Arcadian Functor

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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.


Blogger kneemo said...

An interesting find. Another approach involves extending the Pauli matrices with 2x2 traceless Hermitian quaternionic matrices (See here). We can even extend the Pauli matrices further using the octonions and recover the Massless Dirac equation in 9+1 dimensions (math-ph/9910004). Multiple quaternionic Dirac equations (35 in all) are then obtained by restricting to a quaternionic subalgebra.

March 23, 2008 6:18 AM  
Blogger Kea said...

Ah, kneemo, thanks! Good to hear from you.

March 23, 2008 6:47 AM  

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