Canonical quantization in a spinor substructure of Minkowski space

TL;DR

This work develops a spinor-space canonical quantization framework for Minkowski space by replacing spacetime coordinates with spinor bilinears in a split Clifford algebra $\mathcal{C}l(4,4,\mathbb{R})$ and extending to the octonionic tensor product $\mathcal{O}\otimes_{\mathbb{R}}\mathcal{C}l(n,n,\mathbb{R})$ to realize ten-dimensional Lorentz invariance. It derives the Lorentz algebra for the quantum string from spinor-space Poisson (Clifford) brackets, and shows that a spinor string can possess both integral and half-integral spin states depending on the (anti)commutation relations of modes. The octonionic generalization yields a Lorentz-invariant string action in ten dimensions, supporting the hypothesis that space-time dimensionality is linked to normed division algebras. Overall, the paper provides a covariant, spinor-based quantization perspective with potential implications for high-energy theory and the geometric origin of space-time symmetries.

Abstract

We factorize the space-time coordinates of Minkowski space into Weyl spinors with components in a split Clifford algebra. Poisson brackets are defined for spinor-valued canonical variables and applied to the quantization of point particles and strings. In particular, we obtain the Lorentz algebra for the quantum string, and show that the string supports both integral and half-integral spin states. The Clifford algebra is augmented with the octonions through an R-algebra tensor product, and we apply the results of Manogue, Schray and Dray on octonionic Lorentz transformations to obtain a Lorentz invariant string action in ten dimensions.