A Theory of Quantized Fields Based on Orthogonal and Symplectic Clifford Algebras

TL;DR

This paper develops a novel approach to quantum field theory using orthogonal and symplectic Clifford algebras, linking classical phase space variables to quantum operators and exploring implications for unification and gravity.

Contribution

It introduces a Clifford algebra-based framework for quantization, replacing phase space variables with basis vectors that satisfy quantum commutation relations.

Findings

Basis vectors satisfy Heisenberg equations of motion.

Quantization is achieved by replacing phase space variables with basis vectors.

Approach has potential applications in grand unification and quantum gravity.

Abstract

The transition from a classical to quantum theory is investigated within the context of orthogonal and symplectic Clifford algebras, first for particles, and then for fields. It is shown that the generators of Clifford algebras have the role of quantum mechanical operators that satisfy the Heisenberg equations of motion. For quadratic Hamiltonians, the latter equations are obtained from the classical equations of motion, rewritten in terms of the phase space coordinates and the corresponding basis vectors. Then, assuming that such equations hold for arbitrary path, i.e., that coordinates and momenta are undetermined, we arrive at the equations that contains basis vectors and their time derivatives only. According to this approach, quantization of a classical theory, formulated in phase space, is replacement of the phase space variables with the corresponding basis vectors (operators). The basis vectors, transformed into the Witt basis, satisfy the bosonic or fermionic (anti)commutation relations, and can create spinor states of all minimal left ideals of the corresponding Clifford algebra. We consider some specific actions for point particles and fields, formulated in terms of commuting and/or anticommuting phase space variables, together with the corresponding symplectic or orthogonal basis vectors. Finally we discuss why such approach could be useful for grand unification and quantum gravity.