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45
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5answers
4k views

Trace of a commutator is zero - but what about the commutator of $x$ and $p$?

Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about ...
15
votes
1answer
2k views

Onsager's Regression Hypothesis, Explained and Demonstrated

Onsager's 1931 regression hypothesis asserts that “…the average regression of fluctuations will obey the same laws as the corresponding macroscopic irreversible process". (Here is the links to ...
1
vote
1answer
134 views

Understanding Noether's theorem rigorously

I've known about Noether's theorem for some time and reading some things about it recently I've realised I haven't completely understood it. In that case I've been trying to understand a more rigorous ...
9
votes
1answer
142 views

Explaining causal completion axiom in Haag-Kastler axioms?

There are several variants of the Haag-Kastler axioms for algebraic quantum field theory. Usually one associates an algebra $\mathcal{A}(O)$ to each open region $O$ of spacetime. An often-suggested ...
4
votes
0answers
68 views

Uniqueness of solution in newtonian mechanics

Recently I came across the problem of Norton's dome. I thought of two questions, for which I found no answer. Does there exist a newtonian initial value problem, where the total force on each body ...
5
votes
1answer
66 views

Can I *always* decompose a normalizable function into the discrete Hydrogen spectrum?

This question has been bothering me for a while now: can one reconstruct an arbitrary (normalizable) function $\phi(\mathbf r)$ in $\mathbb R^3$, with only the discrete set of Hydrogen wavefunctions ...
8
votes
2answers
1k views

Sources to learn about Greens functions

For a physics major, what are the best books/references on Greens functions for self-studying? My mathematical background is on the level of Mathematical Methods in the physical sciences by Mary ...
3
votes
1answer
102 views

Number theoretic loophole allows alternative definition of entropy?

A bit about the post I apologize for the title. I know it sounds crazy but I could not think of an alternative one which was relevant. I know this is "wild idea" but please read the entire post. ...
1
vote
1answer
132 views

Why is the logarithm of the number of all possible states of a system differentiable?

Temperature of a system is defined as $$\left( \frac{\partial \ln(\Omega)}{ \partial E} \right)_{N, X_i} = \frac{1}{kT}$$ Where $\Omega$ is the number of all accessible states (ways) for the system. $ ...
10
votes
1answer
778 views

String theory from a mathematical point of view

I have a great interest in the area of string theory, but since I am more focused on mathematics, I was wondering if there is any book out there that covers mathematical aspects of string theory. I ...
42
votes
6answers
4k views

Why does calculus of variations work?

How does it make sense to vary the position and the velocity independently? Edit: Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
2
votes
2answers
50 views

Variant of the Sokhotski–Plemelj theorem

I am aware of the Sokhotski–Plemelj theorem (I have also heard people referring to it as the "Dirac identity") which states that in the limit $\eta\rightarrow 0^+$ $$\frac{1}{x\pm i\eta}=\mathcal ...
9
votes
3answers
899 views

Boundary layer theory in fluids learning resources

I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. ...
65
votes
7answers
7k views

Number theory in Physics [closed]

As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications ...
2
votes
1answer
68 views

Conservation of momentum in infinite square well

This is inspired by Griffiths QM section 2.2, on the infinite square well, which is about how far I've gotten (so, sorry if this is addressed later in the book). For any given starting wavefunction, ...
2
votes
1answer
67 views

Semi-infinite forms?

I am reading Vafa's paper 'Topological Mirros and Quantum Strings'. In this paper, the author says the Hilbert Space of a fermionic string theory corresponds to the space of semi-infinite forms on the ...
3
votes
1answer
29 views

Do all equillibrium points of a discrete mapping show up on the bifurcation diagram?

The question in the title is perhaps vaguely posed, so I'll include the concrete example which is bugging me. Suppose we have a mapping given by $$N_{t+1}=N_t\cdot ...
4
votes
2answers
94 views

Radial quantum number for infinite circular well

For completeness, I will sketch the solution of a particle in an infinite circular well first and then get to my question. I apologize in advance since the introduction is standard undergraduate ...
5
votes
2answers
165 views

What is the correct relativistic distribution function?

General Statement and Questions I am trying to figure out the proper way to model a velocity/momentum distribution function that is correct in the relativistic limit. I would like to determine/know ...
10
votes
1answer
459 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
0answers
38 views

Tension in string

In the derivation of the 1d linear wave equation for small amplitude waves on 1d string, they said at the end that if the density $\rho(x)$ is assumed to be constant, then the Tension $T(t)$ would ...
1
vote
0answers
29 views

Physics : Formula involving density, Mass, temperature and pressure [closed]

A set of dryers has a mass 40kg and density of 16kg/m3 at temperature of 40°C under the pressure 760mmHg. At what temperature will the mass be 50kg with density 20kg/m3 under a pressure of 700mmHg
5
votes
1answer
174 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
2
votes
1answer
71 views

What does unphyisical quantity mean? [closed]

I was reading this Terry Tao article and almost near the end he says "the terms involving infinity do not make particularly rigorous sense, but would be considered orthogonal to the application at ...
1
vote
1answer
48 views

response function and Fourier transform

A response function defined as the kernel of the following integral: $\rho(t) = \int_{-\infty}^t \chi(t,t') E(t')dt'$ (1), where $\chi(t,t')$ is the response function. Physically, it relates ...
4
votes
2answers
174 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 ...
2
votes
1answer
98 views

Derivation of Schwarzschild metric using the full machinery of differential geometry [closed]

How would one derive the Schwarzschild metric using the full machinery of differential geometry, using the component approach as little as possible? Something along these lines: Begin with a manifold ...
0
votes
0answers
39 views

About Green's function in spherical coordinates

Here is a self-contained part of the content from one paper I am currently reading. But there is one point I can't understand. Though there will be some equations, they are easy to follow. In $d$ ...
12
votes
3answers
8k views

Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
2
votes
1answer
65 views

A question about the uniqueness of Riesz representation theorem

I am sorry this question may be too math related. However, I come from physics background and I would like to ask for an physicist's explanation. As far as I know, the Riesz representation theorem ...
2
votes
1answer
123 views

Description of charged sphere with Heaviside function in cylindrical coordinates

I need to describe density of charge of uniformly charged sphere (radius R, total charge Q, position of centre (0,0,0)) with Dirac delta function and Heaviside step function. The hard part is to ...
13
votes
1answer
453 views

What is Motivic mathematics and how is it used in physics?

In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being ...
6
votes
0answers
100 views

Canonical second quantization vs canonical quantization with multisymplectic form in AQFT

First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor. 1) The usual ...
18
votes
5answers
3k views

“Velvet way” to Grassmann numbers

In my opinion, the Grassmann number "apparatus" is one of the least intuitive things in modern physics. I remember that it took a lot of effort when I was studying this. The problem was not in the ...
3
votes
4answers
794 views

Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
6
votes
2answers
344 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 ...
18
votes
1answer
1k 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 ...
4
votes
0answers
79 views

Mathematics of Surface Divergence and Surface Curl

While studying electrodynamics I found two functions - Surface Divergence and Surface Curl - that seem to condense the formulas for superficial discontinuities of the electric and magnetic fields ...
1
vote
1answer
52 views

Physic explanation of some property of heat equation? [closed]

When I compute the heat equation, I found that gross of heat do not change,it is same as our real world.But as I know , the heat equation is found according to the way of heat conduction. Why the ...
0
votes
1answer
44 views

Spectrum of Laplacian on one hemisphere

as is well-known, the spectrum of the Laplace operator on $S^2$, computed via $-\Delta f=\lambda f$, is positive and discrete. What happens to the spectrum if we just take one hemisphere into ...
0
votes
0answers
31 views

The change in time of a concentration in a fluid can be described by Reynolds' theorem. Is that the whole story?

Let $d\in\left\{2,3\right\}$ and $\Omega_t\subseteq\mathbb R^d$ be the bounded set occupied by a fluid at time $t\ge 0$. Moreover, let $\eta_t:\Omega_t\to[0,\infty)$ be the concentration of imaginary ...
0
votes
0answers
66 views

Are there applications of $L_p$ spaces in quantum mechanics?

In quantum mechanics, there a lot of emphasis on $L^2$ spaces since Hilbert spaces describe states in quantum mechanics, so we have $$ \langle \psi | \psi \rangle = \int |\psi^2(x)|\, dx$$ Even ...
3
votes
0answers
44 views

Proof that fixed points of a null field are zero

Suppose we have a scalar field $V$ (which can be acoustic pressure, or a scalar electric potential) that is a solution of the wave equation $$\Box V(x,y,z,t) = 0$$ I am wondering if a fixed ...
17
votes
1answer
325 views

What, to a physicist, are instantons and the Donaldson invariants?

I study gauge theory from a mathematical perspective. To me, one of the most fundamental ideas is the notion of an instanton on a 4-manifold. To be precise, I have a Riemannian 4-manifold and a ...
1
vote
0answers
127 views

Legendre Transformation for multiple variables

I need to show that for $F(x_1, .., x_n)$, the Legendre transformation is, $$G(s_1, ..., s_n) = \sum_{i}^{N} x_i s_i - F$$ where $$s_i = \frac{\partial F}{\partial x_i}$$ and has the property that ...
4
votes
1answer
165 views

Not satisfied with “trick” in zeta function regularization

I am not satisfied with the explanations of $$\sum_n \log \lambda_n = - \frac{d}{ds} \sum_n \lambda_n^{-s}\bigg|_{s=0}$$ "trick" used in zeta function regularization, discussed here and here, or the ...
7
votes
1answer
237 views

$L^{1}$ energy-momentum tensors in general relativity; semi-classical gravity

I was unsure whether to pose this question in a physics or mathematics forum, but it is an interesting idea I have been thinking about for some time. In any (semi-)classical field theory it is often ...
1
vote
0answers
43 views

Is time-1 map of a Hamiltonian vector field on a cylinder always twist?

I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed ...
0
votes
1answer
39 views

Angular momentum partial components of a $k$-dependent pairing potential

I am going over this review on pairing in unconventional superconductors :http://arxiv.org/abs/1305.4609v3 which on page 21 states that for a "regular" function $U(\theta)$, partial components $U_l$ ...
36
votes
0answers
1k views

Linear sigma models and integrable systems

I'm a mathematician who recently became very interested in questions related to mathematical physics but somehow, I already have difficulties in penetrating the literature... I'd highly appreciate any ...