Beginning with the work of Birkhoff and von Neumann, the lattice of closed subspaces of Hilbert space has been studied by mathematicians, physicists, and philosophers. Since the 1960's, work of Gudder and of Kochen has focussed on partial structures extracted from, or related to, the lattice mentioned above. This thesis follows in the same tradition. Starting from a new point of view, a particular sort of partial algebra is constructed. The relations of these structures, quasi boolean algebras, to Gudder's work on various sorts of posets and lattices are studied. Their relation to the partial boolean algebras studied by Kochen is also described. In this connection, his definition of partial boolean algebras is refined. A study of quasi boolean algebras themselves is started. Various properties of these partial algebras are isolated and studied. The theory of relations between these partial algebras is then investigated. This theory is modelled after both standard developments in quantum mechanics proper, and some recent work by Kochen. In particular, a theory of "projections" and "interactions" in the context of these algebras is begun.
Beginning with the work of Birkhoff and von Neumann, the lattice of closed subspaces of Hilbert space has been studied by mathematicians, physicists, and philosophers. Since the 1960's, work of Gudder and of Kochen has focussed on partial structures extracted from, or related to, the lattice mentioned above. This thesis follows in the same tradition. Starting from a new point of view, a particular sort of partial algebra is constructed. The relations of these structures, quasi boolean algebras, to Gudder's work on various sorts of posets and lattices are studied. Their relation to the partial boolean algebras studied by Kochen is also described. In this connection, his definition of partial boolean algebras is refined. A study of quasi boolean algebras themselves is started. Various properties of these partial algebras are isolated and studied. The theory of relations between these partial algebras is then investigated. This theory is modelled after both standard developments in quantum mechanics proper, and some recent work by Kochen. In particular, a theory of "projections" and "interactions" in the context of these algebras is begun.
It is well known that the “quantum logic” approach to the foundations of quantum mechanics is based on the subspace ordering of projectors on a Hilbert space. In this paper, we show that this is a special case of an ordering on partial isometries, introduced by Halmos and McLaughlin. Partial isometries have a natural physical interpretation, however, they are notoriously not closed under composition. In order to take a categorical approach, we demonstrate that the Halmos-McLaughlin partial ordering, together with tools from both categorical logic and inverse categories, allows us to form a category of partial isometries. This category can reasonably be considered a “categorification” of quantum logic – we therefore compare this category with Abramsky and Coecke's “compact closed categories” approach to foundations and with the “monoidal closed categories” view of categorical logic. This comparison illustrates a fundamental incompatibility between these two distinct approaches to the foundations of quantum mechanics. 9.1 Introduction As early as 1936, von Neumann and Birkhoff proposed treating projectors on Hilbert space as propositions about quantum systems (Birkhoff and von Neumann 1936), by direct analogy with classical order-theoretic approaches to logic. Boolean lattices arise as the Lindenbaum-Tarski algebras of propositional logics, and as the set of all projectors on a Hilbert space also forms an orthocomplemented lattice, the operations meet, join , and complement were analogously interpreted as the logical connectives conjunction, disjunction , and negation .
Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” A of A, which is a commutative Rickart C*-algebra in the topos of functors from CA to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A for A = B(H).
Following Birkhoff and von Neumann, quantum logic has traditionally been based\non the lattice of closed linear subspaces of some Hilbert space, or, more generally,\non the lattice of projections in a von Neumann algebra A. Unfortunately, the logical\ninterpretation of these lattices is impaired by their nondistributivity and by various\nother problems. We show that a possible resolution of these difficulties, suggested by\nthe ideas of Bohr, emerges if instead of single projections one considers elementary\npropositions to be families of projections indexed by a partially ordered set C(A) of\nappropriate commutative subalgebras of A. In fact, to achieve both maximal generality\nand ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra\nand that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such\nfamilies of projections form a Heyting algebra in a natural way, so that the associated\npropositional logic is intuitionistic: distributivity is recovered at the expense of the\nlaw of the excluded middle.\nSubsequently, generalizing an earlier computation for n×n matrices, we prove that\nthe Heyting algebra thus associated to A arises as a basis for the internal Gelfand\nspectrum (in the sense of Banaschewski–Mulvey) of the "Bohrification" A of A, which\nis a commutative Rickart C*-algebra in the topos of functors from C(A) to the category\nof sets. We explain the relationship of this construction to partial Boolean algebras\nand Bruns–Lakser completions. Finally, we establish a connection between probability\nmeasure on the lattice of projections on a Hilbert space H and probability valuations\non the internal Gelfand spectrum of A for A = B(H).
Partial Boolean algebras (PBAs) were introduced by Kochen and Specker as an algebraic model reflecting the mutual relationships among quantum-physical yes–no tests. The fact that not all pairs of tests are compatible was taken into special account. In this paper, we review PBAs from two sides. First, we generalise the concept, taking into account also those yes–no tests which are based on unsharp measurements. Namely, we introduce partial MV-algebras, and we define a corresponding logic. Second, we turn to the representation theory of PBAs. In analogy to the case of orthomodular lattices, we give conditions for a PBA to be isomorphic to the PBA of closed subspaces of a complex Hilbert space. Hereby, we do not restrict ourselves to purely algebraic statements; we rather give preference to conditions involving automorphisms of a PBA. We conclude by outlining a critical view on the logico-algebraic approach to the foundational problem of quantum physics.
Quasi-set theory was proposed as a mathematical context to investigate collections of indistinguishable objects. After presenting an outline of this theory, we define an algebra that has most of the standard properties of an orthocomplete orthomodular lattice, which is the lattice of the closed subspaces of a Hilbert space. We call the mathematical structure so obtained I-lattice. After discussing, in a preliminary form, some aspects of such a structure, we indicate the next problem of axiomatizing the corresponding logic, that is, a logic which has I-lattices as its algebraic models. We suggest that the intuitions that the `logic of quantum mechanics' would be not classical logic (with its Boolean algebra), is consonant with the idea of considering indistinguishability right from the start, that is, as a primitive concept. In other words, indiscernibility seems to lead `directly' to I-lattices. In the first sections, we present the main motivations and a `classical' situation which mirrors that one we focus on the last part of the paper. This paper is our first study of the algebraic structure of indiscernibility within quasi-set theory.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.