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Aerts, Diederik; Steirteghem, Bart Van

doi:10.1023/a:1003661015110

Cited by 22 publications

(18 citation statements)

References 16 publications

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“…She proved that an infinite dimensional generalized Hilbert space that contains an orthonormal base is isomorphic with one of the three standard Hilbert spaces [16]. It has meanwhile also been possible to formulate an operational axiom, called 'plane transitivity' on the set of operational propositions that implies Solèr's condition [17], which completes the axiomatics for standard quantum mechanics by means of six axioms, the original five axioms of Piron and plane transitivity as sixth axiom.…”

Section: State Property Spacesmentioning

confidence: 99%

“…The sixth axiom that brings us directly to the structure of one of the three standard Hilbert spaces is much more recent [17].…”

Section: Axiom 6: Plane Transitivitymentioning

confidence: 99%

“…This means that the nonclassical components are all trivial. For the general situation of a physical entity described by (Σ, L, κ) it can be shown that L ω contains no classical properties with respect to Σ ω except 0 and ω, the minimal and maximal element of L ω , and that if (Σ, L, κ) satisfies Axiom 1, 2, 3, 4, 5 and 6 then also (Σ ω , L ω , κ ω ) ∀ω ∈ Ω satisfy Axiom 1, 2, 3, 4, 5 and 6 (see [8,10,17]). …”

Section: Definition 7 (Nonclassical Part)mentioning

confidence: 99%

“…Proof: see [8,10,17] From the previous section follows that if Axiom 1, 2, 3, 4, 5 and 6 are satisfied we can write the state property space (Σ, L, κ) of the physical entity under study as the direct union ∨ ω∈Ω (Σ ω , L ω , κ ω ) over its classical state space Ω of its nonclassical components (Σ ω , L ω , κ ω ), and that each of these nonclassical components also satisfies Axiom 1, 2, 3, 4, 5 and 6. Additionally for each one of the nonclassical components (Σ ω , L ω , κ ω ) no classical properties except 0 and ω exist.…”

Section: Definition 7 (Nonclassical Part)mentioning

confidence: 99%

See 3 more Smart Citations

Towards a General Operational and Realistic Framework for Quantum Mechanics and Relativity Theory

Aerts¹,

Aerts²

2005

Quo Vadis Quantum Mechanics?

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We propose a general operational and realistic framework that aims at a generalization of quantum mechanics and relativity theory, such that both appear as special cases of this new theory. Our framework is operational, in the sense that all aspects are introduced with specific reference to events to be experienced, and realistic, in the sense that the hypothesis of an independent existing reality is taken seriously. To come to this framework we present a detailed study of standard quantum mechanics within the axiomatic approach to quantum mechanics, more specifically the Geneva-Brussels approach, identifying two of the traditional 6 axioms as 'failing axioms'. We prove that these two failing axioms are at the origin of the impossibility for standard quantum mechanics to describe a continuous change from quantum to classical and hence its inability to describe macroscopic physical reality. Moreover we show that the same two axioms are also at the origin of the impossibility for standard quantum mechanics to deliver a model for the compound entity of two 'separated' quantum entities. We put forward that it is necessary to replace these two axioms in order to proceed to the more general theory. Next we analyze the nature of the quantum probability model and show that it can be interpreted as the consequence of the presence of a lack of knowledge on the interaction between the measurement apparatus and the physical entity under consideration. These two insights, the failing axioms and the nature of quantum probability, give rise to a very specific view on the quantum phenomenon of nonlocality. Nonlocality should be interpreted as nonspatiality. This means that an entity in a nonlocal state, like for example the typical EPR state, is 'not inside space'. As a consequence, space is no longer the all embracing theatre of reality, but a structure that has emerged together with the macroscopic material entities that have emerged from the microworld. This clarifies why general relativity theory cannot be used as a basis for the elaboration of the new generalized theory, since in general relativity theory the set of events is taken a priori to be the time-space continuum. Hence in general relativity theory time-space is a basic structure considered to capture all of reality. In our framework we introduce 'happenings' and the 'set of happenings' as constituting reality. A happening is however not identified with a point of time-space, as this is the case for an events of general relativity theory. We analyze different aspects of the new framework, and list the most important problems to be investigated for an elaboration of this framework into a workable and as complete as possible theory.

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Section: State Property Spacesmentioning

confidence: 99%

“…The sixth axiom that brings us directly to the structure of one of the three standard Hilbert spaces is much more recent [17].…”

Section: Axiom 6: Plane Transitivitymentioning

confidence: 99%

Section: Definition 7 (Nonclassical Part)mentioning

confidence: 99%

Section: Definition 7 (Nonclassical Part)mentioning

confidence: 99%

See 2 more Smart Citations

Towards a General Operational and Realistic Framework for Quantum Mechanics and Relativity Theory

Aerts¹,

Aerts²

2005

Quo Vadis Quantum Mechanics?

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“…For sake of completeness we mention other work that has contributed to advances in the approach that is less directly of importance for what we do in this paper, but shows how the approach is developing into other directions as well [52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81].…”

Section: How We Will Proceedmentioning

confidence: 99%

Being and Change: Foundations of a Realistic Operational Formalism

Aerts¹

2002

Probing the Structure of Quantum Mechanics

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The aim of this article is to represent the general description of an entity by means of its states, contexts and properties. The entity that we want to describe does not necessarily have to be a physical entity, but can also be an entity of a more abstract nature, for example a concept, or a cultural artifact, or the mind of a person, etc..., which means that we aim at very general description. The effect that a context has on the state of the entity plays a fundamental role, which means that our approach is intrinsically contextual. The approach is inspired by the mathematical formalisms that have been developed in axiomatic quantum mechanics, where a specific type of quantum contextuality is modelled. However, because in general states also influence contextwhich is not the case in quantum mechanics -we need a more general setting than the one used there. Our focus on context as a fundamental concept makes it possible to unify 'dynamical change' and 'change under influence of measurement', which makes our approach also more general and more powerful than the traditional quantum axiomatic approaches. For this reason an experiment (or measurement) is introduced as a specific kind of context. Mathematically we introduce a state context property system as the structure to describe an entity by means of its states, contexts and properties. We also strive from the start to a categorical setting and derive the morphisms between * Published as: D. Aerts, "Being and change: foundations of a realistic operational formalism", in Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Computation and Axiomatics, eds. D. Aerts, M. Czachor and T. Durt, World Scientific, Singapore (2002). 1 state context property systems from a merological covariance principle. We introduce the category SCOP with as elements the state context property systems and as morphisms the ones that we derived from this merological covariance principle. We introduce property completeness and state completeness and study the operational foundation of the formalism.

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Fundamental Quantum Structures on Hilbert Spaces

Moretti

2019

Fundamental Mathematical Structures of Quantum Theory

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Cited by 22 publications

References 16 publications

Towards a General Operational and Realistic Framework for Quantum Mechanics and Relativity Theory

Towards a General Operational and Realistic Framework for Quantum Mechanics and Relativity Theory

Being and Change: Foundations of a Realistic Operational Formalism

Fundamental Quantum Structures on Hilbert Spaces

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