An observable on a quantum structure is any $σ$ -homomorphism of quantum structures from the Borel $σ$ -algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form $(-\infty,t)$ is sufficient to determine uniquely the whole observable defined on quantum structures like $σ$ -MV-algebras, $σ$ -effect algebras, Boolean $σ$ -algebras, monotone $σ$ -complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.An observable on a quantum structure is any $σ$-homomorphism of quantum structures from the Borel $σ$-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form $(-\infty,t)$ is sufficient to determine uniquely the whole observable defined on quantum structures like $σ$-MV-algebras, $σ$-effect algebras, Boolean $σ$-algebras, monotone $σ$-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.

arXiv:1204.6460Apr 29, 20121 citation

Algebra of Observables

Jean-Louis Basdevant

Graduate texts in physicsJan 1, 20161 citation

Effect algebras, witness pairs and observables

Gejza Jenča

The category of effect algebras is the Eilenberg-Moore category for the monad arising from the free-forgetful adjunction between categories of bounded posets and orthomodular posets. In the category of effect algebras, an observable is a morphism whose domain is a Boolean algebra. The characterization of subsets of ranges of observables is an open problem. For an interval effect algebra E, a witness pair for a subset of S is an object living within E that "witnesses existence" of an observable whose range includes S. We prove that there is an adjunction between the poset of all witness pairs of E and the category of all partially inverted E-valued observables.The category of effect algebras is the Eilenberg-Moore category for the monad arising from the free-forgetful adjunction between categories of bounded posets and orthomodular posets. In the category of effect algebras, an observable is a morphism whose domain is a Boolean algebra. The characterization of subsets of ranges of observables is an open problem. For an interval effect algebra E, a witness pair for a subset of S is an object living within E that "witnesses existence" of an observable whose range includes S. We prove that there is an adjunction between the poset of all witness pairs of E and the category of all partially inverted E-valued observables.

EPiC series in computingJan 23, 20180 citations

The Algebra of Observables

Jean-Louis Basdevant

Graduate texts in physicsJan 1, 20230 citations

Toward an Understanding of Parochial Observables

Benjamin H. Feintzeig

Ruetsche ([2011]) claims that an abstract C*-algebra of observables will not contain all of the physically significant observables for a quantum system with infinitely many degrees of freedom. This would signal that in addition to the abstract algebra, one must use Hilbert space representations for some purposes. I argue to the contrary that there is a way to recover all of the physically significant observables by purely algebraic methods. 1. Introduction2. Preliminaries3. Three Extremist Interpretations 3.1. Algebraic imperialism3.2. Hilbert space conservatism3.3. Universalism4. Parochial Observables 4.1. Parochial observables for the imperialist4.2. Parochial observables for the universalist5. ConclusionRuetsche ([2011]) claims that an abstract C*-algebra of observables will not contain all of the physically significant observables for a quantum system with infinitely many degrees of freedom. This would signal that in addition to the abstract algebra, one must use Hilbert space representations for some purposes. I argue to the contrary that there is a way to recover all of the physically significant observables by purely algebraic methods. 1. Introduction2. Preliminaries3. Three Extremist Interpretations 3.1. Algebraic imperialism3.2. Hilbert space conservatism3.3. Universalism4. Parochial Observables 4.1. Parochial observables for the imperialist4.2. Parochial observables for the universalist5. Conclusion

The British Journal for the Philosophy of ScienceJun 14, 201637 citations

An observable on a quantum structure is any

σ

-homomorphism of quantum structures from the Borel

σ

-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form

(-\infty,t)

is sufficient to determine uniquely the whole observable defined on quantum structures like

σ

-MV-algebras,

σ

-effect algebras, Boolean

σ

-algebras, monotone

σ

-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.