The basic ideas and results behind polyvector-valued gauge field theories and Quantum Mechanics in Noncommutative Clifford spaces are presented. The star products are noncommutative but associative up to second order only. The construction of Noncommutative Clifford-space gravity as polyvector-valued gauge theories of twisted diffeomorphisms in Clifford-spaces would require quantum Hopf algebra...

We suggest Clifford algebra as a useful simplifying language for present quantum dynamics. Clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups. Aggregates using such representations for their permutations obey Clifford statistics. The vectors supporting the Clifford algebras of permutations and rotations are plexors an...

We study optimal synthesis of Clifford circuits, and apply the results to peep-hole optimization of quantum circuits. We report optimal circuits for all Clifford operations with up to four inputs. We perform peep-hole optimization of Clifford circuits with up to 40 inputs found in the literature, and demonstrate the reduction in the number of gates by about 50%. We extend our methods to the opt...

The spinor representation of the Lorentz group does not accept simple generalization with the group GL(4,R) of general linear coordinate transformations. The Dirac equation may be written for an arbitrary choice of a coordinate system and a metric, but the covariant linear transformations of the fourcomponent Dirac spinor exist only for isometries. For usual diagonal Minkowski metric the isomet...

We demonstrate the emergence of the conformal group SO(4,2) from the Clifford algebra of spacetime. A Clifford space (that contains spacetime, in particular) does not contain only points (events), but also lines, surfaces, volumes, etc.. All those geometric objects are very elegantly and compactly described by means of the geometric calculus based on Clifford algebra. A subgroup of the Clifford...

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 Cl...

in the present paper we give a partially negative answer to a conjecture of ghahramani, runde and willis. we also discuss the derivation problem for both foundation semigroup algebras and clifford semigroup algebras. in particular, we prove that if s is a topological clifford semigroup for which es is finite, then h1(m(s),m(s))={0}.

It is shown how one can attain the desired locally causal property of QM in Clifford-spaces despite the spacelike separation of two massive spin-12 particles involved in the Einstein-Podolksy-Rosen (EPR) experiment. This is achieved by proving why the addition laws of the particles poly-vector-valued momentum in Clifford-space is null-like. This is the key reason why it is possible to implement...

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...