### 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.

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

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.

## 3 Comments:

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

www.icms.org.uk/downloads/GandP/Vafa.pdf

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

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

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.

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