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The Quantum Statistical Mechanics in the Analysis of the Context Dependence in Quantum Cognition Studies: May a Quantum Statistical Analysis Connect the Science of Complexity ?

Elio Conte, Ferda Kaleağasıoğlu, Rich Norman

Published Jun 1, 2017• 0 citations
OpenAlex

Abstract

Quantum probability fields are involved during processing of human perception and cognition with a wave function that mathematically is articulated in a proper Hilbert space and conceptually represents the space of all the potential alternatives at a semantic and semiotic level. Each subject has developed a unique mind within his/her correspondent context, hence his/her own wave function requires to be displayed as a complex-valued probability amplitude. Reaction time as well as the time of observation / decision / measurement also represent important variables contributing to our cognitive response and having also proper neurological correlates. Mind is a so complex abstract entity so that we can certainly retain that, generally speaking, the observer and the observed system enter in the domain of what technically are considered to be un-complete specified systems . An elementary wave function remains insufficient to represent a mental state, whereas a quantum wave function, which exhibits intrinsic indetermination and fluctuation of the basic probability amplitudes, may be implemented. In this paper, along with von Neumann’s postulate for non-existing dispersion-free ensembles, a new quantum statistical approach is elaborated to elucidate the peculiar quantum mechanical feature of the context dependent dynamics of human cognitive conceptual processing. In the framework of the present quantum statistical elaboration we introduce for the first time the possibility of using the Pareto distribution as a probability density function. This result links quantum statistical mechanics with the science of complexity .

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Quantum probability fields are involved during processing of human perception and cognition with a wave function that mathematically is articulated in a proper Hilbert space and conceptually represents the space of all the potential alternatives at a semantic and semiotic level. Each subject has developed a unique mind within his/her correspondent context, hence his/her own wave function requires to be displayed as a complex-valued probability amplitude. Reaction time as well as the time of observation / decision / measurement also represent important variables contributing to our cognitive response and having also proper neurological correlates. Mind is a so complex abstract entity so that we can certainly retain that, generally speaking, the observer and the observed system enter in the domain of what technically are considered to be un-complete specified systems . An elementary wave function remains insufficient to represent a mental state, whereas a quantum wave function, which exhibits intrinsic indetermination and fluctuation of the basic probability amplitudes, may be implemented. In this paper, along with von Neumann’s postulate for non-existing dispersion-free ensembles, a new quantum statistical approach is elaborated to elucidate the peculiar quantum mechanical feature of the context dependent dynamics of human cognitive conceptual processing. In the framework of the present quantum statistical elaboration we introduce for the first time the possibility of using the Pareto distribution as a probability density function. This result links quantum statistical mechanics with the science of complexity .
In quantum mechanics a wave function is connected to a quantum system. This abstract entity represents a description of the quantum state of this system. The wave function is a complex-valued probability amplitude and the probabilities for the possible results of measurements made on the system can be derived from it . In the present paper it is demonstrated that the tentative often adopted from scholars to attribute a wave function as representation of a mental state in current quantum cognition studies cannot be accepted. A mental state is a so complex structure , depending contextually at each instant from a so complex structure of inner and subjective elements and psychological and neurological components, to exclude its representation by a simple and elementary wave function, The mental state may be represented by a quantum wave function that, owing of the complex structure of the mental state, has an intrinsic indetermination and fluctuation of the basic probability amplitudes to respond to the basic notion of intrinsecally , incompletely specified system that was introduced by P. T. Landsberg in 1961. As consequence a simple quantum wave function cannot be adopted and instead an averaging statistical formulation is absolutely required. Therefore, the subsequent quantum statistical theory is formulated.
We consider transition from the classical statistical model to the quantum statistical model through ignorance (of huge volume) of information in process of construction of a wave function { a complex probability amplitude. Our approach clarifles relation between classical and quantum statistical models (and hence relation between classical and quantum information theories). In particular, it can be considered as a step toward demystiflcation of quantum theory. The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities.In our approach it is forbidden to consider abstract context independent probabilities: \flrst context and then probability. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. Our formalism can be applied to various domains outside quantum physics, e.g., cognitive sci- ences, psychology, economics, chemistry: roughly speaking in any domain in that it is impossible to extract the whole information about an ontic (or to say realistic) model and hence the epis- temological (experimental or to say observational) model is constructed through huge reduction of information volume. We introduced a notion of prespace { space of states containing complete information about systems under consideration. Classical phase space and quantum Hilbert space are obtained as projections of such a prespace.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.
My discussion is articulated under the neurological as well as the psychological profile. I insist in particular on the view that mental events arise in analogy with quantum probability fields. I review some results obtained on quantum cognition discussing in detail those that we obtained on quantum interference in mental states during perception-cognition in ambiguous figures. Frequently, I use the approach to quantum mechanics by Clifford algebra. I insist in particular on two recent results. The first is the justification that I obtain of the von Neumann postulate on quantum measurement and the second relates my Clifford demonstration on the logical origins of quantum mechanics and thus on the arising feature that quantum mechanics relates conceptual entities. The whole discussion aims me to support the conclusion that we think in a quantum probabilistic manner.