I. V. Volovich

THE WAVE FUNCTION OF THE UNIVERSE AND p-ADIC GRAVITY

A new approach to the wave function of the universe is suggested. The key idea is to take into account fluctuating number fields and present the wave function in the form of a Euler product. For this purpose we define a p-adic generalization of both classical and quantum gravitational theory. Elements of p-adic differential geometry are described. The action and gravitation field equations over the p-adic number field are investigated. p-adic analogs of some known solutions to the Einstein equations are presented. It follows that in quantum cosmology one should consider summation only over algebraic manifolds. The correspondence principle with the standard approach is considered.

Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems

This monograph provides a mathematical foundation to the theory of quantum information and computation, with applications to various open systems including nano and bio systems. It includes introductory material on algorithm, functional analysis, probability theory, information theory, quantum mechanics and quantum field theory. Apart from standard material on quantum information like quantum algorithm and teleportation, the authors discuss findings on the theory of entropy in C*-dynamical systems, space-time dependence of quantum entangled states, entangling operators, adaptive dynamics, relativistic quantum information, and a new paradigm for quantum computation beyond the usual quantum Turing machine. Also, some important applications of information theory to genetics and life sciences, as well as recent experimental and theoretical discoveries in quantum photosynthesis are described.

Anti-Kählerian manifolds

Abstract An anti-Kahlerian manifold is a complex manifold with an anti-Hermitian metric and a parallel almost complex structure. It is shown that a metric on such a manifold must be the real part of a holomorphic metric. It is proved that all odd Chern numbers of an anti-Kahlerian manifold vanish and that complex parallelisable manifolds (in particular the factor space G/D of a complex Lie group G over the discrete subgroup D ) are anti-Kahlerian manifolds. A method of generating new solutions of Einstein equations by using the theory of anti-Kahlerian manifolds is presented.

QUANTUM GROUP GAUGE FIELDS

Gauge fields models with quantum groups playing the role of gauge groups are discussed. An appropriate description of the quantum group SUq(2) is presented. The exponential map for this quantum group is constructed and it is shown that it can be parametrized by elements of quantum superplane. Some applications including quantum group chiral fields are briefly considered. In particular, a solution of the SUq(2) WZNW model is presented and an infinite number of conserved currents for the 2-dimensional quantum group chiral field is constructed.

Semiclassical properties and chaos degree for the quantum Baker’s map

We study the chaotic behavior and the quantum-classical correspondence for the Baker’s map. Correspondence between quantum and classical expectation values is investigated and it is numerically shown that it is lost at the logarithmic timescale. The quantum chaos degree is computed and it is demonstrated that it describes the chaotic features of the model. The correspondence between classical and quantum chaos degrees is considered.

Non-Linear Extensions of Classical and Quantum Stochastic Calculus and Essentially Infinite Dimensional Analysis

It is likely (at least for its proponent) that quantum probability, or more generally algebraic probability shall play for probability a role analogous to that played by algebraic geometry for geometry: many will complain against a loss of immediate intuition, but this is compensated for by an increase in power, the latter being measured by the capacity of solving old problems, not only inside probability theory, or at least of bringing non-trivial contributions to their advancement. The present, reasonably satisfactory, balance between developement of new techniques and problems effectively solved by these new tools should be preserved in order to prevent implosion into a self-substaining circle of problems and the main route to achieve this goal is the same as for classical probability, namely to keep a strong contact with advanced mathematical developement on one side and with real statistical data, wherever they come from, on the other.

TIME MACHINE AT THE LHC

Recently, black hole and brane production at CERNs Large Hadron Collider (LHC) has been widely discussed. We suggest that there is a possibility to test causality at the LHC. We argue that if the scale of quantum gravity is of the order of few TeVs, proton-proton collisions at the LHC could lead to the formation of time machines (spacetime regions with closed timelike curves) which violate causality. One model for the time machine is a traversable wormhole. We argue that the traversable wormhole production cross section at the LHC is of the same order as the cross section for the black hole production. Traversable wormholes assume violation of the null energy condition (NEC) and an exotic matter similar to the dark energy is required. Decay of the wormholes/time machines and signatures of time machine events at the LHC are discussed.

Generalized Quantum Turing Machine and its Application to the SAT Chaos Algorithm

Ohya and Volovich have proposed a new quantum computation model with chaotic amplification to solve the SAT problem, which went beyond usual quantum algorithm. In this paper, we generalize quantum Turing machine, and we show in this general quantum Turing machine (GQTM) that we can treat the Ohya-Volovich (OV) SAT algorithm.

Universal gravitational equations

SummaryWe show that for a wide class of Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the first-order formalism,i.e. treating the metric and the connection as independent variables, leads to «universal» equations. If the dimensionn of space-time is greater than two, these universal equations are Einstein equations for a generic Lagrangian. There are exceptional cases where a bifurcation appears. In particular, bifurcations take place for conformally invariant LagrangiansL =R(sun/2)√g. For 2-dimensional space-time we obtain that the universal equation is the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi-Civita connection of the metric and an additional vector field.

Local realistic representation for correlations in the original EPR-model for position and momentum

It is currently widely accepted, as a result of Bells theorem and related experiments, that quantum mechanics is inconsistent with local realism and there is the so called quantum non-locality. We show that such a claim can be justified only in a simplified approach to quantum mechanics when one neglects the fundamental fact that there exist space and time. Mathematical definitions of local realism in the sense of Bell and in the sense of Einstein are given. We demonstrate that if we include into the quantum mechanical formalism the space–time structure in the standard way then quantum mechanics might be consistent with Einsteins local realism. It shows that loopholes are unavoidable in experiments aimed to establish a violation of Bells inequalities. We show how the space–time structure can be considered from the contextual point of view. A mathematical framework for the contextual approach is outlined.