occasional meanderings in physics' brave new world

## Sunday, April 26, 2009

### M Theory Lesson 273

Abtruse Goose tells us that for a matrix $A$, the exponential satisfies

$e^{F^{-1} A F} = F^{-1} e^{A} F$

We can easily apply this to Koide matrices, which are diagonalised by the Fourier transform matrix $F_3$. It follows that a Koide matrix is an exponential of the matrix The first entry of the circulant $A$ is $\textrm{log} (\sqrt{m_1 m_2 m_3})$. Expressed in terms of the natural scale

$\mu = 25.054309435 \sqrt{\textrm{MeV}}$,

the charged lepton case takes the value

$\textrm{log} (\sqrt{2} + \textrm{cos}(\frac{2}{9}))(\sqrt{2} + \textrm{cos}(\frac{2}{9} + \frac{2 \pi}{3}))(\sqrt{2} + \textrm{cos}(\frac{2}{9} - \frac{2 \pi}{3}))$

which gives us more crazy numbers to play with! The ease of swapping addition for multiplication in the circulant Fourier transform is a sign that the Fourier transform might have something to do with basic arithmetic. Tony Smith said...

Kea, about matrices such as Koide,
have you seen Vafa's contribution to the Atiyah 80th birthday conference ?

The pdf file (54 slides) is at
and
slides 48 and 51
show a 3-generation mass matrix

mu mc mt
md ms mb
me mmu mtau

as

1 e e^2
e e^2 e^3
e^2 e^3 e^4

where e = 0.04 = alpha_GUT
and
m1 : m2 : m3 = alpha_GUT^4 : alpha_GUT^2 : 1

Maybe it might be related to arxiv 0904.3101
or maybe equations 6.25 and 7.1 of 0904.1419
but still
I don't understand what Vafa is doing,
so
I wondered if what he is doing with matrices is close enough to what you are doing that you could explain it ?

A physical question I have is:
it seems to me that Vafa is using GUT physics,
so
does that mean that he believes that GUT theory
is NOT ruled out by experimental observations such as Kamiokande etc. ?

Tony

April 29, 2009 5:07 PM Kea said...

Wow, Tony, that's interesting. Let me take a look...

April 29, 2009 7:47 PM Kea said...

OK, I had a look at the slides. Tony, to me it looks like a pile of G-string waffle about simple symmetric matrices, followed on slide 51 by the statement that the hierarchy should be m1:m2:m3 = a^4:a^2:a^1, which you know we've been talking about for a while without invoking GUTs.

April 29, 2009 8:28 PM