On Fermions in Compact momentum Spaces Bilinearly Constructed with Pure Spinors

In a long series of works the author has demonstrated that the model named the {\it spin-charge-family} theory offers the explanation for all in the {\it standard model} assumed properties of the fermion and boson fields, as well as for many of their so far observed properties if the space-time is $\ge (13 +1)$ while fermions interact with gravity only. In this paper, I briefly report on the so far achievements of the theory. The main contribution demonstrates the offer of th...

In a recent paper, algebraic descriptions for all non-relativistic spins were derived by elementary means directly from the Lie algebra $\specialorthogonalliealgebra{3}$, and a connection between spin and the geometry of Euclidean three-space was drawn. However, the details of this relationship and the extent to which it can be developed by elementary means were not expounded. In this paper, we will reveal the geometric content of the spin algebras by realising them within a ...

Spinor structure and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It is shown that tensor products of biquaternion algebras are associated with the each irreducible representation of the Lorentz group. Space-time discrete symmetries $P$, $T$ and their combination $PT$ are generated by the fundamental automor...

Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spi...

Spinors have played an essential but enigmatic role in modern physics since their discovery. Now that quantum-gravitational theories have started to become available, the inclusion of a description of spin in the development is natural and may bring about a profound understanding of the mathematical structure of fundamental physics. A program to attempt this is laid out here. Concepts from a known quantum-geometrical theory are reviewed: (1) Classical physics is replaced by a...

This contribution presents properties of the second quantized not only fermion fields but also boson fields, if the second quantization of both kinds of fields origins in the description of the internal space of fields with the ''basis vectors'' which are the superposition of odd (when describing fermions) or even (when describing bosons) products of the Clifford algebra operators $\gamma^a$'s. The tensor products of the ''basis vectors'' with the basis in ordinary space form...

Both algebras, Clifford and Grassmann, offer "basis vectors" for describing the internal degrees of freedom of fermions. The oddness of the "basis vectors", transferred to the creation operators, which are tensor products of the finite number of "basis vectors" and the infinite number of momentum basis, and to their Hermitian conjugated partners annihilation operators, offers the second quantization of fermions without postulating the conditions proposed by Dirac, enabling th...

Division algebras have demonstrated their utility in studying non-associative algebras and their connection to the Standard Model through complex Clifford algebras. This article focuses on exploring the connection between these complex Clifford algebras and their corresponding real Clifford algebras providing insight into geometric properties of bivector gauge symmetries. We first generate gauge symmetries in the complex Clifford algebra through a general Witt decomposition. ...

We suggest that the Weil spinors originate from the multi - component fermion fields. Those fields belong to the unusual theory that, presumably, exists at extremely high energies. In this theory there is no Lorentz symmetry. Moreover, complex numbers are not used in the description of its dynamics. Namely, the one - particle wave functions are real - valued, the functional integral that describes the second - quantised theory does not contain the imaginary unit as well. In t...

Dedicated to Ludwig Faddeev on his 80th birthday. Ludwig exemplifies perfectly a mathematical physicist: significant contribution to mathematics (algebraic properties of integrable systems) and physics (quantum field theory). In this note I present an exercise which bridges mathematics (restricted Clifford algebra) to physics (Majorana fermions).