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3
votes
1answer
109 views

Is the Legendre transformation a unique choice in analytical mechanics?

Consider a Lagrangian $L(q_i, \dot{q_i}, t) = T - V$, for kinetic energy $T$ and generalized potential $V$, on a set of $n$ independent generalized coordinates $\{q_i\}$. Assuming the system is ...
3
votes
1answer
165 views

Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation?

I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its ...
1
vote
0answers
37 views

The fate of Poincaré recurrence with the Big Rip

Recently, there has been a lot of talk in the media about the "Big Rip". It most certainly resulted from the paper by Marcelo M. Disconzi and Thomas W. Kephart where they have figured out a ...
10
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10answers
3k views

Physics for mathematicians

How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages ...
3
votes
1answer
69 views

Can the math for physics be expressed without any uncountable sets at all?

I am wondering about this and have wondered about it for a short while. Usually physics is modeled using things based off of the Real Number Line $\mathbb{R}$, which is uncountable. (E.g. we may use ...
4
votes
0answers
104 views

Why the field strength tensor for locomotion at low Reynolds number may be written like that?

I've been studying locomotion at low Reynolds number for some time now and it has been a quite tough problem. I've already asked two questions about the problem here, and now there is this question ...
0
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0answers
38 views

Is there a physical interpretation of the alternating property?

A map from a vector-space to its base field is called "alternating" if each vector with repeated elements is mapped to zero. I've read that symplectic geometry is an important representation of ...
1
vote
0answers
38 views

Is the Cauchy Horizon of Anti deSitter spacetime stable?

The Cosmic Censorship Conjecture are two mathematical conjectures about the structure of spacetime. In particular the so called Strong Cosmic Conjecture asserts heuristically that generically, ...
1
vote
1answer
50 views

Gauge freedom in tetrad

I asked the question in the MathOverflow, but didn't get any response. I thought maybe better luck here. I'm reading the following paper about Petrov type D space times called "Type D vacuum ...
3
votes
2answers
888 views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
0
votes
0answers
40 views

Extension of vector field and fluid velocity

I've been studying locomotion at low Reynolds number with gauge theories reading this paper and on pages 567 and 568 we find the explanation on how to compute the field strength tensor. For simplicity ...
5
votes
1answer
159 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 ...
1
vote
1answer
60 views

What is the required differentiability of the solutions to do QFT?

Given a real scalar field satisfying: $$P\psi=(\square_{g}+m^{2})\psi=0$$ on a globally hyperbolic spacetime ($M,g_{ab}$). One can construct a $C^{*}$-algebra $A(M,g)$ ("the minimal algebra") which ...
2
votes
1answer
56 views

Separability of the Hilbert space: countable orthonormal basis vs. continuous spectrum

Hilbert spaces are mostly assumed to be separable. A Hilbert space is separable if and only if it admits a countable orthonormal basis. How does this fit together with the possible existence of the ...
0
votes
0answers
28 views

Can anybody recommend a book on quantum physics for a mathematically minded student? [duplicate]

Basically what the title says. I'm a math/physics double major going into my second year, but I'm much more math oriented. I'll be taking a quantum class in the fall and I'd like a book that ...
4
votes
1answer
272 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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0answers
45 views

Quantized Banach Spaces

Reading the paper here, it mentions on the very first page that "The requirement of 'closed'-ness is imposed because we want to think of operator spaces as 'quantized (or non-commutative) Banach ...
0
votes
0answers
39 views

Regarding Ampere's Circuital Law

If I am to show that Ampere's Circuital law holds true for any arbitrary closed loop in a plane normal to the straight wire, with its validity already established for the closed loop being a circle of ...
4
votes
1answer
919 views

Representation of operators in quantum mechanics

For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?
7
votes
2answers
607 views

Vector Potential for Magnetic field when the field is not in simply-connected region

According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some ...
5
votes
1answer
100 views

Physical implications of the Gibbs phenomenon for Quantum Mechanics

From Wikipedia: The Gibbs Phenomenon is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth ...
3
votes
2answers
201 views

Why there is no Gibb's phenomenon in QM?

Why we don't see any Gibb's phenomenon in quantum mechanics? EDIT At sharp edges (discontinuities), we usually find ringing. This can be observed in many physical phenomenon (eg. shock waves). ...
8
votes
1answer
380 views

Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
0
votes
1answer
79 views

Field theory in four dimensions

I was reading Schwartz's book on QFT. In chapter 14.5 at p.267, while speaking about path integral he says: [...] the path integral (and field theories more generally) is only known to exist (i.e. ...
0
votes
1answer
84 views

Path dependent phase in quantum mechanics

In elementary treatments of quantum mechanics, we are taught that the wavefunction of a single particle is complex valued ($\Psi : \mathbb{R}^3 \to \mathbb{C}$). In particular, the wavefunction has a ...
5
votes
0answers
108 views

Gauge potential for locomotion at low Reynolds number

I've been studying some approaches with gauge theory to some problems in Mechanics and I've found the problem of self propulsion at low Reynolds number a quite complicated one. The approach I'm asking ...
5
votes
3answers
327 views

Is there a natural (suitable) definition for functional derivative in Curved space time

If $$\delta S = \int \sqrt g F[\phi] \delta \phi\tag{1}$$ Then is it natural to define the functional derivative as follows, $$\frac{\delta S}{\delta \phi} = F[\phi].\tag{2}$$ In particular does ...
4
votes
1answer
70 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 ...
3
votes
2answers
54 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 ...
1
vote
0answers
14 views

Creating an arbitrary state of the quantum simple harmonic oscillator [duplicate]

Suppose $\mathcal{B}=\{\lvert 0\rangle, \lvert 1\rangle, \lvert 2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} ...
12
votes
4answers
903 views

What happened with Hilbert's sixth problem (the axiomatization of physics) after Gödel's work?

I'll write the question but I'm not fully confident of the premises I'm making here. I'm sorry if my proposal is too silly. Hilbert's sixth problem consisted roughly about finding axioms for physics ...
0
votes
0answers
35 views

How to calculate the eigenenergies of a particle in a triangular billiard?

Suppose we take the Dirichlet boundary condition, namely the wave function must vanish on the boundary. How about a general n-polygon?
0
votes
0answers
50 views

How to solve Laplace equation in a domain with one boundary along a curve?

Is there a way to solve the 2D Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =0$ on $0 <x<\infty$ and $0 < y < \infty$, such that the domain is ...
10
votes
5answers
482 views

Motivation for tensor product in Physics

This question is about a mathematical object (the tensor product) but thinking about the motivation that comes from Physics. Algebraists motivate the tensor product like that: "given $k$ vector spaces ...
2
votes
2answers
103 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 $t, q(t)$ ...
3
votes
2answers
140 views

Consequences of compactness in physics

If we understand spacetime as a $4$-dimensional manifold $M$, from the point of view of physics what are the consquences of a subset of it being compact? My point here is simple: in math we usually ...
2
votes
1answer
177 views

Floquet quasienergy spectrum, continuous or discrete?

I haven't got a feeling about Floquet quasienergy, although it is talked by many people these days. Floquet theorem: Consider a Hamiltonian which is time periodic $H(t)=H(t+\tau)$. The Floquet ...
5
votes
1answer
220 views

Symplectic leaves, tori and Poisson manifolds

For classical systems we can define a configuration manifold, whose cotangent bundle is a momentum phase space equipped with a closed, non-degenerate 2-form. Upon the commutative algebra of smooth ...
0
votes
0answers
23 views

Integral of absolute value of spin angular momentum of $N$-body system

There are $N$ particles moving freely in a plane. Let $J(t)$ be the spin angular momentum of the system of particles about its center of mass. (even center of mass keeps changing with time as ...
6
votes
1answer
259 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group?

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
15
votes
1answer
968 views

Motivation for the use of Tsallis entropy

Every now and again I hear something about Tsallis entropy, $$ S_q(\{p_i\}) = \frac{1}{q-1}\left( 1- \sum_i p_i^q \right), \tag{1} $$ and I decided to finally get around to investigating it. I haven't ...
2
votes
1answer
81 views

Learning physics from a strong math background? [closed]

I have a master's degree in operations research and took several PhD level math courses. In particular I'm most familiar with analysis/measure theory, probability theory, and stochastic processes, and ...
2
votes
1answer
65 views

How is Green function in many-body theory introduced?

Normally, for a (linear) operator $L$ and a DE $$ Lu(x) = f(x) $$ the Green function is defined as $$ LG(x,s) = \delta(x-s) $$ and it is found that $$ u(x) = \int G(x,s) f(s) ds $$ is the ...
2
votes
1answer
60 views

Distributional Extension of a Hilbert Space

This question comes from the Complexification section of Thomas Thiemann's Modern Canonical Quantum General Relativity, but I believe it just deals with the foundations of quantum mechanics. The ...
0
votes
0answers
33 views

Energy Oscillations in a One Dimensional Crystal?

Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially interested in ...
1
vote
1answer
116 views

Distributions (generalized functions) over manifolds

I have asked a similar question on the math stackexchange website, but since this type of question might have an answer that is known to physicists better than mathematicians I'm posting the question ...
0
votes
1answer
213 views

What are the boundary conditions for EM waves normally incident on the interface between two dielectric media?

An EM wave, amplitude $E_0$, frequency $\omega_0$, is incident upon a material with relative permittivity (dielectric function) $$\varepsilon \left( z \right) = \left\{ \begin{gathered}{\varepsilon ...
5
votes
2answers
521 views

Teach me Wick's theorem the honest way

Generally speaking the average guy marginally acquainted with quantum field theory or advanced combinatorics describes Wick's theorem as some sort of correspondence between higher order differential ...
13
votes
3answers
721 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}} = ...
2
votes
2answers
60 views

Commuting observables and CSCO's

I've been looking at some basic quantum mechanics all day in an attempt to better my understanding of the subject. While going over the proof that commuting operators are compatible, I started getting ...