We describe a scheme of quantum mechanics in which the Hilbert space and linear operators are only secondary structures of the theory. As primary structures we consider observables, elements of noncommutative algebra, and the physical states, the nonlinear functionals on this algebra, which associate with results of single measurement. We show that in such scheme the mathematical apparatus of the standard quantum mechanics does not contradict a hypothesis on existence of an objective local reality, a principle of a causality and Kolmogorovian probability theory.We describe a scheme of quantum mechanics in which the Hilbert space and linear operators are only secondary structures of the theory. As primary structures we consider observables, elements of noncommutative algebra, and the physical states, the nonlinear functionals on this algebra, which associate with results of single measurement. We show that in such scheme the mathematical apparatus of the standard quantum mechanics does not contradict a hypothesis on existence of an objective local reality, a principle of a causality and Kolmogorovian probability theory.

arXiv:quant-ph/0404045Apr 7, 20040 citations

The Logico-Algebraic Approach to Quantum Mechanics

Springer eBooksJan 1, 1979160 citations

A Topos for Algebraic Quantum Theory

Chris Heunen, Nicolaas P. Landsman, Bas Spitters

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos ${\mathcal{T}(A)}$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra ${\underline{A}}$ . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum ${\underline{\Sigma}(\underline{A})}$ in ${\mathcal{T}(A)}$ , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on ${\underline{\Sigma}}$ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from ${\underline{\Sigma}}$ to Scott’s interval domain. Noting that open subsets of ${\underline{\Sigma}(\underline{A})}$ correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos ${\mathcal{T}(A)}$ .These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos $${\mathcal{T}(A)}$$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra $${\underline{A}}$$ . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum $${\underline{\Sigma}(\underline{A})}$$ in $${\mathcal{T}(A)}$$ , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on $${\underline{\Sigma}}$$ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from $${\underline{\Sigma}}$$ to Scott’s interval domain. Noting that open subsets of $${\underline{\Sigma}(\underline{A})}$$ correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos $${\mathcal{T}(A)}$$ .These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.

arXiv:0709.4364Communications in Mathematical PhysicsJul 20, 200982 citations

The Logico-Algebraic Approach to Quantum Mechanics

C. A. Hooker

Springer eBooksJan 1, 1975114 citations

{\mathcal{T}(A)}

in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra

{\underline{A}}

. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum

{\underline{\Sigma}(\underline{A})}

{\mathcal{T}(A)}

, which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on

{\underline{\Sigma}}

, and self-adjoint elements of A define continuous functions (more precisely, locale maps) from

{\underline{\Sigma}}

to Scott’s interval domain. Noting that open subsets of

{\underline{\Sigma}(\underline{A})}

correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos

{\mathcal{T}(A)}

.These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.