534 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
On possible violation of the CHSH Bell inequality in a classical context
It has been shown that there is a small possibility to experimentally violate
the CHSH Bell inequality in a 'classical' context. The probability of such a
violation has been estimated in the framework of a classical probabilistic
model in the language of a random-walk representation.Comment: 9 pages, 1 figur
Classical Signal Model for Quantum Channels
Recently it was shown that the main distinguishing features of quantum
mechanics (QM) can be reproduced by a model based on classical random fields,
so called prequantum classical statistical field theory (PCSFT). This model
provides a possibility to represent averages of quantum observables, including
correlations of observables on subsystems of a composite system (e.g.,
entangled systems), as averages with respect to fluctuations of classical
(Gaussian) random fields. In this note we consider some consequences of PCSFT
for quantum information theory. They are based on the observation \cite{W} of
two authors of this paper that classical Gaussian channels (important in
classical signal theory) can be represented as quantum channels. Now we show
that quantum channels can be represented as classical linear transformations of
classical Gaussian signa
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 -adic valued probabilities. The latter model is discussed in details.
The introduction of -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)
An analog of Heisenberg uncertainty relation in prequantum classical field theory
Prequantum classical statistical field theory (PCSFT) is a model which
provides a possibility to represent averages of quantum observables, including
correlations of observables on subsystems of a composite system, as averages
with respect to fluctuations of classical random fields. PCSFT is a classical
model of the wave type. For example, "electron" is described by electronic
field. In contrast to QM, this field is a real physical field and not a field
of probabilities. An important point is that the prequantum field of e.g.
electron contains the irreducible contribution of the background field, vacuum
fluctuations. In principle, the traditional QM-formalism can be considered as a
special regularization procedure: subtraction of averages with respect to
vacuum fluctuations. In this paper we derive a classical analog of the
Heisenberg-Robertson inequality for dispersions of functionals of classical
(prequantum) fields. PCSFT Robertson-like inequality provides a restriction on
the product of classical dispersions. However, this restriction is not so rigid
as in QM. The quantum dispersion corresponds to the difference between e.g. the
electron field dispersion and the dispersion of vacuum fluctuations. Classical
Robertson-like inequality contains these differences. Hence, it does not imply
such a rigid estimate from below for dispersions as it was done in QM
The quantum measurement process in an exactly solvable model
An exactly solvable model for a quantum measurement is discussed which is
governed by hamiltonian quantum dynamics. The -component of a
spin-1/2 is measured with an apparatus, which itself consists of magnet coupled
to a bath. The initial state of the magnet is a metastable paramagnet, while
the bath starts in a thermal, gibbsian state. Conditions are such that the act
of measurement drives the magnet in the up or down ferromagnetic state
according to the sign of of the tested spin. The quantum measurement goes
in two steps. On a timescale the off-diagonal elements of the
spin's density matrix vanish due to a unitary evolution of the tested spin and
the apparatus spins; on a larger but still short timescale this is made
definite by the bath. Then the system is in a `classical' state, having a
diagonal density matrix. The registration of that state is a quantum process
which can already be understood from classical statistical mechanics. The von
Neumann collapse and the Born rule are derived rather than postulated.Comment: 7 pages revtex, 2 figure
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
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