An illustration of canonical quantumclassical dynamics
Abstract
Using the example of the harmonic oscillator, we illustrate the use of hybrid dynamical brackets in analyzing quantumclassical interaction. We only assume that a hybrid dynamical bracket exists, is bilinear, and reduces to the pure quantum/classical bracket when acting on pure quantum/classical variables. Any hybrid bracket obeying these natural requirements will produce the same dynamics for pure classical or quantum variables, given a hybrid Hamiltonian. Backreaction is manifested in the evolution of a nonvanishing commutator between classical variables. The more massive the classical system is, the less it is affected by backreaction. Interestingly, we show that while pure variables evolve to violate the pure canonical relations, they always obey the hybrid canonical relations. The dynamics of hybrid variables, on the other hand, is shown to require a fully specified and consistent hybrid bracket, otherwise evolution cannot be defined.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 DOI:
 10.48550/arXiv.2107.01341
 arXiv:
 arXiv:2107.01341
 Bibcode:
 2021arXiv210701341A
 Keywords:

 Quantum Physics
 EPrint:
 6 pages, 1 figure