Hyperquaternionic Unitary Symplectic Groups: A Unifying Tool for Physics

Abstract

The mathematical tools of physics, based on group theory, are in permanent evolution. Major covariance groups are the orthogonal, unitary and symplectic groups. These groups are generally expressed in terms of real and complex matrices. Here we shall develop a new representation of the unitary symplectic groups USp(n) in terms of Clifford algebras constituted by tensor products of quaternion algebras called hyperquaternions. Concise expressions of the generators are obtained and a concrete example USp(4) is provided. Isomorphic quaternion matrix representations will also be used in the applications. The first application concerns classical mechanics. The Hamiltonian formalism, Poisson brackets and canonical transforms are related to the unitary symplectic groups. The 1D and 2D harmonic oscillators are examined within that framework. The second application concerns quantum mechanics. The Schrödinger and Heisenberg equations are derived in a new hyperquaternionic unitary symplectic way, the complex imaginary i being replaced by the quaternion k in phase space. The 1D and 2D quantum harmonic oscillators are treated within that formalism. Allowing a representation of both classical and quantum mechanics, it is hoped that the hyperquaternion algebras might deepen our mathematical comprehension of the foundational principles of physics.