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 ...
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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 ...
Tomas's user avatar
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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 ...
Pulcinella's user avatar
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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 ...
mcmathy's user avatar
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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 ...
carfog's user avatar
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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 $\zeta(z)\in(0,\infty]$ ...
0xbadf00d's user avatar
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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 ...
Fetchinson0234's user avatar
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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\{\...
Young22's user avatar
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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 ...
MathMath's user avatar
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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 \...
Matha Mota's user avatar
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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 ...
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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 ...
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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 ...
Hollis Williams's user avatar
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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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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 $a$ ...
Frederik Ravn Klausen's user avatar
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1 answer
316 views

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 ...
Frederik Ravn Klausen's user avatar
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273 views

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 ...
Andi Bauer's user avatar
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1 answer
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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 ...
Piyush Grover's user avatar
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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 ...
Frederik Ravn Klausen's user avatar
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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 ...
 Analyst 's user avatar
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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 ...
Chris Janjigian's user avatar
15 votes
1 answer
674 views

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 ...
Confused Physicist's user avatar
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1 answer
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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 ...
user142929's user avatar
6 votes
2 answers
404 views

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 ...
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1 answer
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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 ...
Mtheorist's user avatar
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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. ...
Daniel Castro's user avatar
1 vote
0 answers
91 views

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},\...
Alec's user avatar
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2 votes
1 answer
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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 ...
Medo's user avatar
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6 votes
2 answers
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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 ...
display llvll's user avatar
4 votes
1 answer
133 views

"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 ...
Yonah Borns-Weil's user avatar
19 votes
1 answer
2k views

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 ...
Anton Hilado's user avatar
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2 votes
0 answers
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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, ...
Pritam Bemis's user avatar
7 votes
1 answer
679 views

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 ...
MathMath's user avatar
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4 votes
1 answer
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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 ...
MathMath's user avatar
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6 votes
0 answers
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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 ...
MathMath's user avatar
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4 votes
1 answer
286 views

Sharpest version of semiclassical Calderon-Vaillancourt theorem

Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
Yonah Borns-Weil's user avatar
2 votes
1 answer
326 views

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 ...
Shaq155's user avatar
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3 votes
1 answer
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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 ...
LeechLattice's user avatar
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1 vote
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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 ...
DJA's user avatar
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6 votes
1 answer
262 views

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 ...
MathMath's user avatar
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2 votes
0 answers
81 views

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{\...
J_P's user avatar
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15 votes
2 answers
2k views

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 ...
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