 Author
 Title
 Orthogonality and quantum geometry: Towards a relational reconstruction of quantum theory
 Supervisors
 Cosupervisors
 Award date
 9 September 2015
 Number of pages
 246
 ISBN
 9789064648946
 Document type
 PhD thesis
 Faculty
 Interfacultary Research Institutes
 Institute
 Institute for Logic, Language and Computation (ILLC)
 Abstract

This thesis is an indepth mathematical study of the nonorthogonality relation between the (pure) states of quantum systems. In Chapter 2, I define quantum Kripke frames, the protagonists of this thesis. A quantum Kripke frame is a Kripke frame in which the binary relation possesses some simple properties of the nonorthogonality relation in quantum theory. The structure of quantum Kripke frames is studied extensively from a geometric perspective. In the meantime, several kinds of projective geometries are discovered to be Kripke frames in disguise. In Chapter 3, maps between quantum Kripke frames are studied. I define continuous homomorphisms between quantum Kripke frames and study extensively their properties. Chapter 4 concerns the automated reasoning of quantum Kripke frames. I prove that the firstorder theory of quantum Kripke frames is undecidable. Moreover, I characterize the firstorder definable, biorthogonally closed subsets in a special kind of quantum Kripke frame. Chapter 5 is a pilot study of the transition probabilities between the states of quantum systems. They are the quantitative, more finegrained version of the nonorthogonality relation. I define probabilistic quantum Kripke frames and quantum transition probability spaces, whose definitions capture some essential properties of the transition probabilities in quantum theory. Some elementary but useful results about these two kinds of structures are proved.
 Note
 Research conducted at: Universiteit van Amsterdam
Series: ILLC dissertation series DS201503  Permalink
 http://hdl.handle.net/11245/1.484537
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