Abstract
Starting with a quantum logic (a σ-orthomodular poset) L. a set of probabilistically motivated axioms is suggested to identify L with a standard quantum logic L(H) of all closed linear subspaces of a complex, separable, infinite-dimensional Hilbert space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Mackey,The Mathematical Foundations of Quantum Mechanics (Benjamin, New York, 1963).
M. Maczyński, “A remark on Mackey's axiom system for quantum mechanics,”Bull. Acad. Pol. Sci. 15, 583–587 (1967).
V. Varadarajan,Geometry of Quantum Theory (Van Nostrand, Princeton, New Jersey, 1968).
P. Pták and S. Pulmannová,Orthomodular Structures as Quantum Logics (Kluwer, Dordrecht, 1991).
E. Beltrametti and G. Cassinelli,The Logic of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1981).
S. P. Gudder, “A superposition principle in physics,”J. Math. Phys. 11, 1037–1040 (1970).
P. Dirac,The Principles of Quantum Mechanics (Clarendon, Oxford, 1980).
R. Mayet and S. Pulmannová, “Nearly orthosymmetric ortholattices and Hilbert spaces,” this issue.
J. Hedlíková and S. Pulmannová, “Orthogonality spaces and atomistic orthocomplemented lattices,”Czech. J. Math. 41, 8–23 (1991).
C. Piron, “Axiomatique quantique,”Helv. Phys. Acta 37, 439–468 (1964).
F. Maeda and S. Maeda,Theory of Symmetric Lattices (Springer, New York, 1970).
W. Wilbur, “On characterizing the standard quantum logics,”Trans. Am. Math. Soc. 233, 265–282 (1977).
A. Gleason, “Measures on the closed subspaces of a Hilbert space,”J. Math. Mech. 6, 885–894 (1957).
N. Zierler, “Axioms for nonrelativistic quantum mechanics,”Pacific J. Math. 19, 1151–1169 (1961).
N. Zierler, “On the lattice of closed subspaces of Hilbert space,”Pacific J. Math. 19, 583–586 (1966).
S. P. Gudder and C. Piron, “Observables and the field in quantum mechanics,”J. Math. Phys. 12, 1583–1588 (1971).
S. P. Gudder, “Quantum logics, physical space, position observables, and symmetry,”Rep. Math. Phys. 4, 193–202 (1973).
M. Maczyński, “Hilbert space formalism of quantum mechanics without the Hilbert space axiom,”Rep. Math. Phys. 3, 209–219 (1972).
M. Maczyński, “The field of real numbers in axiomatic quantum mechanics,”J. Math. Phys. 14, 1469–1471 (1973).
P. Delyiannis, “Vector space models of abstract quantum logics,”J. Math. Phys. 14, 249–253 (1973).
A. Cirelli and P. Cotta-Ramusino, “On the isomorphism of a quantum logic with the logic of the projections in a Hilbert space,”Int. J. Theor. Phys. 8, 11–19 (1973).
A. Cirelli, P. Cotta-Ramusino, and E. Novati, “On the isomorphism of a quantum logic with the logic of the projections in a Hilbert space II,”Int. J. Theor. Phys. 11, 135–144 (1974).
H. Szambien, “Characterization of projection lattices of Hilbert spaces,”Int. J. Theor. Phys. 25, 939–944 (1986).
H. Szambien, “Topological projective geometries,”J. Geometry 26, 163–171 (1986).
M. MacLaren, “Atomic orthocomplemented lattices,”Pacific J. Math. 14, 597–612 (1964).
G. Birkhoff and J. von Neumann, “The logic of quantum mechanics,”Ann. Math. 37, 823–834 (1936).
R. Mayet, “Orthosymmetric ortholattices,”Proc. Am. Math. Soc. 114, 295–306 (1992).
H. Gross and M. Künzi, “On a class of orthomodular quadratic spaces,”Enseign. Math. 31, 187–212 (1985).
L. Bunce and J. Wright, “Quantum logics and convex geometry,”Comm. Math. Phys. 101, 87–96 (1985).
J. Jauch,Foundations of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1968).
J. Jauch and C. Piron, “On the structure of quantal proposition systems,”Helv. Phys. Acta 42, 842–848 (1969).
C. Piron,Foundations of Quantum Mechanics (Benjamin, Reading, Massachusetts, 1976).
K. Bugajska and S. Bugajski, “On the axioms of quantum mechanics,”Bull. Acad. Pol. Sci., Math., Astron., Phys. 20, 231–234 (1972).
M. Navara, “A note on the axioms of quantum mechanics,”Acta Polytechnica, ČVUT, Prague 15, 5–8 (1988).
G. Rüttimann, “Jauch-Piron states,”J. Math. Phys. 18, 189–193 (1977).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pulmannová, S. Quantum logics and hilbert space. Found Phys 24, 1403–1414 (1994). https://doi.org/10.1007/BF02283040
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02283040