Mixing quantum and classical mechanics and uniqueness of Planck's constant
Abstract
Observables of quantum or classical mechanics form algebras called quantum or classical Hamilton algebras (HAs), respectively (Grgin and Petersen 1974 J. Math. Phys. 15 764, Sahoo 1977 Pramana 8 545). We show that the tensor product of two quantum HAs, each characterized by a different Planck's constant (PC), is an algebra of the same type characterized by yet another PC. The algebraic structure of mixed quantum and classical systems is then analysed by taking the limit of vanishing PC in one of the component algebras. This approach provides new insight into failures of various formalisms dealing with mixed quantumclassical systems. It shows that in the interacting mixed quantumclassical description, there can be no backreaction of the quantum system on the classical system. A natural algebraic requirement involving restriction of the tensor product of two quantum HAs to their components proves that PC is unique.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 January 2004
 DOI:
 10.1088/03054470/37/3/031
 arXiv:
 arXiv:quantph/0301044
 Bibcode:
 2004JPhA...37..997S
 Keywords:

 algebraic structureHamilton AlgebraComposition of algebrasPlanck's Constant;
 Quantum Physics;
 Mathematical Physics;
 Mathematics  Mathematical Physics
 EPrint:
 revised version accepted for publication in J.Phys.A:Math.Phys