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0
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1answer
38 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
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0answers
74 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
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2answers
423 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
256 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 ...
28
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5answers
2k 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 ...
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2answers
416 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 ...
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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 ...
49
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
1answer
88 views

Continuous basis of eigenvectors

In Quantum Mechanics, when using Dirac's formalism one of its features is the expansion of state vectors into continuous basis of eigenvectors of unbounded self-adjoint operators. Let $\mathcal{H}$ be ...
2
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0answers
46 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
373 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.
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2answers
69 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 ...
10
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3answers
4k 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
97 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
370 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 ...
5
votes
6answers
417 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
57 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
87 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
41 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
139 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
415 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
81 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
33 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
29 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
103 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
96 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
80 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
931 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
22 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
29 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
97 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 ...
1
vote
0answers
35 views

Proof of the nonintegrability of the Henon-Heiles problem?

The Henon-Heiles potential is $$ U(x,y ) = \frac{1}{2} (x^2 + y^2 + 2 x^2 y - \frac{2}{3} y^3) .$$ This is a two degree-of-freedom system. The full Hamiltonian is $$ H = p_x^2 + p_y^2 + U(x,y ) ...
6
votes
1answer
191 views

Functional derivatives as distributions

I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ...
6
votes
1answer
260 views

Why are the quantum observables defined on opens sets a presheaf and not a sheaf?

In local quantum field theory or AQFT one can mathematically describe over each open set $U$ of a spacetime $M$ the quantum states or observables of the theory. This structure is commonly referred as ...
2
votes
1answer
89 views

How to prove that sum converges to integral using density of states?

Essentially, I would like to prove $$ \sum_k f(k) \to \int f(k) \rho dE \tag{1}$$ where $$ \rho = \frac{dk}{dE} \tag{2}$$ is the density of states and $k \to \infty$. The model is that there is a ...
50
votes
2answers
3k views

Intuitively, why are bundles so important in Physics?

I've seem the notion of bundles, fiber bundles, connections on bundles and so on being used in many different places on Physics. Now, in mathematics a bundle is introduced to generalize the ...
0
votes
2answers
82 views

Examples of measure zero sets in physics [closed]

The problem with Riemann integration of functions like ${\chi }_{ℚ}(x)$ (a function which takes the value of 1 if $x\in ℚ$, and 0 otherwise) is usually used to show the need for defining the Lebesgue ...
2
votes
1answer
99 views

Intuition about Momentum Maps

I'm studying Classical Mechanics and there is one object that appeared recently on the book I'm not being able to get a physical intuition about it. The mathematical definition goes as follows: Let ...
4
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0answers
95 views

Free probability in Physics

Recently I have started reading some materials on non-commutative probability. IN this area mathematicians sometimes consider quantum theory as a non-commutative version of classical probability, with ...
5
votes
1answer
102 views

State space of interacting theories

Haag's theorem states that in general, an interacting quantum field and the corresponding free field have unitary-inequivalent state space representations. I would like to have an example of a state ...
0
votes
1answer
81 views

Lippmann-Schwinger equation and $T$ expansion

Lippmann-Schwinger equation, in operator form, is: $$ T=V+V\frac{1} {E-H_0+i \hbar \varepsilon} T=:V+V\Theta_0T, $$ where $H_{tot}=H_0+{V}$ is the hamiltonian ($H_0$ is the free particle hamiltonian ...
0
votes
0answers
58 views

How does the functional Bethe Ansatz work?

This question is a bit general and less specific. I've understood the algebraic Bethe Ansatz so far, but since we have a completeness problem Sklyanin suggested a different method, the so called ...
2
votes
0answers
63 views

Flow between two infinite plates [closed]

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
0
votes
1answer
76 views

How to deal with eigenvectors which are not square integrable?

In Quantum Mechanics there is one type of situation I'm still unsure on how to deal with. First of all, I want to make clear I'm trying to understand how to deal with this rigorously. What I'm talking ...
5
votes
1answer
84 views

Can we reconstruct 1D potentials in QM from the spectrum? [duplicate]

Knowing the potential, we can find the spectrum of the Schrödinger operator. The converse question is: Knowing the spectrum, can we reconstruct the potential? As an example, a harmonic potential has ...
2
votes
0answers
49 views

What does complexification mean for our particles in physics

As gauge group let's consider the popular $SO(10)$ group. The fundamental representation $\pi$ of the corresponding Lie algebra $\mathfrak{so}(10)$ is $10$ dimensional $$ \pi: \mathfrak{so}(10) ...
5
votes
2answers
90 views

Conservation Laws and Symmtery

The toughest of topics in physics, like Quantum Mechanics, Relativity, String theory, can be explained in layman words and many have done so. Though there is no substitute to the understanding a ...
1
vote
1answer
160 views

How does height of a parachute affect air resistance compared to circumference or diameter?

I'm trying to find out how much a double in height (making it more ovular or oblong in shape) of the parachute affects air resistance compared to a double in circumference or diameter. Can someone ...
1
vote
0answers
59 views

Wightman axioms always imply triviality in 4D?

Someone mentioned to me in passing that it had been proven that the Wightman axioms are over-restrictive in four dimensions and provably always result in trivial correlators (or maybe a trivial ...
2
votes
1answer
150 views

Divergence Theorem, mathematical approach to Gauss's Law?

Let $D$ be a compact region in $\mathbb{R}^3$ with a smooth boundary $S$. Assume $0 \in \text{Int}(D)$. If an electric charge of magnitude $q$ is placed at $0$, the resulting force field is ...