Abstract
In the framework of the topos approach to quantum mechanics a kind of global valuation is introduced and studied. It allows us to represent certain features related to the logical consequences of properties about quantum systems when its phase space is endowed with an intuitionistic structure.
Similar content being viewed by others
References
Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–843
Balbes R, Dwinger Ph (1974) Distributive lattices. University of Missouri Press, Columbia
Dalla Chiara ML, Giuntini R, Sergioli G (2014) Probability in quantum computation and in quantum computational logics. Math Struct Comput Sci 14:e240306 (2014)
de Ronde C, Freytes H, Domenech G (2014) Interpreting the modal Kochen–Specker theorem: possibility and many worlds in quantum mechanics. Stud Hist Philos Mod Phys 45:11–18
Domenech G, Freytes H (2005) Contextual logic for quantum systems. J Math Phys 46:012102
Domenech G, Freytes H, de Ronde C (2006) Scopes and limits of modality in quantum mechanics. Ann Phys 15:853–860
Domenech G, Freytes H, de Ronde C (2009) Many worlds and modality in the interpretation of quantum mechanics: an algebraic approach. J Math Phys 50:072108
Döring A, Isham CJ (2008) A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory. J Math Phys 49:053516
Döring A, Isham CJ (2010) What is a thing? Topos theory in the foundations of physics. In: Coecke B (ed) New structures for physics, Lecture Notes for Physics, 813. Springer, Berlin, pp 753–937
Freyd PJ (1972) Aspects of topoi. Bull Aust Math Soc 7:1–76
Freytes H, de Ronde C, Domenech G (2009) Modal type othomodular logic. Math Log Q 55:287–299
Freytes H, de Ronde C, Domenech G (2014) Physical properties as modal operators in the topos approach to quantum mechanics. Found Phys 44:1357–1368
Heunen C, Landsman N, Spitters B (2009) A topos for algebraic quantum theory. Comm Math Phys 291:63–110
Isham C (2010) Topos methods in the foundations of physics. In: Halvorson H (ed) Deep beauty: understanding the quantum world through mathematical innovation. Cambridge University Press, Cambridge, pp 187–206
Johnstone PT (1982) Stone spaces., Cambridge studies in advanced mathematics 3Cambridge University Press, Cambridge
Kalman JA (1958) Lattices with involution. Trans Am Math Soc 87:485–491
Kalmbach G (1983) Ortomodular lattices. Academic Press, London
Kochen S, Specker E (1967) On the problem of hidden variables in quantum mechanics. J Math Mech 17:59–87
Lawvere FW (1970) Quantifiers and sheaves. Actes Congres Int Math Tome 1:329–334
Macnab DS (1981) Modal operators on Heyting algebras. Algebra Univ 12:5–29
Maeda F, Maeda S (1970) Theory of symetric lattices. Springer, Berlin
Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. L. Dalla Chiara, R. Giuntini, E. Negri and S. Smets.
Rights and permissions
About this article
Cite this article
Freytes, H., de Ronde, C. & Domenech, G. Semilattices global valuations in the topos approach to quantum mechanics. Soft Comput 21, 1373–1379 (2017). https://doi.org/10.1007/s00500-015-1780-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-015-1780-8