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1answer
102 views

Can binary sequences generated from ergodic maps be chaotic?

Briefly, the way symbols are generated is: Consider a one-dimensional chaotic map $T: [0,1]→[0,1]$ and a time series $\{x_n\}_{n=1}^N$ generated with this map. Define a threshold $A$ and a ...
2
votes
1answer
78 views

How do we take the limit of this quantum operation?

I am wondering how to take the following limit: \begin{align} L= \lim_{\tau \to \infty} \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} dy \, \left(1 - \frac{1}{\sqrt{ \pi} \sigma } ...
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0answers
133 views

Cones with deficit angle 2$\pi$ and euler characteristics

I've managed to confuse myself with cones and deficit angles. Let's consider a conical defect in 2 dimensions. So the metric is the usual one in polar coordinates, $$ ds^2 = dr^2 + r^2 d\phi^2,$$ ...
1
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0answers
90 views

What are the implications of integrating the Poisson bracket? [closed]

Reading Ref. 1 I admit I am a little lost in some places. I was hoping that someone in this area could explain the basic premise of integrating the Lie bracket and further is it connected to nlab's ...
4
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0answers
90 views

Gromov-Witten invariants

I'm a mathematician studying Schubert calculus, and I'm out to compute the Gromov-Witten invariants of the complete flag manifold. Well, I actually already know how to compute them, but only in a way ...
1
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1answer
56 views

Are there Non-conformal maps encountered in Physics?

We always encounter Conformal maps in Physics, may be they are easier to study, but are there Non-Conformal transformations encountered in Physics anywhere? if they are encountered, where are they ...
2
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1answer
49 views

which is correct way to find error in degree of linear polarization?

I have a set of "degree of polarization (DOP)" values for a star. Assume there are 10 DOP values in the set. DOP is defined as square root of sum of squares of fractional Stokes parameters, namely q ...
7
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5answers
742 views

Curvature of Conical spacetime

Inspired by: Angular deficit The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward ...
3
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0answers
37 views

Action for solution of general nth order differential equation [duplicate]

Suppose I want to find solution to a general nth order differential equation. (If I am right about the logic then) one might say that the solution $y\equiv y(x)$ is that function for which the ...
34
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6answers
895 views

Motion described by $m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12 }x}{\mathrm{d}t^{\frac12}}$

What kind of motion would a (preferably dimensionless for simplicity) body do if the force acted on it was proportional to the semi-derivative of displacement, i.e. $$m \frac{\mathrm{d}^2 ...
5
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1answer
103 views

Renormalization and Conway/Surreal Numbers

In the final chapter of his book "An Interpretive Introduction to Quantum Field Theory", Paul Teller writes about three interpretations of renormalization in quantum field theory. In particular, ...
35
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11answers
2k views

Examples of number theory showing up in physics [duplicate]

Are there any interesting examples of number theory showing up unexpectedly in physics? This probably sounds like rather strange question, or rather like one of the trivial to ask but often unhelpful ...
2
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2answers
96 views

Orbifold with discrete torsion

I'm trying to understand some of the early works of Vafa and Witten [1-3]. The way I look at orbifolds is they are the quotient space $M/G$. This is simply a quotient manifold when the action of $G$ ...
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0answers
72 views

Signs of Grandi's series $1-1+1-1+1-1+1+\ldots$ in the real world?

I'm talking about the convergence of the series $1-1+1-1+1-1+1+\ldots$ to $1/2$. I was discussing with some friends (we study physics) and I argued that Cesaro summation is a fair extension of the ...
2
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0answers
48 views

Some mathematical questions around lattice QFT [closed]

I have three quite mathematical questions in modern QFT. 1) Why it's supposed that N=2 SUSY Yang-Mills probably cannot be put on a lattice? 2) What is the recent status of lattice approach to ...
3
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0answers
114 views

From Newton to Kepler without infinitesimals

I've read some interesting calculus-free proofs of at least parts of the derivation of Kepler's Laws from Newton's gravitational force. One is of course Feyman's "Lost Lecture" (which was already ...
3
votes
2answers
180 views

Infinitesimal input, macroscopic output

I must admit that I never got well how physicists handle infinitesimal quantities, mainly because of my education as a mathematician. So the following lines (taken from the preface of Berezin and ...
4
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1answer
97 views

Closure relation for degenerate eigenkets

Consider an observable in quantum mechanics, with a degenerate eigenvalue in a continuous spectrum. Is it possible for such an eigenvalue to have a finite degeneracy? If the degeneracy is infinite, ...
4
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0answers
76 views

Underlying C*Algebra operators in standard quantum mechanics?

Linearity in standard quantum mechanics (QM) is the key to making the math possible in this field, but the presence of nonlinear operators in QM is what is more generally dealt with. Working with the ...
15
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5answers
6k views

Fourier transform of the Coulomb potential

When trying to find the Fourier transform of the Coulomb potential $$V(\mathbf{r})=-\frac{e^2}{r}$$ one is faced with the problem that the resulting integral is divergent. Usually, it is then argued ...
2
votes
2answers
223 views

Detailed conditions for symmetries of Lagrangian

Edit: To clarify the question, I am asking why we are justified in calling a continuous symmetry a symmetry of a system when it changes the Lagrangian by a total derivative of a function of $t, q(t)$ ...
1
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1answer
87 views

Error in Kac's “Vertex algebra for beginners” proof that a Wightman QFT gives rise to a vertex algebra?

Given $$ i[Q_k,\Phi_a(x)]=((x_0^2-x_1^2)\partial_{x_k}-2\eta_k x_k E - 2\Delta_a \eta_k x_k) \Phi_a(x), \quad (1.1.8) $$ applying a coordinate change $t= x_0-x_1$, $\bar{t}= x_0+x_1$ and defining $$ ...
0
votes
0answers
160 views

Alternative expression for Bethe vectors of su(2) XXX chain

As is well known, the eigenvectors of the transfer matrix for the su(2) XXX spin chain of length $L$ are given by acting on the ground state $|\uparrow^L\rangle$ with all spins point up with the ...
7
votes
1answer
117 views

Constructing differential equation from arbitrary Hamiltonian

Suppose I begin with the time-independent Schrodinger equation $$ \left(-\frac{1}{2m}\partial_x^2 + V(x)\right)\psi_n(x) = E_n\psi_n(x), $$ ordinarily we specify the function $V$ and then solve for a ...
31
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8answers
5k views

Classical mechanics without coordinates book

I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ...
0
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0answers
30 views

Determine the pressure difference required to drive a prescribed constant volume flux $Q$ through a gap

Determine the pressure difference $P_{1}-P_{2}$ required to drive a prescribed constant volume flux $Q$ per unit width through a gap of thickness $\delta$ of length $L$. To do this introduce ...
0
votes
1answer
69 views

Transformations of states in quantum mechanics

In Classical Mechanics we usually describe the possible configurations of a system by points on a smooth manifold $M$ which is the configuration manifold of the system. In that case, when we talk ...
13
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0answers
96 views

Can one classify partial differential equations according to the causality properties of their solutions (and if yes, then how)?

Recently, I bumped into this interesting comment by Valter Moretti which made me wonder about the following, more general question (to which I suspect the answer is affirmative): Can we easily tell, ...
4
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1answer
176 views

On the definition of hyperbolic conservation law

There is a Wikipedia article and an article by A. Bressan that introduce the notion of hyperbolic conservation law. I'm not used to this area, so I have some questions about the definition. I will ...
2
votes
1answer
137 views

What's the difference between classical and quantum vector superposition?

$(1)$Since quantum-mechanical states between two consecutive measurements are represented as superposition of orthonormal basis vectors in a vector space, at first glance it seems like it makes sense ...
0
votes
0answers
25 views

How to determine time of dephasing?

Let's assume that I have an oscillating value A. After some time the oscillations are being damped so the diagram of A is like on the picture below: Now how to determine when does the A is reduced ...
0
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0answers
81 views

Topology of Anti-de Sitter manifold with black hole

I'm interested in understanding the topology of space-time with a black hole. In other words how does having a black hole affect quantities such as the fundamental group, de-Rham cohomologies, Euler ...
6
votes
3answers
178 views

Necessary and sufficient conditions for a function to be the Wigner function of state

For any quantum state defined with a continuous position, the Wigner function is a quasiprobability distribution on phase space. It has many properties, such as that its marginal are probability ...
74
votes
12answers
24k views

Best books for mathematical background?

What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory? Some subjects off the top of my head that probably need covering: ...
32
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6answers
2k views

In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?

A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential. Is there a one to one correspondence between the potential and its spectrum? If the ...
8
votes
3answers
1k views

Group Cohomology and Topological Field Theories

I have a two-part question: First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories". Here they give a general definition for ...
0
votes
1answer
55 views

Is hysteresis essential for a memory system or material?

I want to know whether its essential for a memory system or material to have hysteresis between two of its variable ? If Yes, what can be the general relation of volatility of a memory with its ...
1
vote
0answers
88 views

Minkowski metric and Null tetrad metric

I'm starting with the Newman-Penrose formalism and have a very basic question that I'm very confused about. The standard Minkoswki metric is $\eta_{ab}=\mathrm{diag}(-1,1,1,1)$. Is then the null ...
9
votes
2answers
461 views

What is the algebraic property that corresponds to a topological term?

Warning: This question will be fairly ill-posed. I have spent a lot of time trying to make it better posed without success, so please bear with me. A single $SU(2)$ spin may be represented by the ...
6
votes
1answer
296 views

Must bounded operators have normalisable eigenfunctions and discrete eigenvalues?

When we have bound states, to my knowledge, we have states that are normalisable and a discrete energy spectrum. However, in the case of scattering states that have a continuous energy spectrum, the ...
30
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5answers
3k views

Is the converse of Noether's first theorem true: Every conservation law has a symmetry?

Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Is the converse true: Any conservation law of a physical ...
19
votes
2answers
422 views

Geometric picture behind quantum expanders

A $(d,\lambda)$-quantum expander is a distribution $\nu$ over the unitary group $\mathcal{U}(d)$ with the property that: a) $|\mathrm{supp} \ \nu| =d$, b) $\Vert \mathbb{E}_{U \sim \nu} U \otimes ...
8
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3answers
1k views

What really is a Dirac delta function?

Yesterday a friend asked me what a Dirac delta function really is. I tried to explain it but eventually confused myself. It seems that a Dirac delta is defined as a function that satisfies these ...
51
votes
10answers
5k views

Is there something similar to Noether's theorem for discrete symmetries?

Noether's theorem states that, for every continuous symmetry of an action, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any ...
2
votes
0answers
53 views

Simulation of oscillator with frequency dependent damping

What would be the equation for the frequency dependent damping of harmonic oscillator? Is there something like: $$ \ddot{x}+2\delta\dot{x}+\omega_0^2x = \frac{F}{m}f(t) $$ with frequency dependent ...
5
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2answers
432 views

Resources for theory of distributions (generalized functions) for physicists

I am looking for tutorials, articles or books containing theory of distributions in context of mathematical physics. Please suggest.
1
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2answers
78 views

Configuration manifold of a rigid body

As I know, a rigid body is a set of $N$ particles in three-dimensional space subject to the following constraint: if $b_1,\dots,b_N\in \mathbb{R}^3$ are the initial positions of the particles and if ...
11
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3answers
5k views

Integral of the product of three spherical harmonics

Does anyone know how to derive the following identity for the integral of the product of three spherical harmonics?: \begin{align}\int_0^{2\pi}\int_0^\pi ...
3
votes
0answers
102 views

Dirac Delta in definition of Green function

For a inhomogeneous differential equation of the following form $$\hat{L}u(x) = \rho(x)$$ solution can be written in terms of the Green function $$u(x) = \int dx' G(x;x')\rho(x')$$ such that ...
3
votes
3answers
428 views

Hilbert space of a quantum system

The first postulate of Quantum Mechanics as I've learned can be stated as: The states of a quantum system can be described by vectors in a Hilbert space. I've seem also some people also ...