164 research outputs found
Quantum stochastic equation for the low density limit
A new derivation of quantum stochastic differential equation for the
evolution operator in the low density limit is presented. We use the
distribution approach and derive a new algebra for quadratic master fields in
the low density limit by using the energy representation. We formulate the
stochastic golden rule in the low density limit case for a system coupling with
Bose field via quadratic interaction. In particular the vacuum expectation
value of the evolution operator is computed and its exponential decay is shown.Comment: Replaced with version published in J. Phys. A. References are adde
Derivation of the particle dynamics from kinetic equations
We consider the microscopic solutions of the Boltzmann-Enskog equation
discovered by Bogolyubov. The fact that the time-irreversible kinetic equation
has time-reversible microscopic solutions is rather surprising. We analyze this
paradox and show that the reversibility or irreversibility property of the
Boltzmann-Enskog equation depends on the considered class of solutions. If the
considered solutions have the form of sums of delta-functions, then the
equation is reversible. If the considered solutions belong to the class of
continuously differentiable functions, then the equation is irreversible. Also,
we construct the so called approximate microscopic solutions. These solutions
are continuously differentiable and they are reversible on bounded time
intervals. This analysis suggests a way to reconcile the time-irreversible
kinetic equations with the time-reversible particle dynamics. Usually one tries
to derive the kinetic equations from the particle dynamics. On the contrary, we
postulate the Boltzmann-Enskog equation or another kinetic equation and treat
their microscopic solutions as the particle dynamics. So, instead of the
derivation of the kinetic equations from the microdynamics we suggest a kind of
derivation of the microdynamics from the kinetic equations.Comment: 18 pages; some misprints have been corrected, some references have
been adde
A stochastic golden rule and quantum Langevin equation for the low density limit
A rigorous derivation of quantum Langevin equation from microscopic dynamics
in the low density limit is given. We consider a quantum model of a microscopic
system (test particle) coupled with a reservoir (gas of light Bose particles)
via interaction of scattering type. We formulate a mathematical procedure (the
so-called stochastic golden rule) which allows us to determine the quantum
Langevin equation in the limit of large time and small density of particles of
the reservoir. The quantum Langevin equation describes not only dynamics of the
system but also the reservoir. We show that the generator of the corresponding
master equation has the Lindblad form of most general generators of completely
positive semigroups
Rolling in the Higgs Model and Elliptic Functions
Asymptotic methods in nonlinear dynamics are used to improve perturbation
theory results in the oscillations regime. However, for some problems of
nonlinear dynamics, particularly in the case of Higgs (Duffing) equation and
the Friedmann cosmological equations, not only small oscillations regime is of
interest but also the regime of rolling (climbing), more precisely the rolling
from a top (climbing to a top). In the Friedman cosmology, where the slow
rolling regime is often used, the rolling from a top (not necessary slow) is of
interest too.
In the present work a method for approximate solution to the Higgs equation
in the rolling regime is presented. It is shown that in order to improve
perturbation theory in the rolling regime turns out to be effective not to use
an expansion in trigonometric functions as it is done in case of small
oscillations but use expansions in hyperbolic functions instead. This regime is
investigated using the representation of the solution in terms of elliptic
functions. An accuracy of the corresponding approximation is estimated.Comment: Latex, 36 Pages, 8 figures, typos correcte
Four-Loop Cusp Anomalous Dimension From Obstructions
We introduce a method for extracting the cusp anomalous dimension at L loops
from four-gluon amplitudes in N=4 Yang-Mills without evaluating any integrals
that depend on the kinematical invariants. We show that the anomalous dimension
only receives contributions from the obstructions introduced in hep-th/0601031.
We illustrate this method by extracting the two- and three-loop anomalous
dimensions analytically and the four-loop one numerically. The four-loop result
was recently guessed to be f^4 = - (4\zeta^3_2+24\zeta_2\zeta_4+50\zeta_6-
4(1+r)\zeta_3^2) with r=-2 using integrability and string theory arguments in
hep-th/0610251. Simultaneously, f^4 was computed numerically in hep-th/0610248
from the four-loop amplitude obtaining, with best precision at the symmetric
point s=t, r=-2.028(36). Our computation is manifestly s/t independent and
improves the precision to r=-2.00002(3), providing strong evidence in favor of
the conjecture. The improvement is possible due to a large reduction in the
number of contributing terms, as well as a reduction in the number of
integration variables in each term.Comment: 23 pages, revtex; v2,v3: minor typos fixed and references adde
Time Machine at the LHC
Recently, black hole and brane production at CERN's Large Hadron Collider
(LHC) has been widely discussed. We suggest that there is a possibility to test
causality at the LHC. We argue that if the scale of quantum gravity is of the
order of few TeVs, proton-proton collisions at the LHC could lead to the
formation of time machines (spacetime regions with closed timelike curves)
which violate causality. One model for the time machine is a traversable
wormhole. We argue that the traversable wormhole production cross section at
the LHC is of the same order as the cross section for the black hole
production. Traversable wormholes assume violation of the null energy condition
(NEC) and an exotic matter similar to the dark energy is required. Decay of the
wormholes/time machines and signatures of time machine events at the LHC are
discussed.Comment: 12 pages, LATEX, comments and references adde
Quantization of the Riemann Zeta-Function and Cosmology
Quantization of the Riemann zeta-function is proposed. We treat the Riemann
zeta-function as a symbol of a pseudodifferential operator and study the
corresponding classical and quantum field theories. This approach is motivated
by the theory of p-adic strings and by recent works on stringy cosmological
models. We show that the Lagrangian for the zeta-function field is equivalent
to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of
the Riemann zeta-function. Quantization of the mathematics of Fermat-Wiles and
the Langlands program is indicated. The Beilinson conjectures on the values of
L-functions of motives are interpreted as dealing with the cosmological
constant problem. Possible cosmological applications of the zeta-function field
theory are discussed.Comment: 14 pages, corrected typos, references and comments adde
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