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

Generalized probabilities taking values in non-Archimedean fields and topological groups

We develop an analogue of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov's method of axiomatization of probability theory: main distinguishing features of frequency probabilities are taken as axioms in the measure-theoretic approach. We also present a review of non-Kolmogorovian probabilistic models including models with negative, complex, and $p$-adic valued probabilities. The latter model is discussed in details. The introduction of $p$-adic (as well as more general non-Archimedean) probabilities is one of the main motivations for consideration of generalized probabilities taking values in topological groups which are distinct from the field of real numbers. We discuss applications of non-Kolmogorovian models in physics and cognitive sciences. An important part of this paper is devoted to statistical interpretation of probabilities taking values in topological groups (and in particular in non-Archimedean fields)

Pseudodifferential operators on ultrametric spaces and ultrametric wavelets

A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable complex valued functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators, acting on complex valued functions on these ultrametric spaces is introduced. We show that these operators are diagonal in the introduced ultrametric wavelet bases, and compute the corresponding eigenvalues. We introduce the ultrametric change of variable, which maps the ultrametric spaces under consideration onto positive half-line, and use this map to construct non-homogeneous generalizations of wavelet bases.Comment: 19 pages, LaTe

Asymptotical behavior of one class of pp-adic singular Fourier integrals

We study the asymptotical behavior of the $p$-adic singular Fourier integrals J_{\pi_{\alpha},m;\phi}(t) =\bigl< f_{\pi_{\alpha};m}(x)\chi_p(xt), \phi(x)\bigr> =F\big[f_{\pi_{\alpha};m}\phi\big](t), \quad |t|_p \to \infty, \quad t\in \bQ_p, where f_{\pi_{\alpha};m}\in {\cD}'(\bQ_p) is a {\em quasi associated homogeneous} distribution (generalized function) of degree $\pi_{\alpha}(x)=|x|_p^{\alpha-1}\pi_1(x)$ and order $m$, $\pi_{\alpha}(x)$, $\pi_1(x)$, and $\chi_p(x)$ are a multiplicative, a normed multiplicative, and an additive characters of the field \bQ_p of $p$-adic numbers, respectively, \phi \in {\cD}(\bQ_p) is a test function, $m=0,1,2...$, \alpha\in \bC. If $Re\alpha>0$ the constructed asymptotics constitute a $p$-adic version of the well known Erd\'elyi lemma. Theorems which give asymptotic expansions of singular Fourier integrals are the Abelian type theorems. In contrast to the real case, all constructed asymptotics have the {\it stabilization} property