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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
votes
1answer
93 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
votes
0answers
75 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 ...
13
votes
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
215 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
vote
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
155 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 ...
30
votes
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
votes
0answers
29 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
67 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 ...
9
votes
0answers
86 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
votes
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
132 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
23 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
votes
0answers
79 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
173 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
votes
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
53 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
87 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
446 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
291 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
votes
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
420 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
votes
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
51 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
votes
2answers
423 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
vote
2answers
77 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
votes
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
101 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
423 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 ...
6
votes
6answers
465 views

Why does time evolution operator have the form $U(t) = e^{-itH}$?

Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through $$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$ and ...
0
votes
1answer
63 views

Exercise book recommendations to accompany “Lectures on Quantum Mechanics for Mathematics Students” by Faddeev and Yakubovskii?

I'm currently going through "Lectures on Quantum Mechanics for Mathematics Students" by Faddeev and Yakubovskii, which is great. But there are no exercises. Is there a book you'd recommend with ...
2
votes
1answer
115 views

How to determine the trace and determinat of a differential operator?

How to determine the trace and determinant of the operator like $\Box$ or $\nabla^2$ etc. But first of all how to find the same for the simpler operator $\frac{d}{dx}$? I proceeded as follows. What ...
2
votes
0answers
44 views

Have Witten-type TQFT's nonconservation of energy and momentum in interactions?

Witten-type topological quantum field theories are based on cohomology theories. Every observable must lie in a cohomology class. May be $G$ a geometric field. Then every observable expectation value ...
3
votes
2answers
200 views

How the position operator and the position basis are correctly defined?

In Quantum Mechanics, if one deals with wave functions, the Hilbert space in question is $L^2(\mathbb{R}^n)$ for a particle in $n$-dimensions, and the position operator corresponding to the $i$-th ...
2
votes
1answer
416 views

Differentiation and delta function

Need help doing this simple differentiation. Consider 4 d Euclidean(or Minkowskian) spacetime. \begin{equation} \partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ? \end{equation} where $a_\mu$ is a constant ...
1
vote
0answers
85 views

Does Cauchy horizons in AdS have dual picture in the dual Cft?

The AdS/Cft correspondance has kindle interest in anti-de Sitter and asymptotically AdS spacetimes which are non globally hyperbolic. That means Cauchy horizon forms in these spacetimes. Moreover, ...
2
votes
0answers
34 views

Difference between vacuum and pseudovacuum vector?

What exactly is the difference between the vacuum and pseudovacuum vector? In my case the ground state of a system is the vacuum vector and by letting operators act on that vacuum vector magnons are ...
1
vote
0answers
30 views

Do the fourier components of a periodic function need to satisfy the boundary conditions imposed on it?

I cannot understand the opening statement of the second derivation of Bloch's theorem in the solid state physics book by Ashcroft and Mermin (Chapter 8, eqn 8.30). We start with the observation ...
2
votes
1answer
107 views

How to carry out the perturbation expansion of an anharmonic oscillator to high orders?

I think this is a standard problem in quantum mechanics. Consider the anharmonic oscillator $E \psi = \left(- \frac{1}{2} \frac{\partial^2}{\partial^2 x } + \frac{1}{2}x^2 + \epsilon x^4 \right) ...
2
votes
2answers
104 views

Is it possible to define the logarithm of creation/annihilation operators?

This is just a curiosity I had recently. I am going to let the reader interpret the context of the question, in their own way!
3
votes
2answers
92 views

Is there an intuitive explanation to the fact that the solutions to the time-independent Schrödinger equation form a complete basis?

We were always told that the solutions to the time-independent Schrödinger equation form a complete eigenbasis for the space of all functions (all functions?) but I never understood why this is the ...
13
votes
3answers
980 views

$\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = ...
0
votes
0answers
23 views

How to compute remnant discrete charges?

When a $U(1)$ gauge symmetry is broken by Higgs field with $U(1)$ charge $g$, we have a remnant $\mathbb{Z}_q$ symmetry. How can I compute the corresponding charges under this remnant group for ...
0
votes
0answers
33 views

Calculating Phase Behavior from the Gyration Tensor

So the spread of the eigenvalues of the gyration tensor describes the distribution of monomers inside a polymer coil. This tells me that if I look at the level-spacing distribution of the eigenvalues ...
1
vote
0answers
101 views

How to do continuum approximation?

Assume you have $N$ matrix fields $T_{j}$ on a 1d lattice with lattice constant unity. Now consider a sum like the following (you can think of the traces as supertraces), and subject it to a continuum ...