An Essay on the Completion of Quantum Theory. II: Unitary Time Evolution

TL;DR

This paper develops a geometric framework for quantum mechanics by defining a unitary setting with a Cayley transform interpretation, linking Hamiltonian flows to Lie group structures.

Contribution

It introduces a geometric, group-theoretic approach to quantum time evolution, extending the theory with intrinsic data and a Cayley transform perspective.

Findings

Unitary time evolution corresponds to flows of invariant vector fields.

The geometric space can be seen as a Cayley transform of Hermitian operator space.

A natural octahedral group relates to the Cayley transform and triple of elements.

Abstract

In this second part of the `essay on the completion of quantum theory' we define the {\em unitary setting of completed quantum mechanics}, by adding as intrinsic data to those from Part I (arXiv:1711.08643) the choice of a north pole N and south pole S in the geometric space. Then we explain that, in the unitary setting, a complete observable corresponds to a right (or left) invariant vector field (Hamiltonian field) on the geometric space, and {\em unitary time evolution} is the flow of such a vector field. This interpretation is in fact nothing but the Lie group-Lie group algebra correspondence, for a geometric space that can be interpreted as the Cayley transform of the usual, Hermitian operator space. In order to clarify the geometric nature of this setting, we realize the Cayley transform as a member of a natural octahedral group that can be associated to any triple of pairwise transversal elements.