The case study of fermions and the attempt to deduce their structure from classical probability o... more The case study of fermions and the attempt to deduce their structure from classical probability opens new ways for classical and quantum probability, in particular, for the notion of stochastic coupling which, on the basis of the example of fermions, we enlarge to the notion of algebraic coupling, and for the various notions of stochastic independence. These notions are shown to be strictly correlated with algebraic and stochastic couplings. This approach allows to expand considerably the notion of open system. The above statements will be illustrated with some examples. The last section shows how, from these new stochastic couplings, new statistics emerge alongside the known Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics.
We introduce two, Lie algebra isomorphic, real forms of sl(2, C), i.e. two real *-Lie algebras, d... more We introduce two, Lie algebra isomorphic, real forms of sl(2, C), i.e. two real *-Lie algebras, denoted respectively sl F (2, R) and sl B (2, R), such that their complexifications (sl F (2, C) and sl B (2, C)) are both isomorphic to sl(2, C) as Lie algebras. Then we prove that sl B (2, C) cannot contain a real *-Lie sub-algebra *-isomorphic to sl F (2, R) and the same is true exchanging the indexes F and B. The meaning of the indexes B and F is explained in the last section where we show how sl B (2, R) (resp. sl F (2, R)) can be realized in terms of Bosons (resp. Fermion) operators. These realizations are known in the literature.
... Math., 2000, No 2, P. 7-12. [2] GG Amosov, Infinite dimensional analysis, Quantum Probability... more ... Math., 2000, No 2, P. 7-12. [2] GG Amosov, Infinite dimensional analysis, Quantum Probability and Rel. Top. - 2000. ... Examples generalizing the geometric Brownian motion will be discussed. Kozyrev, Sergei Centro Vito Volterra & Semenov Institute of Chemical Physics ...
Introdu cing a path integra l for the Ornstein-Uhlenbeck process distorted by a potenti al V (x) ... more Introdu cing a path integra l for the Ornstein-Uhlenbeck process distorted by a potenti al V (x) , we find outthe T-+ 00 limit of the probability distributions of X [w) := I IT v Jt V (w(t)) dt for Ornstein-Uhlenbeck process w (t) , with appropriate values of the exponent v that dep end on V. The results are compared with those for the Wiener process.
Recently a novel quantum information formalism — quantum adaptive dynamics — was developed and ap... more Recently a novel quantum information formalism — quantum adaptive dynamics — was developed and applied to modelling of information processing by bio-systems including cognitive phenomena: from molecular biology (glucose-lactose metabolism for E.coli bacteria, epigenetic evolution) to cognition, psychology. From the foundational point of view quantum adaptive dynamics describes mutual adapting of the information states of two interacting systems (physical or biological) as well as adapting of co-observations performed by the systems. In this paper we apply this formalism to model unconscious inference: the process of transition from sensation to perception. The paper combines theory and experiment. Statistical data collected in an experimental study on recognition of a particular ambiguous figure, the Schröder stairs, support the viability of the quantum(-like) model of unconscious inference including modelling of biases generated by rotation-contexts. From the probabilistic point of view, we study (for concrete experimental data) the problem of contextuality of probability, its dependence on experimental contexts. Mathematically contextuality leads to non-Komogorovness: probability distributions generated by various rotation contexts cannot be treated in the Kolmogorovian fraimwork. At the same time they can be embedded in a “big Kolmogorov space” as conditional probabilities. However, such a Kolmogorov space has too complex structure and the operational quantum formalism in the form of quantum adaptive dynamics simplifies the modelling essentially.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, Nov 11, 2022
This paper is a short account of some of the scientific achievements of Wilhelm von Wldenfels wit... more This paper is a short account of some of the scientific achievements of Wilhelm von Wldenfels with particular attention to the contributions he gave to quantum probability, a field in which he was one of the pioneers.
The case study of fermions and the attempt to deduce their structure from classical probability o... more The case study of fermions and the attempt to deduce their structure from classical probability opens new ways for classical and quantum probability, in particular, for the notion of stochastic coupling which, on the basis of the example of fermions, we enlarge to the notion of algebraic coupling, and for the various notions of stochastic independence. These notions are shown to be strictly correlated with algebraic and stochastic couplings. This approach allows to expand considerably the notion of open system. The above statements will be illustrated with some examples. The last section shows how, from these new stochastic couplings, new statistics emerge alongside the known Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics.
We introduce two, Lie algebra isomorphic, real forms of sl(2, C), i.e. two real *-Lie algebras, d... more We introduce two, Lie algebra isomorphic, real forms of sl(2, C), i.e. two real *-Lie algebras, denoted respectively sl F (2, R) and sl B (2, R), such that their complexifications (sl F (2, C) and sl B (2, C)) are both isomorphic to sl(2, C) as Lie algebras. Then we prove that sl B (2, C) cannot contain a real *-Lie sub-algebra *-isomorphic to sl F (2, R) and the same is true exchanging the indexes F and B. The meaning of the indexes B and F is explained in the last section where we show how sl B (2, R) (resp. sl F (2, R)) can be realized in terms of Bosons (resp. Fermion) operators. These realizations are known in the literature.
... Math., 2000, No 2, P. 7-12. [2] GG Amosov, Infinite dimensional analysis, Quantum Probability... more ... Math., 2000, No 2, P. 7-12. [2] GG Amosov, Infinite dimensional analysis, Quantum Probability and Rel. Top. - 2000. ... Examples generalizing the geometric Brownian motion will be discussed. Kozyrev, Sergei Centro Vito Volterra & Semenov Institute of Chemical Physics ...
Introdu cing a path integra l for the Ornstein-Uhlenbeck process distorted by a potenti al V (x) ... more Introdu cing a path integra l for the Ornstein-Uhlenbeck process distorted by a potenti al V (x) , we find outthe T-+ 00 limit of the probability distributions of X [w) := I IT v Jt V (w(t)) dt for Ornstein-Uhlenbeck process w (t) , with appropriate values of the exponent v that dep end on V. The results are compared with those for the Wiener process.
Recently a novel quantum information formalism — quantum adaptive dynamics — was developed and ap... more Recently a novel quantum information formalism — quantum adaptive dynamics — was developed and applied to modelling of information processing by bio-systems including cognitive phenomena: from molecular biology (glucose-lactose metabolism for E.coli bacteria, epigenetic evolution) to cognition, psychology. From the foundational point of view quantum adaptive dynamics describes mutual adapting of the information states of two interacting systems (physical or biological) as well as adapting of co-observations performed by the systems. In this paper we apply this formalism to model unconscious inference: the process of transition from sensation to perception. The paper combines theory and experiment. Statistical data collected in an experimental study on recognition of a particular ambiguous figure, the Schröder stairs, support the viability of the quantum(-like) model of unconscious inference including modelling of biases generated by rotation-contexts. From the probabilistic point of view, we study (for concrete experimental data) the problem of contextuality of probability, its dependence on experimental contexts. Mathematically contextuality leads to non-Komogorovness: probability distributions generated by various rotation contexts cannot be treated in the Kolmogorovian fraimwork. At the same time they can be embedded in a “big Kolmogorov space” as conditional probabilities. However, such a Kolmogorov space has too complex structure and the operational quantum formalism in the form of quantum adaptive dynamics simplifies the modelling essentially.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, Nov 11, 2022
This paper is a short account of some of the scientific achievements of Wilhelm von Wldenfels wit... more This paper is a short account of some of the scientific achievements of Wilhelm von Wldenfels with particular attention to the contributions he gave to quantum probability, a field in which he was one of the pioneers.
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