missing something. I may also be describing a well-known model that I cannot find the name for, so any comment/suggestion is ...
4
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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 ...
6
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Nonlinear-PDE arising from flat conformal Chebyshev nets
Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...
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NSR superstring as a map of supermanifolds
On one hand, I know that the NSR superstring is described by a map $\Phi: \Sigma \to X$, where $\Sigma$ is a supermanifold with local coordinates $(\sigma,\theta)=(\sigma^0,\sigma^1 | \bar{\theta},\...
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Examples of ODEs with complex constant coefficients and applications to physics?
This question is asked on stackexchange: Are there examples for ODEs with complex coefficients with applications in physics?
but received no answers. I am reposting it here on the hope that it catches ...
6
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Poincaré recurrence and its implications for statistical physics and the arrow of time
A very important theorem in mathematical physics is Poincaré’s recurrence theorem.
As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for ...
4
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1
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"Open systems" version of Stone's Theorem for one-parameter groups of quantum operations
Let $H$ be a Hilbert space, which we interpret as a space of quantum states.
If $U(t):H\to H$ is a unitary norm-continuous one-parameter group with $U(0)=I$, (essentially) Cauchy's functional ...
19
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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 ...
2
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Fourier transform harmonic oscillator eigenstates
The normalized eigenfunctions of the quantum harmonic oscillator are
$$\psi_{n}(x)= \frac{1}{\sqrt{2^n n!}} e^{-x^2/2}H_n(x),$$
where $n \in \mathbb N_0$ and $H_n$ is the $n$-th Hermite polynomial, ...
7
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Reference request for $\phi^{4}_{d}$ theory - where to begin?
When I started studying the basics of $\phi^{4}_{d}$, I looked for papers or lecture notes which would give me some general ideas about the topic and which would construct and/or prove the basic ...
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Cluster expansion, Mayer expansion and perturbative renormalization group
This is a second part of my previous question, which I decided to split into two parts not to mix up different topics at one giant question.
Again, according to V. Rivasseau (section 1.5 of ...
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What is a large field problem?
I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify.
On page 2, Rivasseau talks about the large field problem and, if I understood it ...
4
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1
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Sharpest version of semiclassical Calderon-Vaillancourt theorem
Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
2
votes
1
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Property implies finite propagation speed
Let $u(x, t)$ be a (non-negative, bounded) function on $\mathbb{R}^{n}\times [0, +\infty)$ and suppose that $u$ satisfies some time-independent PDE, e.g. $\partial_{t}u=\Delta_{p}u$. Let us assume ...
3
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What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution?
The Tracy–Widom distributions admit many interpretations.
One of them is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) = x^2$, then in the ...
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physical interpretation of ruelle probablity cascades (SK model)
Background: the Parisi formula gives an exact expression for the free energy of the SK model. The formula (at least the upper-bound) can be derived by looking at the free energy, and then replacing ...
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The role of estimates in field theories
I have been taking a look at some papers in constructive quantum field theory and I got the impression that there is a systematic of estimating things like e.g the effective action or the free energy ...
2
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Pullbacks of LCS-valued distributions
Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\...
15
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Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh
In upcoming work of Ben-Zvi-Sakellaridis-Venkatesh, (see for instance these notes or this lecture) some important aspects of the Langlands correspondence are stated in the language of topological ...