Element contents
The Classical–Quantum Correspondence
Published online by Cambridge University Press: 16 December 2022
Summary
This Element provides an entry point for philosophical engagement with quantization and the classical limit. It introduces the mathematical tools of C*-algebras as they are used to compare classical and quantum physics. It then employs those tools to investigate philosophical issues surrounding theory change in physics. It discusses examples in which quantization bears on the topics of reduction, structural continuity, analogical reasoning, and theory construction. In doing so, it demonstrates that the precise mathematical tools of algebraic quantum theory can aid philosophers of science and philosophers of physics.
- Type
- Element
- Information
- Online ISBN: 9781009043557Publisher: Cambridge University PressPrint publication: 19 January 2023
References
Alfsen, E. and Shultz, F. (2001). State Spaces of Operator Algebras. Birkhauser, Boston, MA.CrossRefGoogle Scholar
Aliprantis, C. and Border, K. (1999). Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin.Google Scholar
Anzai, H. and Kakutani, S. (1943). Bohr compactifications of a locally compact abelian group I. Proceedings of the Imperial Academy, 19(9):476–480.Google Scholar
Ashtekar, A. and Isham, C. (1992). Inequivalent observable algebras: Another ambiguity in field quantisation. Physics Letters B, 274(3-4):393–398.Google Scholar
Baez, J., Segal, I., and Zhou, Z. (1992). Introduction to Algebraic and Constructive Quantum Field Theory. Princeton University Press, Princeton, NJ.CrossRefGoogle Scholar
Barrett, T. W. (2020). Structure and equivalence. Philosophy of Science, 87(5):1184–1196.Google Scholar
Batterman, R. (1991). Chaos, quantization, and the correspondence principle. Synthese, 89(2):189–227.CrossRefGoogle Scholar
Batterman, R. (1995). Theories between theories: Asymptotic limiting intertheoretic relations. Synthese, 103(2):171–201.Google Scholar
Batterman, R. (1997). “Into a mist”: Asymptotic theories on a caustic. Studies in the History and Philosophy of Modern Physics, 28(3):395–413.CrossRefGoogle Scholar
Batterman, R. (2002). The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence. Oxford University Press, Oxford.Google Scholar
Batterman, R. (2005). Response to Belot’s “Whose devil? Which details?” Philosophy of Science, 72(1):154–163.Google Scholar
Batterman, R. (2018). Autonomy of theories: An explanatory problem. Nous, 52(4):858–873.CrossRefGoogle Scholar
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D. (1978a). Deformation theory and quantization. I. Deformations of symplectic structures. Annals of Physics, 111(1):61–110.CrossRefGoogle Scholar
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D. (1978b). Deformation theory and quantization. II. Physical applications. Annals of Physics, 111(1):111–151.CrossRefGoogle Scholar
Belov-Kanel, A., Elishev, A., and Yu, J.-T. (2021). On automorphisms of the tame polynomial automorphism group in positive characteristic. https://arxiv.org/abs/2103.12784.Google Scholar
Berry, M. (1994). Asymptotics, singularities and the reduction of theories. In Skyrms, B., Prawitz, D., and Westerståhl, D. (eds.), Logic, Methodology and Philosophy of Science, IX: Proceedings of the Ninth International Congress of Logic, Methodology, and Philosophy of Science, Uppsala, Sweden Aug 7–14, 1991, pages 597–607.Google Scholar
Bieliavsky, P. and Gayral, V. (2015). Deformation Quantization for Actions of Kählerian Lie Groups, volume 236 of Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI.Google Scholar
Binz, E., Honegger, R., and Rieckers, A. (2004a). Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space. Journal of Mathematical Physics, 45(7):2885–2907.Google Scholar
Binz, E., Honegger, R., and Rieckers, A. (2004b). Field-theoretic Weyl quantization as a strict and continuous deformation quantization. Annales de l’Institut Henri Poincaré, 5:327–346.Google Scholar
Bokulich, A. (2008). Reexamining the Quantum-Classical Relation: Beyond Reductionism and Pluralism. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Bordemann, M. and Waldmann, S. (1998). Formal GNS construction and states in deformation quantization. Communications in Mathematical Physics, 195:549–583.CrossRefGoogle Scholar
Boyd, R. N. (1973). Realism, underdetermination, and a causal theory of evidence. Noûs, 7(1):1.CrossRefGoogle Scholar
Boyd, R. (1989). What realism implies and what it does not. Dialectica, 43(1–2):5–29.CrossRefGoogle Scholar
Brading, K. and Landry, E. (2006). Scientific structuralism: Presentation and representation. Philosophy of Science, 73(5):571–581.Google Scholar
Bratteli, O. and Robinson, D. (1987). Operator Algebras and Quantum Statistical Mechanics, volume 1. Springer, New York.CrossRefGoogle Scholar
Bratteli, O. and Robinson, D. (1996). Operator Algebras and Quantum Statistical Mechanics, volume 2. Springer, New York.Google Scholar
Browning, T. and Feintzeig, B. (2020). Classical limits of symmetry invariant states and the choice of algebra for quantization. Letters in Mathematical Physics, 110(7):1835–1860.CrossRefGoogle Scholar
Browning, T., Feintzeig, B., Gates, R., Librande, J., and Soiffer, R. (2020). Classical limits of unbounded quantities by strict quantization. Journal of Mathematical Physics, 61(11):112305.Google Scholar
Brunetti, R., Dütsch, M., and Fredenhagen, K. (2009). Perturbative algebraic quantum field theory and the renormalization groups. Advances in Theoretical and Mathematical Physics, 13(5):1541–1599.Google Scholar
Brunetti, R., Fredenhagen, K., and Ribeiro, P. L. (2019). Algebraic structure of classical field theory: Kinematics and linearized dynamics for real scalar fields. Communications in Mathematical Physics, 368(2):519–584.Google Scholar
Buchholz, D. (1996a). Phase space properties of local observables and structure of scaling limits. Annales de l’Institut Henri Poincaré, 64(4):433–459.Google Scholar
Buchholz, D. (1996b). Quarks, gluons, colour: Facts or fiction. Nuclear Physics B, 469(1-2):333–353.Google Scholar
Buchholz, D. (2017). The resolvent algebra for oscillating lattice systems: Dynamics, ground and equilibrium states. Communications in Mathematical Physics, 353(2):691–716.Google Scholar
Buchholz, D. (2018). The resolvent algebra of non-relativistic Bose fields: Observables, dynamics and states. Communications in Mathematical Physics, 362(3):949–981.Google Scholar
Buchholz, D. and Fredenhagen, K. (2020a). A C*-algebraic approach to interacting quantum field theories. Communications in Mathematical Physics, 377(2):947–969.Google Scholar
Buchholz, D. and Fredenhagen, K. (2020b). Classical dynamics, arrow of time, and genesis of the Heisenberg commutation relations. Expositiones Mathematicae, 38(2):150–167.CrossRefGoogle Scholar
Buchholz, D. and Grundling, H. (2008). The resolvent algebra: A new approach to canonical quantum systems. Journal of Functional Analysis, 254(11):2725–2779.Google Scholar
Buchholz, D. and Grundling, H. (2015). Quantum systems and resolvent algebras. In Blanchard, B. and Fröhlich, J. (eds.), The Message of Quantum Science: Attempts Towards a Synthesis, pages 33–45. Springer, Berlin.Google Scholar
Buchholz, D. and Størmer, E. (2015). Superposition, transition probabilities and primitive observables in infinite quantum systems. Communications in Mathematical Physics, 339(1):309–325.Google Scholar
Buchholz, D. and Verch, R. (1995). Scaling algebras and renormalization group in algebraic quantum field theory. Reviews in Mathematical Physics, 7(8):1195.Google Scholar
Buchholz, D. and Verch, R. (1998). Scaling algebras and renormalization group in algebraic quantum field theory. II. Instructive examples. Reviews in Mathematical Physics, 10(6):775–800.Google Scholar
Bueno, O. (1999). What is structural empiricism? Scientific change in an empiricist setting. Erkenntnis, 50(1):59–85.Google Scholar
Butterfield, J. (2011a). Emergence, reduction, and supervenience: A varied landscape. Foundations of Physics, 41:920–959.CrossRefGoogle Scholar
Butterfield, J. (2011b). Less is different: Emergence and reduction reconciled. Foundations of Physics, 41:1065–1135.CrossRefGoogle Scholar
Callender, C. and Huggett, N. (eds.) (2001). Physics Meets Philosophy at the Planck Scale. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Cao, T. Y. (2003). Structural realism and the interpretation of quantum field theory. Synthese, 136(1):3–24.Google Scholar
Clifton, R. and Halvorson, H. (2001). Are Rindler quanta real? Inequivalent particle concepts in quantum field theory. British Journal for the Philosophy of Science, 52(3):417–470.Google Scholar
Crowther, K. (2021). Defining a crisis: The role of principles in the search for a theory of quantum gravity. Synthese, 198 (Suppl 14), 3489–3516.CrossRefGoogle Scholar
Curd, M., Cover, J., and Pincock, C. (2013). Philosophy of science: The central issues. 2nd edition. W. W. Norton, New York.Google Scholar
Darrigol, O. (1992). From c-Numbers to q-Numbers. University of California Press, Berkeley.Google Scholar
Dewar, N. (2017). Interpretation and equivalence; or, equivalence and interpretation. Unpublished. http://philsci-archive.pitt.edu/13234/1/EI-2.pdf.Google Scholar
Dewar, N. (2022). Structure and Equivalence. Cambridge University Press, Cambridge, UK.Google Scholar
Dirac, P. (1930). The Principles of Quantum Mechanics. Oxford University Press, Oxford.Google Scholar
Douglas, M. (2004). Report on the status of the Yang-Mills Millenium Prize Problem. www.claymath.org/sites/default/files/ym2.pdf.Google Scholar
Duncan, A. and Janssen, M. (2007a). On the verge of Umdeutung in Minnesota: Van Vleck and the correspondence principle. Part one. Archive for History of Exact Sciences, 61(6):553–624.CrossRefGoogle Scholar
Duncan, A. and Janssen, M. (2007b). On the verge of Umdeutung in Minnesota: Van Vleck and the correspondence principle. Part two. Archive for History of Exact Sciences, 61(6):625–671.Google Scholar
Duncan, A. and Janssen, M. (2012). (Never) Mind your p’s and q’s: Von Neumann versus Jordan on the foundations of quantum theory. The European Physical Journal H, 38(2):175–259.Google Scholar
Duncan, A. and Janssen, M. (2019). Constructing Quantum Mechanics: Volume 1: The Scaffold: 1900–1923. Oxford University Press, Oxford.Google Scholar
Duncan, A. and Janssen, M. (2022). Quantization conditions, 1900–1927. In Freire, O. (ed.), The Oxford Handbook of the History of Quantum Interpretations, pages 76–94. Oxford University Press, Oxford.Google Scholar
Emch, G. (1972). Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley, New York.Google Scholar
Emch, G. (1983). Geometric dequantization and the correspondence problem. International Journal of Theoretical Physics, 22(5):397–420.Google Scholar
Fannes, M. and Verbeure, A. (1974). On the time evolution automorphisms of the CCR-algebra for quantum mechanics. Communications in Mathematical Physics, 35(3):257–264.Google Scholar
Fedosov, B. (1996). Deformation Quantization and Index Theory. Akademie Verlag, Berlin.Google Scholar
Feintzeig, B. (2015). On broken symmetries and classical systems. Studies in the History and Philosophy of Modern Physics, 52, Part B:267–273.CrossRefGoogle Scholar
Feintzeig, B. (2016). Unitary inequivalence in classical systems. Synthese, 193(9):2685–2705.Google Scholar
Feintzeig, B. (2017). On theory construction in physics: Continuity from classical to quantum. Erkenntnis, 82(6):1195–1210.Google Scholar
Feintzeig, B. (2018a). On the choice of algebra for quantization. Philosophy of Science, 85(1):102–125.Google Scholar
Feintzeig, B. (2018b). The classical limit of a state on the Weyl algebra. Journal of Mathematical Physics, 59:112102.Google Scholar
Feintzeig, B. (2018c). Toward an understanding of parochial observables. British Journal for the Philosophy of Science, 69(1):161–191.CrossRefGoogle Scholar
Feintzeig, B. (2022). Reductive explanation and the construction of quantum theories. British Journal for Philosophy of Science, 73(2):457–486.Google Scholar
Feintzeig, B., Manchak, J., Rosenstock, S., and Weatherall, J. (2019). Why be regular? Part I. Studies in the History and Philosophy of Modern Physics, 65:122–132.Google Scholar
Feintzeig, B. and Weatherall, J. (2019). Why be regular? Part II. Studies in the History and Philosophy of Modern Physics, 65:133–144.Google Scholar
Feintzeig, B. H. (2020). The classical limit as an approximation. Philosophy of Science, 87(4):612–539.Google Scholar
Feintzeig, B. H., Librande, J., and Soiffer, R. (2021). Localizable particles in the classical limit of quantum field theory. Foundations of Physics, 51(2):49.CrossRefGoogle Scholar
Feyerabend, P. (1962). Explanation, reduction, and empiricism. In Feigl, H. and Maxwell, G. (eds.), Minnesota Studies in the Philosophy of Science: Scientific Explanation, Space, and Time, volume 3, pages 28–97. University of Minnesota Press, Minneapolis.Google Scholar
Fine, A. (1988). The Shaky Game: Einstein, Realism, and the Quantum Theory. University of Chicago Press, Chicago.Google Scholar
Forman, P. (1971). Weimar culture, causality, and quantum theory, 1918–1927: Adaptation by German physicists and mathematicians to a hostile intellectual environment. Historical Studies in the Physical Sciences, 3:1–115.Google Scholar
Fraser, D. (2020a). The development of renormalization group methods for particle physics: Formal analogies between classical statistical mechanics and quantum field theory. Synthese, 197:3027–3063.CrossRefGoogle Scholar
Fraser, D. (2020b). The non-miraculous success of formal analogies in quantum theories. In French, S. and Saatsi, J., editors, Scientific Realism and the Quantum. Oxford University Press, Oxford.Google Scholar
Fraser, D. and Koberinski, A. (2016). The Higgs mechanism and superconductivity: A case study of formal analogies. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 55:72–91.Google Scholar
Fredenhagen, K. and Rejzner, K. (2015). Perturbative construction of models in algebraic quantum field theory. In Brunetti, R., Dappiaggi, C., Fredenhagen, K., and Yngvason, J., editors, Advances in Algebraic Quantum Field Theory, pages 31–74. Springer, Cham.Google Scholar
Fredenhagen, K. and Rejzner, K. (2016). Quantum field theory on curved spacetimes: Axiomatic framework and examples. Journal of Mathematical Physics, 57(3):031101.Google Scholar
French, S. (2012). Unitary inequivalence as a problem for structural realism. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 43(2):121–136.Google Scholar
French, S. and Ladyman, J. (2003). Remodelling structural realism: Quantum physics and the metaphysics of structure. Synthese, 136(1):31–56.Google Scholar
French, S. and Saatsi, J. (2006). Realism about structure: The semantic view and nonlinguistic representations. Philosophy of Science, 73(5):548–559.Google Scholar
Frigg, R. and Votsis, I. (2011). Everything you always wanted to know about structural realism but were afraid to ask. European Journal for Philosophy of Science, 1(2):227–276.CrossRefGoogle Scholar
Gotay, M. J. (1980). Functorial geometric quantization and Van Hove’s theorem. International Journal of Theoretical Physics, 19(2):139–161.Google Scholar
Gotay, M. J. (1999). On the Groenewold–Van Hove problem for
. Journal of Mathematical Physics, 40(4):2107–2116.Google Scholar
Gracia-Bondía, J. M. (1992). Generalized Moyal quantization on homogeneous symplectic spaces. Contemporary Mathematic, 134:93–114.Google Scholar
Groenewold, H. (1946). On the principles of elementary quantum mechanics. Physica, 12(7):405–460.CrossRefGoogle Scholar
Grundling, H. (1997). A group algebra for inductive limit groups: Continuity problems of the canonical commutation relations. Acta Applicandae Mathematicae, 46:107–145.Google Scholar
Grundling, H. and Neeb, K.-H. (2009). Full regularity for a C*-algebra of the Canonical Commutation Relations. Reviews in Mathematical Physics, 21(5):587–613.Google Scholar
Gutt, S. (1983). An explicit
-product on the cotangent bundle of a Lie group. Letters in Mathematical Physics, 7(3):249–258.Google Scholar
Haag, R. (1993). Local quantum physics and models. Communications in Mathematical Physics, 155(1):199–204.Google Scholar
Haag, R. and Kastler, D. (1964). An algebraic approach to quantum field theory. Journal of Mathematical Physics, 5(7):848–861.Google Scholar
Haag, R. and Ojima, I. (1996). On the problem of defining a specific theory within the frame of local quantum physics. Annales de L’Institut Henri Poincare-physique Theorique, 64(4):385–393.Google Scholar
Halvorson, H. (2001a). On the nature of continuous physical quantities in classical and quantum mechanics. Journal of Philosophical Logic, 30:27–50.Google Scholar
Halvorson, H. (2001b). Reeh–Schlieder defeats Newton–Wigner: On alternative localization schemes in relativistic quantum field theory. Philosophy of Science, 68(1):111–133.Google Scholar
Halvorson, H. (2004). Complementarity of representations in quantum mechanics. Studies in the History and Philosophy of Modern Physics, 35(1):45–56.Google Scholar
Halvorson, H. (2007). Algebraic quantum field theory. In Butterfield, J. and Earman, J., editors, Handbook of the Philosophy of Physics, volume 1, pages 731–864. Elsevier, New York.CrossRefGoogle Scholar
Halvorson, H. (2016). Scientific theories. In Humphreys, P., editor, The Oxford Handbook of Philosophy of Science. Oxford University Press, Oxford.Google Scholar
Halvorson, H. and Clifton, R. (2002). No place for particles in relativistic quantum theories? Philosophy of Science, 69(1):1–28.Google Scholar
Hesse, M. (1952). Operational definition and analogy in physical theories. British Journal for Philosophy of Science, 2(8):281–294.CrossRefGoogle Scholar
Hesse, M. (1953). Models in physics. British Journal for Philosophy of Science, 4(15):198–214.Google Scholar
Hesse, M. (1961). Forces and Fields: The Concept of Action at a Distance in the History of Physics. Dover, Mineola.Google Scholar
Hesse, M. (1970). Models and Analogies in Science. University of Notre Dame Press Notre Dame, IN.Google Scholar
Hewitt, E. (1953). Linear functionals on almost periodic functions. American Mathematical Society, 74(2):303–322.Google Scholar
Honegger, R. and Rieckers, A. (2005). Some continuous field quantizations, equivalent to the C*-Weyl quantization. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 41(1):113–138.Google Scholar
Honegger, R., Rieckers, A., and Schlafer, L. (2008). Field-theoretic Weyl deformation quantization of enlarged Poisson algebras. Symmetry, Integrability and Geometry: Methods and Applications, 4:047–084.Google Scholar
Howard, D. (1986). What makes a classical concept classical? Toward a reconstruction of Niels Bohr’s philosophy of physics. In Niels Bohr and Contemporary Philosophy, pages 201–229. Kluwer Academic Publishers, New York.Google Scholar
Hudetz, L. (2019). Definable categorical equivalence. Philosophy of Science, 86(1):47–75.CrossRefGoogle Scholar
Jacobs, C. (2021). The coalescence approach to inequivalent representation: Pre-QM
parallels. The British Journal for the Philosophy of Science. DOI: https://doi.org/10.1086/715108.Google Scholar
Jaffe, A. and Witten, E. (2000). Quantum Yang–Mills theory. www.claymath.org/sites/default/files/yangmills.pdf.Google Scholar
Jammer, M. (1966). The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York.Google Scholar
Kadison, R. and Ringrose, J. (1997). Fundamentals of the Theory of Operator Algebras. American Mathematical Society, Providence, RI.Google Scholar
Kaschek, D., Neumaier, N., and Waldmann, S. (2009). Complete positivity of Rieffel’s deformation quantization by actions of
. Journal of Noncommutative Geometry, 3(3):361–375.Google Scholar
Kay, B. (1979). A uniqueness result in the Segal-Weinless approach to linear Bose fields. Journal of Mathematical Physics, 20(8):1712–3.Google Scholar
Kay, B. (1985). The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes. Communications in Mathematical Physics, 100(1):57–81.Google Scholar
Kay, B. and Wald, R. (1991). Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon. Physics Reports, 207(2):49–136.CrossRefGoogle Scholar
Kirchberg, E. and Wasserman, S. (1995). Operations on continuous bundles of C*-algebras. Mathematische Annalen, 303(4):677–697.Google Scholar
Kontsevich, M. (2003). Deformation quantization of Poisson manifolds. Letters in Mathematical Physics, 66(3):157–216.Google Scholar
Kuhn, T. (1970[1962]). The Structure of Scientific Revolutions. 2nd edition. University of Chicago Press, Chicago.Google Scholar
Kuhn, T. S. (1984). Revisiting Planck. Historical Studies in the Physical Sciences, 14(2):231–252.Google Scholar
Kuhn, T. S. (1987). Black-Body Theory and the Quantum Discontinuity, 1894–1912. The University of Chicago Press, Chicago.Google Scholar
Ladyman, J. (1998). What is structural realism? Studies in History and Philosophy of Science Part A, 29(3):409–424.Google Scholar
Landsman, N. P. (1990a). C*-algebraic quantization and the origin of topological quantum effects. Letters in Mathematical Physics, 20:11–18.Google Scholar
Landsman, N. P. (1990b). Quantization and superselection sectors I. Transformation group C*-algebras. Reviews in Mathematical Physics, 2(1):45–72.Google Scholar
Landsman, N. P. (1993). Strict deformation quantization of a particle in external gravitational and Yang–Mills fields. Journal of Geometry and Physics, 12(2):93–132.Google Scholar
Landsman, N. P. (1998a). Mathematical Topics between Classical and Quantum Mechanics. Springer, New York.Google Scholar
Landsman, N. P. (1998b). Twisted Lie group C*-algebras as strict quantizations. Letters in Mathematical Physics, 46:181–188.Google Scholar
Landsman, N. P. (1999). Lie groupoid C*-algebras and Weyl quantization. Communications in Mathematical Physics, 206(2):367–381.CrossRefGoogle Scholar
Landsman, N. P. (2003). Quantization as a functor. In Voronov, T., editor, Quantization, Poisson Brackets and beyond, pages 9–24. Contemporary Mathematics, 315, AMS. DOI: https://doi.org/10.1090/conm/315.Google Scholar
Landsman, N. P. (2007). Between classical and quantum. In Butterfield, J. and Earman, J., editors, Handbook of the Philosophy of Physics, volume 1, pages 417–553. Elsevier, New York.Google Scholar
Landsman, N. P. (2013). Spontaneous symmetry breaking in quantum systems: Emergence or reduction? Studies in the History and Philosophy of Modern Physics, 44(4):379–394.Google Scholar
Landsman, N. P. (2017). Foundations of Quantum Theory: From Classical Concepts to Operator Algebras. Springer, Cham.Google Scholar
Landsman, N. P. and Reuvers, R. (2013). A flea on Schrödinger’s cat. Foundations of Physics, 43(3):373–407.Google Scholar
Laudan, L. (1980). Why was the logic of discovery abandoned? In Nickles, T., editor, Scientific Discovery, Logic, and Rationality, pages 173–183. Dordrecht Reidel, Dordrecht.Google Scholar
Laudan, L. (1981). A confutation of convergent realism. Philosophy of Science, 48(1):19–49.Google Scholar
Lee, R.-Y. (1976). On the C*-algebras of operator fields. Indiana University Mathematics Journal, 25(4):303.Google Scholar
Mackey, G. (1949). A theorem of Stone and von Neumann. Duke Mathematical Journal, 16(2):313–326.Google Scholar
Mackey, G. W. (1968). Induced Representations of Groups and Quantum Mechanics. W. A. Benjamin, Inc., New York.Google Scholar
Malament, D. (1996). In defense of dogma – why there cannot be a relativistic quantum mechanical theory of (localizable) particles. In Clifton, R., editor, Perspectives on Quantum Reality. Kluwer Academic Publishers, Amsterdam.Google Scholar
Manuceau, J., Sirugue, M., Testard, D., and Verbeure, A. (1974). The smallest C*-algebra for the canonical commutation relations. Communications in Mathematical Physics, 32:231–243.Google Scholar
Marsden, J. and Weinstein, M. (1974). Reduction of symplectic manifolds with symmetry. Reports on Mathematical Physics, 5(1):121–30.Google Scholar
Martinez, A. (2002). An Introduction to Semiclassical and Microlocal Analysis. Springer, New York.Google Scholar
Mazzucchi, S. (2009). Mathematical Feynman Path Integrals and Their Applications. World Scientific, Hackensack, NJ.Google Scholar
Moyal, J. E. (1949). Quantum mechanics as a statistical theory. Mathematical Proceedings of the Cambridge Philosophical Society, 45(1):99–124.Google Scholar
Murray, F. J. and von Neumann, J. (1936). On rings of operators. The Annals of Mathematics, 37(1):116.Google Scholar
Nagel, E. (1998). Issues in the logic of reductive explanations. In Curd, M. and Cover, J., editors, Philosophy of Science: The Central Issues, pages 905–921. W. W. Norton & Co., New York.Google Scholar
Nickles, T. (1973). Two concepts of intertheoretic reduction. Journal of Philosophy, 70:181–201.Google Scholar
Nickles, T. (1985). Beyond divorce: Current status of the discovery debate. Philosophy of Science, 52(2):177–206.Google Scholar
Peskin, M. E. and Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Perseus Books, New York.Google Scholar
Petz, D. (1990). An Invitation to the Algebra of Canonical Commutation Relations. Leuven University Press, Leuven.Google Scholar
Post, H. (1971). Correspondence, invariance, and heuristics: In praise of conservative induction. Studies in the History and Philosophy of Modern Science, 2(3):213–255.Google Scholar
Primas, H. (1998). Emergence in exact natural sciences. Acta Polytechnica Scandinavica, 91:83–98.Google Scholar
Psillos, S. (2001). Is structural realism possible? Philosophy of Science, 68(S3):S13–S24.Google Scholar
Radder, H. (1991). Heuristics and the generalized correspondence principle. British Journal for the Philosophy of Science, 42(2):195–226.Google Scholar
Reed, M. and Simon, B. (1975). Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, New York.Google Scholar
Reichenbach, H. (1938). Experience and Prediction. An Analysis of the Foundations and the Structure of Knowledge. University of Chicago Press, Chicago.Google Scholar
Rejzner, K. (2016). Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Springer, New York.Google Scholar
Rejzner, K. (2019). Locality and causality in perturbative algebraic quantum field theory. Journal of Mathematical Physics, 60(12):122301.Google Scholar
Rieffel, M. (1989). Deformation quantization of Heisenberg manifolds. Communications in Mathematical Physics, 122(4):531–562.Google Scholar
Rieffel, M. (1993). Deformation quantization for actions of
. Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI.Google Scholar
Rohrlich, F. (1990). There is good physics in theory reduction. Foundations of Physics, 20(11):1399–1412.Google Scholar
Romano, D. (2016). Bohmian classical limit in bounded regions. In Felline, L., Ledda, A., Paoli, F., and Rossanese, E., editors, New Directions in Logic and the Philosophy of Science, volume 3 of SILFS, pages 303–318. London, College Publications.Google Scholar
Rosaler, J. (2015a). “Formal” versus “empirical” approaches to quantum-classical reduction. Topoi, 34(2):325–338.Google Scholar
Rosaler, J. (2015b). Is de Broglie–Bohm theory specially equipped to recover classical behavior? Philosophy of Science, 82(5):1175–1187.Google Scholar
Rosaler, J. (2016). Interpretation neutrality in the classical domain of quantum theory. Studies in the History and Philosophy of Modern Physics, 53:54–72.Google Scholar
Rosaler, J. (2018). Generalized Ehrenfest relations, deformation quantization, and the geometry of inter-model reduction. Foundations of Physics, 48(3):355–385.Google Scholar
Rosenstock, S., Barrett, T., and Weatherall, J. (2015). On Einstein algebras and relativistic spacetimes. Studies in the History and Philosophy of Modern Physics, 52B:309–316.Google Scholar
Rosenstock, S. and Weatherall, J. (2016). A categorical equivalence between generalized holonomy maps on a connected manifold and principal connections on bundles over that manifold. Journal of Mathematical Physics, 57(10):102902.Google Scholar
Ruetsche, L. (2002). Interpreting quantum field theory. Philosophy of Science, 69(2):348–378.Google Scholar
Ruetsche, L. (2003). A matter of degree: Putting unitary inequivalence to work. Philosophy of Science, 70(5):1329–1342.Google Scholar
Ruetsche, L. (2006). Johnny’s so long at the ferromagnet. Philosophy of Science, 73(5):473–486.Google Scholar
Ruetsche, L. (2011). Interpreting Quantum Theories. Oxford University Press, New York.Google Scholar
Scheibe, E. (1986). The Comparison of scientific theories. Interdisciplinary Science Reviews, 11(2):148–152.Google Scholar
Segal, I. (1963). Mathematical Problems of Relativistic Physics. American Mathematical Society, Providence, RI.Google Scholar
Slawny, J. (1972). On factor representations and the C*-algebra of canonical commutation relations. Communications in Mathematical Physics, 24(2):151–170.Google Scholar
Stanford, P. K. (2003). No refuge for realism: Selective confirmation and the history of science. Philosophy of Science, 70(5):913–925.Google Scholar
Steeger, J. and Feintzeig, B. (2021a). Extensions of bundles of C*-algebras. Reviews in Mathematical Physics, 33(8):2150025.Google Scholar
Steeger, J. and Feintzeig, B. H. (2021b). Is the classical limit “singular”? Studies in History and Philosophy of Science Part A, 88:263–279.Google Scholar
Stein, H. (1981). “Subtler forms of matter” in the period following Maxwell. In Cantor, G. and Hodge, M., editors, Conceptions of Ether: Studies in the History of Ether Theories 1740–1900, pages 309–340. Cambridge University Press, Cambridge, UK.Google Scholar
Stein, H. (1987). After the Baltimore Lectures: Some philosophical reflections on the subsequent development of physics. In Achinstein, P. and Kargon, R., editors, Kelvin’s Baltimore Lectures and Modern Theoretical Physics, pages 375–398. Massachusetts Institute of Technology Press, Cambridge, MA.Google Scholar
Sternberg, S. (1977). Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang–Mills field. Proceedings of the National Academy of the Sciences, 74(12):5253–4.Google Scholar
Summers, S. (1999). On the Stone–von Neumann uniqueness theorem and its ramifications. In Redei, M. and Stoeltzner, M., editors, John von-Neumann and the Foundations of Quantum Physics, pages 135–152. Kluwer, Dordrecht.Google Scholar
Swanson, N. (2019). Deciphering the algebraic CPT theorem. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 68:106–125.Google Scholar
Teller, P. (1979). Quantum mechanics and the nature of continuous physical quantities. Journal of Philosophy, 76(7):345–361.Google Scholar
Thébault, K. P. Y. (2016). Quantization as a guide to ontic structure. The British Journal for the Philosophy of Science, 67(1):89–114.Google Scholar
van Hove, L. (1951). Sur le problème des relations entre les transformations unitaires de la mécanique quantique et les transformations canoniques dela mécanique classique. Academie Royale de Belgique, Bulletin Classe des Sciences Memoires, 5(37):610–620.Google Scholar
van Nuland, T. D. H. (2019). Quantization and the resolvent algebra. Journal of Functional Analysis, 277(8):2815–2838.Google Scholar
von Neumann, J. (1932). Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton, NJ. English translation published 1955.Google Scholar
Waldmann, S. (2005). States and representations in deformation quantization. Reviews in Mathematical Physics, 17(1):15–75.Google Scholar
Waldmann, S. (2010). Positivity in Rieffel’s strict deformation quantization. In XVIth International Congress on Mathematical Physics, Prague, Czech Republic, August 3–8,2009. With DVD, pages 509–513. World Scientific, Hackensack, NJ.Google Scholar
Waldmann, S. (2016). Recent developments in deformation quantization. In Finster, F., Kleiner, J., Röken, C., Tolksdorf, J., editors, Quantum Mathematical Physics. Birkhäuser, Cham.Google Scholar
Waldmann, S. (2019). Convergence of star products: From examples to a general framework. EMS Surveys in Mathematical Sciences, 6(1):1–31.Google Scholar
Wallace, D. (2001). Emergence of particles from bosonic quantum field theory. Unpublished. arXiv: quant-ph/0112149v1.Google Scholar
Wallace, D. (2006). In defence of naiveté: The conceptual status of Lagrangian quantum field theory. Synthese, 151(1):33–80.Google Scholar
Wallace, D. (2011). Taking particle physics seriously: A critique of the algebraic approach to quantum field theory. Studies in the History and Philosophy of Modern Physics, 42(2):116–125.Google Scholar
Weatherall, J. (2021). Why not categorical equivalence? In Madarász, Judit, Székely, Gergely, editors, Hajnal Andréka and István Németi on Unity of Science, p. 427–451, Springer Cham.CrossRefGoogle Scholar
Weatherall, J. O. (2019a). Part 1: Theoretical equivalence in physics. Philosophy Compass, 14(5).Google Scholar
Weatherall, J. O. (2019b). Part 2: Theoretical equivalence in physics. Philosophy Compass, 14(5).Google Scholar
Weinless, M. (1969). Existence and uniqueness of the vacuum for linear quantized fields. Journal of Functional Analysis, 4(3):350–379.Google Scholar
Weinstein, M. (1978). A universal phase space for particles in Yang–Mills fields. Letters in Mathematical Physics, 2:417–20.Google Scholar
Wigner, E. (1959). Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra. Academic Press, Cambridge, MA.Google Scholar
Williams, P. (2018). Scientific realism made effective. British Journal for Philosophy of Science, 70(1):209–237.Google Scholar
Worrall, J. (1989). Structural realism: The best of both worlds? Dialectica, 43(1–2):99–124.Google Scholar
Yaghmaie, A. (2021). Deformation quantization as an appropriate guide to ontic structure. Synthese, 198:10793–10815.Google Scholar
Zahar, E. (1983). Logic of discovery or psychology of invention? British Journal for Philosophy of Science, 34(3):243–61.Google Scholar
Zalamea, F. (2018). The twofold role of observables in classical and quantum kinematics. Foundations of Physics, 48(9):1061–1091.Google Scholar
Zworski, M. (2012). Semiclassical Analysis. American Mathematical Society, Providence, RI.Google Scholar
- 7
- Cited by