occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Friday, August 29, 2008

### M Theory Lesson 219

In this old post on $E_{8}$ Carl Brannen proposes recovering the self referential nature of the algebra, or at least some deformed version of it, using a finite tower of composite states built from density operators.

Such studies use ternary (eg. qutrit) as well as qubit logic. Mixtures of qubits and qutrits occur in the Kasteleyn type recursive operators. For example, starting with $4 \times 4$ Dirac matrices inserted in a $3 \times 3$ block, one allows a deformation parameter $\frac{\pi}{12}$, a 24th root of unity. Since 24 is not a prime number, it naturally factorises into $3 \times 2^{3}$, where one factor of 2 corresponds to the doubling of roots, which also occurred in the modular relation for $B_{3}$.

Similarly, the dimension of $E_{8}$ factorises into $31 \times 2^{3}$, as noted by Kostant; see this post on the j invariant, speaking of which, as terms of a theta function, the $q$ expansion lists distances of vectors on a lattice. Amazingly, in dimension 24 there are exactly 24 nice lattices. One of these is a triple of $E_{8}$ lattices, and another example is the Leech lattice, which is secretly related to all the 24 dimensional lattices.