# An approach to the foundations of quantum mechanics, using the concept of a filter as a primitive / Hans Kummer

Type de document : MonographieCollection : Queen's papers in pure and applied mathematics, 70Langue : anglais.Pays : Canada.Éditeur : Kingston, Ontario : Queen's University, 1984Description : 1 vol. (pagin. multiple) ; 28 cmISSN : 0079-8797.Bibliographie : Bibliogr. p. [128-129].Sujet MSC : 46Nxx, Functional analysis - Miscellaneous applications of functional analysis17C65, Nonassociative rings and algebras - Jordan algebras, Jordan structures on Banach spaces and algebras

46H70, Functional analysis - Topological algebras, normed rings and algebras, Banach algebras, Nonassociative topological algebras

46B42, Functional analysis - Normed linear spaces and Banach spaces; Banach lattices, Banach lattices

81P10, Quantum theory, Logical foundations of quantum mechanics; quantum logic

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CMI Salle R | 81 KUM (Browse shelf) | Available | 09101-01 |

It is shown that the Jordan structure of non-relativistic quantum theory can be deduced, at least in the finite-dimensional case, from four physically clear postulates. The axiomatic approach to the foundations of quantum mechanics developed here takes the concept of a filter as a starting point. More precisely, the primitive notions of the theory are: probability, filter, composition of filters, and stochastic selection of filters. All filters are assumed to be built from a countable set of primitive filters by composition and stochastic selection. Next, two physically transparent axioms are introduced, which allow the author to define the concept of a system (Definition 3.2). With every system there is associated a quadruple, the so-called σ-quadruple (Definition 3.10), whose first member is a separable complete order unit space. Then, two additional axioms are formulated, which lead to the effect that this space becomes isomorphic to the order unit space underlying a Jordan- Banach algebra, at least in the finite-dimensional case (Corollary 4.23). (Zentralblatt)

Bibliogr. p. [128-129]

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