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**Me**

Here's the first part of an interview. I used it as an excuse to say what I've been doing all these years. I also talk about my uncle Albert Baez, who got me interested in physics in the first place - and what I'm working on right now.

I hope it's interesting even if you care more about math and physics. There's a lot here about quantum gravity, category theory and some of my hobbies, like the octonions. But I hope it's pretty easy to read! If you have questions, fire away.

For example: what are the octonions?

They're a number system where you can add, subtract, multiply and divide. Such number systems only exist in 1, 2, 4, and 8 dimensions: you’ve got the real numbers, which form a line, the complex numbers, which form a plane, the quaternions, which are 4-dimensional, and the octonions, which are 8-dimensional.

The octonions are the biggest, but also the weirdest. For example, multiplication of octonions violates the associative law: (xy)z is not equal to x(yz).

So the octonions sound completely crazy at first, but they turn out to have fascinating connections to string theory and other things. They’re pretty addictive, and if became a decadent wastrel I would spend even more time on them!

There’s a concept of “integer” for the octonions, and integral octonions form a lattice, a repeating pattern of points, in 8 dimensions. This is called the

**E8 lattice**. Each point in this lattice has 240 nearest neighbors, which form a beautiful shape called the

**E8 root polytope**.

The picture here is not me: it's the E8 root polytope. It's projected down to the plane, of course, and a bunch of points are directly in front of others, so you don't see 240 of them.

View 20 previous comments

- +John Baez Thank you very much!, well I have three laws(where A=exp(tW))

1)V'(Ax)=V(x)+tWV(x)+O(t^2)

2) V'(Ax) solve the differential equation Y'=WY with Y(0)=V(x)

and

3) V'(Ax)=AV(x)

I see that 2) and 3) are equivalent and that 3) implies 1),, but i need to think a little more about the role of 1) because I would like that 1) (with a little modification perhaps) implies 3)..Dec 29, 2016 - Given the rather mild assumption that V is an analytic function, 1) is equivalent to this pair of conditions

V'(Ax) = V(x) when t = 0

(d/dt) V'(Ax) = WV(x) when t = 0Dec 29, 2016 - +John Baez, Thanks! That's right, 1) is equivalent to that pair of conditions, but I see that pair doesn't imply 3) because the curve AV(x) is not the only solution of that pair of equation..Dec 30, 2016
- Indeed, equation 1) only says something interesting about what's happening at t = 0.Dec 30, 2016
- +John Baez I think 1) plus something like "V'(Ax) behave similar to a one-parameter group" would imply 3)..Dec 30, 2016
- +John Baez Hello Professor.. Happy new year!..well I could prove that the components of the Dirac spinor and Weil spinor transform as a Lorentz vector using the following "infinitesimal" proposition (which have a simple rigorous proof )

if exp(At)C_vexp(Bt)=S_uv C_u are equal at first order when t=0 then they are equal for all t. (C_u are lambda matrices or Pauli matrices)

for now that is what I wanted, Thanks for your help...Jan 6, 2017

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