Particular boundary condition ensures that a fermion in d=1+5, compactified on a finite disk, manifests in d=1+3 as massless spinor with a charge 1/2, mass protected and chirally coupled to the gauge field

The genuine Kaluza-Klein-like theories--with no fields in addition to gravity--have difficulties with the existence of massless spinors after the compactification of some space dimensions \cite{witten}. We proposed in previous paper a boundary condition for spinors in d=(1+5) compactified on a flat disk that ensures masslessness of spinors (with all positive half integer charges) in d=(1+3) as well as their chiral coupling to the corresponding background gauge gravitational field. In this paper we study the same toy model, proposing a boundary condition allowing a massless spinor of one handedness and only one charge (1/2) and infinitely many massive spinors of the same charge, allowing disc to be curved. We define the operator of momentum to be Hermitean on the vector space of spinor states--the solutions on a disc with the boundary.Comment: 15 page

Quantum gates and quantum algorithms with Clifford algebra technique

We use our Clifford algebra technique, that is nilpotents and projectors which are binomials of the Clifford algebra objects $\gamma^a$ with the property $\{\gamma^a,\gamma^b\}_+ = 2 \eta^{ab}$, for representing quantum gates and quantum algorithms needed in quantum computers in an elegant way. We identify $n$-qubits with spinor representations of the group SO(1,3) for a system of $n$ spinors. Representations are expressed in terms of products of projectors and nilpotents. An algorithm for extracting a particular information out of a general superposition of $2^n$ qubit states is presented. It reproduces for a particular choice of the initial state the Grover's algorithm.Comment: 9 pages, revte

How Clifford algebra helps understand second quantized quarks and leptons and corresponding vector and scalar boson fields, opening a new step beyond the standard model

This article presents the description of the internal spaces of fermion and boson fields in d-dimensional spaces, with the odd and even “basis vectors” which are the superposition of odd and even products of operators γa. While the Clifford odd “basis vectors” manifest properties of fermion fields, appearing in families, the Clifford even “basis vectors” demonstrate properties of the corresponding gauge fields. In d≥(13+1) the corresponding creation operators manifest in d=(3+1) the properties of all the observed quarks and leptons, with the families included, and of their gauge boson fields, with the scalar fields included, making several predictions. The properties of the creation and annihilation operators for fermion and boson fields are illustrated on the case d=(5+1), when SO(5,1) demonstrates the symmetry of SU(3)×U(1)