1
vote
2answers
249 views

decomposition of Hilbert space into tensor product $L^2([0,\tfrac{1}{2}]) \otimes L^2([\tfrac{1}{2},1]) \simeq L^2([0,1])$

The definition of entanglement entropy in Quantum Field Theory involves decompositing a Hilbert space into a tensor product $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$. As an example, is it ...
1
vote
0answers
183 views

An integral with Gamma functions (Part 2)

I was wondering if there is a generalization of the integral discussed here to a case like, \begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ...
3
votes
1answer
143 views

Using Fourier Transform to speed up calculation of forces following an inverse square law

Suppose I have $n$ electric point charges in, say, two dimensions. Is there any algorithm (and I have a hunch that it might be related to the Fourier transform) to compute the net forces that act on ...
11
votes
3answers
1k views

Fourier transforms of functions not in $L^2.$

This is probably something five-year-old physicists know, but here goes: Is there a standard methodology for computing Fourier transforms of things like $\log |x|$? Wolfram Alpha will happily give an ...
0
votes
2answers
551 views

Do you know this form of an uncertainty principle?

I hope this question is focused enough - it's not about real problem I have, but to find out if anyone knows about a similar thing. You probably know the Heisenberg uncertainty principle: For any ...
2
votes
1answer
729 views

fourier transform on an interval?

Hello, May I ask how to define the fourier transform of a function that is not defined on the whole real line? For example, what is the fourier transform of $\frac{1}{\sqrt{x}}$? And what is the ...
0
votes
2answers
1k views

fundamental solution of radial wave equation

i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
1
vote
3answers
544 views

Fourier Transforms restricted to mass shell

Hello, I am stuck with the following (hopefully not too trivial) problem. I want to know, if the map $${\cal D}(\mathbb{R}^2)\to L^2(H_m,d\Omega_m)\qquad f \mapsto \hat{f}|_{H_m}$$ has dense range. ...
5
votes
2answers
2k views

Can I relate the L1 norm of a function to its Fourier expansion?

I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
2
votes
2answers
637 views

Constraints on the Fourier transform of a constant modulus function

Considering the function $f:\mathbb{R} \to \mathbb{C}$, with $\left| f(x) \right|=1$ for all $x\in \mathbb{R}$. Considering $g:\mathbb{R} \to \mathbb{C}$ with ...