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Kinematic problem (initial acceleration)

"A ball is rolling over a soccer field with constant velocity Vb (> 0) under an angle of 45° with respect to the goal line. It is starting at the corner of the field (distance d beside the middle of ...
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0answers
19 views

Existence of effective Hamiltonian for variational ansatz

I recently saw on Wikipedia that in perturbation theory one can define in a systematic way effective Hamiltonian $\hat{H}_{\rm eff}$ that produces the same kind of states and energies as if one ...
3
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0answers
23 views

Homotopy Theory for Topological Insulators

I'm trying to understand topological insulators in terms of homotopy invariants. I understand that in 2 spatial dimensions, we have Chern insulators since $$\pi_2(S^2) = \mathbb{Z}$$ One subtlety that ...
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1answer
47 views

Spectral covers and a specific exact short sequence

I have a question about the spectral cover construction of Friedman, Morgan, and Witten (typically used to map a description in heterotic string theory into F-theory). I realise this is a highly ...
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0answers
45 views

Why are topological phases described by modular tensor categories?

After some reading, I have an inuitive idea what topological phases of matter are. But where is the connection to modular tensor categories? Is there fundamental literature where this is covered? ...
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2answers
69 views

Completeness relation for coherent states of the quantum harmonic oscillator

For the Quantum harmonic oscillator with energy eigenstates $|n\rangle$ one defines a coherent state for every complex number $z$ by setting (note that the normalization varies across the literature) $...
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0answers
47 views

Normalizable eigenvectors of the inverted harmonic oscillator

Consider the inverted harmonic potential $V(x) = - x^2 $. Does the corresponding Hamiltonian $$ H = p^2 - x^2 $$ have any normalizable eigenstate? How about $$ H = p^2 - x^4 ? $$ Any good ...
6
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1answer
53 views

Examples of short-range correlated gapless systems

I thought this must have been asked before, but couldn't find it through search. It was proved by Hastings and Koma in arXiv:math-ph/0507008, given a Hamiltonian satisfying certain locality ...
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0answers
115 views

Does this divergent series appear anywhere in physics?

Question I recently managed to analytically continue certain divergent series. I was hoping if anyone could tell me if this expression appeared somewhere in physics: $$ \implies \lim_{k \to \infty} ...
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1answer
88 views

Is the electric force a vector or a vector field?

The electric force (Coulomb's law) on a point charge $Q_2$ due to $Q_2$: \begin{gather*} \mathbf{F}_{12}=\frac{Q_1Q_2}{4\pi\epsilon_0}\frac{\mathbf{r}_2-\mathbf{r}_1}{|\mathbf{r}_2-\mathbf{r}_1|^3} \...
7
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0answers
113 views

Metric transformation, polygons and gravitons

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\...
6
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0answers
171 views

Objective time derivative that is not a Lie derivative

Summary Led by an interest into the concept of "Material Objectivity", I am asking myself: Are there objective time rates that are not Lie derivatives? The long read I am trying to understand the ...
3
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1answer
84 views

Exotic differentiable structures in physics

When reading a bit on exotic spheres and exotic $\mathbb{R}^4$'s, I came across some papers of Carl H. Brans and Torsten Asselmeyer-Maluga: "Exotic differentiable structures and general relativity" (...
4
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1answer
75 views

Is the flow of a viscous fluid in free space under no pressure gradient always laminar?

Consider a (Newtonian) incompressible viscous fluid in three spatial dimensions, whose velocity field $\mathbb{v}=\mathbb{v}(x,y,z,t)$ moves according to the Navier-Stokes equations $$\tag{1}\label{...
4
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3answers
88 views

How can I show that inversion is continuously connected to a reflection?

From Ex 3.1 in the TASI lectures on the conformal bootstrap: http://arxiv.org/abs/1602.07982 the problem is the inversion map (with Euclidean signature) $$ I\colon x^\mu \mapsto \frac{x^\mu}{x^2} $$ ...
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0answers
47 views

Proof of Tridecompositional Uniqueness Theorem (Elby and Bub, 1994)

I'm looking for a proof of the Tridecompositional Uniqueness Theorem (Elby and Bub, 1994). Could someone help me? References: Maximilian Schlosshauer, arXiv:quant-ph/0312059; p.12-13. A. Elby &...
7
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2answers
114 views

Properties of spectrum of a self-adjoint operator on a separable Hilbert space

So, if I understand it correctly, the spectrum of a self-adjoint operator on a Hilbert space $H$ consists of two parts: $ \newcommand{\ket}[1]{\,\lvert{#1}\rangle} \newcommand{\op}[1]{\hat{#1}} $ ...
2
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0answers
60 views

Jet bundles for physicists

In order to make Classical Field Theory rigorous we need the idea of jet bundles. I've seem some books on the subject, but most of them are aimed at mathematicians and tend to go quite deep in the ...
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0answers
10 views

Collision laws (formulas) for colliding discs with rotation

Say you have two discs with fixed radii $r_{1}, r_{2}$, positions $q_{1}(t), q_{2}(t)$, momenta $p_{1}(t), p_{2}(t)$, which both depend on time $t$ and fixed masses $m_{1}, m_{2}$. They are assumed to ...
1
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0answers
54 views

Classical Grand canonical partition function derivation

Consider a classical grand canonical ensemble. Let $S_r$ be the reservoir entropy. Suppose it could be expanded at first order: $$S_r \approx S_r(E_t,N_t) + \frac{\mathrm dS_r}{\mathrm dE_i} \cdot ...
12
votes
1answer
250 views

Significance of the exception to Gleason's Theorem when n = 2

Gleason's Theorem famously asserts that (appropriately defined) measures on the lattice of a complex Hilbert space can be implemented by density operators via the trace operation, except in the case ...
2
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0answers
58 views

Rigorous way of box normalisation

This is follow up from an answer to my previous question about unitarity in rigged Hilbert space. As it turns out, that there is no idea of unitarity in rigged Hilbert space (hence no meaningful QM ...
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7answers
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Why is a Hermitian operator a “quantum random variable”?

To me, as a stupid mathematician, a random variable is a measurable function from some probability space $(\Omega, \sigma, \mu)$ to $(\Bbb{R}, B(\Bbb{R}))$. This makes sense. You have outcomes, events,...
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1answer
85 views

Wick renormalization

I'm trying to understand the Wick renormalization in the framework of the Ito integral. I saw the Wick theorem as presented on Wikipedia in a QFT course and I would like to understand how that is ...
1
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1answer
115 views

Functional Analysis for Quantum Mechnanics [duplicate]

I have completed three sequences of courses in QM, and I'm very much eager to to do the functional analysis of QM on my own in my spare time. Can someone suggest some books? I like books with ...
2
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0answers
46 views

The super Grassmannian $G_{2|2}(4|4)$

In the paper, the super Grassmannian $G_{2|2}(4|4)$ is defined by (12)--(18). An element of $G_{2|2}(4|4)$ can be written as a $(2 | 2) \times (4 | 4)$ matrix of full rank modulo the left action by ...
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1answer
93 views

Steady state heat equation intuition

I am learning a bit of fourier analysis, with an interest in physics as well. I originally posted this question on the math stack exchange, but perhaps you physicists have more experience in these ...
4
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1answer
87 views

Lack of Maslov index in the path integral formalism

Introduction Consider Feynman's famous path integral formula \begin{equation} K(x_a,x_b) = \int \mathcal{D}[x(t)] \exp \left[ \frac{i}{\hbar} \int_{t_a}^{t_b} dt \, \mathcal{L}(x(t),\dot{x}(t),t) \...
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1answer
45 views

renormalization and set theory

I am in high school, thus most of what I know about the topics I ask comes from popular science books, which risks asking some dumb questions, because I rarely understand the math behind. It is my ...
0
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1answer
30 views

How many sheets for the Green function?

The Hamiltonian of a particle in a 1D potential is $$H = H_0 + V(x) . $$ Here $H_0 = p^2/2m$ is the free part. It is known that the Green function $$ G_0(E) = \frac{1}{E - H_0 } $$ has a cut ...
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1answer
41 views

Multiscale master equation for dividing cells

I have a conceptual problem trying to build a master equation for dividing cells which have a certain surface receptor. Each cell has its own receptor dynamics, and they divide with receptor-dependent ...
1
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0answers
90 views

Rigorous derivation of general relativity from first principles

What is the minimal set of axioms required to derive the mathematical formulation of General Relativity from first principles? What are these first principles? How does such a derivation go step by ...
1
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0answers
66 views

Meaning of a Dirac delta speed in 1d shallow-water equations

A 1d thin layer of a fluid of constant density in hydrostatic balance can be modeled by shallow-water equations: $$\begin{align*}\partial_tu+\partial_x\left(v+\frac{u^2}{2}\right)&=0,\\ \...
3
votes
3answers
167 views

Mathematical Formulation of Classical Spacetime

I have seen two formulations of Classical Mechanics: Newtonian spacetime (learned it from the lectures of Professor Frederic P. Schuller): Definition: A Newtonian spacetime is a quintuple $(M, \...
4
votes
1answer
83 views

What, exactly, is a “delta function p-form” as used in the theory of branes?

In string theory, when dealing with branes, the following happens: We rewrite a worldvolume action $S = \int_{\Sigma_{p+1}} \omega^{(p+1)}$ of a $D_p$-brane as an integral over the whole $\mathbb{R}^{...
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0answers
23 views

Random walk of a polymer in an annular space - filling in some steps

In Muthukumar, M., 2003. Polymer escape through a nanopore, J. Chem. Phys., 118, 5174–5184, the author solves the diffusion equation for the probability $P(\vec{r},\vec{r_0},N)$ that the ends of a ...
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1answer
132 views

Question on quotient groups and SLOCC [closed]

I have a math-physics question, which is based on an interest in SLOCC systems for black hole entanglement. The Cartan decomposition of a group $G$ such that $H = G/K$ is such that the derivation or ...
5
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0answers
79 views

Is there an algorithm to diagonalize a matrix using gauge transformations

I have two matrices $U(\lambda, x,t)$ and $V(\lambda, x,t)$, where $\lambda$ is a parameter, which belong to the $sl(2)$ algebra, and satisfy the zero-curvature equation $$ \partial_t U - \partial_x V ...
0
votes
1answer
59 views

Velocity from the cumulative distribution function of the Boltzmann distribution

I want to get a Boltzmann distribution of the $v_x$, $v_y$ and $v_z$ velocity components (please, notice that the distribution is one-dimensional). To do so, I need the cumulative distribution ...
1
vote
3answers
64 views

Rotation matrix for aligning x-axis in an arbitrary direction

I want to align the x-axis of my coordinate system, with an arbitrary direction in space $\hat{n}$. About which axis should I rotate? Ceratinty rotation about x-axis or $\hat{n}$-axis will not serve ...
0
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0answers
25 views

Strain Flow in Fluid Mechanics

I am working on hydrodynamic instabilities and was looking for non axisymmetric vortex models that are not Stuart or Taylor green vortices. What I came across was multipolar vortices where a uniform ...
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0answers
58 views

How to calculate the contour integration with branch point? [closed]

The question come from a Mutusbara Sum like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ it equal a contour integral around Imaginary ...
7
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1answer
139 views

How to properly use Perturbation Theory in classical systems?

Context: If we consider a particle in upwards motion near the Earth's surface and acted by a quadratic drag we get the non-linear eom: $$\frac{dv}{dt}=-g-\frac{b}{m}v^2.$$ We can solve it ...
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2answers
59 views

Is there any general theorem which specifies conditions where the critical solution of an action is unique (for given boundary conditions)? [duplicate]

Consider a classical mechanical system with generalized coordinates $q_i$, $i \in \{1,\dots\,n\}$. And Lagrangian $L$. Given a path $\gamma$ (with coordinates $\gamma_i$) and two times $t_1$ and $t_2$ ...
2
votes
2answers
64 views

Bulk-to-Boundary propagator

How can I show that the bulk-to-boundary propagator $$ K(z,x;x')~=~\frac{z^{\Delta}}{[z^2+(x-x')^2]^{\Delta}} \tag{1} $$ goes as a delta function near the boundary $$ K(z,x;x')~\sim ~z^{d-\Delta}\...
10
votes
2answers
640 views

Non-separable solutions of the Schroedinger equation

I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method. If we suppose that the solutions ...
3
votes
1answer
80 views

Wave speed of a hanging rope

Let us consider a homogeneous rope hanging from the ceiling. I will call the vertical direction $x$ and the horizontal displacement $y$. When we apply the second Newton's Law to a portion of mass $\...
1
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1answer
54 views

Anomalies and determinant bundle curvature

I heard that anomalies and curvature of determinant bundle are related. Namely, curvature of determinant bundle is related to Chern-Simons form (which are involved in description of gauge anomalies). ...
2
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0answers
58 views

(Causal) Set notation round brackets vs square brackets? [closed]

In many (quite old) papers & books I have been reading recently in the causal theory of general relativity (e.g. On the structure of causal spaces, Kronheimer & Penrose, 1967) I find sets ...
2
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0answers
47 views

What is the charge density for line and surface charges?

In electrostatics it is common to see line, surface and volumetric charges being described differently. A line distribution is a function defined on the line, a surface charge distribution is a ...