Principle of General Q Covariance

Abstract

In this paper the physical implications of quaternion quantum mechanics are further explored. In a quanternionic Hilbert space ${\mathcal{H}}_Q$ , the lattice of subspaces has a symmetry group which is isomorphic to the group of all co‐unitary transformations in ${\mathcal{H}}_Q$ . In contrast to the complex space ${\mathcal{H}}_C$ (ordinary Hilbert space), this group is connected, while for ${\mathcal{H}}_C$ it consists of two disconnected pieces. The subgroup of transformations in ${\mathcal{H}}_Q$ which associates with every quaternion q of magnitude 1, the correspondence ψ → qψq⁻¹ for all $\mathrm{\psi \in }{\mathcal{H}}_Q$ (called Q conjugations), is isomorphic to the three‐dimensional rotation group. We postulate the principle of Q covariance: The physical laws are invariant under Q conjugations. The full significance of this postulate is brought to light in localizable systems where it can be generalized to the principle of general Q covariance: Physical laws are invariant under general Q conjugations. Under the latter we understand conjugation transformations which vary continuously from point to point. The implementation of this principle forces us to construct a theory of parallel transport of quaternions. The notions of Q‐covariant derivative and Q curvature are natural consequences thereof. There is a further new structure built into the quaternionic frame through the equations of motion. These equations single out a purely imaginary quaternion η(x) which may be a continuous function of the space—time coordinates. It corresponds to the i in the Schrödinger equation of ordinary quantum mechanics. We consider η(x) as a fundamental field, much like the tensor g_μν in the general theory of relativity. We give here a classical theory of this field by assuming the simplest invariant Lagrangian which can be constructed out of η and the covariant Q connection. It is shown that this theory describes three vector fields, two of them with mass and charge, and one massless and neutral. The latter is identifiable with the classical electromagnetic field.