2
votes
1answer
46 views

Mellin-Barnes (MB) integrals and hypergeometic functions

I'm trying to understand a step in arXiv:1104.2661. Equation 3.4 reads, \begin{equation} ...
1
vote
1answer
93 views

Inverse Fourier transform of Yukawa potential (troubles with Mathematica)

It can be proved that the potential $\frac{e^{-u|r|}}{|r|}$ has Fourier transform $\frac{4\pi}{u^2+q^2}$. Now, I'm trying to go backwards and do the inverse Fourier transform but I'm running into ...
5
votes
2answers
199 views

Evaluate $1$-loop contribution to the $4$-point Green's function

I am trying to evaluate the following integral \begin{equation} I = \int \frac{d^d p_\text{E}}{(2 \pi)^d} \frac{1}{(p_\text{E}^2+m^2)((q_\text{E}-p_\text{E})^2 + m^2)} \tag{1} \end{equation} where ...
0
votes
1answer
101 views

Making a cut trough a center of mass, can the masses of the pieces be equal?

Let's say point $P$ is the center of mass of an irregularly shaped object. If I make a straight cut trough point $P$ and split the object in two, is it possible for the two pieces to have the same ...
3
votes
3answers
234 views

Physical significance of getting an non-integrable function in an equation

I just found out during my Calculus course in High School, that there exist functions which cannot be integrated. Then I thought that I come across a lot of integrals while solving Physics ...
3
votes
2answers
204 views

Extension to continuous in proofs of rigid body mechanics

I'm studying rigid body mechanics and I've seen several proofs of properties related to total angular momentum, kinetic energy, etc. that all regard discrete set of points. For example, to show that ...
4
votes
2answers
129 views

Twistor Function for Coulomb Field

In an article by Penrose in Hughston and Ward "Advances in Twistor Theory", it is claimed that the twistor function $$ f(Z^\alpha) = \log{\frac{Z^1Z^2 - Z^0Z^3}{Z^2Z^3}}$$ produces an anti-self-dual ...
3
votes
2answers
240 views

Gaussian type integral with negative power of variable in integrand

How can we compute the integral $\int_{-\infty}^\infty t^n e^{-t^2/2} dt$ when $n=-1$ or $-2$? It is a problem (1.11) in Prof James Nearing's course Mathematical Tools for Physics. Can a situation ...
-1
votes
1answer
177 views

What will happen when measuring unmeasurable object?

There is a set called Vitali Set which is not Lebesgue measurable. Analogously, there also exists a Vitali set $Y$ in $\mathbb R^3$ which is a subset of $[0,1]^3$ and $|Y\cap q|=1$ for all $q\in ...
1
vote
1answer
216 views

What is the definition of density as a function?

(Before I start, I don't know which tag is suitable for this post. Please retag my post if it bothers you.) Let's say there is a string on $[0,1]$ with a mass given by $m(x)$. ($m(x)$ means the mass ...
4
votes
1answer
493 views

A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free ...
5
votes
3answers
397 views

Why we use $L_2$ Space In QM?

I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space ...
15
votes
3answers
889 views

When is Lebesgue integration useful over Riemann integration in physics?

Riemann integration is fine for physics in general because the functions dealt with tend to be differentiable and well behaved. Despite this, it's possible that Lebesque integration can be more ...
7
votes
3answers
985 views

Principal value integral

I am reading A. Zee, QFT in a nutshell, and in appendix 1 he has: Meanwhile the principal value integral is defined by: $$\int dx\,{\cal P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0} \int ...