On an interpolative Schr\"{o}dinger equation and an alternative classical limit

Abstract

We introduce a simple deformed quantization prescription that interpolates the classical and quantum sectors of Weinberg's nonlinear quantum theory. The result is a novel classical limit where $\hbar$ is kept fixed while a dimensionless mesoscopic parameter, $\lambda\in[0,1]$, goes to zero. Unlike the standard classical limit, which holds good up to a certain timescale, ours is a precise limit incorporating true dynamical chaos, no dispersion, an absence of macroscopic superpositions and a complete recovery of the symplectic geometry of classical phase space. We develop the formalism, and discover that energy levels suffer a {\em generic perturbation\/}. Exactly, they become $E(\lambda^{2}\hbar)$, where $\lambda = 1$ gives the standard prediction. Exact interpolative eigenstates can be similarly constructed. Unlike the linear case, these need no longer be orthogonal. A formal solution for the interpolative dynamics is given, and we exhibit the free particle as one exactly soluble case. Dispersion is reduced, to vanish at $\lambda = 0$. We conclude by discussing some possible empirical signatures, and explore the obstructions to a satisfactory physical interpretation.