On Fermions in Compact momentum Spaces Bilinearly Constructed with Pure Spinors

We discuss fermions for arbitrary dimensions and signature of the metric, with special emphasis on euclidean space. Generalized Majorana spinors are defined for $d=2,3,4,8,9$ mod 8, independently of the signature. These objects permit a consistent analytic continuation of Majorana spinors in Min-kowski space to euclidean signature. Compatibility of charge conjugation with complex conjugation requires for euclidean signature a new complex structure which involves a reflection ...

The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of $\text{U}(1)$, $\SU(2)$, and...

We further explore the idea that physics takes place in Clifford space which should be considered as a generalization of spacetime. Following the old observation that spinors can be represented as members of left ideals of Clifford algebra, we point out that the transformations which mix bosons and fermions could be represented by means of operators acting on Clifford algebra-valued (polyvector) fields. A generic polyvector field can be expanded either in terms of bosonic, or...

A solution to the 50 year old problem of a spinning particle in curved space has been recently derived using an extension of Clifford calculus in which each geometric element has its own coordinate. This leads us to propose that all the laws of physics should obey new polydimensional metaprinciples, for which Clifford algebra is the natural language of expression, just as tensors were for general relativity. Specifically, phenomena and physical laws should be invariant under ...

Clifford algebras and Majorana conditions are analyzed in any spacetime. An index labeling inequivalent $\Gamma$-structures up to orthogonal conjugations is introduced. Inequivalent charge-operators in even-dimensions, invariant under Wick rotations, are considered. The hermiticity condition on free-spinors lagrangians is presented. The constraints put by the Majorana condition on the free-spinors dynamics are analyzed. Tables specifying which spacetimes admit lagrangians wit...

Although intrinsic spin is usually viewed as a purely quantum property with no classical analog, we present evidence here that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of relativistic classical mechanics that uses spinors. Spinors and projectors arise naturally in the Clifford's geometric algebra of physical space and not only provide powerful tools...

We factorize the space-time coordinates of Minkowski space into Weyl spinors with components in a split Clifford algebra. Poisson brackets are defined for Clifford-valued canonical variables and applied to the quantization of the point particle and string. In particular, we obtain the Lorentz algebra for the quantum string, and show that the string supports both integral and half-integral spin. Commutation and anti-commutation relations are derived for the amplitudes of the s...

The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold ($C$-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. We assume that $C$-space is the true space in which physics takes...

It is shown that classical Clifford algebras are group algebras of cyclic subgroups of arrowy rermutations. It is established that Euclidean 3-space, Pauli and Dirac algebras and groups of global guage transformations are corollary from the geometry of 8-dimensional vacuum and 9-dimensional cosmos.

While general relativity provides a complete geometric theory of gravity, it fails to explain the other three forces of nature, i.e., electromagnetism and weak and strong interactions. We require the quantum field theory (QFT) to explain them. Therefore, in this article, we try to geometrize the spinor fields. We define a parametric coordinate system in the tangent space of a null manifold and show that these parametric coordinates behave as spinors. By introducing a complex ...