The quantal algebra and abstract equations of motion

TL;DR

This paper introduces the quantal algebra as a unified abstract framework combining classical and quantum mechanics, highlighting its structural similarities to gravitational and electromagnetic field equations.

Contribution

It generalizes equations of motion using an abstract derivation in the quantal algebra, connecting algebraic identities to fundamental physical laws.

Findings

The Jacobi identity resembles the Bianchi identity in general relativity.

The algebra's properties reflect electromagnetic tensor antisymmetry.

The structure contains spacetime local properties without explicit postulation.

Abstract

The quantal algebra combines classical and quantum mechanics into an abstract structurally unified structure. The structure uses two products: one symmetric and one anti-symmetric. The local structure of spacetime is contained in the quantal algebra without having been postulated. We will introduce an abstract derivation concept and generalize classical and quantum mechanics equations of motion to abstract equations of motion in which the anti-symmetric product of the quantal algebra plays a key role. We will express the defining identities of the quantal algebra in terms of the abstract derivation. In this form the first defining identity (the Jacobi identity) is analogous in form to the Bianchi identity in general relativity which is one set of gravitational field equations for the curvature tensor. The identity for the unit element is equivalent to the important property that the metric is covariantly constant. Similarly, the Jacobi identity is analogous in form to the homogeneous Maxwell equations. The anti-symmetric product of the quantal algebra is reflected in the antisymmetry of the electromagnetic tensor.