Presymmetry. II

The association between laboratory procedures and self-adjoint operators (observables), usually implicit and allusive, is made explicit by the introduction of two collections of nonmathematical objects; the observation procedures and the state-preparing procedures. By a process of idealization and extrapolation from empirical facts, these two collections are turned into mathematical sets ("hardware spaces") with the structure of a star algebra $\mathcal{O}$ and a convex linear set $\mathcal{S}$, respectively. There is a many-one mapping from $\mathcal{O}$ into the space $\mathfrak{U}$ of observables (the operators on Hilbert space), and physical laws or solutions of equations of motion are embodied in the mapping $\Phi :\mathcal{O}\to \mathfrak{U}$. Certain material motions of observation instruments and state-preparing instruments induce automorphisms of the hardware spaces but in general, not automorphisms of $\mathfrak{U}$. An extension of space-time invariance theory to accelerations [$\mathbf{x}\to \mathbf{x}+\Sigma 2^{\infty }{\mathrm{}(n!)\mathrm{}}^{-1\mathrm{}}{\mathbf{r}}_n{t}^n$], to external symmetry-breaking fields, and to subsystems under the influence of other subsystems becomes possible. The main results of this paper are a derivation of Newton's second law and of the gravitational equivalence principle for nonrelativistic quantum mechanics from invariance and causality principles.