occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Wednesday, March 19, 2008

### The Dirac Code II

In a new post, Carl Brannen compares Rowlands' nilpotents with the idempotents of the density operator formalism. Rowlands says on slide 22, "this is intriguingly close to twistor algebra", in reference to 4 complex variables arising from a combination of his quaternions ($1$, $I$, $J$, $K$) and multivariate vectors ($i$, $u$, $v$, $w$). This results in 64 possible products of 8 units, which may be generated, for example, by the combinations

$iK$, $uI$, $vI$, $wI$, $1J$

namely 5 in number, as the Dirac gamma matrices. Rowlands then writes the Dirac equation in the form

$[ iK \frac{\partial}{\partial t} + Iu \frac{\partial}{\partial x} + Iv \frac{\partial}{\partial y} + Iw \frac{\partial}{\partial z} + iJ m ] \psi = 0$

thereby associating the quaternion units $I$, $J$ and $K$ with momentum, mass and energy. The nilpotency appears for the amplitude $A$ when trying to interpret $\psi$ as a plane wave solution. See the slides for extensions of these ideas. For example, requiring $iKE + Ip + Jm$ to be nilpotent, we obtain the expression $E^{2} = p^{2} + m^{2}$ of special relativity. It is OK to put $c = 1$ here, because we work in the one time approximation.

From the perspective of M Theory, even novel algebras are merely representative of the meta-algebraic categorical axioms (Rowlands eliminates equations on slide 40), but analogous number theoretic structures, such as those arising from the $\mathbb{F}_{3}$ matrices for the quaternions in a Langlands type context, contain an even richer potential for interpreting operators in a measurement context, where numbers are the inevitable outcome.