The theory of the Clifford algebra of a vector space with a given symmetric bilinear form is rather well-understood: One of the basic properties of the Clifford algebra gives an explicit basis for it in terms of a basis of the underlying vector space (Theorem 1 below), and another one provides a canonical vector space isomorphism between the Clifford algebra and the exterior algebra of the same...

We want in this chapter to regard Clifford algebras as natural generalizations of the complex number system. First let us note that if z is a complex number then zz = ‖z‖. Now for a quaternion q we also have qq = ‖q‖. In this way quaternions may be regarded as a generalization of the complex number system. It seems natural to ask if one can extend basic results of one complex variable analysis ...

A straightforward two-line derivation of the Bekenstein-Hawking Area-Entropy relation in any dimension is shown based on Shannon’s information theory and Clifford algebras required by the New Relativity Principle. Recently we have proposed that a New Relativity principle may be operating in Nature which could reveal important clues to find the origins of M theory [1]. We were forced to introduc...

We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras –as Clifford algebras– by different filtrations resp. induced gradings. The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however...

This is an elementary introduction to the representation theory of finite semigroups. We illustrate Clifford–Munn correspondence between representations a semigroup and its maximal subgroups. The emphasis throughout on naturally occurring examples.

A Clifford algebra has three major multiplications: inner product, outer product and geometric product. Accordingly, the same Clifford algebra has three versions: Clifford vector algebra, which features on inner products and outer products of multivectors; Clifford bracket algebra, which features on pseudoscalars and inner products of vectors; Clifford geometric algebra, which features on geome...

This paper expands the earlier paper [30] and presents foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theory based on the representation theory of SL2(R) group. We describe here geometries of corresponding domains. The principal rôle is played by Clifford algebras of matching types. In this paper we also generalise the Fillmor...

Automorph class theory formalism is developed for the case of integral nonsingular quadratic forms in an odd number of variables. As an application, automorph class theory is used to construct a lifting of similitudes of quadratic Z-modules of arbitrary nondegenerate ternary quadratic forms to morphisms between certain subrings of associated Clifford algebras. The construction explains and gene...

The efficiency of compiling high-level quantum algorithms into instruction sets native to quantum computers defines the moment in the future when we will be able to solve interesting and important problems on quantum computers. In my work I focus on the new methods for compiling single qubit operations that appear in many quantum algorithms into single qubit operations natively supported by sev...

In Clifford analysis, one studies spin-invariant differential operators on spaces of arbitrary dimension m. At the heart of the classical theory lies the well-known Dirac operator, which finds its origin in physics [5]: the Dirac equation describes the behaviour of electrons in the 4-dimensional spacetime. Our aim is to study generalizations of this operator, the so-called higher spin Dirac ope...