2013/12/15 by K. R. W. Jones, Jones, K. R. W.
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #math-ph #math.DS #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1312.4195
42 pages
arxiv created 2013/12/15 · arxiv updated 2013/12/17
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 ℏ is kept fixed while a dimensionless mesoscopic parameter, λ∈[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(λ2ℏ), where λ= 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 λ= 0. We conclude by discussing some possible empirical signatures, and explore the obstructions to a satisfactory physical interpretation.