Recall that Leinster’s Euler characteristic for 1-categories could take on rational values. But we need more numbers than that. How might numbers be extended into the complex plane? We suspect that at some point higher categories will be involved.

For a finite directed graph, $\chi$ is simply given by $V – E$. For example, for the five arrow graph

we have that $\chi = -1$. Observe that this is the simplest way to obtain an Euler characteristic of -1 using circuit free graphs. An extra arrow is needed to make the diagram a category. What would it mean to take a square root of this diagram? Numbers must be represented by diagrams, so we must ask this question. One guess would be to call a triangle, with a 2-arrow in the interior, the number $i$. Let’s think about it. Everybody remembers Euler’s relation

$e^{i \pi} = -1$

Thinking of the logarithm instead, can we turn the multiplication (composition) into addition?

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## Doug said,

February 22, 2007 @ 1:38 am

Hi Kea

I agree with Carl and you about neutrinos [M Theory Lesson 15], but consider:

1 – the equations of John Bahcall, expert in calculating neutrino fluxes and spectra with Standard Solar Model, but no Nobel

http://www.sns.ias.edu/~jnb/

2 – The Nobel Prize in Physics 2002

Presentation Speech by Professor Per Carlson of the Royal Swedish Academy of Sciences, December 10, 2002. [Translation of the Swedish text.]

“… The most common particle in the universe, the neutrino, is created in huge quantities during these reactions. These remarkable particles pass unaffected through the enormous mass of the sun and reach us on the earth. Each second, 60 billion neutrinos pass through each square centimeter of the earth’s surface! Ray Davis and later Masatoshi Koshiba took on the challenge of catching solar neutrinos. However, the neutrinos are very difficult to detect because they only interact very weakly with matter. Only one out of every 1,000 billion neutrinos from the sun is stopped when they pass through the earth …”

http://nobelprize.org/nobel_prizes/physics/laureates/2002/presentation-speech.html

3 – NOVA | The Ghost Particle | Case of the Missing Particles | PBS

“NARRATOR: so the only difference between the atmospheric neutrinos coming from above and those R: For the chargeless neutrino, the solid rock of the Earth is just empty space, from below was the time they had been traveling before they reached the detector, which meant that contrary to all theory, neutrinos must have a sense of time.

DAVE WARK [HEP, Imperial College, London]: Just that fact, just that fact that the “ you that neutrinos have mass, because they tell you a neutrino knows how far it’s gone. And the only way it can know how far it’s gone is if its clock isn’t stopped, which means it can’t be traveling at the speed of light, which means it must have a mass.

[and later]

DAVE WARK: In the Big Bang, we would have made huge numbers of neutrinos. And if neutrinos have mass, it is possible that the matter in the universe today arose because of the decay of massive neutrinos created in the early universe. So we may be the grandchildren of neutrinos. All the matter that makes us up may have arisen purely through the decay of neutrinos.”

http://www.pbs.org/wgbh/nova/neutrino/

4 – 29 Comments on “arxiv Find: Dark Matter and Sterile Neutrinos”,

Sean, Cosmic Variance

RE: Dark Matter and Sterile Neutrinos, Peter L Biermann, Faustin Munyaneza

http://arxiv.org/abs/astro-ph/0702173

Comment #11, Doug on Feb 11th, 2007 at 10:29 am

This is a highly speculative comment.

Neutrinos may be able to link QM and GR through stochastic dynamic noncooperative game theory.

QM may run on ‘neutrino time’.

Neutrinos travel near the speed of light, which if Einstein is correct, so time is considerably slower than anthropic time used in GR.

Currently anthropic time appears to be used simultaneously for QM and GR from yoctoseconds to light years].

Perhaps some form of Itzhak Bars Two-Time Physics (2T-physics) is appropriate?

http://physics1.usc.edu/%7Ebars/research.html#2T

Events such as supernovas, GRB and comet collisions with planets are types of statistical mechanics [SM] in GR, which except for scale, are not unlike the SM of QM.

5 – David Hestenes paper on ‘The Kinematic Origin of Complex Wave Functions” may apply to both QM and GR simultaneously as a Zitterbewegung Interaction at neutrino rather than anthropic time.

http://www.physicsforums.com/showthread.php?t=156838

## Mahndisa S. Rigmaiden said,

February 22, 2007 @ 10:42 pm

02 22 07

Kea: I am not sure if I understand what you are asking. If you mean get an additive expression for exp(ix) Why not just expand it in its Taylor Series to get sines and cosines out of it?

I am not as advanced in category theory as you are, so maybe I am misinterpreting what you are asking. Pls clarify at your leisure. Thanks:)

## CarlBrannen said,

February 23, 2007 @ 9:27 am

Hey all, another look at the fermions, this one taking a better analysis of the bosons and the standard model Lagrangian (the key I’m missing), but missing most of what I know about the generations.