Hostname: page-component-6766d58669-bkrcr Total loading time: 0 Render date: 2026-05-17T18:03:17.546Z Has data issue: false hasContentIssue false
  • English
  • Français

Bohmian Insights into Quantum Chaos

Published online by Cambridge University Press:  01 April 2022

James T. Cushing
James T. Cushing*
Affiliation:
University of Notre Dame
*
†Send requests for reprints to the author, Department of Physics, University of Notre Dame, Nieuwland Science Hall, Notre Dame, IN 46556-5670.
Get access Rights & Permissions [Opens in a new window]

Abstract

The ubiquity of chaos in classical mechanics (CM), as opposed to the situation in standard quantum mechanics (QM), might be taken as speaking against QM being the fundamental theory of physical phenomena. Bohmian mechanics (BM), as a formulation of quantum theory, may clarify both the existence of chaos in the quantum domain and the nature of the classical limit. Two interesting possibilities are (i) that CM and classical chaos are included in and underwritten by quantum mechanics (BM) or (ii) that BM and CM simply possess a common region of (noninclusive) overlap. In the latter case, neither CM nor QM alone would be sufficient, even in principle, to account for all of the physical phenomena we encounter. In this talk I shall summarize and discuss the implications of some recent work on chaos and on the classical limit within the framework of BM.

Information

Type
Philosophy of Physics and Chemistry
Information
Philosophy of Science , Volume 67 , Issue S3: Supplement. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part II: Symposia Papers Sep., 2000 , 2000 , pp. S430 - S445
Copyright
Copyright © 2000 by the Philosophy of Science Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

‡

Several of the ideas discussed here are developed further in Cushing and Bowman 1999 and in Bowman 2000. I also want to thank Gary Bowman for extended discussions on the subject of this essay.

References

Ammann, H., Gray, R., Shvarchuck, I., and Christensen, N. (1998), “Quantum Delta-Kicked Rotor: Experimental Observation of Decoherence”, Physical Review Letters 80: 4111–4115.CrossRefGoogle Scholar
Batterman, R. W. (1991), “Chaos, Quantization, and the Correspondence Principle”, Synthese 89: 189–227.CrossRefGoogle Scholar
Batterman, R. W. (1993), “Quantum Chaos and Semiclassical Mechanics”, in Hull, D., Forbes, M., and Okruhlik, K. (eds.), PSA 1992, vol. 2. East Lansing, MI: Philosophy of Science Association, 50–65.Google Scholar
Belot, G. (1998), “Quantum Chaos”, Philosophy of Science, this issue.Google Scholar
Berry, M. V. (1987), “The Bakerian Lecture. Quantum Chaology”, in Berry, M. V., Percival, I. C., and Weiss, N. O. (eds.), Dynamical Chaos. Princeton: Princeton University Press, 183–198.Google Scholar
Berry, M. V. (1989), “Quantum Chaology, Not Quantum Chaos”, Physica Scripta 40: 325–326.CrossRefGoogle Scholar
Berry, M. V. (1994), “Asymptotics, Singularities and the Reduction of Theories', in Prawitz, D., Skyrms, B., and Westerståhl, D. (eds.), Logic, Methodology and Philosophy of Science IX. Uppsala: Elsevier Science B. V., 597–607.Google Scholar
Bobik, J. (1998), Aquinas on Matter and From and the Elements. Notre Dame, IN: University of Notre Dame Press.CrossRefGoogle Scholar
Bohm, D. (1952a), “A Suggested Interpretation of Quantum Theory in Terms of ‘Hidden’ Variables, I and II”, Physical Review 85: 166–193.CrossRefGoogle Scholar
Bohm, D. (1952b), “Reply to Criticism of a Causal Re-Interpretation of the Quantum Theory”, Physical Review 87: 389–390.CrossRefGoogle Scholar
Bowman, G. E. (2000), “Wave Packets, Quantum Chaos and the Classical Limit of Bohmian Mechanics”. Unpublished Ph.D. dissertation, University of Notre Dame.Google Scholar
Brun, T. A. (1995), “An Example of the Decoherence Approach to Quantum Dissipative Chaos”, Physics Letters A 206: 167–176.CrossRefGoogle Scholar
Casati, G. (ed.) (1985), Chaotic Behavior in Quantum Systems. New York: Plenum Publishing Co.CrossRefGoogle Scholar
Casati, G., Chirikov, B. V., Izrailev, F. M., and Ford, J. (1979), “Stochastic Behavior of a Quantum Pendulum Under a Periodic Perturbation”, in G. Casati and J. Ford (eds.), Stochastic Behavior in Classical and Quantum Hamiltonian Systems (Lecture Notes in Physics 93). New York: Springer-Verlag, 334–352.CrossRefGoogle Scholar
Caves, C. M. and Schack, R. (1997), “Unpredictability, Information, and Chaos”, University of New Mexico preprint.3.0.CO;2-W>CrossRefGoogle Scholar
Chirikov, B. V. (1991), “Time-Dependent Quantum Systems”, in M.-J. Giannoni, A. Voros, and J. Zinn-Justin (eds.), Chaos and Quantum Physics (Les Houches, Session 52). Amsterdam: North-Holland Publishing Co., 443–446.Google Scholar
Cushing, J. T. (1991), “Quantum Theory and Explanatory Discourse: Endgame for Understanding?”, Philosophy of Science 58: 337–358.CrossRefGoogle Scholar
Cushing, J. T. (1994), Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony. Chicago: University of Chicago Press.Google Scholar
Cushing, J. T. and Bowman, G. E. (1999), “Bohmian Mechanics and Chaos” in Butterfield, J., and Pagonis, C. (eds.), From Physics to Philosophy. Cambridge: Cambridge University Press, 90–107.CrossRefGoogle Scholar
Deutsch, D. (1997), The Fabric of Reality. London: Allen Lane.Google Scholar
Dewdney, C. and Malik, Z. (1996), “Measurement, Decoherence and Chaos in Quantum Pinball”, Physics Letters A 229: 183–188.CrossRefGoogle Scholar
Dürr, D., Goldstein, S., and Zanghì, N. (1992a), “Quantum Equilibrium and the Origin of Absolute Uncertainty”, Journal of Statistical Physics 67: 843–907.CrossRefGoogle Scholar
Dürr, D., Goldstein, S., and Zanghì., N. (1992b), “Quantum Chaos, Classical Randomness, and Bohmian Mechanics”, Journal of Statistical Physics 68: 259–270.CrossRefGoogle Scholar
Faisal, F. H. M. and Schwengelbeck, U. (1995), “Unified Theory of Lyapunov Exponents and a Positive Example of Deterministic Quantum Chaos”, Physics Letters A 207: 31–36.CrossRefGoogle Scholar
Ford, J. (1988), “Quantum Chaos, Is There Any?”, in Bai-Lin, H. (ed.), Directions in Chaos, vol. 2. Teaneck, NJ: World Scientific Publishing Co., 128–147.CrossRefGoogle Scholar
Ford, J. (1989), “What Is Chaos That We Should Be Mindful of It?”, in Davies, P. (ed.), The New Physics. Cambridge: Cambridge University Press, 348–371.Google Scholar
Ford, J. and Mantica, G. (1992), “Does Quantum Mechanics Obey the Correspondence Principle? Is It Complete?”, American Journal of Physics 60: 1086–1098.CrossRefGoogle Scholar
Frisk, H. (1997), “Properties of the Trajectories in Bohmian Mechanics”, Physics Letters A 227: 139–142.CrossRefGoogle Scholar
Goldstein, H. (1950), Classical Mechanics. Reading, MA: Addison-Wesley Publishing Co.Google Scholar
Guckenheimer, J. and Homes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag.CrossRefGoogle Scholar
Guglielmo, I. and Pettini, M. (1996), “Regular and Chaotic Quantum Motions”, Physics Letters A 212: 29–38.Google Scholar
Gutzwiller, M. C. (1990), Chaos in Classical and Quantum Mechanics. New York: Springer-Verlag.CrossRefGoogle Scholar
Gutzwiller, M. C. (1992), “Quantum Chaos”, Scientific American 226(1): 78–84.CrossRefGoogle Scholar
Haake, F. (1991), Quantum Signatures of Chaos. Berlin: Springer-Verlag.Google Scholar
Habib, S., Shizume, K., and Zurek, W. H. (1998), “Decoherence, Chaos, and the Correspondence Principle”, Physical Review Letters 80: 4361–4365.CrossRefGoogle Scholar
Hirsch, M. W. and Smale, S. (1974), Differential Equations, Dynamical Systems, and Linear Algebra. New York: Academic Press.Google Scholar
Holland, P. R. (1993), The Quantum Theory of Motion. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Holland, P. R. (1996), “Is Quantum Mechanics Universal?”, in Cushing, J. T., Fine, A., and Goldstein, S. (eds.), Bohmian Mechanics and Quantum Theory: An Appraisal. Dordrecht: Kluwer Academic Publishers, 99–110.CrossRefGoogle Scholar
Iacomelli, G. and Pettini, M. (1996), “Regular and Chaotic Quantum Motions”, Physics Letters A 212:29–38.CrossRefGoogle Scholar
Izrailev, F. M. (1990), “Simple Models of Quantum Chaos: Spectrum and Eigenfunctions”, Physics Reports 196: 299–392.CrossRefGoogle Scholar
Konkel, S. and Makowski, A. J. (1998), “Regular and Chaotic Causal Trajectories for the Bohm Potential in a Restricted Space”, Physics Letters A 238: 95–100.CrossRefGoogle Scholar
Kronz, F. M. (1998), “Nonseparability and Quantum Chaos”, Philosophy of Science, this issue.CrossRefGoogle Scholar
Lichtenberg, A. J. and Lieberman, M. A. (1992), Regular and Chaotic Dynamics. New York: Springer-Verlag.CrossRefGoogle Scholar
Malik, Z. and Dewdney, C. (1995), “Quantum Mechanics, Chaos and the Bohm Theory”, University of Portsmouth preprint.Google Scholar
Nakamura, K. (1993), Quantum Chaos. Cambridge: Cambridge University Press.Google Scholar
Pan, J.-W., Bouwmeester, D., Weinfurter, H., and Zeilinger, A. (1998), “Experimental Entanglement Swapping: Entangling Photons That Never Interacted”, Physical Review Letters 80: 3891–3894.CrossRefGoogle Scholar
Parmenter, R. H. and Valentine, R. W. (1995), “Deterministic Chaos and the Causal Interpretation of Quantum Mechanics”, Physics Letters A 201: 1–8. [Erratum 213(1996), 319.]CrossRefGoogle Scholar
Parmenter, R. H. and Valentine, R. W. (1997), “Chaotic Causal Trajectories Associated With a Single Stationary State of a System of Noninteracting Particles”, Physics Letters A 227: 5–14.CrossRefGoogle Scholar
Pattanayak, A. K. and Brumer, P. (1997a), “Chaos and Lyapunov Exponents in Classical and Quantal Distribution Dynamics”, Physical Review E 56: 5174–5177.CrossRefGoogle Scholar
Pattanayak, A. K. and Brumer, P. (1997b), “Exponentially Rapid Decoherence of Quantum Chaotic Systems”, Physical Review Letters 79: 4131–4134.CrossRefGoogle Scholar
Rasband, S. N. (1990), Chaotic Dynamics of Nonlinear Systems. New York: John Wiley & Sons.Google Scholar
Rohrlich, F. (1988), “Pluralistic Ontology and Theory Reduction in the Physical Sciences”, British Journal for the Philosophy of Science 39: 295–312.CrossRefGoogle Scholar
Schack, R. and Caves, C. M. (1993), “Hypersensitivity to Perturbations in the Quantum Baker's Map”, Physical Review Letters 71: 525–528.CrossRefGoogle ScholarPubMed
Schack, R., D'Ariano, G. M., and Caves, C. M. (1994), “Hypersensitivity to Perturbation in the Quantum Kicked Top”, Physical Review E 50: 972–987.CrossRefGoogle ScholarPubMed
Schwengelbeck, U. and Faisal, F. H. M. (1995), “Definition of Lyapunov Exponents and KS Entropy in Quantum Dynamics”, Physics Letters A 199: 281–286.CrossRefGoogle Scholar
Sengupta, S. and Chattaraj, P. K. (1996), “The Quantum Theory of Motion and Signatures of Chaos in the Quantum Behaviour of a Classically Chaotic System”, Physics Letters A 215: 119–127.CrossRefGoogle Scholar
Spiller, T. P. and Ralph, J. F. (1994), “The Emergence of Chaos in an Open Quantum System”, Physics Letters A 194: 235–240.CrossRefGoogle Scholar
Tabor, M. (1989), Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: John Wiley & Sons.Google Scholar
Valentini, A. (2000), Pilot-Wave Theory: An Alternative Approach to Modern Physics. Berlin: Springer-Verlag (forthcoming).Google Scholar
Wilkinson, P. B., Fromhold, T. M., Eaves, L., Sheard, F. W., Miurs, N., and Takamasu, T. (1996), “Observation of ‘Scarred’ Wavefunctions in a Quantum Well with Chaotic Electron Dynamics”, Nature 380: 608–610.CrossRefGoogle Scholar