Hostname: page-component-7bb8b95d7b-lvwk9 Total loading time: 0 Render date: 2024-10-05T00:40:20.309Z Has data issue: false hasContentIssue false

Horizontal Models: From Bakers to Cats

Published online by Cambridge University Press:  01 January 2022

Abstract

At the center of quantum chaos research is a particular family of models known as quantum maps. These maps illustrate an important “horizontal” dimension to model construction that has been overlooked in the literature on models. Three ways in which quantum maps are being used to clarify the relationship between classical and quantum mechanics are examined. This study suggests that horizontal models may provide a new and fruitful framework for exploring intertheoretic relations.

Type
Research Article
Copyright
Copyright © 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.)

Footnotes

I would especially like to thank Jim Cushing, Steve Tomsovic, Jon Keating, and an anonymous referee for helpful comments on an earlier draft of this paper.

References

Arnold, Vladimir I., and Avez, André (1968), Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin, Inc.Google Scholar
Balazs, Nandor L., and Voros, André (1989), “The Quantized Baker’s Transformation”, The Quantized Baker’s Transformation 190:131.Google Scholar
Batterman, Robert (1991), “Chaos, Quantization, and the Correspondence Principle”, Chaos, Quantization, and the Correspondence Principle 89:189227.Google Scholar
Batterman, Robert (1995), “Theories Between Theories: Asymptotic Limiting Intertheoretic Relations”, Theories Between Theories: Asymptotic Limiting Intertheoretic Relations 103:171201.Google Scholar
Belot, Gordon (2000), “Chaos and Fundamentalism”, Chaos and Fundamentalism 67 (Proceedings): S454S465.Google Scholar
Belot, Gordon, and Earman, John (1997), “Chaos Out of Order: Quantum Mechanics, the Correspondence Principle and Chaos”, Chaos Out of Order: Quantum Mechanics, the Correspondence Principle and Chaos 28:147182.Google Scholar
Berry, Michael V. (1989), “Quantum Chaology, Not Quantum Chaos”, Quantum Chaology, Not Quantum Chaos 40:335336.Google Scholar
Berry, Michael V. (1991), “Some Quantum-to-Classical Asymptotics”, in Giannoni, Marie-Joya, Voros, André, and Zinn-Justin, Jean (eds.) Chaos and Quantum Physics (Les Houches Session LII). Amsterdam: North Holland, 251303.Google Scholar
Berry, Michael V., Balazs, Nandor L., Tabor, Michael, and Voros, André (1979), “Quantum Maps”, Quantum Maps 122:2663.Google Scholar
Bokulich, Alisa. (2001), Philosophical Perspectives on Quantum Chaos: Models and Interpretations. Ph. D. diss., University of Notre Dame.Google Scholar
Cartwright, Nancy (1999), The Dappled World: A Study of the Boundaries of Science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Cartwright Nancy, Towfic Shomar, and Suárez, Mauricio (1995), “The Tool-Box of Science”, in Herfel, William, Krajewski, Wladyslaw, Niiniluoto, Ilkka, and Wójcicki, Ryszard (eds.), Theories and Models in Scientific Processes (Poznan Studies in the Philosophy of the Sciences and the Humanities, 44). Amsterdam and Atlanta: Rodopoi, 137149.Google Scholar
Casati Giulio, Boris Chirikov, Izrailev, Felix, and Ford, Joseph (1979), “Stochastic Behaviour of a Quantum Pendulum Under a Periodic Perturbation”, Stochastic Behaviour of a Quantum Pendulum Under a Periodic Perturbation 93:334352.Google Scholar
Casati, Giulio, and Chirikov, Boris (1995), “The Legacy of Chaos in Quantum Mechanics”, in Casati, Giulio, and Chirikov, Boris (eds.), Quantum Chaos: Between Order and Disorder. Cambridge: Cambridge University Press, 353.CrossRefGoogle Scholar
Ford Joseph, Giorgio Mantica, and Ristow, Gerald (1991), “The Arnol’d Cat: Failure of the Correspondence Principle”, The Arnol’d Cat: Failure of the Correspondence Principle D 50:493520.Google Scholar
Ford, Joseph, and Mantica, Giorgio (1992), “Does Quantum Mechanics Obey the Correspondence Principle?”, Does Quantum Mechanics Obey the Correspondence Principle? 60:10861098.Google Scholar
Frisch, Mathias (1998), Theories, Models, and Explanation. Ph. D. diss., University of California, Berkeley.Google Scholar
Grempel Daniel, Richard Prange, and Fishman, Shmuel (1984), “Quantum Dynamics of a Nonintegrable System”, Quantum Dynamics of a Nonintegrable System A 29:16391647.Google Scholar
Hannay, John, and Berry, Michael V. (1980), “Quantization of Linear Maps on a Torus-Fresnel Diffraction by a Periodic Grating”, Quantization of Linear Maps on a Torus-Fresnel Diffraction by a Periodic Grating D 1:267290.Google Scholar
Hughes, R.I.G. (1997), “Models and Representation”, Models and Representation 64 (Proceedings): S325S336.Google Scholar
Izrailev, Felix (1990), “Simple Models of Quantum Chaos: Spectrum and Eigenfunctions”, Simple Models of Quantum Chaos: Spectrum and Eigenfunctions 196:299392.Google Scholar
Keating, Jonathan (1991), “Asymptotic Properties of the Periodic Orbits of the Cat Maps”, Asymptotic Properties of the Periodic Orbits of the Cat Maps 4:277307.Google Scholar
Leplin, Jarrett (1980), “The Role of Models in Theory Construction”, in Nickles, Thomas (ed.), Scientific Discovery, Logic and Rationality. Dordrecht: D. Reidel Publishing Company, 267283.CrossRefGoogle Scholar
Moore Fred, John Robinson, Bharucha, Cyrus, Sundaram, Bala, and Raizen, Mark (1995), “Atom Optics Realization of the Quantum δ-Kicked Rotor”, Atom Optics Realization of the Quantum δ-Kicked Rotor 75:45984601.Google Scholar
Morrison, Margaret, and Morgan, Mary (1999), “Models as Mediating Instruments”, in Morgan, Mary and Morrison, Margaret (eds.), Models as Mediators: Perspectives on Natural and Social Science. Cambridge: Cambridge University Press, 1037.CrossRefGoogle Scholar
Morton, Adam (1993), “Mathematical Models: Questions of Trustworthiness”, Mathematical Models: Questions of Trustworthiness 44:659674.Google Scholar
Nagel, Ernest ([1961] 1979), The Structure of Science: Problems in the Logic of Scientific Explanation. Indianapolis: Hackett Publishing.Google Scholar
Nickles, Thomas (1973), “Two Concepts of Intertheoretic Reduction”, Two Concepts of Intertheoretic Reduction 70(7): 181201.Google Scholar
O’Connor, Patrick, and Tomsovic, Steven (1991), “The Unusual Nature of the Quantum Baker’s Transformation”, The Unusual Nature of the Quantum Baker’s Transformation 207:218264.Google Scholar
O’Connor Patrick, Steven Tomsovic, and Heller, Eric (1992), “Semiclassical Dynamics in the Strongly Chaotic Regime: Breaking the Log Time Barrier”, Semiclassical Dynamics in the Strongly Chaotic Regime: Breaking the Log Time Barrier D 55:340357.Google Scholar
Redhead, Michael (1980), “Models in Physics”, Models in Physics 31:145163.Google Scholar
Rohrlich, Fritz (1988), “Pluralistic Ontology and Theory Reduction in the Physical Sciences”, Pluralistic Ontology and Theory Reduction in the Physical Sciences 39:295312.Google Scholar
Saraceno, Marcos (1989), “Classical Structures in the Quantized Baker Transfomation”, Classical Structures in the Quantized Baker Transfomation 190:3760. Reprinted in Giulio Casati and Boris Chirikov (1995), Quantum Chaos: Between Order and Disorder. Cambridge: Cambridge University Press, 483–506.Google Scholar
Sepúlveda Miguel-Angel, Steven Tomsovic, and Heller, Eric (1992), “Semiclassical Propagation: How Long Can It Last?”, Semiclassical Propagation: How Long Can It Last? 69:402405. Reprinted in Giulio Casati, and Boris Chirikov (1995), Quantum Chaos: Between Order and Disorder. Cambridge: Cambridge University Press, 447–450.Google Scholar
Tomsovic, Steven, and Heller, Eric (1993), “Long-Time Semiclassical Dynamics of Chaos: The Stadium Billiard”, Long-Time Semiclassical Dynamics of Chaos: The Stadium Billiard E 47:282299.Google Scholar