Generalized Ehrenfest Relations, Deformation Quantization, and the Geometry of Inter-model Reduction
- 355 Downloads
This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit \(\hbar \rightarrow 0\), and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models.
KeywordsQuantum Classical Deformation quantization Ehrenfest’s Theorem Reduction
This work was supported by the DFG Research Unit “The Epistemology of the Large Hadron Collider” (grant FOR 2063). The author wishes to thank Robert Harlander and Erhard Scholz for helpful comments.
- 2.Bacciagaluppi, G.: The role of decoherence in quantum mechanics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Cambridge University Press, Cambridge (2012). 2012 editionGoogle Scholar
- 9.Landsman, N.P.: Between classical and quantum. In: Butterfield, J., Earman, J. (eds.) Philosophy of Physics (Handbook of the Philosophy of Science), vol. 1. Elsevier, Amsterdam (2007)Google Scholar
- 15.Tao, T.: Some notes on Weyl quantisation (2012)Google Scholar
- 16.Woit, P: Quantum theory, groups and representations: an introduction (under construction) (2016)Google Scholar
- 19.Rosaler, J.: Inter-theory relations in physics: case studies from quantum mechanics and quantum field theory. Dissertation, Ph.D. thesis, University of Oxford (2013)Google Scholar
- 20.Wallace, D.: The emergent multiverse: quantum theory according to the Everett interpretation. Oxford University Press (2012)Google Scholar
- 21.Giunti, M.: Emulation, reduction, and emergence in dynamical systems (2006)Google Scholar