Replica symmetry breaking related to a general ultrametric space III: the case of general measure

Family of replica matrices, related to general ultrametric spaces with general measures, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed. Replica symmetry breaking solution is found.Comment: 21 page

Contextual viewpoint to quantum stochastics

We study the role of context, complex of physical conditions, in quantum as well as classical experiments. It is shown that by taking into account contextual dependence of experimental probabilities we can derive the quantum rule for the addition of probabilities of alternatives. Thus we obtain quantum interference without applying to wave or Hilbert space approach. The Hilbert space representation of contextual probabilities is obtained as a consequence of the elementary geometric fact: $\cos$-theorem. By using another fact from elementary algebra we obtain complex-amplitude representation of probabilities. Finally, we found contextual origin of noncommutativity of incompatible observables

Contextual approach to quantum mechanics and the theory of the fundamental prespace

We constructed a Hilbert space representation of a contextual Kolmogorov model. This representation is based on two fundamental observables -- in the standard quantum model these are position and momentum observables. This representation has all distinguishing features of the quantum model. Thus in spite all ``No-Go'' theorems (e.g., von Neumann, Kochen and Specker,..., Bell) we found the realist basis for quantum mechanics. Our representation is not standard model with hidden variables. In particular, this is not a reduction of quantum model to the classical one. Moreover, we see that such a reduction is even in principle impossible. This impossibility is not a consequence of a mathematical theorem but it follows from the physical structure of the model. By our model quantum states are very rough images of domains in the space of fundamental parameters - PRESPACE. Those domains represent complexes of physical conditions. By our model both classical and quantum physics describe REDUCTION of PRESPACE-INFORMATION. Quantum mechanics is not complete. In particular, there are prespace contexts which can be represented only by a so called hyperbolic quantum model. We predict violations of the Heisenberg's uncertainty principle and existence of dispersion free states.Comment: Plenary talk at Conference "Quantum Theory: Reconsideration of Foundations-2", Vaxjo, 1-6 June, 200

Genetic code on the dyadic plane

We introduce the simple parametrization for the space of codons (triples of nucleotides) by 8\times 8 table. This table (which we call the dyadic plane) possesses the natural 2-adic ultrametric. We show that after this parametrization the genetic code will be a locally constant map of the simple form. The local constancy of this map will describe degeneracy of the genetic code. The map of the genetic code defines 2-adic ultrametric on the space of amino acids. We show that hydrophobic amino acids will be clustered in two balls with respect to this ultrametric. Therefore the introduced parametrization of space of codons exhibits the hidden regularity of the genetic code.Comment: Some gap in the construction was fixe

Contextualist viewpoint to Greenberger-Horne-Zeilinger paradox

We present probabilistic analysis of the Greenberger-Horne-Zeilinger (GHZ) scheme in the contextualist framework, namely under the assumption that distributions of hidden variables depend on settings of measurement devices. On one hand, we found classes of probability distributions of hidden variables for that the GHZ scheme does not imply a contradiction between the local realism and quantum formalism. On the other hand, we found classes of probability distributions of hidden variables for that the GHZ scheme still induce such a contradiction (despite variations of distributions). It is also demonstrated that (well known in probability theory) singularity/absolute continuity dichotomy for probability distributions is closely related to the GHZ paradox. Our conjecture is that this GHZ-coupling between singularity/absolute continuity dichotomy and incompatible/compatible measurements might be a general feature of quantum theory.Comment: By taking into account contextualism of probabilities, i.e., dependence on complexes of experimental physical conditions, we resolve GHZ-parado

Noncommutative probability in classical systems

Two examples of the situation when the classical observables should be described by a noncommutative probability space are investigated. Possible experimental approach to find quantum-like correlations for classical disordered systems is discussed. The interpretation of noncommutative probability in experiments with classical systems as a result of context (complex of experimental physical conditions) dependence of probability is considered

Quantum field inspired model of decision making: Asymptotic stabilization of belief state via interaction with surrounding mental environment

This paper is devoted to justification of the quantum-like model of the process of decision making based on theory of open quantum systems: decision making as decoher- ence. This process is modeled as interaction of a decision maker, Alice, with a mental (information) environment R surrounding her. Such an interaction generates “dissipation of uncertainty” from Alice’s belief-state ρ ( t ) into R and asymptotic stabilization of ρ ( t ) to a steady belief-state. The latter is treated as the decision state. Mathematically the problem under study is about finding constraints on R guaranteeing such stabilization. We found a partial solution of this problem (in the form of sufficient conditions). We present the corresponding decision making analysis for one class of mental environments, so-called “almost homogeneous environments”, with the illustrative examples: a) behavior of electorate interacting with the mass-media “reservoir”; b) consumers’ persuasion. We also comment on other classes of mental environments