# Generalized Ehrenfest Relations, Deformation Quantization, and the Geometry of Inter-model Reduction

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## Abstract

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.

## Keywords

Quantum Classical Deformation quantization Ehrenfest’s Theorem Reduction## Notes

### Acknowledgements

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.

## References

- 1.Anderson, P.W., et al.: More is different. Science
**177**(4047), 393–396 (1972)ADSCrossRefGoogle Scholar - 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
- 3.Ballentine, Leslie E: Quantum Mechanics: A Modern Development. World scientific, Singapore (1998)CrossRefMATHGoogle Scholar
- 4.Berndt, R.: An Introduction to Symplectic Geometry, vol. 26. American Mathematical Society, Park City (2001)MATHGoogle Scholar
- 5.Hilgevoord, J., Uffink, J.B.M.: More certainty about the uncertainty principle. Eur. J. Phys.
**6**(3), 165 (1985)CrossRefGoogle Scholar - 6.Inonu, E., Wigner, E.P.: On the contraction of groups and their representations. Proc. Natl. Acad. Sci. U.S.A.
**39**(6), 510 (1953)ADSMathSciNetCrossRefMATHGoogle Scholar - 7.Joos, E., Zeh, D., Kiefer, C., Giulini, D., Kupsch, J., Stamatescu, I.-O.: Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd edn. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
- 8.Landsman, N.P.: Mathematical Topics Between Classical and Quantum Mechanics. Springer monographs in mathematics. Springer, Berlin (1998)CrossRefMATHGoogle 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
- 10.Landsman, N.P.: Foundations of Quantum Theory: from Classical Concepts to Operator Algebras. Springer, Berlin (2017)CrossRefMATHGoogle Scholar
- 11.Pesci, A.I., Goldstein, R.E., Uys, H.: Hermitization and the poisson bracket-commutator correspondence as a consequence of averaging. J. Phys. A
**39**(4), 789 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar - 12.Rieffel, M.A.: Quantization and \(\text{ c }^{\wedge *}\)-algebras. Contemp. Math.
**167**, 67–67 (1994)MathSciNetGoogle Scholar - 13.Sagle, A.-A., Walde, R.-E.: Introduction to Lie Groups and Lie Algebras. Stony Brook, New York (1974)MATHGoogle Scholar
- 14.Schlosshauer, M.A.: Decoherence and the Quantum-To-Classical Transition. Springer, Berlin (2008)MATHGoogle 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
- 17.Zurek, Wojciech H.: Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time. Phys. Scr.
**1998**(T76), 186 (1998)MathSciNetCrossRefMATHGoogle Scholar - 18.Rosaler, J.: Local reduction in physics. Stud. Hist. Philos. Sci. B: Stud. Hist. Philos. Mod. Phys.
**50**, 54–69 (2015)ADSMathSciNetMATHGoogle 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
- 22.Butterfield, J.: Laws, causation and dynamics at different levels. Interface Focus
**2**(1), 101–114 (2012)CrossRefGoogle Scholar - 23.Yoshimi, J.: Supervenience, dynamical systems theory, and non-reductive physicalism. Br. J. Philos. Sci.
**63**(2), 373–398 (2011)MathSciNetCrossRefMATHGoogle Scholar