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CENTER LEO APOSTELfor Interdisciplinary Studies |
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CENTER LEO APOSTELfor Interdisciplinary Studies |
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The working hypothesis in CLEA's worldview construction project states that reality is a compound of emergently related ontological layers, approximately described by the scientific disciplines . It is therefore, in the layered structure of reality approach, essential to describe the transition between different layers by means of models that detail such a transition. One of the examples of such a transition that is taken from the natural sciences and lends itself particularly well to both qualitative and quantitative analysis is the quantum-classical transition. We aim at a general description of the transition between respective domains of reality, employing concepts and abstractions of the dynamical and structural features of the quantum classical transition as a generic model of the interaction of an entity and its supervening context. The contextual dynamical evolution is interpreted as an actualisation of potentiality states of general entities in the layered structure model. This leads to a study of (i) the concomitant associated potentiality ontology, in which a rupture with classical determinism is apparent, and (ii) contextuality connected to the change that an abstract or physical 'system' will suffer when exposed to different 'environments', as an evolution dynamics.
The nature of the interface between the classical and quantum mechanical layers of reality remains clouded, especially within the context of the observation of quantum mechanical systems, a process whose very description necessitates the use of concepts from both layers. This apparent dichotomy is echoed in several ways, each of which has been given due attention by CLEA within the framework of a generalised operational approach developed in close cooperation with the FUND research group that studies general quantum formalisms from a purely quantum mechanical perspective. We represent the spectrum of results under separate headings.
An operational quantum formalism was elaborated, with roots in quantum axiomatics and quantum logic, which is capable of describing quantum-like situations of a more general nature than the one described by standard quantum mechanics [2, 45, 131, 260]. The basic mathematical structure was identified to be that of a state property system [6, 75, 86, 138, 241, 259, 286, 288]. The postulated basic notions are: the set of states S_of the entity under consideration, the set of properties L of this entity, and the Cartan map K, which maps each property to the set of states that makes this property actual. Probability and measurement context are not primitive notions of the state property system, because the determination of context and probability from states and properties is, by Gleason's theorem, a characteristic of standard quantum mechanics. Once the generalised quantum structures are used for other domains (and even for meso and macrophysical entities), this determination no longer holds, so that we must introduce probability as well as context as primitive notions. This leads to the categorical structure of a state context property system we have called 'SCOP' (see also subsection (iii) on probability). Hence a state property system (S__L, K) is formally a predecessor of SCOP, used in the CLEA concept theory (see III, 2, i), which illustrates the importance of the formal quantum research put forward in this section for the CLEA research on the cognition theme (III, 2, i). Necessary categorical elaborations are worked out in [89, 114, 141, 142, 165]. Key publications of this general operational formalism are [2, 45, 110, 131, 260], while philosophical considerations that stem from this approach can be found in [47, 48, 49, 68, 108].
The observation process produces two seemingly very different descriptions of dynamical processes. The proposal to regard both as aspects of contextual interaction was the key to their successful unification and gave rise to an extremely rich and flexible operational framework for describing these interactions. The resulting paradigm, 'Context-Driven Actualisation of Potential', was later to become one of the main research pillars of CLEA with a truly interdisciplinary reach. The introduction of contextual interaction for entities of a particular domain implies the introduction of states of potentiality for the formal description of these entities. This insight clarifies our successful application of quantum models: states of potentiality correspond to the superposition states of standard quantum mechanics. Due to the complexity of the contextual dynamical processes that take place in the different domains, it is not the mathematical formalism of standard quantum mechanics that we employ for our objectives, but rather mathematical formalisms of generalisations of the quantum formalism that are used in the axiomatic approach to quantum mechanics [2, 5, 6, 47, 49, 68, 75, 108, 131, 138, 259, 260]. We are able to derive the continuous part of the dynamics from a representation of a one-parameter group of time translations in the group of automorphisms, applied to the generalised quantum structures. In this respect, CAP allows linear evolutions, but also intrinsically non-linear evolutions incorporating the quantum collapse and other types of non-linear change [53, 143, 158, 171, 172, 173, 174, 175, 180]. The CAP approach enables us to unite these different forms of change -including quantum mechanical evolution, classical mechanical evolution, biological evolution and cultural evolution- into one formal theory of how entities evolve [276, 277]. This proves the importance of the formal study of these structures for the global worldview project (see III, 3) (i)).
A striking difference between the classical and the quantum mechanical layer is the nature of the origin of probability that is generally attributed to each layer. The paradigm of Kolmogorovian probabilities relating to classical descriptions and projection-based probability related to quantum mechanics was challenged in two ways. First, by demonstrating that lack of knowledge models can represent quantum probabilities [10, 11, 14, 15, 87, 107, 139, 160, 170], and second, by constructing models with a fully classical physical description that do not fit the Kolmogorov axioms [99, 107, 133, 164]. Case studies of more general probability models that do not fit in either paradigm can be found in [3, 99, 133, 164, 183]. It turns out that there even exist models with a quantum logic that yet lead to a probability model that is not fully quantum [111], showing highly non-trivial aspects of classification. These models show the need for an abstract probability theory that follows from these general considerations and employs probability as fundamental concept. We believe we have identified a viable paradigm for such a theory, embodied by the abstract inclusion of a set of deterministic observers, along with a probability measure on this set. We named this approach Interactive Probability Theory (IPT). The proposed theory is simple and yet general in the sense that it effectively models probabilities of all situations (quantum, classical and intermediate) described in the aforementioned publications. Though it is still an active area of investigation, the most relevant publications in the period 1998-2002 are [3, 10, 164]. In close relation to the problem of the origin of the probabilities is the problem of determination and classification of families of probabilities within a given framework. A central question in this respect is whether a given set of probabilities can be represented by a local realistic model. This problem only admits a general solution, if it is presented in an idealised world. In actual experiments, things become significantly more complex, and the matter needs to be decided on a case-by-case basis. On a more abstract level, the problem is intimately connected with the characterisation of compoundness (see subsection iv) and has been dedicated substantial effort at CLEA: [81, 84, 87, 95, 107, 129, 130, 139]. Several of these publications have been received with great interest by the relevant research community.
A very important area of research is related to the problem of compound systems. Whereas in a classical setting, interacting entities still have separable properties, in quantum mechanics they do not, giving rise to the phenomenon of entanglement. This is a direct consequence of the respective state space construction for compound entities. In a classical case, compound systems are described in the Cartesian product space, whereas in quantum mechanics this role is fulfilled by the tensor product space. Studies of compoundness and product structure can be found in [6, 8, 9, 37, 105, 112, 128, 167, 168]. From the perspective of CAP, see subsection (ii), entanglement means co-change of the states of the subsystems of the compound system under influence of a context that affects only one of the subsystems. This is exactly the type of contextual influence that is present in the case of concepts, and this is why we were able to solve the combination problem for concepts (see III, 2, (i)) through the use of the tensor product of quantum structures, which highlights the relevance and importance of the study of these formal structures to CLEA research on cognition (see III, 2) (i)). It has to be stressed that formal mathematical models of interactions between systems and their environments will practically always lead to nonlinear dynamical systems. This statement is true for physics, sociology and ecology. Therefore much of the theoretical effort was addressed at understanding the issue of compound systems in the context of generalised quantum formalisms [76, 143, 172, 174]. Yet another but closely related investigation was focused on concrete formal models for which one of the main drawbacks was the insufficient availability of appropriate mathematical techniques in the literature. In particular, the works [21, 28, 36, 53, 102, 115, 143, 145, 158, 171, 175, 180] were focused entirely on technical aspects associated with solving nonlinear mathematical models.
The search for properties of entities in the quantum and classical layers gives rise to two quite distinct types of logic. This was studied in relation to many valued logics, fuzzy set theory and category theory in [20, 30, 62, 63, 98, 99, 118, 127, 183, 252, 291, 292]. The subtle differences between the binary operations of the corresponding logics are studied in [98, 99, 109]. Another research area that requires both classical and quantum mechanical concepts is that of quantum computation and information, highlighting the need for a general approach that does not suffer this dichotomy. As the more general approaches that we present question the simple classifications often found in the literature, it is important to check whether certain quantum-like properties can be found in classical computers, and whether there exist intermediate situations of quantum information processing [7, 13, 16, 30, 93, 144, 242, 252, 263, 264].
Created by Alex Riegler · Last update: 20 October 2008