Abstract
A statistical theory (ST) is based on a formalized structure in which peculiar classes of objects are interpreted as representing the primitive physical concepts of states, consisting of classes of preparations of individual samples of the physical entity under well defined and repeatable conditions; observables or physical measurable quantities (or magnitudes); a probability function.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aerts, D.: 1983, ‘Classical theories and non classical theories as special cases of a more general theory’, J. Math. Phys. 24, 2441.
Beitrametti, E. and Cassinelli, G.: 1976, ‘Logical and mathematical structures of quantum mechanics’, Rivista Nuovo Cimento 6, 321.
Beitrametti, E. and Cassinelli, G.: 1981, The logic of Quantum Mechanics, Addison-Wesley, Reading, Mass.
Birkhoff, G. and Von Neumann, J.: 1936, ‘The logic of quantum mechanies’, Ann. Math. 37, 823.
Bub, J.: 1973, ‘On the completeness of Quantum Mechanics’, in Contemporary Research in the Foundations and Philosophy of Quantum Theory (ed. by C.A. Hooker), Reidel, Dordrecht
Bub, J.: 1974, The Interpretation of Quantum Mechanics, Reidel, Dordrecht
Cattaneo, G., Dalla Chiara, M.L., and Giuntini, R.: 1991, ‘Fuzzy-intuitionistic quantum logics’, in print in Studia Logica.
Cattaneo, G., Oalla Pozza, C., Garola, C., and Nisticó, G.: 1988, ‘On the logical foundations of the Jauch Piron approach to quantum physics’, Int. J. Theor. Phys. 27, 1313.
Cattaneo, G., Garola, C., and Nisticó, G.: 1989, ‘Preparation—effect versus question—proposition structures’, Phys. Essays 2,197.
Cattaneo, G. and Giuntini, R.: 1991, ’solution of two open problems on Hilbert unsharp quantum physics’, preprint
Cattaneo, G., and Marino, G.: 1988, ‘Non usual orthocomplementations on partially ordered sets and fuzziness’, Fuzzy Sets Syst. 25, 107.
Cattaneo, G. and Nisticó, G.: 1985, ‘Complete effect-preparation structures: Attempt of an unification of two different approaches to axiomatic quantum mechanics’, Il Nuovo Cim. 908, 161.
Cattaneo, G. and Nisticó, G.: 1989, ‘Brouwer-Zadeh posets and three vldued Lukasiewicz posets’, Fuzzy Sets Syst. 33, 165.
Dalla Chiara, M.L. and Giuntini, R.: 1989, ‘Paraconsistent quantum logics’, Found. Phys. 19, 891.
Foulis, D.J., Piron, C., and Randali, C.H.: 1983, ‘Realism, operationalism, and quantum mechanics’, Found. Phys. 13, 813.
Foulis, D.J. and Randali, C.H.: 1971, ‘Lexicographic orthogonality’, J. Combo Theo. 11, 152.
Foulis, D.J. and Randali, C.H.: 1972, ‘Operational statistics, I, Basic concepts’, J. Math. Phys. 13, 1667.
Foulis, D.J. and Randali, C.H.: 1974a, ‘The empirical logic approach to the physical sciences’, in Foundations of Quantum Mechanics and Ordered Linear spaces (ed. by A. Harkärnper and H. Neumann), Lecture Notes in Physics, Vol. 29, Springer-Verlag, Berlin.
Foulis, D.J. and Randldl, C.H.: 1974b, ‘Empiricallogic and quantum mechanics’, Synthese 29,84.
Foulis, D.J. and Randali, C.H.: 1984, ‘A note on misunderstanding ofPiron’s Axioms of Quantum Mechanics’, Found. Phys. 14, 65.
Garola, C.: 1980, ‘Propositions and orthocomplementation in quantum logic’, Int. J. Theor. Phys. 19,369.
Garola, C.: 1985, ‘Embedding of posets into lattices in quantum logic’, Int. J. Theor. Phys. 24,423.
Garola, C. and Solombrino, L.: 1983, ‘Yes-no experiments and ordered structures in quantum physics’, Il Nuovo Cim. 778, 87.
Giles, R.: 1970, ‘Foundations forquantum mechanics’, J. Math. Phys. 11, 2139.
Giuntini, R.: 1990, ‘Brouwer-Zadeh logic and the operational approach to quantum mechanics’, Found. Phys. 20, 701.
Giuntini, R.: 1991, ‘A semantical investigation on Brouwer-Zadeh logic’, J. Phil. Logic. 20,411.
Gudder, S.P.: 1965, ’spectral methods for a generalized probability theory’, Trans. Amer. Math. Soc. 119,428.
Gudder, S.P.: 1966, ‘Uniqueness and existence properties of bounded observables’, Pac. J. Math. 19,81.
Gudder, S.P.: 1967, ’Systems of observables in axiomatic quantum mechanics’,J. Math. Phys. 8,2109.
Gudder, S.P.: 1968, ‘Joint distributions of observables’, J. Math. Mech. 18, 335.
Gudder, S.P.: 1969, ‘On the quantum logic approach to quantum mechanics’, Comm. Math. Phys. 12,1.
Gudder, S.P.: 1970, ‘Axiomatic quantum mechanics and generalized probability theory’, in Probabilistic Methods in Applied Mathematics, Vol. 2, (ed. by A. Bharucha-Reid), Academic Press, NY.
Hadjisavvas, N., Thieffine, F., and Mugur-Schschter, M.: 1980, ’Study of Piron’s system of questions and propositions’, Found. Phys. 10, 751.
Hadjisavvas, N. and Thieffine, F.: 1984, ‘Piron’s axioms for quantum mechanics, a reply to Foulis and Randall’, Found. Phys. 14, 83.
Jammer, M.: 1974, The Philosophy of Quantum Mechanics, J. Wiley & Sons, NY.
Jauch, J.M.: 1968, Foundations of Quantum Mechanics, Addision-Wesley, Reading, Mass.
Jauch, J.M.: 1971, ‘Foundations of quantum mechanics’, in Foundations of Quantum Mechanics (ed. by B. D’Espagnat), Academic Press, NY.
Jauch, J.M. and Piron, C.: 1969, ‘On the structure of quantal proposition systems’, Helv. Phys. Acta 42, 842.
Jauch, J.M. and Piron, C.: 1970, ‘What is quantum logic?’, in Quanta (ed. by P.G.O. Freund, C.J. Goebel and Y. Nambu), Chicago University Press, Chicago.
Kraus, K.: 1983, States, Effects, and Operations, Lecture Notes in Physics, Vol. 190, Springer Verlag, Berlin.
Kraus, K.: 1986, ‘The classical behaviour of measuring instruments’, in Fundamental Aspects of Quantum Theory (ed. by V. Gorini and A. Frigerio), Plenum, NY.
Ludwig, G.: 1971, vThe measuring process and an axiomatic foundation of quantum mechanics’, in Foundations of Quantum Mechanics (ed. by B. D’Espagnat), Academic Press, NY.
Ludwig, G.: 1974, ‘Measuring and preparing processes’, in Foundations of Quantum Mechanics and Ordered Linear Spaces (ed. by A. Hartkämper and H. Neumann), Lecture Notes in Physics, Vol. 29, Springer-Verlag, Berlin.
Ludwig, G.: 1977, ‘A theoretical description of single microsystems’, in The Uncertainty Principle and Foundations of Quantum Physics (ed. by W.C. Price and S.S. Chissick), J. Wiley & Sons, NY.
Ludwig, G.: 1981, ‘Quantum theory as a theory of interactions between macroscopic systems which can be described objectively’, Erkenntnis 16, 359.
Ludwig, G.: 1983, Foundations of Quantum Mechanics, Vol. I, Springer-Verlag, NY.
Mackey, G.W.: 1963, The Mathematical Foundations of Quantum Mechanics, Benjamin, NY.
Maczynski, M.J.: 1973, ‘The orthogonality pustulate in axiomatic quantum mechanics’, Int. J. Theor. Phys. 8, 353.
Maczynski, M.J.: 1974, ‘Functional properties of quantum logics’, Int. J. Theor. Phys. 11, 149.
Mielnik, B.: 1976, ‘Quantum logic: is it necessarily orthocomplemented?, in Quantum Mechanics, Determinism, Causality and Particle(ed. by M. Halo et al.), Reidel, Dordrecht
Piron, C.: 1964, ‘Axiomatique quantique’, Helv. Phys. Acta 37, 439.
Piron, C.: 1972, ’Survey of general quantum physics’, Found. Phys. 2, 287.
Piron, C.: 1976a, Foundations of Quantum Physics, Benjamin, Reading, Mass
Piron, C.: 1976b, ‘On the foundations of quantum physics’, in Quantum Mechanics, Determinism, Causality and Particles (ed. by M. Hato et al.), Reidel, Dordrecht
Piron, C.: 1977, ‘A first lecture in quantum mechanics’, in Quantum Mechanics, a Half Century After (ed. by J. Leite Lopes and M. Paty), Reidel, Dordrecht
Piron, C.: 1978, ‘La description d’un systeme physique et le presupposè de la theorie classique’, Ann. de La Fondation L. de Broglie 3, 131.
Piron, C.: 1981, ‘Ideal measurement and probability in quantum mechanics’, Erkenntnis 16,397.
Pool, J.C.T.: 1968a, ‘Baer*-semigroups and the logic of quantum mechanics’, Comm. Math. Phys. 9,118.
Pool, J.C.T.: 1968b, ’Semimodularity and the logic of quantum mechanics’, Comm. Math. Phys. 9,212.
Randali, C.H. and Foulis, D.J.: 1973, ‘Operational statistics, II, Manual of operations and their logic’, J. Math. Phys. 14, 1472.
Randali, C.H. and Foulis, O.J.: 1977, ‘The operational approach to quantum mechanies’, in The Logico-Algebraic Approach to Quantum Mechanics, Vol. III (ed. by C.A. Hooker), Reidel, Dordrecht
Randali, C.H. and Foulis, O.J.: 1983, ‘Properties and operational propositions in quantum mechanics’, Found. Phys. 13, 843.
Thieffine, F., Hadjivsavvas, N., and Mugur-Schächter, M.: 1981, ’supplement to a critique of Piron’s system of questions and propositions’, Found. Phys. 11, 645.
Thieffine, F.: 1983, ‘Compatible complement in Piron’s system and ordinary modallogic’, Leu. Nuovo Cim. 36, 377.
van Fraassen, B.: 1973, ’Semantical analysis of quantum logic’, in Contemporary Research in the Foundations of Quantum Theory (ed. by C.A. Hooker), Reidel, Dordrecht
van Fraassen, B.: 1974, ‘The labyrinth of quantum logics’, in Boston Studies in the Philosophy of Science, Vol. XIII (ed. by R.S. Cohen and M.W. Wartofsky), Reidel, Dordrecht
Varadarajan, V.S.: 1962, ‘Probability in physics and a theorem on simultaneous observability’, Comm. Pure Appl. Math. 15, 189.
Varadarajan, V.S.: 1968, Geometry of Quantum Theory, Vol. I, Van Nostrand, Princeton, 1968.
Von Neumann, J.: 1932, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton (Translated from 1932).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Cattaneo, G. (1993). The ‘Logical’ Approach to Axiomatic Quantum Theory. In: Corsi, G., Chiara, M.L.D., Ghirardi, G.C. (eds) Bridging the Gap: Philosophy, Mathematics, and Physics. Boston Studies in the Philosophy of Science, vol 140. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2496-6_12
Download citation
DOI: https://doi.org/10.1007/978-94-011-2496-6_12
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5101-9
Online ISBN: 978-94-011-2496-6
eBook Packages: Springer Book Archive