On Classical and Quantum Cryptography

Lectures on classical and quantum cryptography. Contents: Private key cryptosystems. Elements of number theory. Public key cryptography and RSA cryptosystem. Shannon`s entropy and mutual information. Entropic uncertainty relations. The no cloning theorem. The BB84 quantum cryptographic protocol. Security proofs. Bell`s theorem. The EPRBE quantum cryptographic protocol.Comment: 27 page

On Holographic Thermalization and Dethermalization of Quark-Gluon Plasma

We estimate the ratio of the thermalization time over the freeze-out time using a holographic AdS-Vaidya model for quark-gluon plasma formed in the heavy ion collisions. In the model the process of thermalization is described as formation of the black hole in AdS space while dethermalization process, related with the freeze-out, as the black hole evaporation due to the Hawking radiation that is modeled by the Vaidya metric with a negative mass. In this model the thermalization takes place only at small scales and absent in the infrared region. At small scales the system tends towards a state of the thermal equilibrium only for the short time after which the processes of dethermalization starts. In this simple model the dethermalization time has a low bound about 7 fm/c which is consistent with experimental data.Comment: 15 pages, 5 figure

Cosmological Daemon

Classical versions of the Big Bang cosmological models of the universe contain a singularity at the start of time, hence the time variable in the field equations should run over a half-line. Nonlocal string field theory equations with infinite number of derivatives are considered and an important difference between nonlocal operators on the whole real line and on a half-line is pointed out. We use the heat equation method and show that on the half-line in addition to the usual initial data a new arbitrary function (external source) occurs that we call the daemon function. The daemon function governs the evolution of the universe similar to Maxwell`s demon in thermodynamics. The universe and multiverse are open systems interacting with the daemon environment. In the simplest case the nonlocal scalar field reduces to the usual local scalar field coupled with an external source which is discussed in the stochastic approach to inflation. The daemon source can help to get the chaotic inflation scenario with a small scalar field.Comment: LATEX, 31 pages, Sect. "Inflation and External Sources" and refs. are adde

Matrix Theory in Curved Space

According to the Matrix theory proposal of Banks, Fischler, Shenker and Susskind M-theory in the infinite momentum frame is the large N limit of super Yang-Mills theory in a flat background. To address some physical issues of classical gravity such as gravitational collapse and cosmological expansion we consider an extension of the BFSS proposal by defining M-theory in curved space as the large N limit of super Yang-Mills theory in a curved background. Motivations and possible implications of this extension are discussed.Comment: 7 pages, Latex, Talk presented at the Birthday Conference dedicated to A.Arvilski, February, 1998; one ref. adde

Noncommutative Gauge Fields on Poisson Manifolds

It is shown by Connes, Douglas and Schwarz that gauge theory on noncommutative torus describes compactifications of M-theory to tori with constant background three-form field. This indicates that noncommutative gauge theories on more general manifolds also can be useful in string theory. We discuss a framework to noncommutative quantum gauge theory on Poisson manifolds by using the deformation quantization. The Kontsevich formula for the star product was given originally in terms of the perturbation expansion and it leads to a non-renormalizable quantum field theory. We discuss the nonperturbative path integral formulation of Cattaneo and Felder as a possible approach to construction of noncommutative quantum gauge theory on Poisson manifolds. Some other aspects of classical and quantum noncommutative field theory are also discussed.Comment: 19 pages, 3 figures. Invited lecture given by I.V. Volovich at the Madeira workshop on Noncommutative Infinite Dimensional Analysis, July 1999; typos correcte

Note on the Unruh Effect

It was suggested by Unruh that a uniformly accelerated detector in vacuum would perceive a noise with a thermal distribution. We obtain a representation of solutions of the wave equation in two dimensions suitable for the Rindler regions. The representation includes the dependence on a parameter. The Unruh field corresponds to a singular limit of the representation.Comment: 5 pages, Late

Quantum Decoherence and Higher Order Corrections to the Large Time Exponential Behaviour

There exists the well known approximate expression describing the large time behaviour of matrix elements of the evolution operator in quantum theory: =exp(at)+... This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf-Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: =exp(At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.Comment: 30 pages, 9 figure

Anisotropic Asymptotics and High Energy Scattering

Recently E.Verlinde and H.Verlinde have suggested an effective two-dimensional theory describing the high-energy scattering in QCD. In this report we attempt to clarify some issues of this suggestion. We consider {\it anisotropic asymptotics} of correlation functions for scalar and gauge theories in four dimensions. Anisotropic asymptotics describe behaviour of correlation functions when some components of coordinates are large as compare with others components. It is occurred that (2+2) anisotropic asymptotics for 4-points functions are related with the well known Regge regime of scattering amplitudes. We study an expansion of correlation functions with respect to the rescaling parameter $\lambda$ over a part of variables (anisotropic $\lambda$-expansion). An effective theory describing the anisotropic limit of free scalar field contains two 2 dim conformal theories. One of them is a conformal theory in configuration space and another one is a conformal theory in momentum space. In some special cases ,in particular for the Wilson line correlators in gauge theories, the leading term of the anisotropic expansion involves only one of the conformal theories and it can be described by an effective theory with an action being a dimensional reduction of the original action.Comment: 16 pages, LATEX, 3 figures, Invited talk at the Conference Quarks 94, Vladimir, May, 199

Knots and Matrix Models

We consider a matrix model with d matrices NxN and show that in the limit of large N and d=0 the model describes the knot diagrams. The same limit in matrix string theory is also discussed. We speculate that a prototypical M(atrix) without matrix theory exists in void.Comment: 9 pages, LaTex, 6 figure

The Master Field for QCD and qq-Deformed Quantum Field Theory

The master fields for the large $N$ limit of matrix models and gauge theory are constructed. The master fields satisfy to standard equations of relativistic field theory but fields are quantized according to a new rule. To define the master field we use the Yang-Feldman equation with a free field quantized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra however there is a non-Abelian gauge symmetry. For the construction of the master field it is essential to work in Minkowski space-time and to use the Wightman correlation functions. The BRST quantization of the master field for gauge theory and a loop equation are considered.Comment: More comments about a new gauge theory and more references are added, 14 pages, late