187 research outputs found

### 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 $p$-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

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