Quantum Potential - Re-interpretation in Terms of Clifford Algebras

Re-interpretation in Terms of Clifford Algebras

B. Hiley and R. E. Callaghan re-interpret the role of the Bohm model and its notion of quantum potential in the framework of Clifford algebra, taking account of recent advances that include the work of David Hestenes on spacetime algebra. They show how, within a nested hierarchy of Clifford algebras, for each Clifford algebra an element of a minimal left ideal and an element of a right ideal representing its Clifford conjugation can be constructed, and from it the Clifford density element (CDE), an element of the Clifford algebra which is isomorphic to the standard density matrix but independent of any specific representation. On this basis, bilinear invariants can be formed which represent properties of the system. Hiley and Callaghan distinguish bilinear invariants of a first kind, of which each stands for the expectation value of an element of the algebra which can be formed as, and bilinear invariants of a second kind which are constructed with derivatives and represent momentum and energy. Using these terms, they reconstruct the results of quantum mechanics without depending on a particular representation in terms of a wave function nor requiring reference to an external Hilbert space. Consistent with earlier results, the quantum potential of a non-relativistic particle with spin (Pauli particle) is shown to have an additional spin-dependent term, and the momentum of a relativistic particle with spin (Dirac particle) is shown to consist in a linear motion and a rotational part. The two dynamical equations governing the time evolution are re-interpreted as conservation equations. One of them stands for the conservation of energy; the other stands for the conservation of probability and of spin. The quantum potential plays the role of an internal energy which ensures the conservation of total energy.