Hyperquaternions: A New Tool for Physics

Abstract

A hyperquaternion formulation of Clifford algebras in n dimensions is presented. The hyperquaternion algebra is defined as a tensor product of quaternion algebras H (or a subalgebra thereof). An advantage of this formulation is that the hyperquaternion product is defined independently of the choice of the generators. The paper gives an explicit expression of the generators and develops a generalized multivector calculus. Due to the isomorphism H ⊗ H m(4, R), hyperquaternions yield all real, complex and quaternion square matrices. A hyperconjugation is introduced which generalizes the concepts of transposition, adjunction and transpose quaternion conjugate. As applications, simple expressions of the unitary and unitary symplectic groups are obtained. Finally, the hyperquaternions are compared, in the context of physical applications, to another algebraic structure based on octonions which has been proposed recently.

Quaternion Algebra on 4D Superfluid Quantum Space-Time: Can Dark Matter Be a Manifestation of the Superfluid Ether?

“…These matrices were successfully applied at studying the spin neutron resonance and demonstrated perfect abilities [59][60][61]. These matrices form the quaternion group H 2 [52][53][54][55]. One can consider the quaternion algebra on 4D space-time as a lateral branch of Penrose's twister program.…”

“…The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science." The quaternions are hypercomplex numbers [50][51][52][53][54][55][56][57][58] forming a vector space of dimension four over the field of real numbers:…”

Hyperquaternion Conformal Groups

“…Quaternions, constituting the quaternion algebra (H), are a set of four real numbers [7,9,10,12,13,23]…”

“…Moore was to call Lipschitz's algebras "hyperquaternions" and gave a canonical decomposition of Euclidean rotations which has been extended to pseudo-Euclidean rotations by the authors [9]. In recent papers, we have applied hyperquaternions to express the unitary, unitary symplectic groups and the Poincaré groups with dual hyperquaternions [7,9,10]. Though dual hyperquaternions provide a correct representation of the Poincaré groups, the introduction of a dual element (squaring to zero) might appear as somewhat unnatural.…”

“…Here, by adding two dimensions to the initial space, we shall introduce such elements within the hyperquaternion framework thereby extending the Poincaré groups to the nD conformal groups. Concerning the isomorphism of hyperquaternions with standard Clifford algebras, a modern proof has been given in [10]. Hyperquaternions yield an intrinsically defined tensor product (without generators), a unique multivector structure (after a generator choice) as well as simple expressions of the generators.…”