# Why nature made a choice of Clifford and not Grassmann coordinates.

The fields are discussed, the internal degrees of freedom of which are expressed by either the Grassmann or the Clifford "coordinates". Since both "coordinates" fulfill anticommutation relations, both fields can be second quantized so that their creation and annihilation operators fulfill the requirements of the commutation relations for fermion fields. However, while the internal spin, determined by the generators of the Lorentz group of the Clifford objects $S^{ab}$ and $\tilde{S}^{ab}$ (in the spin-charge-family theory $S^{ab}$ determine the spin degrees of freedom and $\tilde{S}^{ab}$ the family degrees of freedom) have half integer spin, ${\cal {\bf S}}^{ab}$ (expressible with $S^{ab} + \tilde{S}^{ab}$) have integer spin. Nature "made" obviously a choice of the Clifford algebra, at least in the so far observed part of our universe. We discuss the quantization - first and second - of the fields, the internal degrees of freedom of which are functions of the Grassmann coordinates $\theta$ and their conjugate momentum and the fields, the internal degrees of freedom of which are functions of the Clifford $\gamma^{a}$. Inspiration comes from the spin-charge-family theory with the action for fermions in $d$-dimensional space is $ \int \; d^dx \; E\;\frac{1}{2}\, (\bar{\psi} \, \gamma^a p_{0a} \psi) + h.c.$, $p_{0a} = f^{\alpha}{}_a p_{0\alpha} +\frac{1}{2E}\, \{ p_{\alpha}, E f^{\alpha}{}_a\}_- $, $ p_{0\alpha}=$ $ p_{\alpha} - \frac{1}{2} S^{ab} \omega_{ab \alpha} - \frac{1}{2} \tilde{S}^{ab} \tilde{\omega}_{ab \alpha}$. We write the basic states of the Grassmann fields and the Clifford fields as a function of products of either Grassmann or Clifford objects, trying to understand "the choice of nature". We look for the action for free fields in Grassmann and Clifford space to understand why Clifford algebra "wins" in the competition for the physical degrees of freedom.

Publisher URL: http://arxiv.org/abs/1802.05554

DOI: arXiv:1802.05554v1

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