In 1953, Freudenthal noticed that rank one projections are a special case of operators that satisfy Q*Q=0 (up to nonzero scalar multiple), where ‘ * ‘ is a generalized cross product (Freudenthal product). In fact, the more modern definitions of projective space use this nilpotent relation to describe points.

]]>That certainly sounds like an interesting paper! Thanks for the link.

]]>Is the bimonster a bimonoid?

I found a definition for wreath product but am having difficulty visualizing this entity. Are you familiar with it?

I have become interested in error correcting Golay codes.

More at PF RE Witten paper.

Golay codes [GC12 and GC24] and the bimonster: [especially Appendix A]

The complex Lorentzian Leech lattice and the bimonster by Tathagata Basak

Comments: 24 pages, 3 figures, revised and proof corrected. Some small results added. to appear in the Journal of Algebra

Subj-class: Group Theory; Number Theory

MSC-class: 11H56, 20F55

Abstract: We find 26 reflections in the automorphism group of the Lorentzian Leech lattice L over Z[exp(2*pi*i/3)] that form the Coxeter diagram seen in the presentation of the bimonster. We prove that these 26 reflections generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the refletion group.

http://arxiv.org/abs/math.GR/0508228

Carl Basically, Parity is an idempotent operator. See Sakurai for more elucidation.

]]>Hello Kea and Carl:

I could not post earlier so I responded to Carl’s statement here.

Quantum states are usually represented by state vectors, and these can’t be put into the idempotency relation. However, density matrices can, and pure states satisfy the equation.

As an operator, idempotency is another word for “projection operator”, so you can see it as a way of picking out a subset of the available states.

I like to think of quantum mechanics as defining a relationship between an initial state and a final state (while refusing to define what happens in betweeen). In this sense, an idempotent can be thought of as something that stays the same; the final is the same as the initial.

When I told Dr. Hestenes that I was modeling particles based on idempotents, he said that most people used nilpotents instead. He went on to say that there is a close relation between nilpotents and idempotents.

You need niplotents in fermion state vectors because you want the Pauli exclusion principle to apply. I’m convinced that the Pauli exclusion principle is just a low energy (i.e. << Planck mass) approximation, but at our energies that means that it is perfect. In my understanding of physics, there is no quantum vacuum, it’s just a mathematical convenience used for calculations. So there is no foundational need to require nilpotency in order to enforce the Pauli exclusion principle among the creation operators. I see the nilpotency as resulting only from the whole thing being an effective theory only. On the subject of parity. The usual parity operator is a mapping from states to states. For this, I don’t think that P^2 = 1 is always obtained. Instead, you will end up with an arbitrary phase. There is another way of thinking about parity (mentioned in Landau and Lipshitz QED I think), and that is to define the parity operator as taking an eigenvalue of +1 on particles with an even internal parity, and -1 on particles with odd internal parity. Then parity does satisfy P^2 = 1, and the observed internal states are eigenstates of parity. The operators that satisfy Q^2 =1 are closely connected to the idempotents by the following prescription: \rho = 0.5(1 +- Q) Then \rho (for either sign) is idempotent. And the idempotents are eigenstates of Q with eigenvalues +- 1. And you can take Q as the rare definition of parity.

]]>*As I was walking among the fires of Hell, delighted with the enjoyments of Genius; which to Angels look like torment and insanity, I collected some of their Proverbs: thinking that as the sayings used in a nation, mark its character, so the Proverbs of Hell, shew the nature of Infernal wisdom better than any description of buildings or garments.*

When I came home; on the abyss of the five senses, where a flat sided steep frowns over the present world, I saw a mighty Devil folded

in black clouds, hovering on the sides of the rock, with corroding

fires he wrote the following sentence now percieved by the minds of men, and read by them on earth:

*How do you know but ev’ry Bird that cuts the airy way, Is an immense world of delight, clos’d by your senses five?*

from **The Marriage of Heaven and Hell** (1790)

Though we called your friend from his bed this night,

he could not speak for you,

For the race is run by one and one

and never by two and two.

That is, souls have to be saved (or lost) one at a time. I wrote Koide a two page letter that spoke to what he alone was doing.

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