Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Wednesday, March 19, 2008

The Dirac Code III

The Pauli exclusion principle for solutions to Dirac's equation, and nilpotency in general, are summed up by the expression

$D^{2} = 0$

for an operator $D$. Mathematicians really like this expression. It immediately brings to mind the (co)homological dimension raising and lowering operators. And since we really want to do M Theory, why not skip the boring manifold de Rham theory (including equivariant cohomology, for that matter) and go straight to a universal motivic cohomology? After all, our particle states are to be represented by knotty diagrams with interpretations in higher logos logic, so the cohomology would naturally be universal (and we're supposed to be doing Quantum Gravity, dammit).

Dimension shifts are categorical. That's why we try so hard to view cardinalities of sets (such as particle counts) in the context of higher topos axiomatics. Thus we don't even know how to count to 3 until we reach the land of tricategories and their multicategorical analogues. Fortunately, as J. W. Gray showed some time ago, by dimension 3, weak $n$-categories reveal a remarkable surprise: the ability to contain dimension altering operations.

Rowlands talks about simple fermions, for which two spin states form the basis of the quantum logic. The squareness of nilpotency, as opposed to the more general $D^{n} \simeq 0$, may be viewed as a consequence of exclusion in two steps, arising from the spin quantum numbers. A mass analogue therefore suggests the study of $D^{3} \simeq 0$. In the context of generalised cohomology, this asks for an enormous extension of the idea of cohomology itself, which relies on concepts such as duality, as opposed to triality. If the categorical structure was significantly extended with each dimension, such as via a concept of $n$-ordinal category, the nilpotent case should be sufficiently rich to reproduce the cohomology theory in question. And there would be so much more.


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