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2
votes
1answer
80 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
56 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 ...
1
vote
0answers
47 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 $d$ ...
2
votes
1answer
117 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 ...
2
votes
1answer
73 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 ...
7
votes
1answer
150 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 ...
4
votes
2answers
132 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 ...
4
votes
0answers
143 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 ...
2
votes
1answer
59 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 ...
1
vote
1answer
145 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. $ ...
0
votes
1answer
59 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 account?...
0
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0answers
36 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
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0answers
69 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
45 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 (non-...
21
votes
1answer
395 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
54 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 ...
1
vote
1answer
47 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 $U_l$ ...
0
votes
0answers
40 views

Expansion of comparator

Currently I am working on Pesking Schroeder Section 15.1 and trying to understand the expansion given in (15.5), which is $$ U(x+\epsilon n, x) = 1 - i\,e\,\epsilon\,n^{\mu}\,A_{\mu}(x)+O(\epsilon^2) $...
13
votes
1answer
1k views

Equation of a torus

In the recent paper http://arxiv.org/abs/1509.03612, page 37. They say that a torus can be described by the equation $$y^2=x(z-x)(1-x)$$ where $x$ is a coordinate on the base $\mathbb{P}_1$. Could ...
2
votes
1answer
68 views

Maxwell Boltzmann distribution: Going from momentum to energy

I am learning about the Maxwell Boltzmann distribution, and am trying to convert the equation from momentum into energy, but I'm stuck on changing $d^n p$ into $dE$. I have the equation: $$ E=\frac{|\...
1
vote
1answer
55 views

Optics Diffraction Grating Plot

I was unsure whether to post this in physics stackexchange or mathematica stackexchange, so I posted it in both. I'm trying make an intensity plot for a diffraction grating that contains 100 lines/mm....
2
votes
1answer
122 views

In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise?

In the algebraic formulation of quantum mechanics, consider $\mathcal{B}(\mathcal{H})$ as the set of all bounded operators on $\mathcal{H}$ (with involution, norm, etc.), which form a C*-algebra $C$. ...
13
votes
4answers
927 views

Can momentum have a complex expectation value?

I'm making examples of wave functions to incorporate in a QM exam. I came up with the following wave function, which gives me some troubles: $$\psi(x,0) = \begin{cases} A(a-x), & -a \leq x \leq a\...
1
vote
0answers
76 views

Experiments to resolve dillema between continuity and discrete

Which experiments/experimental methods are suggested to resolve an alternative about the structure of our universe space and time - is it continuous or is it discrete in a very small scale, especially ...
2
votes
1answer
48 views

Time dependent electric field: Mathematical expansion for local electric field

In many articles and books I see that local electric field is expanded as $$\vec E_0(\vec r(t)) = \vec E_0(\vec R_0) − (\vec a(t) \cdot \nabla) \vec E_0(\vec R_0) \cos(\Omega t) + \ldots $$ For ...
1
vote
1answer
204 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 ...
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 } \int_{-\tau/2}^{\tau/2}...
1
vote
0answers
144 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
vote
0answers
95 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
votes
0answers
91 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 ...
9
votes
1answer
150 views

Resources for algebraic topology in condensed matter physics

I wanted to know if anyone had any good introductions on algebraic topology for the theoretical physicist? I am particularly interested in applications to condensed matter physics, but would be happy ...
1
vote
1answer
59 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 ...
3
votes
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 ...
5
votes
2answers
270 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 ...
2
votes
1answer
50 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 ...
1
vote
0answers
73 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
votes
0answers
50 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
votes
0answers
115 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 ...
4
votes
1answer
102 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, ...
3
votes
0answers
80 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 ...
5
votes
1answer
118 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, ...
2
votes
2answers
104 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 $G$ ...
0
votes
0answers
166 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 ...
0
votes
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
votes
0answers
109 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, ...
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 ...
2
votes
1answer
154 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
82 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 ...
12
votes
1answer
194 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 ...