The
beta function $B(a,b) = \frac{\Gamma (a) \Gamma (b)}{\Gamma (a + b)}$
has an integral representation, which can be found by using polar coordinates in the expression
$m! n! = \int_{0}^{2 \pi} \int_{0}^{\infty} e^{- r^2} | r \textrm{cos} \theta |^{2m + 1} | r \textrm{sin} \theta |^{2n + 1} r \textrm{d} r \textrm{d} \theta$
and pulling out the $r$ factors to obtain
$2 (m + n + 1)! \int_{0}^{\frac{\pi}{2}} \textrm{cos}^{2m+1} \theta \textrm{sin}^{2n+1} \theta \textrm{d} \theta$.
It could be fun to play around with higher dimensional analogues of such integrals using, for instance,
Euler angles. In a series of
papers,
Kholodenko looks at multidimensional analogues, and he concludes that Veneziano-like amplitudes are capable of reproducing spectra for both open and closed strings. This involves a study of amplitudes using period integrals for Fermat hypersurfaces. On an historial note, he mentions the original paper of Chowla and Selberg on elliptic integrals. Recall that Chowla is the guy who introduced Montgomery to Dyson
at tea.
The Veneziano amplitude takes the form
$V(s,t,u) = V(s,t) + V(s,u) + V(t,u)$
for $V(a,b) = B (- \alpha (a) , - \alpha (b))$. When the condition
$\alpha (s) + \alpha (t) + \alpha (u) = -2$
holds, the amplitude takes the form
$V(s,t,u) = \frac{\zeta (1 + 0.5 \alpha (s)) \zeta (1 + 0.5 \alpha (t)) \zeta (1 + 0.5 \alpha (u))}{\zeta (- 0.5 \alpha (s)) \zeta (- 0.5 \alpha (t)) \zeta (- 0.5 \alpha (u))}$.
Kholodenko discusses the generalised condition
$\alpha (s) m + \alpha (t) n + \alpha (u) l + k = 0$
where the Veneziano case is recovered when $k$ is associated to the degree $N$ of the Fermat surface, while the Shapiro-Virasoro condition arises for degree $2N$.
More recently, Kholodenko has been
looking at honeycombs, which
were used by
Terence Tao and
Allen Knutson to solve the long outstanding
Horn conjecture on the spectra of $n \times n$ Hermitean matrices.
We see that an observer at Now looks back to the past, and the infinite future looks back to Now. How much simpler can it be?
We see that an observer at Now looks back to the past, and the infinite future looks back to Now. How much simpler can it be?
