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International Conference on Theoretical Physics 2011
Moscow, Russia
20-23 of June 2011
Moscow State Open University
The entropy gain and the Choi-Jamiolkowski correspondence for infinite-dimensional quantum evolutions
S. Holevo
Steklov Mathematical Institute, Russia
holevo@mi.ras.ru
In the first part of the talk we discuss the entropy gain for infinite-dimensional quantum evolutions. We show that unlike finite-dimensional case where the minimal entropy gain is always nonpositive, there are many channels with positive minimal entropy gain. We present the new lower bound and compute the minimal entropy gain for a broad class of Bosonic Gaussian channels by proving that the infimum is attained on the Gaussian states. The second part of the talk is devoted to the Choi-Jamiolkowski correspondence between channels and states in the infinite-dimensional case in the form close to one used in quantum information theory. In particular, we obtain explicit expression for the Choi-Jamiolkowski operator defining a general nondegenerate Bosonic Gaussian evolution and compute its norm.
On Superstitions and Errors in Understanding the Relativity Theory
Nikolai V. Mitskievich
Universidad de Guadalajara, Mexico
nmitskie@gmail.com
In this lecture, with strictly given proofs, we show that several basic axioms, including those used in generally accepted foundations of Special and General Relativity theory, do merely represent regrettable misunderstandings whose elimination by no means does falsify these theories, but it helps to understand more profoundly their contents and opens new horizons for the development of Theoretical Physics. This situation is illustrated by two examples which are accepted to be obvious truths from which were admittedly deduced both Special, as well as General Relativity Theories: in the first case it is shown that the so-called Galileo transformations have in fact the same contents as the Lorentz transformations, and in the second case that the Principle of Equivalence of Inertial and Gravitating Masses is not fulfilled in General Relativity Theory trivially yielding an unavoidable relativistic generalization of this Equivalence Principle which radically differs from the generally accepted formulation of this Principle, not reducing to addition of small corrections to this old and primitive formulation.
Quantum Effects in Photosynthesis and
Entropy Decreasing
Igor V. Volovich
Steklov Mathematical Institute
Russian Academy of Sciences
Gubkin St. 8, 119991, Moscow, Russia
email:volovich@mi.ras.ru
Photosynthesis changes the energy from the sun into chemical energy and is vital for life on Earth. Study of photosynthesis is of a fundamental importance not only for pure science but also for applications. If researchers could learn how to move energy with such precision and efficiency over comparable distance as Nature does in photosynthesis, then enormous leaps in the development of cheap organic solar cell technology would ensue.
Previously the role of quantum effects in the photosynthesis at the room temperature was ruled out because of the quantum decoherence. However, a remarkable recent experiment (Scholes et al.) has shown that quantum mechanics might be involved in the process of photosynthesis in some marine algae even at the room temperature, see [1] for a discussion.
In this talk, based on [2], it will be suggested that the phenomenon of the enhancement of the transport of excitons in photosynthesis might be related with the decreasing (not increasing!) of entropy for the solutions of the master equation for some the complete positive trace-preserving noisy quantum channels. A constructive role of noise in quantum computations was mentioned earlier and a new paradigm for quantum computations which goes beyond the quantum Turing machine was suggested, see [1]. Note also that it was found by Caruso et al that the quantum capacity for a quantum channel in the quantum network dynamics can be enhanced by introducing dephasing noise.
References
[1] M. Ohya and I. Volovich, Mathematical Foundations of Quantum Information
and Computation and Its Applications to Nano- and Bio-systems, Springer, 2011.
[2] S. Iriyama, M. Ohya, K. Sato and I. Volovich, Photosynthetic anthenna and
entropy decreasing, TUS preprint, 2010 (to be published).
The mechanism of tunneling and formation of bound pairs of electrons
Martin Rivas
University of the Basque Country, Spain
martin.rivas@ehu.es
The classical description of elementary spinning particles shows that the center of mass and center of charge of an elementary particle are different points. This separation is half Compton's wave length and because of this the interaction of two electrons with their spins parallel can produce a bound pair provided the internal phase is opposite and the relative velocity of their centers of mass is below a certain limit. It is also this separation which justifies that an electron under a potential barrier can cross it with an energy below the top of the potential provided the spin is properly oriented and the barrier has a narrow range. This can justify the spin polarized tunneling effect.
References
M. Rivas Kinematical Theory of spinning particles, Classical and quantum mechanical formalism of elementary particles, Fundamental Theories of Physics Series, Vol 116, Kluwer Academic Publishers and Springer
author web-page:http://tp.lc.ehu.es/martin.htm
Soliton Configurations in Generalized Mie Electrodynamics
Yu.P. Rybakov
Peoples’ Friendship University
Department of Theoretical Physics
E-mail: soliton4@mail.ru
We
consider
the
generalization
of
the
G.
Mie
electrodynamics
including
8-spinor
field
source
and
higher
degrees
of
the
Mie
invariant
.
Peculiar
topological
properties
of
8-spinors
are
distinguished
and
expressed
via
the
existence
of
the
remarkable
8-squares
F.
Brioschi
identity
permitting
to
obtain
the
natural
8-spinor
unification
of
the
Skyrme
baryons
model
and
the
Faddeev
leptons
model,
these
particles
being
considered
as
the
topological
solitons.
We
construct
the
two
types
of
the
soliton-like
configurations
admitted
by
the
model:
charged
static
ones
and
luxons,
i.e.
neutral
photon-like
solitons.
Non-classical soliton structures in dynamics
M. A. Aguero
Department of Physics, Faculty of Science, Universidad Autonoma del Estado de Mexico, 50000 Toluca, Mexico,
maagueorg@uaemex.mx
We have analyzed the improved Dauxoi-Peyrard-Bishop model that takes into consideration the inclusion of nonlinear interaction between adjacent pair of bases. The study of displacements along the Hydrogen bonds of DNA shows the appearance of nonlinear structures named crowdons, cuspons and peakons. These solutions exist in certain domain of the main parametric space of the model that determines a priori the velocity of the traveling structures. Nevertheless these structures would allow us to catch some peculiarities of the denaturation process. The crowdon perturbations should be considered the natural counterparts that cure the appearance of denaturation.
Are electrons pointlike or extended?
Alexander Burinskii, Moscow,
NSI Russian Academy of Sciences
bur@ibrae.ac.ru
We give a brief review of the old and recent models of the extended electron, in particular the toroidal ringlike (stringlike) models considered from diverse posits of view by many authors (Parson (1916), A.Compton (1919-21), H. Hoenl 1938 and many others). After the great success of QED, and experiments on the deep inelastic scattering, these old models were considered as obsolete. Meanwhile, the QED does not take into account gravity, and moreover, there is great problem with its consistence with gravity. On the other hand, there are many evidences that black holes are akin to elementary particles (G. `t Hooft, A.Sen, F.Wilczek), and the Kerr-Newman solution has given new evidences in support of the old stringlike extended model of the electron. After Carter’s observation (1968) that the KN solution has g=2 as that of the Dirac electron there appears new activity on the model of spinning electron consistent with gravity. Singular ring of the KN solution takes the form of the lightlike circular string of the Compton size [1]. The KN model of an extended electron was started by W.Israel (1969), and from diverse point of view was considered by author in [2] as a model of a `microgeon with spin’. C.Lopez (1984) developed the Israel model (there were also the works by Arcos and Pereira (2004), T. Nieuwenhuizen (2006), Dymnikova (2006), and others, some of the refs. are given in [3].) In the paper [3] we showed that the regularization of the KN solution by the Higgs field leads to a model of the extended spinning electron consistent with gravitaty, and again there appeared a circular string of the Compton size on the border of the KN source, reproducing the old toroidal ring models. We show now that the lowest excitation of the KN soliton creates a singular node, which may be exhibited as a pointlike structure of the consistent with gravity extended KN electron.
References
[1] A. Burinskii, "Some Properties of the Kerr Solution to Low-energy String Theory," Phys. Rev. D 52 (1995) 5826 [arXiv:hep-th/9504139].
[2] A.Ya. Burinskii, «Микрогеон со спином». ЖЭТФ, т.66 (1974) 406-411; translation in: Sov. Phys. JETP, 39 (1974) 193.
[3] A. Burinskii, “Regularized Kerr-Newman Solution as a Gravitating Soliton"J. Phys. A: Math. Theor. 43 (2010) 392001, [arXiv: 1003.2928].
Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin
Carlo Di Franco
Physics Department, University College Cork, Ireland
cdifranco@caesar.ucc.ie
The two-dimensional Grover quantum walk has raised the interests of the scientific community, as it can be used in order to implement the two-dimensional Grover search algorithm [1]. During this talk, I will demonstrate that the non-localized case of the spatial density probability of the Grover walk can be obtained using only a two-dimensional coin space and a quantum walk in alternate directions [2]. To prove formally this equivalence, I will illustrate how the coefficients of the Grover walk in the non-localized case can be mapped to the coefficients of the alternate walk state for a particular instance of the coin initial conditions.
One of the key properties of quantum walks is their ability to evolve disentangled states into entangled ones and to efficiently generate entanglement in experimentally feasible systems [3]. Controlled entanglement generation has currently a place at the forefront of research, as it is a fundamental resource in quantum computation and cryptography and therefore a pre-requisite for the construction of reliable devices for quantum information processing [4]. I will present an analysis of the behavior of the coin-position as well as the x-y spatial entanglement in the proposed scheme with respect to the Grover one. I will show that this experimentally simpler scheme allows to entangle the two orthogonal directions of the walk more efficiently.
Finally, I will discuss a possible physical implementation of the proposed walk, along the lines of a recent experimental realization of a linear quantum walk of a single neutral atom in a spin-dependent one-dimensional optical lattice [5].
References
[1] N. Shenvi, J. Kempe, and K. B. Whaley, Phys. Rev. A 67, 052307 (2003); A. Ambainis, J. Kempe, and A. Rivosh, in Proc. 16th ACM-SIAM SODA, Vancouver (SIAM, Philadelphia, USA, 2005), p. 1099; A. Tulsi, Phys. Rev. A 78, 012310 (2008).
[2] C. Di Franco, M. McGettrick, Th. Busch, arXiv:1010.2470 (2010), accepted to be published on Phys. Rev. Lett.
[3] S. E. Venegas-Andraca and S. Bose, arXiv:0901.3946 (2009); S. K. Goya and C. M. Chandrashekar, J. Phys. A 43, 235303 (2010).
[4] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
[5] M. Karski, L. Forster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede and A. Widera, Science 325, 174 (2009).
Densyty Matrices of the nuclear Shall Model
Deveikis
Vytautas Magnus University, Lithuania
a.deveikis@if.vdu.lt
The initio no-core nuclear shell-model approach is based on calculation of wave functions for description of many particle systems [1]. However it is well known that long series expansion of exact wave function in shell model ones is plagued with a number of serious convergence problems. In the light of ever-increasing model space size, the more promising approach for calculation of identical particle systems may be based on translationally invariant density matrices instead of wave functions. The approach based on density matrices may considerably reduce the size of calculations and memory demand. Moreover the translationally invariant density matrices may be calculated in antisymmetric but not translationally basis, so the sophisticated calculation of translationally invariant coefficients of fractional parentage may be completely avoided.
The presented two-particle translationally invariant density matrices are defined as two-particle density matrices integrated over centre-of-mass position vector of two last particles and complemented with isospin variables [2]. The procedures for calculation of two-particle translationally invariant density matrices were developed and implemented in computer code. The theoretical formulation have been illustrated by calculation of translationally invariant density matrices for Ex=0,1,2,3,4 excitations in the case of A=6 and JT=21 nucleus.
References
[1] Navr/SPAN> P., Quaglioni S., Stetcu I., Barrett B.R. Recent developments in no-core shell-model calculations // J. Phys. G: Nucl. Part. Phys. 2009. V.36, 083101, P. 1–54.
[2] Deveikis A., Kamuntavicius G.P. Intrinsic density matrices of the nuclear shell model // Lithuanian J. Phys. 1996. V.36, No. 2, P. 83–95.
Darboux Transformations for Generalized Schr"odingerEquations
A.A. Suzko and E.P. Velicheva (JINR)
suzko@jinr.ru
The generalized Darboux transformations are constructed for Schr"odinger equations with a position-dependent effective mass and with linearly energy-dependent potentials.
The point canonical method and the intertwining relation technique are used to obtain a family of exact solutions for this type of equations. Some examples are given for different forms of mass functions.
The Smooth Skew Product in the Plane with Ramified Continuum as the Global Chaotic Attractor Containing Nonchaotic Invariant Subsets
L,S. Efremova
Nizhny Novgorod State University, Russia
lefunn@gmail.com
Using the notions of the NT>-function and of functions suitable for the NT>-function [1], we construct the example of the C^1-smooth skew product in the closed unit square with the one-dimensional ramified continuum as the global attractor such that the following properties are valid:
(1) the
set
of
ramification
points
of
the
global
attractor
has
continuum
cardinality,
and
the
order
of
any
ramification
point
equals
3;
(2) the
cardinality
of
the
set
of
points
of
local
connectedness
of
,
just
as
the
cardinality
of
the
set
of
points
which
are
not
points
of
local
connectedness
of
,
equals
continuum;
(3) although
the
topological
entropy
of
the
map
on
is
positive
(in
this
sense
is
a
chaotic
attractor),
nevertheless
contains
invariant
(under
some
iterations
of
the
map)
closed
intervals
with
nonchaotic
behavior
of
trajectories,
where
the
topological
entropy
of
the
map
equals
zero
[2], [3].
This research was supported in the part by the Federal Target Program "Scientific and Scientific-Pedagogical Personnel of Innovative Russia" (2009 -- 2011) of the Federal Education Agency (Project No. NK -13/9).
References
1. Efremova L. S., Space of C^1-smooth skew products of maps of an interval, Theor. and Math. Physics 164 (2010), no. 3, 1208--1214.
2. Efremova L. S, Example of the smooth skew product in the plane with the one-dimensional ramified continuum as the global attractor, ESAIM: PROCEEDINGS (2011).
3. Efremova L. S., Differential properties and attracting sets of a simplest skew product of interval maps, Sbornik: Mathematics 201 (2010), no. 6, 873-907.
Quantum Mechanics as Asymptotics of Classical Diffusion Processes for Waves in the Phase Space
E.M. Beniaminov
Moscow State Gumenitary University
ebeniamin@yandex.ru
We consider the process of diffusion scattering of a wave function given on the phase space. In this process the heat diffusion is considered only along momenta. We write down the modified Kramers equation describing this situation. In this model, the usual quantum description arises as asymptotics of this process for large values of resistance of the medium per unit of mass of particle.It is shown that in this case the process passes several stages. During the first short stage, the wave function goes to one of ``stationary'' values. At the second long stage, the wave function varies in the subspace of``stationary'' states according to the Schrodinger equation. Further, dissipation of the process leads to decoherence, and any superposition of states goes to one of eigenstates of the Hamilton operator. At the last stage, the mixed state of heat equilibrium (the Gibbs state) arises due to the heat influence of the medium and the random transitions among the eigenstates of the Hamilton operator. Besides that, it is shown that, on the contrary, if the resistance of the medium per unit of mass of particleis small, then in the considered model, the density of distribution of probability satisfies the standard Liouville equation, as in classical statistical mechanics.
The dynamics of binary alternatives for a discrete pregeometry
Alexey L. Krugly
Scientific Research Institute for System Analysis of the Russian Academy of Science
akrugly@mail.ru
An example of a discrete pregeometry on a microscopic scale is introduced. The model is a directed dyadic acyclic graph. This is the particular case of a causal set. The causal set is a locally finite partially ordered set. The dynamics of this model is a stochastic sequential growth dynamics. New vertexes are added one by one. The probability of this addition of a new vertex depends on the structure of existed graph. The particular case of the dynamics is considered. This dynamics is based on binary alternatives. Each directed path is considered as a sequence of outcomes of binary alternatives. The probabilities of a stochastic sequential growth have quadratic dependence on these paths. There is a matrix form of the dynamics. A matrix of amplitudes of causal connection is introduced for the description of binary alternatives. We have linear equations for the matrix of amplitudes during sequential growth of a graph. These equations are represented by evolution operators. The probabilities of a stochastic sequential growth have quadratic dependence on the matrix of amplitudes.
Integral charge quark super symmetry
U. V. S. SESHAVATHARAM1 & Prof. S. LAKSHMINARAYANA2
1DIP QA Engineer, Lanco Industries Ltd, Srikalahasti-517641, A.P, India.
2Department Of Nuclear Physics, Andhra University,
Visakhapatnam-530003, AP, India.
‘Quark flavor’ is a property of ‘strong interaction charge’ and nowhere connected with ‘fermions’ or ‘bosons’. There exists nature friendly ‘integral charge quark flavors’. If a ‘charged quark flavor’ rests in a ‘fermionic container’ it is a ‘quark fermion’. Similarly if a ‘charged quark flavor’ rests in a ‘bosonic container’ it is a ‘quark boson’. Strong interaction charge contains ‘multiple flavors’ and can be called as the ‘hybrid charge quark’. No 3 quark fermions couples together to form a baryon and no 2 quark fermions couples together to form a meson. In super symmetry, quark fermion and quark boson mass ratio is Si = 2.262218404 but not unity. Quark fermions convert into quark baryons and effective quark fermions convert into effective quark baryons. Similarly quark bosons convert into quark mesons. Effective quark baryons generates charged and unstable multi flavor baryons and light quark bosons couples with these charged baryons and generates doublets or triplets. Any two oppositely charged quark mesons generates neutral and unstable mesons.
T.F. Kamalov
Physics Department, Moscow State Open University
TimKamalov@gmail.com
The problem of axiomatization of physics formulated by Hilbert as early as 1900 and known as the Sixth Problem of Hilbert is nowadays even more topical than at the moment of its formulation. Axiomatic inconsistency of classic, quantum, and geometrized relativistic physics of the general relativistic theory does not in the least fade away, but on the contrary, becomes more pronounced each year. This naturally evokes the following questions: 1. Is it possible, without drastically changing the mathematics apparatus, to set up the axiomatics of physics so as to transform physics, being presently a multitude of unmatched theories with inconsistent axiomatics, into an integrated science? 2. Is it possible, maybe through expanding their scopes, to generalize of transform the existing axiomatics into an integral system of axioms in such a manner that existing axiomatics of inconsistent theories would follow there from as a particular case?
Dark energy: Astronomical aspects
A.D. Chernin
Sternberg Astronomical Institute, Moscow University, Russia
Dark
energy
is
the
mysterious
form
of
cosmic
energy
that
produces
antigravity
and
accelerates
the
global
expansion
of
the
universe.
It
was
first
discovered
in
observations
of
the
Hubble
expansion
flow
with
the
use
of
type
Ia
supernovae
at
horizon-size
distances
of
more
than
1000 megaparsec
(Mpc)
(1 Mpc
is
equal
to
3.26 million
light-years).
These
and
other
studies,
especially
the
observations
of
the
cosmic
microwave
background
(CMB)
anisotropy,
indicate
that
the
global
dark
energy
density
is
(0.75±0.05)x10PAN>^2PAN>
kilograms
per
cubic
meter
(kg/m^3).
It
contributes
nearly
3/4 the
total
energy
of
the
universe.
According
to
the
simplest,
straightforward
and
quite
likely
interpretation,
dark
energy
is
described
by
the
Einstein
cosmological
constant.
If
this
is
so,
dark
energy
is
the
energy
of
the
cosmic
vacuum
with
the
equation
of
state
.
Here
,
are
the
dark
energy
density
and
pressure
which
are
both
constant
in
time
and
uniform
in
space
(the
speed
of
light
c=1).
The
interpretation
implies
that
although
dark
energy
betrayed
it
existence
through
its
effect
on
the
universe
as
a
whole,
it
exists
everywhere
in
space
with
the
same
density
and
pressure.
We
have
found
dynamical
effects
of
dark
energy
in
our
closest
galactic
neighborhood
using
systematic
observations
of
distances
and
motions
of
galaxies
in
the
Local
Group
and
in
the
flow
around
it
carried
out
with
the
Hubble
Space
Telescope.
Kerr-Schild Way to Quantum Gravity
Alexander Burinskii
Moscow, NSI, Russian Academy of Sciences
Controversies between quantum theory and gravity are well known. It is expected that resolution of these controversies could bring to solution of the one of the principal problems of the modern theoretical physics – the unified theory of quantum gravity. The black holes (BH) are the most convenient objects for studying this inconsistency. One of the sources of the inconsistency is related with general covariance of General Relativity which conflicts with the Fourier transform. In this respect the Kerr-Schild (KS) form of the BH solutions represents exclusive interest. Being based on a congruence of twistor null lines in Minkowski space-time, the KS geometry is strongly linked to the Minkowski background [1], and the coordinate freedom of General Relativity turns out to be maximally suppressed in the KS representation. It allows one to use a special “twistor version” of the Fourier transform [2] in the curved KS space-time, which is necessary for Quantum treatments. In the work [3,4] we described a class of the exact Kerr-Schild (KS) solutions for electromagnetic (EM) excitations of the Kerr-Newman (KN) black hole and their back-reaction on the metric and horizon. It has been shown that there are no smooth harmonic excitations on the KS background, and the typical EM excitations of the KS geometry take the form singular beams which have very strong back-reaction on metric and the BH horizon. There were also obtained the fluctuating beamlike solutions, which deform the BH horizon topologically. The fluctuating KS geometry leads to a fluctuating horizon, which allows matter to escape the BH. The obtained KS geometry of fluctuating beams differs drastically from the usual smooth stationary gravity and was classified as a pre-quantum geometry [4]. Now we can specify the transfer to the usual classical gravity. It was shown in [4,5] that the beamlike solutions are determined by twistorial structure of the Kerr-Schild (KS) geometry which is built of a time-oriented congruence of the lightlike geodesics, forming a time-oriented vacuum (ket) state |in >. The congruences are determined by the Kerr Theorem via a special generating function F(T) of the projective twistor variables T=(Y, AN>-Yv, u+Y AN>*), and for the idealized KN background function F is quadratic in Y. We notice that any process of the measurement of a physical observable ‘G’ breaks this idealization, since the measurer breaks topology of the initial KN space-time. In accordance with the Kerr theorem [6], the measurer creates extra sheets of the space-time with an extra twistor structure which has an opposite-oriented congruence (directed from the measurer to BH) and may be identified with the dual (bra-) vacuum state <out| . The measurement is related with formation of an amplitude of probability < out| G |in > and involves the dual (bra-) state < out| which is complex conjugate to state |in > and described by a reverse time-evolution. Therefore, the obtained classical Kerr-Schild geometry of fluctuating twistor beams may be associated with a ket |in > pre-quantum vacuum state, while the transfer to the classical level of the real physical observables is related with an interplay with the dual (bra-) vacuum state <out| , in accordance with the known principles of quantum theory.
References
[1] G.C. Debney, R.P. Kerr and A. Schild, J. Math. Phys., 10, 1842 (1969).
[2] E. Witten, Comm. Math. Phys. 252, 189 (2004).
[3] A. Burinskii, First Award of GRF 2009, Gen. Rel. Grav. 41, 2281 (2009).
[4] A. Burinskii, J. Phys.: Conf. Ser. 222, 012044 (2010), arXiv:1001.0332[hep-th].
[5] A. Burinskii, Theor. Math. Phys. 163 (3), 782-787, (2010), arXiv:1001.0332.
[6] A. Burinskii, Grav. Cosmol. 11, 301 (2005);
On Quantization in Gravity Theory
M.L. Fil’chenkov, Yu.P. Laptev
Institute of Gravitation and Cosmology,
Peoples’ Friendship University of Russia
fmichael@mail.ru
Gravity theory is shown to be consistent with quantum theory. Gravitation is considered within the framework of General Relativity as well as Newtonian theory. Quantum theory is presented on the level of quantum mechanics and quantum field theory. Quantization is performed following perturbation theory for weak gravitational fields and by nonperturbative methods for strong ones.
It is widely believed that quantum gravity does not exist at all, which is correct if by this an impossibility is meant of constructing the theory by perturbative methods at high energies, because the gravitational field is not normalizable in this case. Since in the theories for other fields only perturbative methods are used, there arises an illusion about impossibility of quantizing the gravitational field, in principle.
We consider the levels of quantization as follows:
Quantization in a given gravitational field or a given Riemannian space-time including quantum mechanics and quantum field theory.
Quantization of a curved space-time or a gravitational field including perturbative approaches, e.g. quantization of a weak gravitational field and superstring theory, as well as nonperturbative ones, e.g. quantum geometrodynamics and loop quantum gravity.
In the framework of field-theoretic and geometric approaches a set of problems has been considered. Although a unified approach is not found, nevertheless it is possible to obtain some results for spherically symmetric and cosmological models being of interest for relativistic astrophysics and cosmology.
Dirac's scalar field as the metric tensor component and the cosmological constant problem
O. V. Babourova1, B. N. Frolov2
Moscow State Pedagogical University
Department of Physics and Computer Technologies
In
the
Poincare-Weyl
gauge
theory
of
gravitation
developed
by
the
authors
[1], additional
scalar
(Dirac)
field
appears
as
an
essential
geometrical
component
of
the
metric
tensor,
and
the
spacetime
has
the
geometrical
structure
of
Weyl-Cartan
space.
The
Lagrange
density
of
the
theory
can
be
found
in
[2]. It
includes
Lagrangians
quadratic
in
torsion
and
nonmetricity,
and
also
a
proper
Lagrangian
of
,
in
which
an
effective
cosmological
constant
determined
by
describes
the
energy
of
physical
vacuum
(the
dark
energy).
It
is
accepted
in
modern
cosmology
that
the
dark
energy
(described
by
the
cosmological
constant)
is
of
dominant
importance
in
dynamics
of
the
universe.
The
major
unsolved
problem
of
modern
fundamental
physics
is
very
large
difference
of
around
120 orders
of
magnitude
between
a
very
small
value
of
Einstein's
cosmological
constant
,
which
can
be
estimated
on
the
basis
of
modern
observations
in
cosmology,
and
theoretical
calculation
in
quantum
field
theory
of
quantum
fluctuation
contributions
to
the
vacuum
energy
[3].
In homogeneous and isotropic spacetime for the spatially flat FRW metric and for the inflation stage (when the densities of ordinary matter and dark matter are very small), the field equations have the consequences
(1)
where
the
coefficients
are
calculated
via
the
coupling
constants
of
.
In
this
case
the
theory
becomes
similar
to
the
generalized
Brans–Dicke
theory
with
Dirac
scalar
field.
Then
the
field
equations
lead
to
the
system
of
equations,
which
has
two
families
of
solutions
for
the
definite
values
of
the
couple
constants
of
the
.
The
first
one
has
exponential
form
and
has
been
realized
at
the
beginning
of
the
universe
evolution,
when
the
effective
cosmological
constant
has
been
very
large:
,
,
.
(1a)
The second family of the solutions can be realized at the last period of inflation:
,
,
.
(2)
where
is
an
arbitrary
constant
of
integration,
and
a
power
can
be
chosen
to
fulfill
physical
requirements.
This
‘power
law’
inflation
is
more
suitable
for
the
last
stage
of
inflation,
because
of
the
problem
of
the
smooth
transition
from
the
inflation
stage
to
the
Friedman
stage
of
universe
evolution.
Thus the exponential solution (1) can explain the exponential decrease in time at very early universe of the dark energy (the energy of physical vacuum), describing by the effective cosmological constant. This solves the problem of cosmological constant.
This research work has been performed in the framework of the Federal Purposeful Program “Research and Pedagogical Personnel of Innovative Russia” for 2009-2013.
References
1. Babourova O. V., Frolov B. N., Zhukovsky V. Ch. Phys. Rev. D . 2006. V. 74. P. 064012-1-12 (gr-qc/0508088, 2005).
2. Babourova O. V., Frolov B. N., Kostkin R. S. Dirac's scalar field as an effective component of the dark energy and an evolution of the cosmological ‘constant’. 2011. e-Print: gr-qc/1102.2901.
3. Weinberg S. Rev. Mod. Phys. 1989. V. 61. No 1. P. 1-23.
Self-coordinated system of equations for interacting
electromagnetic and quadratic bispinor fields
Vladimir V. Kassandrov
Institute of Gravitation and Cosmology,
Peoples’ Friendship University of Russia, Moscow
Nina V. Markova
Department of Mathematics,
Peoples’ Friendship University of Russia, Moscow
In Ref.[1] we had shown that any solution to free Dirac equation can be obtained via differentiation from a doublet of scalar fields both obeying free Klein-Gordon equation. This is possible owing to the known factorization property of free Dirac equation. As a consequence, one can obtain then a whole hierarchy of solutions to both Dirac and Klein-Gordon equations. Besides, we had demonstrated that canonical spinor transformations (and even more general ones!) follow as a result of combination of Lorentz transformations for Dirac operator and internal symmetry transformation intermixing the components of the scalar field doublet.
Unfortunately,
the
procedure
cannot
be
explicitly
generalized
to
the
case
of
scalar-spinor
fields
interacting
with
electromagnetic
or
gravitational
ones.
This
is
obviously
related
to
the
loss
of
factorization
property
of
the
Klein-Gordon
operator
in
these
cases.
However,
many
interesting
consequences
of
the
above
construction
can
be
preserved
by
consideration
of
the
quadric
Dirac
equation
(QDE)
instead
of
ordinary
linear
Dirac
equation
(LDE)
itself.
In
the
case
of
a
bispinor
field
interacting,
say,
with
an
electromagnetic
one,
the
QDE
operator
is
well
known
to
factorize
into
a
product
of
two
canonical
(i.e.
with
)
Dirac
equations.
Moreover,
in
the
case
of
a
fixed
external
4-potential
the
above
scheme
reproduces
the
results
of
orthodox
relativistic
quantum
mechanics,
e.g.
in
the
problem
of
hydrogen
atom
spectrum
etc.
(for
this,
see
below,
Eq.
(5)).
On the other hand, the interchange of LDE and QDE leads to a quite new theory in the general case of self-consistently interacting electromagnetic and bispinor fields. In the case of LDE one deals in fact with a classical analogue of the canonical system of equations (for corresponding field operators) accepted in quantum electrodynamics. Alternatively, in the case of QDE one comes to a principally novel system which can be set by the following Lagrangian:
,
(1)
where
is
the
electromagnetic
field
strength
tensor,
the
set
of
four
Dirac
matrices,
and
the
last
Pauli
interaction
term
is
necessary
to
guarantee
the
factorization
property.
System
of
field
Eqs.
corresponding
to
(1) takes
the
form:
(2a)
(2b)
and
one
more
equation
for
the
Dirac-conjugated
bispinor
.
Now
it
is
a
simple
exercise
to
convince
oneself
that
Eq.(2a)
may
be
equivalently
represented
as
(3)
i.e. in an ordinary factorized form. Then for a newly defined bispinor field
(4)
the canonical Dirac linear equation will be fulfilled,
(5)
Let
us
now
notice
some
peculiarities
of
the
above
self-coordinated
system
of
Eqs.
(2a,2b).
In
the
first
place,
one
has
to
distinguish
therein
the
Dirac
current
4-vector
associated
usually
with
positive
definite
probability
density
of
a
Dirac
particle
from
the
electromagnetic
current
4-vector
defined
by
Maxwell
Eqs.(2b).
First
one
is
obviously
conserved
if
it
is
defined
via
the
auxiliary
bispinor
(4); second
is
conserved
as
a
result
of
Maxwell
Eqs.(2b)
themselves.
Thus
one
has
more
conserved
quantities
in
the
framework
of
the
presented
model
than
in
the
canonical
case.
Moreover,
if
one
would
attempt
to
substitute
the
quantum
electrodynamics
(operator)
system
for
that
given
by
Eqs.(2a),(2b),
he
is
not
obliged
to
introduce
two
independent
bispinor
fields
– for
electrons
and
positrons,
respectively,
but
could
hope
to
describe
both
particles
by
only
one
bispinor
field
satisfying
the
QDE
and
system
(2a,2b)
as
a
whole!
This
opportunity
is
just
related
to
indefinite
sign
of
the
charge
density
in
the
r.h.s.
of
Eq.(2b).
We, however, are interested in a purely classical consideration of particle-like (``soliton-like’’) solutions to self-coordinated system of Eqs.(2a,2b). It is noteworthy to mention that such solutions have been studied for the case of the canonical Dirac-Maxwell system for minimally interacting fields in Refs.2-4 and others. Despite some interesting exact results, e.g. the obtained general relationship between admissible spin and charge of such soliton-like formations (Ref.2), conclusive status of the Dirac-Maxwell system of Eqs. can be considered as rather low. Indeed, the main difficulty for exact examination of such a model was the impossibility of separation of variables and isolation of individual spherical harmonics. As to (numerically) obtained properties of the Maxwell-Dirac solitons, all found representatives of these possess negative proper energy (as it was obtained previously in the model of interacting scalar and electromagnetic fields, Ref. 5).
For the model based on the QDE, the energy of solitons is not positively defined too. Nonetheless, a special investigation is needed to determine the true sign of the solitons’ rest energy. Search of solitons for the self-coordinated system of Eqs. (2a),(2b) is currently in process. Some preliminary results will be presented at the report.
References
1. Kassandrov V.V. // Gravit. and Cosmol, 14, No.1, 2008, p.53.
2. Kassandrov V.V., Terletskii Ya.P. // In: Problems in Quantum Physics, M., Peoples’ Friend. Univ. Press, 1977, p. 39 (in Russian).
3. Kassandrov V.V. // Vestnik Peoples’ Friend. Univ. Russia, Fizika, 3(1), 1995, p.168 (in Russian).
4. Wakano M. // Prog. Theor. Phys., 35, 1966, p. 1117.
5. Rosen N. // Phys. Rev., 55, 1939, p.94;
Menius A.C., Rosen N. // Phys. Rev., 62, 1942, p. 436.
Ricci scalar describes both particle and field densities
I.E. Bulyzhenkov
Lebedev Physics Institute RAS
ibw@sci.lebedev.ru
Ricci scalar density can be used for geometrization of the extended radial particle together with geometrization of its radial field. The right hand side of the 1915 Einstein equation can be dropped for such continuous matter. Static metric solutions for nonempty (material) space are free from Schwarzschild singularities. General Relativity can be developed as a selfcontained theory by accepting quantitative equivalence of mass-energy densities associated with the inertial particle and with its gravitational field.
A new model of the Eath athmosphere with strong electric fields described by means of the Yang-Mills theory
A.S. Rabinowitch, S.Yu. Abakumov
Moscow State University of Instrument Construction and Information Sciences
20 Stromynka str., Moscow 107996, Russia
rabial@mail.ru, ab_sergei@mail.ru
We study a model of the Earth atmosphere in which the influence of strong electric fields is taken into account. The model is a generalization of the standard one and can be described by the following equation of equilibrium of the atmosphere:
, (1)
where
is
the
atmospheric
pressure
for
a
certain
latitude
and
longitude,
is
a
distance
from
the
Earth
center,
is
the
Earth
radius,
is
the
atmospheric
mass
density,
is
the
free
fall
acceleration
at
the
Earth
surface,
is
the
Earth
electric
field
and
is
the
density
of
the
atmospheric
electric
charge.
Eq. (1) without the electric field term corresponds to the standard atmospheric model. As follows from computations, this model is applicable for altitudes not more than 150 km. For higher altitudes its predictions substantially differ from experimental data. This shows the importance of the electric field term in Eq. (1). In order to describe strong electric fields in the atmosphere, we use a nonlinear generalization of the classical theory of electricity proposed in Refs. [1 – 3] which is based on the Yang-Mills equations with SU(2) symmetry.
To determine values of the parameters of the suggested model and then to compute solutions of Eq. (1) and compare them with experimental data, we use the empirical model MSIS-E-90 [4] based on data derived from spacecrafts.
Applying our model, we have computed the distributions of the atmospheric mass density and pressure for different geographic coordinates, dates and day times. The computations show that the obtained numerical results are in a good agreement with experimental data. This can be regarded as a serious argument in favor of our model.
References
1. Rabinowitch A.S. Bulletin of PFUR, ser. Phys., 2005, No. 13, pp. 68-77.
2. Rabinowitch A.S. Russ. J. Math. Phys., 2008, Vol. 15, No. 3, pp. 389-394.
3. Rabinowitch A.S. Nonlinear Physical Fields and Anomalous Phenomena. New York, Nova
Science Publishers, 2009.
4. http://omniweb.gsfc.nasa.gov/vitmo/ .
Physics of rotating and expanding black hole universe
U. V. S. SESHAVATHARAM
DIP QA Engineer, Lanco Industries Ltd, Srikalahasti-517641, A.P, India.
Throughout its journey, universe follows strong gravity. Planck particle can be considered as the baby universe. A simple derivation is given for rotating black hole’s temperature. When the rotation speed approaches ‘light speed’, temperature approaches Hawking’s Black hole temperature. Appling this idea to the cosmic black hole, it is noticed that, there is ‘no cosmic temperature’ if there is ‘no cosmic rotation’. Starting from the Planck scale, it is assumed that- “universe is a ‘rotating and expanding’ black hole”. Another key assumption is that, “at any time, cosmic black hole rotates with light speed ”. For this cosmic sphere as a whole, while in light speed rotation, ‘rate of decrease’ in temperature or ‘rate of increase’ in cosmic red shift is a measure of ‘rate of cosmic expansion”. Measured isotropic CMBR temperature 2.7250 Kelvin indicates that, at present “rate of decrease” in temperature is practically ‘zero’ and ‘rate of expansion’ is practically ‘zero’. If present CMBR temperature is 2.725 degree Kelvin, present value of obtained angular velocity is 2.17 x 10-18 rad/sec = 67 Km/sec/Mpc. Present ‘cosmic mass density’ and ‘cosmic time’ are fitted with the natural logarithm of ratio of cosmic volume and planck particle’s volume.
Investigation of closed orbits of nucleons and antinucleons moving in nonlinear fields
A.S. Rabinowitch, M.A. Kramskoy
Moscow State University of Instrument Construction and Information Sciences
20 Stromynka str., Moscow 107996, Russia
rabial@mail.ru, k.m.a@rambler.ru
We study movements of nucleons and antinucleons round nuclei under the action of their nuclear and electric fields. As is known, the classical Yukawa theory cannot describe nonlinear properties of nuclear forces [1]. That is why we use its nonlinear generalization proposed in Refs. [2 – 4]. In this nonlinear theory the relativistic movement of a hadron is described by the following equation:
, (1)
where
,
is
the
nuclear
potential,
is
the
electromagnetic
strengths,
is
the
proton
mass
at
rest,
are
the
mass
at
rest
and
electric
charge
of
the
moving
hadron,
respectively,
are
its
space-time
coordinates
of
the
Minkowski
geometry,
and
.
We solve Eqs. (1) for moving nucleons and antinucleons round nuclei using the polar coordinates. Then these equations can be reduced to a system of two nonlinear differential equations of the second order for the radial coordinate and polar angle of a particle. Numerical computations for the system of equations are performed. They show that protons and neutrons can move in closed orbits round only light nuclei. At the same time antiprotons can move in closed orbits round also medium and heavy nuclei. Besides, such orbits can be sufficiently remote from nuclei. Then the annihilation is impossible and quasi-nuclei with rotating antiprotons can appear. The conditions of forming closed orbits and quasi-nuclei are investigated in both cases of non-relativistic and relativistic antiprotons.
References
Ericson T., Weise W. Pions and Nuclei. Oxford, Clarendon Press, 1988.
Rabinowitch A.S. Int. J. Theor. Phys., 1994, Vol. 33, No 10, pp. 2049-2056.
Rabinowitch A.S. Int. J. Theor. Phys., 1997, Vol. 36, No 2, pp. 533-544.
Rabinowitch A.S. Nonlinear Physical Fields and Anomalous Phenomena. New York,
Nova Science Publishers, 2009.
Avogadro's Strong Nuclear Gravity, Super Symmetry and grand unification
U. V. S. SESHAVATHARAM1 & Prof. S. LAKSHMINARAYANA2
1DIP QA Engineer, Lanco Industries Ltd, Srikalahasti-517641, A.P, India.
Email: seshavatharam.uvs@gmail.com
2Department Of Nuclear Physics, Andhra University,
Visakhapatnam-530003, AP, India.
It
is
suggested
that
‘strong
nuclear
gravitational
constant’
is
(Avogadro
number)2
times
the
‘classical
gravitational
constant’
.
It
can
be
suggested
that
is
a
consequence
of
the
existence
of
.
plays
a
vital
role
in
the
construction
of
atom,
elementary
particles
and
the
charged
nuclear
space-time
curvature.
Based
on
‘strong
nuclear
gravity’
and
‘super
symmetry’
it
is
noticed
that
is
a
secondary
physical
constant.
Previously
proposed
2 strong
interaction
fermion
rest
masses
revised
to
105.3226825 MeV
and
11460.81321 MeV.
Super
symmetric
fermion-boson
mass
ratio
is
revised
to
2.262218404. Nuclear
charge
radius
is
fitted.
Charged
lepton
rest
masses
fitted
accurately
and
4th
heavy
lepton
is
predicted
at
42262 MeV.
Up
and
down
quark
mass
ratio
is
equal
to
.
Quark
masses,
nucleons
rest
masses
and
nuclear
binding
energy
constants
are
fitted
accurately.
QCD
scale,
strong
coupling
constant
and
Fermi's
weak
coupling
constant
are
fitted.
It
can
be
suggested
that
top
quark
boson
and
electroweak
boson
both
are
same.
boson
constitutes
2 oppositely
charged
bosons.
Finally
it
can
be
suggested
that
nuclear
and
particle
physics
can
be
studied
in
the
view
of
`Avogadro’s
strong
nuclear
gravity
and
super
symmetry.
Thermal noise and coating optimization in multilayer dielectric mirrors
Nikita Kondratiev
MSU
noxobar@mail.ru
Optical multilayer coatings of high-reflective mirrors significantly determine properties of Fabry Perot resonators. Thermal (Brownian) noise in these coatings produce excess phase noise which can seriously degrade the sensitivity of high-precision measurements using these cavities. In particular it is one of the main limiting factors at the current stage in laser gravitational-wave detectors (for example project LIGO). We present a method to calculate this effect accurately and analyze different strategies to diminish it by optimizing the coating.
Traditionally the effect of the Brownian noise is calculated as if the beam is reflected from the very surface of the mirror's coating. However, the beam penetrates the coating and Brownian expansion of the layers leads to dephasing of interference in the coating and consequently to additional change in reflected amplitude and phase. Fluctuations in the thickness of a layer change the strain in the medium and hence, due to photoelastic effect, change the refractive index of this layer. This additional effect should also be considered. It is possible to reduce the noise by changing the total number and thicknesses of high and low refractive layers preserving the reflectivity. We show how an optimized coating may be constructed analytically rather than numerically as before. We also check the possibility to use internal resonant layers, optimized cap layer and double mirrors to decrease the thermal noise.
Neutrino Telescopes in ocean and in Antarctica
Vladimir A. Zhukov
Institute for Nuclear Research, RAS
_vlzhukov@mail.ru
Significant progress was made in the field of neutrino physics in the last 30 years. Solved the problem of solar neutrinos, neutrino oscillations are found and registered neutrinos from supernova. There is a serious development of neutrino astronomy. Stage of prototype testing of deep neutrino telescope is completed. Methods of deployment at sea neutrino telescopes worked on pilot arrays in the Mediterranean (NESTOR,ANTARES), in Baikal (NT200) and in Antarctica (AMANDA). Construction of large-scale neutrino telescope with a working volume of 1 cubic km in the Mediterranean launched. Telescope of this size (ICE CUBE) has already been built in the ice of Antarctica. Upper limit in the flux of cosmic high energy neutrinos received.
Axially-Symmetric and Closed-String in the Skyrme Gauge Model
Yu. P. Rybakov, E. Benavente Ramirez
Department of Theoretical Physics,
Peoples Friendship University of Russia,
6, Miklukho- Maklaya str., Moscow, 117198, Russia
The
structure
of
axially-symmetric
fields
in
the
gauge
Skyrme
SU(2)
model
is
studied.
The
Hamiltonian
and
the
topological
charge
are
constructed
for
the
corresponding
invariant
class.
The
internal
discrete
group
of
the
model
is
found,
thus
impliying
essential
simplification
of
the
energy
functional.Within
the
scope
of
the
Skyrme
SU(2)
gauge
model
we
consider
closed
chiral
strings
(vortices),
яthe
closure
radius
being
supposed
large
with
respect
to
the
characteristic
transversal
scale
determined
by
the
model
parameters.
In
this
approximation
the
chiral
and
gauge
fields
inside
the
vortex,
as
well
as
its
energy,
can
be
estimated
as
functions
of
the
topological
charge
Q.
Pairwise Interaction Potential Parameters of Alkali Halide Crystals I-Static Crystal Method
S. Sh. Soulayman*, J. Attiyah** and A. Molhem** , S. Yunusova ***
*) Higher Institute of Applied Sciences and Technology, Department of Applied Physics, Damascus, P. O. Box 31983, Syria
**) Al-Baath University, Faculty of Sciences, Department of Physics,
Homs, P.O.Box77, Syria
***) Department of Theoretical Physics, Peoples` Friendship University of Russia
6, Miklukho-Maklaya str., Moscow, 117198, Russia
ssoulayman@hiast.edu.sy
In this work a methodology suitable for determination the parameters of the potential of pairwise interactions in the alkali halide crystals is presented. The method of static crystal was applied for several known forms of the potential. Different types of functions viz. Pauling, Born-Mayer-Huggins, Varshni-Shukla and Woodcock have been used for the Born repulsive part of energy. The Van der Waals energy due to the dipole-dipole and dipole-quadruple interactions are also considered. A comparison between the obtained results and those available in literature values is carried out.
Supersymmetry in Quantum Optics
V.A. Andreev
P.N. Lebedev Physical Institute, Moscow, Russia
The realization of Witten’s quantum mechanics superssymmetrical cheme in quantum optics is constructed. We consider the two-level systems interacting with one or two bosonic modes. They are described by the Jaynes-Cummings Hamiltonian and its generalizations. It is shown that some of such Hamiltonians form superssymmetrical pairs and can be considered as components of one superssymmetrical Hamiltonian.
Quantum description of complex systems and physical constructivism
Y.Ozhigov
MSU of M.V.Lomonosov, Russia
The core of quantum informatics is the theory and experiments on QuantumComputer. This still hypothetic device bears the outstanding role for theoretical physics, because it represents the most general model of a complex system on the quantum level that could include the more profound understanding of the life. We are standing at the beginning of new discipline: quantum physics of complex systems, in which it will be possible to understand how living things behave. But to analyze this topic we first of all must develop the new mathematical apparatus, because the traditional formalism of quantum mechanics fits only for the simple quantum effects like interference of one - two quantum particles. The only possibility to build this new apparatus is to use constructive mathematics that means to rest on algorithms instead of abstract algebra. We show how to include decoherence in the constructive model of quantum computer and how to represent the real evolutions in Hilbert space without addressing to hypothetic scalable quantum computer in sense of Deutsch and Di Vincenzo. We describe how computer programs look, which model quantum systems with many particles if to account as complex quantum entanglement as the limited classical memory can contain. Constructive rebuilding of quantum theory resolves some old problems, like the absence of "hidden variables" turning it to the exact theorem; it also puts in some order our representation of "quantum" and "classical" worlds that makes possible to focus on the practically important work - building of computer programs simulating real world on the quantum level.
Electron Trapping in Weakly Coupled Concentric Quantum Rings
I. Filikhin, S.G. Matinyan, J. Nimmo, and B. Vlahovic
North Carolina Central University, Durham, NC, USA
We are investigating electron wave function localization in double concentric quantum rings (DCQR)[1] when a perpendicular magnetic field is applied. In weakly coupled double quantum ring, a possible situation occurs when the single electron energy levels associated with different rings may cross. Degeneracy is avoided by anti-crossing of corresponding levels of DCQR. We show that in this DCQR the electron spatial transition between the rings occurs due to the electron level anti-crossing (see, for instant, [2]). The anti-crossing of the levels with different radial quantum numbers provides conditions for the electron tunneling between rings. DCQRs are composed of GaAs in an Al0.70Ga0.30As substrate [1]. To study electronic structure of DCQR, the single sub-band effective mass approach was used with energy dependence of the electron effective mass [3]. Results of numerical simulation for the electron transition are presented for the DCQR with the geometry parameters corresponding to experimentally fabricated DCQR in [1]. Estimation for energy gap between anti-crossing levels is performed to show the energy gap dependence on distance between rings and radial quantum numbers of the states. The last defines spreading of the electron wave function in DCQR. The adequacy of the model is confirmed by comparing obtained results with PL data. Effect of the trapping of an electron in the inner QR of the DCQR (or QD inside QR) may be interesting from the point of view of quantum computing.
This work is supported by the NSF (HRD-0833184) and NASA (NNX09AV07A).
References
[1] T. Mano at al. Nano Letters 5, 425, 2005; T. Kuroda et al. Phys. Rev. B 72, 205301, 2005.
[2] V. Arsoski, M. Tadic and F.M. Peeters, Acta Physica Polonica, 117, 733, 2010.
[3] I. Filikhin, V. M. Suslov and B. Vlahovic, Phys. Rev. B 73, 205332, 2006.
Novel Quantum Approach to the Heisenberg Uncertainty Principle
Itzhak Orion Ph.D. and Michael Laitman Ph.D.*
Ben-Gurion University of the Negev, POB 653, Beer-Sheva,84105 ISRAEL
* Ashlag Research Institute, POB 1552, Ramat-Gan 52115 ISRAEL
Quantum physics perception deals with three basic principles: particle-wave duality, the Heisenberg uncertainty and the wave function interpretation.
We present here a new approach for the uncertainty principle as an outcome of a different quantization order level. Particles and waves are represented the same in quantum physics. Therefore, the particle has to possess wave properties, which leads to the uncertainty problem. In our previous published paper, about particle-wave duality, we proposed that wave properties can be observed only if a bunch of particles are in a state of EoF (Equivalence of Form). Only under this condition the particles would be connected as a group (Kevutsa) holding wave properties.
The uncertainty principle is based on wave-package qualities, where frequency differences between the wave-envelope and its position is produced by a constant.
We claim in this paper that the particle is actualized out of the waves in a wave's dispersion process that gives the particle its’ properties from the wave- package for a certain energy (similar to photons). Other states, or other particles, are possible to be actualized due to the other information structures folded in the wave-package. The appearance of state levels in the atom follows an internal order (like quantum numbers in a range). In atomic or nuclear systems, the propagation operator is extracting the informatics of the wave-package that put into reality a process of photon absorption, from a previous state toward a new state population.
Behind the apparent uncertainty, there is organized information to be potentially expressed, as a particle or a state, with a probable intensity to be realized, while the whole picture of the quantum states for a system is complete.
We propose a new name to this approach: Informational Quantization.
Causality and probability in quantum mechanics
D.A. Slavnov
MSU
slavnov@goa.bog.msu.ru
We discuss the causality problem in quantum theory. We show that there exists a formulation of quantum theory that, on one hand, preserves the mathematical apparatus of the standard quantum mechanics and, on the other hand, ensures the satisfaction of the causality condition for each individual event including the measurement procedure.
Accuracy features for quantum tomography
Yu. I. Bogdanov, I.D. Bukeev
Institute of Physics and Technology, Russian Academy of Sciences,
A throughout study of statistical characteristics of fidelity in different protocols of quantum tomography is given. We consider the protocols based on geometry of platonic solids and semiregular polyhedrons such as fullerene. Characteristics of fidelity in different protocols are compared to the theoretical level of the minimum possible level of fidelity loss. Tomography of pure and mixed states in Hilbert spaces of different dimension is studied.
Platonic solids are used to provide the most symmetrical and uniform distribution of quantum states on the Bloch sphere. States, which set projection quantum measurements, are defined by vectors directed from the center of Bloch sphere to corresponding centers of solid's faces. Therefore the number of solid's faces determines the number of quantum measurement protocol's rows and this number is respectively: 4 for tetrahedron, 6 for cube, 8 for octahedron, 12 for dodecahedron and 20 for icosahedron. Since these five described solids cover the whole set of platonic solids, search for quantum measurement protocols which possess high symmetry and number of rows more than 20 makes us to refer to semiregular polyhedrons which are called Archimedean polyhedrons. As examples of such polyhedrons were chosen fullerene (truncated icosahedron), which determines the quantum protocol with 32 measurements, and polyhedron dual to fullerene (Pentakis dodecahedron), which defines the quantum protocol with 60 rows (according to the number of it's faces or, what is actually the same, to the number of fullerene's vertices).
Fidelity
comparison
of
considered
protocols
with
maximum
available
fidelity
shows
that
as
the
number
of
regular
and
semiregular
polyhedrons'
faces
increases
considered
fidelity
rapidly
converges
to
the
theoretical
limit
(in
addition
an
uniformity
of
fidelity
distribution
on
Bloch
sphere
increases
fast).
In
this
work
is
shown
that
accuracy
of
suggested
protocols
is
much
higher
in
comparison
with
an
accuracy,
which
provide
protocols
not
possessing
high
symmetry.
Considered method is generalized on the case of multi-qubit state tomography and accepts the reconstruction of not only pure states but mixed states of arbitrary rank, too.
Developed method is addressed to increase the fidelity and efficiency of quantum tomography procedures. Results of this work could be used for control of states procedures' debugging in quantum cryptography and for the realization of quantum computer's logic gates.
Semiconductor quantum ring in strong lateral electrostatic fields
V.A.Harutyunyan
State Engineering University of Armenia, Gyumri Branch, 2 M.Mkrtchyan St., 3103 Gyumri, Republic of Armenia
volhar@mail.ru
During two last decades electronic and optical properties of low-dimensional semiconductor structures have been studied both experimentally and theoretically. Along with long-known systems like quantum wells (quntum films), quantum wires, quantum dots and superlattices, the novel confined structures called quntum rings (QR’s) attract much attention [1].
In this report the specificity of single-particle states of charge carriers in semiconductor quantum ring in the presence of strong lateral homogeneous electrostatic field is examined theoretically.The finiteness of the ring thickness in both radial and perpendicular to the radial plane direction is taken into account. The explicit forms of wave functions and of energy spectrum of charge carriers in a quantum ring in the presence of strong uniform field are obtained.
It is shown, that a strong external field creates a new deep potential well; because of this, along with the quantum confinement in the radial direction, the charge carriers in the QR are additionally localized also along their angular motion. Instead of the rotation in the QR circle the particle under action of a strong external field vibrates now in a narrow angular cone of the azimuth variable. The localization cones of opposite charges are disposed at opposite edges of the QR’s diameter directed along the external homogeneous field.
As an example of application of these results we will consider the optical transitions in QR in the presence of external strong electrostatic field. Relevant characteristics for interband and intersubband electro-optical transitions in the ring are calculated analytically in this report. Particularly, it is shown that absorption intensities and threshold frequencies of electro-optical transitions depend explicitly on geometrical sizes of sample, on intensities of external fields and on effective masses of charge carriers. These theoretical results can be effectively used for the experimental observation of optical transitions in quantum ring, and for the controlled variation of electro-optical parameters of the sample.
References
1. T. Chacraborty, Adv. In Sol. St. Phys. 43, 71 (2003); T.Chacraborty and P.Pietilainen, Phys.
Rev. 50, 8460 (1994); B.Szafran, Phys. Rev. B77, 205313 (2008).
Information Transfer Constraints in Quantum Measurements
S.Mayburov
Lebedev Institute of Physics
Moscow, Russia, RU – 117924
Any measurement process includes the transfer of information from the measured object S to the information system O, which stores and processes it; thus, any measuring system (MS) can be regarded as the information channel. It was found earlier that due to the severe constraints induced by Heisenberg commutation relations, the capacity of quantum channels channels is relatively small [1], so the resulting information losses in MS can be significant and influence the measurement outcome [1]. Here the influence of such constraints on the measurement outcomes will be studied for simple MS model [2].
To
check
their
effect,
we
considered
the
model
measurement
of
dichotomic
S
observable
AN>
performed
by
MS,
which
includes
the
detector
D
and
O,
both
of
them
are
regarded
as
the
quantum
objects
[1]. The
measurement
of
two
S
ensembles
is
considered;
EPAN>
includes
the
pure
states
which
are
the
superposition
of
AN>
eigenstates
with
amplitudes
,
another
ensemble
EPAN>
is
the
probabilistic
mixture
of
such
eigenstates
with
the
same
AN>.
First,
we
analyze
the
information
transfer
during
S,
D
and
D,
O
interactions,
neglecting
D,
O
decoherence
by
their
environement.
In
this
case
Heisenberg
constraints
The
comparison
of
final
MS
states
for
such
ensembles
demonstrates
that
the
information
about
the
purity
of
incoming
S
ensemble
isn't
transferred
to
O;
for
ensembles
it
is
described
by
the
expectation
value
of
S
observable
AN>conjugated
to
AN>.
As
the
result,
O
can't
discriminate
the
pure
and
mixed
S
ensembles
with
the
same
AN>
[3]. Apllying
the
quantum
formalism
of
system
self-descripiton
[4], it
is
shown
that
such
losses
induce
the
appearance
of
randomness
in
the
measurement
of
S
pure
ensemble
EPAN>,
so
that
in
the
individual
events
O
would
detect
at
random
O
'pointer'
values
,
which
correspond
to
the
outcomes
for
O
measurement
of
incoming
[5]. Observed
by
O
in
the
individual
events,
which
correspond
to
the
collapse
of
S
pure
state.
For
the
start,
the
analysis
of
D,
O
decoherence
effect
by
the
environment
was
performed
, neglecting
the
influence
of
Heisenberg
constraints.
Concerning
the
decogerence
influence,
it's
shown
that
by
itself,
due
to
the
unitarity
of
decoherence
interactions,
it
can't
result
in
the
appearance
of
randomness
in
the
measurement
outcomes.
However
the
account
of
the
decoherence
dynamics
stabilizes
the
obtained
MS
states
additionally.
References
1. S.Lloyd, Phys. Rev. A 56, 1613 (1997).
2. S.Mayburov Int. J. Quant. Inf. 5, 279 (2007)
3. S.Mayburov (2010), Quant-ph: 1005.3691
4. Breuer T., Synthes, 107, 1-16 (1996)
5. S.Mayburov, in Frontiers of Fundamental physics , vol. 1018 AIP Conf. Proccedings (Mellville, N-Y, 2008), pp. 33-39.
Decay of multiple quantum coherences in a system of equivalent spins
S.I.Doronin, E.B.Fel'dman and A.I.Zenchuk
Institute of Problems of Chemical Physics RAS, Chernogolovka, Moscow reg., Russia
The decay rate of a highly correlated quantum system is an important parameter in view of the development of quantum communication and computing systems, where the decoherence time must be long enough in order to complete at least a single operation. We study the decay of multiple quantum coherence intensities in systems with a large number of equivalent spins. Since interactions between any two spins are described by the same constant of the dipole-dipole interaction, we are able to describe the dynamics in the systems containing hundreds of spins. As a result, we may obtain, for instance, the profiles of MQ NMR coherence intensities generated in a large system of equivalent spins, which is a real problem for other quantum systems. Considering the decay process we, first of all, obtain the decay rates of MQ NMR coherences and, second, reveal a method for creation of the coherence cluster of desirable size, which may be usefull in view of the development of quantum computation, where large clusters of correlated spins are important, for instance, for realization of quantum registers. Along with the standard preparation period of the MQ NMR experiment, we also consider a modification of the preparation period where short evolution periods generating MQ NMR coherences concatenate with short periods of coherence decay. This mechanism seemed out to be more flexible in preparation of the coherence clusters having different sizes.
Section of Applied Physics
От механики материальной точки к механике систем
В.М. Сомсиков
Институт ионосферы, Алма-Ата, 050020, Казахстан
E-mail:vmsoms@rambler.ru
http://sites.google.com/site/peosrussian/
Ньютоновская механика построена на основе модели тел в виде материальной точки (МТ). Такой идеализации отвечает второй закон Ньютона, согласно которому ускорение МТ пропорционально действующей на нее силе [1, 2]. Силы, совершающие работу по перемещению МТ, потенциальны, т.е. работа равна их интегралу по траекториям. Отсюда следует закон сохранения энергии. Из второго закона Ньютона вытекает обратимость динамика МТ.
На самом деле в природе все тела обладают структурой и поэтому имеют внутреннюю энергию, обусловленную относительными движениями их элементов. Следовательно, при движении тел во внешнем поле сил, в общем случае, у них будет меняться не только энергия движения, но и внутренняя энергия. Т.е. часть энергии, которая называется диссипативной, уйдет на возбуждение его внутренних степеней свободы. Это означает, что взаимодействия и движения реальных структурированных тел определяются, как изменением их энергии движения, так и изменением внутренней энергии. На практике учет диссипативных членов выполняется эмпирическим образом, путем дополнения уравнения движения Ньютона силами трения. Их работа определяет диссипативную часть энергии, которая идет во внутреннюю энергию тела и рассеивается в среде [2]. Коэффициент трения берется из эксперимента. Т.о, в рамках классической механики, построенной на основе идеалистических моделей МТ и твердых тел, строгого описания динамики тел с учетом изменения их внутренней энергии, нет. В этом отношении классическая механика не является завершенной. Отсюда возникает задача в рамках законов Ньютона построить механику тел, обладающих структурой. Т.е. нужно получить уравнение движения тела с учетом изменения его внутренней энергии. Чтобы эту задачу решить в рамках законов Ньютона в качестве такого тела нами была взята система, состоящая и достаточно большого числа потенциально взаимодействующих МТ. Назовем эту систему структурированной частицей (СЧ).
Уравнение диффузии с учетом свободного пробега
Коганов А. В.
Научно-исследовательский институт системных исследований
(НИИСИ РАН), Москва, Россия
koganow@niisi.msk.ru
Рассматривается
вопрос
о
построении
релятивистки
согласованного
уравнения
теплопроводности
и
диффузии.
Рассмотрим
классическое
уравнение
диффузии,
одномерный
случай
,
индекс
означает
частную
производную.
Перейдём
к
разностному
уравнению
с
учётом
средних
параметров
времени
и
длины
свободного
пробега:
это
малые,
но
конечные
параметры:
.
Выразим
конечные
разности
через
ряд
Тейлора
(считаем
гладкость
достаточной)
, и
отбросим
члены
третьего
порядка
малости:
.
Однородный
случай:
;
.
Это
уравнение
волнового
типа
с
диффузионной
компонентой
в
форме
трения
(диссипации).
Обозначим
.
Тогда
.
Обозначим
,
где
— скорость
среды
диффузии
относительно
наблюдателя.
Для
обеспечения
Лоренц-инвариантности
уравнения
требуется
,
но
тогда
не
инвариантен
коэффициент
при
.
Согласно
теории
размерности,
поскольку
,
ожидаемое
преобразование
.
Требование
инвариантности
уравнения
диффузии
в
СТО
не
выполнимо.
Но
тогда
можно
потребовать
релятивистской
согласованности:
это
означает,
что
диффузия
идёт
с
субсветовыми
скоростями
во
всех
системах
отсчёта.
Уравнение
характеристик
уравнения
.
Это
задаёт
конус
.
Обнуление
функции
Грина
вне
конуса
характеристик
следует
из
того,
что
для
гиперболического
уравнения
в
частных
производных
выполнено
условие
единственности
решения
граничной
задачи
Коши,
а
нулевое
решение
удовлетворяет
нулевым
исходным
данным.
Торможение заряженной частицы внутри диэлектрического цилиндра
Г.М. Филиппов
Чебоксарский политехнический институт (филиал) МГОУ
Вид
собственных
колебаний
среды
в
значительной
степени
определяется
ее
поляризационными
свойствами.
При
этом
в
каждом
из
характерных
частотных
диапазонов
поляризационные
свойства
могут
как
способствовать
проникновению
поля
в
ограниченную
среду,
так
и
препятствовать
этому.
Для
разных
типов
полей
этот
процесс
протекает
различным
образом.
Например,
аксиально-несимметричные
компоненты
полей
в
однородном
цилиндре,
вообще
говоря,
не
могут
быть
разделены
на
независимые
Е-
и
Н-
волны
[1,2]. Для
аксиально-симметричных
компонент
такое
разделение
оказывается
возможным.
В
зависимости
от
частоты
электромагнитных
полей
поляризационные
свойства
сред
имеют
довольно
сложное
поведение.
В
области
малых
частот,
в
которой
длина
волны
в
диэлектрике
(если
показатель
преломления
)
меньше
длины
волны
в
вакууме,
может
наблюдаться
полное
внутреннее
отражение
при
выходе
волны
из
среды
в
вакуум.
Как
известно,
на
этом
эффекте
основана
работа
волоконной
оптики.
Проблема подразделения электромагнитных полей на поперечные (вихревые) и продольные (потенциальные) компоненты возникает тогда, когда в силу тех или иных соображений, диктуемых постановкой задачи, эти компоненты приходится рассматривать отдельно. Чаще всего данная потребность возникает при рассмотрении явлений поляризации и излучения в неоднородных средах. Реже, но иногда и в случае движения заряженных частиц в вакууме может возникнуть необходимость выделить вихревые компоненты полей в отдельную категорию. В качестве примера можно указать на известный эффект Ааронова-Бома, в котором проявляются нелокальные свойства микромира: фаза волновой функции электрона явным образом зависит от векторного потенциала электромагнитного поля, который, как мы хорошо знаем, является калибровочно-неинвариантной величиной. Нарушение калибровочной инвариантности теории могло бы привести в конце концов к заключению о ее неадэкватности. В данном случае нарушения нет, поскольку фаза зависит только от вихревой части потенциала, которая является калибровочно-инвариантной. Продолжая рассуждения еще на один шаг вперед, приходим к выводу, что вихревая часть векторного потенциала электромагнитного поля является наблюдаемой и измеряемой физической величиной, в отличие от невихревой, продольной части того же потенциала.
Движение материальной точки в центрально-симметричном гравитационном поле
В. С. Яковенко
Филиал Московского Государственного Открытого Университета в г. Перевозе
При
описании
движения
материальной
точки
в
центрально-
симметричном
гравитационном
поле
тяготения,
независимо
от
применяемых
законов,
найти
кинетическое
уравнение
движения,
т.е.
зависимость
радиуса-вектора
от
времени
очень
сложная, а порой и не выполнимая задача.
Найти же обратную зависимость 
в некоторых случаях возможно, например,
при радиальном движении материальной
точки.
Для решения этой задачи использовались законы классической механики и «Общей теории относительности» как вне шварцшильдовой сферы (сферы с радиусом равным гравитационному радиусу rg центрального тела), так и внутри неё.
При
нахождении зависимости
по законам классической механики
использовался второй закон Ньютона,
как дифференциальное уравнение
относительно скорости, и закон сохранения
механической энергии при нулевой
начальной скорости материальной точки.
Зависимость собственного времени ONT>r) движения материальной точки в гравитационном поле, описываемом метрикой Шварцшильда, найдена с использованием уравнения Гамельтона-Якоби. Из сравнения t(r) и ONT>r) следует, что зависимость времени падения материальной точки, найденная по законам классической механики t(r) совпадает с зависимостью собственного времени падения ONT>r), определяемой по формулам «Общей теории относительности» вне шварцшильдовой сферы.
При нахождении времени падения внутри шваршильдовой сферы (r <= rg ) метрика пространства-времени получена из метрики Шварцшильда с использованием преобразований координат
с/FONT>t+
, R=ct+
с
последующим устранением особенности
метрики при r=rg
выбором f(r)=
.
Метрика при r<=rg при радиальном падении материальной точки
имеет вид:
ds2
=
,
т.е. является нестационарной. Поэтому покоющиеся относительно выбранной системы отсчета (при R=const) материальные точки будут падать к центру поля. Уравнение мировых линий при заданных значениях r определяется выражением
ONT>R
)=
).
Если в начальный момент времени материальная точка находилась на расстоянии r0 от центра поля, то оставаясь неподвижной относительно выбранной системы отсчета, она будет двигаться к центру и время её движения от r0 до r по собственным часам
ONT>r)=
.
В частности, если материальная точка в начальный момент находится на поверхности шварцшильдовой сферы (r0=rg), то она упадет в центр поля (r=0) за время
ONT>
,
т.е.
будет двигаться со средней скоростью
,
большей скорости света. Зависимость
скорости материальной точки от радиуса-
вектора определяется из преобразований
координат при 
и определяется формулой: ��с=-с
. В центре поля скорость материальной
точки становится бесконечно большой.
Таким образом, при нестационарной метрике пространства-времени скорость материальной точки может превышать скорость света.
Incomplete derivation, critical analysis and applications of Generalized form >E =Ac2>m
Ajay Sharma
Fundamental Physics Society India
Einstein‘s Sep. 1905 paper [1] in which PAN>L=PAN>mc2 (light energy –mass equation) is derived, is not completely studied; and is only valid under handpicked or super--special conditions of involved parameters. For example a luminous body can emit numerous waves of light of different energy at different angles. But Einstein has taken just two waves of equal energy emitted just in opposite directions. All other possible variables were neglected, and calculations are done only under classical conditions. Further the origin of PAN>E=PAN>mc2 from PAN>L=PAN>mc2 is completely speculative in nature. This derivation (under general conditions) contradicts the law of conservation of matter/energy. In simple words it implies that when light energy is emitted then its mass must increase. The decrease in mass can be equal to L/c2 or less or more than L/c2. Thus self contradictions also exist in Einstein’s derivation. If numerical values are given in derivation of equation (v=1cm/s), then result is mb =ma i.e. no equation is obtained which involves c2 or L. Thus conversion factor c2 is arbitrarily brought in the picture.These are the limitations of the derivation.
Einstein
only
took
super
special
values
of
parameters.
Einstein
derived
PAN>L=PAN>mc2
under
special
conditions
and
speculated
from
it
PAN>E=PAN>mc2
without
mathematical
derivation.
The
same
derivation
also
gives
L=
PAN>mc2
or
L
=APAN>mc2,
where
A
is
coefficient
of
proportionality.
Thus
Generalized
Mass
Energy
inter
conversion
equation
is
derived
in
other
way
as
PAN>E
=Ac2PAN>m,
where
A
is
coefficient
of
proportionality.
There
are
numerous
values
of
coefficients
of
proportionality
in
the
existing
physics,
hence
A
is
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According to PAN>E=PAN>mc2 in burning of wood if 10-9 kg matter vanishes then energy emitted ( 9x107 J ) can derive a truck of mass 1,000kg to distance of 90 km. It is not confirmed, yet. Contrary to general beliefs PAN>E = PAN>mc2 , is not quantitatively confirmed in Hiroshima and Nagasaki atomic explosions. The energy emitted in nuclear fission of U235 and Pu239 is found 20-60 MeV less than Q value. It can be explained with help of PAN>E =Ac2PAN>m with value of A less than one. The generalized equation implies as mass is changed to other energies, it can also be changed to gravitational energy. PAN>E = PAN>mc2 is unable explain how Big Bang took place and the pre-big bang origin of the universe. But these aspects can be explained by PAN>E =A PAN>mc2 , with assumption that universe started its life from zeroans and transformed itself to energy and subsequently to primeval atom. Then big bang took place due to extreme conditions of temperature, pressure and gravitation. This perception is consistent with NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) data, interpreted by Erickcek. Also applications of PAN>E =Ac2PAN>m, explain the binding energy of deuteron and universal equality of masses of nucleons. PAN>E = PAN>mc2 cannot explain both these observatioins simultaneously. The applications based upon SPAN>E =A PAN>mc2 can be extended in nuclear physics and cosmology.
On temporal nature of Dark Energy
S.I. Kuznetsov
The problem of dark energy closely linked with the interpretation of redshift in the spectra of distant galaxies. The temporal interpretation of cosmological redshift is offered. This implies that photon oscillates along the temporal axis while being in translational motion in space. The source of the photon moves in the space of the past receding progressively from current time point. The velocity of this movement in the expanding Universe is governed by the Hubble’s law. In each temporal cycle photon undergoes reemission from its source. Change in the photon frequency (redshift) is caused by its back reradiation from the receding source. It can be determined by taking into account the relativistic Doppler effect. The temporal interpretation allows to obtain the relationship between redshift parameter z and apparent luminosity L with no adjustable parameters.
Проблема обратимости и необратимости в подходе Н.Н. Боголюбова для осциллятора
Ю.Г. Рудой
Российский университет дружбы народов, 117198, Москва, ул. Миклухо-Маклая,6. Кафедра теоретической физики.
rudikar@mail.ru
Проблема обратимости и необратимости физических явлений входят в число наиболее важных и трудных проблем, постановка и решение которых берет начало в трудах классиков физической науки – прежде всего таких как Гиббс, Ланжевен, Фоккер, Планк.
Наиболее последовательный подход был сформулирован Н.Н. Боголюбовым в малоизвестной монографии 1945 года (подробно см. [1]). По форме подход Боголюбова имеет сходство с подходом Ланжевена, но по физическому смыслу он ближе к подходу Гиббса. Исходным является при этом вид функции Гамильтона H или тесно связанного с ней дифференциального оператора Лиувилля L.
По сравнению с Гиббсом, Боголюбовым был сделан важный шаг вперед, допускающий наличие в величинах Н и L статистических параметров. Эти параметры учитывают воздействие на объект со стороны внешнего окружения, которое способно существенно повлиять на эволюцию объекта в фазовом пространстве – прежде всего, сделать эту эволюцию необратимой.
В подходе Боголюбова уравнения динамики могут быть записаны (и, следовательно, хотя бы в принципе решены) в явном виде с помощью систем дифференциальных уравнений – причем обыкновенных, а не стохастических, как в подходе Ланжевена.
Статистический характер внешнего воздействия учитывается последующим усреднением по явно входящим и имеющим ясный физический смысл статистическим параметрам. Например, эту роль могут играть случайные начальные фазы внешней силы при физически естественных предположениях о характере вероятностного распределения этих случайных параметров.
[1] Ю.Г. Рудой. Вклад Н.Н. Боголюбова в неравновесную статистическую механику // Элементарные частицы и атомное ядро.-2010.-Том 41.- № 7.
Приближенное расширение Лоренцевой симметрии до конформной для массивных частиц в пределе сверхвысоких энергий
Ю.Г.Рудой, И.А. Вернигора
Российский университет дружбы народов
rudikar@mail.ru, wenera 83@mail.ru
Вот уже более полувека в астрофизике космических лучей (КЛ) в области сверхвысоких энергий (порядка 1018–1020 эВ) остается открытой проблема существования предела Грейзена – Зацепина – Кузьмина (ГЗК), предположенного практически одновременно и независимо в работах [1] и [2]. Несмотря на длительное время, прошедшее с момента предсказания предела ГЗК, его экспериментальный статус все еще остается не вполне определенным (см., например, [3]). Поэтому начиная с работы Киржница и Чечина [4] (см. также Коулмен и Глэшоу [5] и более поздние работы [6,7]) – рассматривалась возможность, оставаясь в рамках «обычной» физики, найти ресурсы ее «деформации», допускающие (по крайней мере, в принципе) преодоление предела ГЗК. Согласно [4], подобную роль могло бы сыграть отклонение от лоренцевской кинематики при очень высоких значениях лоренц-фактора NT>=Е/E0/FONT>10:1011.
Однако в [4] этот подход был сформулирован на полуфеноменологическом уровне и, на наш взгляд, нуждается в более надежном математическом обосновании, что и является целью данной работы. Наша исходная идея состоит в том, что при сверхвысоких значениях энергии и лоренц-фактора NT> в кинематике любых массивных частиц с необходимостью появляются «конформные» поправки по степеням 1/NT> (отсутствующие для безмассовых частиц в пределе 1/NT>=0). Здесь имеет место аналогия с хорошо известными релятивистскими (лоренцевскими) поправками по степеням малой величины NT>-1 (точнее, по степеням NT>=v/c ~(NT>-1)1/2 «1, с – скорость света) к обычной галилеевской кинематике.
Теоретической основой подобного подхода служит постепенное расширение группы допустимых преобразований с ростом значений NT>: от группы Галилея G10 к группе Лоренца – Пуанкаре Р10 и далее к конформной группе Вейля – Фока С15, причем число точных инвариантов в этом процессе сокращается. Тем не менее, предполагается возможность не только точного, но и приближенного нарушения той или иной симметрии, а также применимость теории возмущений по соответствующим малым параметрам NT>-1 и NT>/FONT>NT>.
В частности, при малых NT>=Е0/E следует заменить лоренцев скаляр IL(р;Е0)=Е2-р2=Е02, инвариантный при любых значениях Е0 и NT>, на конформный скаляр IС(р;Е0)=С(NT>)IL(р;Е0), который является точным инвариантом только в безмассовом случае Е0=0, NT>=0. Именно последнее обстоятельство и указывает естественный путь построения теории возмущений для С(NT>) по малому параметру NT> при переходе от группы P10 к группе С15.
Группа P10 определяется 10 параметрами и является тензорным произведением группы лоренцевых вращений L6 и группы трансляций Т4(с), определяемой произвольным постоянным 4-вектором с. Группа С15 включает дополнительно к P10 одномерную подгруппу D1 дилатаций с одним скалярным параметром NT>>0, а также подгруппу «специальных» конформных преобразований, или преобразований Мёбиуса-Вейля-Фока, С4(с)=RТ4(с)R, где R – нелинейное (зависящее от P) преобразование инверсии RРР/P2, вообще не содержащее каких-либо параметров и обладающее инволютивным свойством R2=R.
Под действием преобразований С4(с) любой 4-вектор Р переходит в
PС=С(Р;с)Р+NT>Р(Р;с), NT>Р(Р;с)=С(Р;с)сР2; PС2= С(Р;с)Р2, (1)
причем конформный множитель С(Р;с) дается выражением
С(Р;с)=[NT>(Р;с)]-1, NT>(Р;с)=1-2сР+с2Р2=с2(P-с/с2)2. (2)
Для поставленных в данной работе целей достаточно ограничиться физически выделенным выбором с=-аР (а>=0) с последующим переходом к однородным координатам: от вектора 4-импульса Р к вектору 4-скорости V=P/E=(1,v=p/E) (это удобно, поскольку V2=NT>2). Показано, что за счет выбора конформного параметра в виде а=NT>*, где NT>*=ЕПланка/Епротона/FONT>18, NT>=О(1), действительно удается преодолеть предел ГЗК для протонов, причем несмотря на различие подходов, полученные результаты в целом согласуются с [4-8].
[1] K. Greisen, Phys. Rev. Lett. 16, 748-750 (1966).
[2] Г.Т. Зацепин, В.А. Кузьмин, Письма в ЖЭТФ, 4, 114-116 (1966).
[3] В.С. Березинский, Труды ВККЛ-31, Москва, МГУ, 5-9 июля 2010.
[4] Д.А. Киржниц, В.А.Чечин, Ядерная физика, 15, 1051-1058 (1972).
[5] S.Coleman, S.L. Glashow, Phys. Rev. D. 59, 116008 (1999).
[6] L.Gonzalez –Mestres, in Proc. 26th ICRC, 1999, arXiv: physics/0003080v1.
[7] S.T. Scully, F.W. Stecker, arXiv: astro-ph/0811.2230v4.
[8] T. Jacobson, S. Liberati, D. Mattingly, arXiv: hep-ph/0407370v1.
Случайные блуждания и фрактальные вероятностные распределения
Ю.Г. Рудой
Российский университет дружбы народов, кафедра теоретической физики
О.А. Котельникова
Московский государственный университет им. М.В. Ломоносова, Физический фак-т, кафедра магнетизма
olgakot@magn.ru
Процессы эволюции многих объектов как в естествознании, так и в гуманитарных науках являются случайными. Одним из способов математического моделирования таких процессов является их реализация в виде последовательности случайных блужданий («скачков»), или дискретных мартовских цепей. Тогда случайный процесс эволюции объекта в его пространстве состояний полностью определяется свойствами вероятности перехода, характеризующей единичный «скачок» (флуктуацию).
Длительное время считалось, что случайные блуждания адекватно описывают лишь обычную диффузию, или броуновское движение объекта в его пространстве состояний. Это верно лишь в том случае, если для вероятности перехода данного процесса характерно «короткодействие» и «одномасштабность», когда длина скачка строго фиксирована и ограничена (флуктуации малы, процесс эволюции можно считать непрерывным).
Ситуация, однако, радикально изменяется, если вероятность перехода обладает «дальнодействием» и «многомасштабностью», так что возможны разрывные скачки – большие флуктуации, или «длинные перелеты Леви». Иначе говоря, процесс эволюции описывается набором неограниченно возрастающих возможных длин единичного скачка и, соответственно, набором неограниченно убывающих вероятностей их реализации.
Подробно рассмотрен т.н. «пример Монтролла», когда характеристической функцией предельного стационарного (после достаточного числа скачков) вероятностного распределения является функция Больцано - Вейерштрасса. В этом случае распределение является либо экспоненциальным типа Гаусса, либо степенным (фрактальным) типа Леви – Хинчина. Показано, что переход от гауссова к негауссовому режиму имеет пороговый характер, а также вычислена фрактальная размерность этого распределения.
1baburova@orc.ru
2frolovbn@orc.ru
