4,295 research outputs found
Parallel algorithm with spectral convergence for nonlinear integro-differential equations
We discuss a numerical algorithm for solving nonlinear integro-differential
equations, and illustrate our findings for the particular case of Volterra type
equations. The algorithm combines a perturbation approach meant to render a
linearized version of the problem and a spectral method where unknown functions
are expanded in terms of Chebyshev polynomials (El-gendi's method). This
approach is shown to be suitable for the calculation of two-point Green
functions required in next to leading order studies of time-dependent quantum
field theory.Comment: 15 pages, 9 figure
Noise Kernel in Stochastic Gravity and Stress Energy Bi-Tensor of Quantum Fields in Curved Spacetimes
The noise kernel is the vacuum expectation value of the (operator-valued)
stress-energy bi-tensor which describes the fluctuations of a quantum field in
curved spacetimes. It plays the role in stochastic semiclassical gravity based
on the Einstein-Langevin equation similar to the expectation value of the
stress-energy tensor in semiclassical gravity based on the semiclassical
Einstein equation. According to the stochastic gravity program, this two point
function (and by extension the higher order correlations in a hierarchy) of the
stress energy tensor possesses precious statistical mechanical information of
quantum fields in curved spacetime and, by the self-consistency required of
Einstein's equation, provides a probe into the coherence properties of the
gravity sector (as measured by the higher order correlation functions of
gravitons) and the quantum nature of spacetime. It reflects the low and medium
energy (referring to Planck energy as high energy) behavior of any viable
theory of quantum gravity, including string theory. It is also useful for
calculating quantum fluctuations of fields in modern theories of structure
formation and for backreaction problems in cosmological and black holes
spacetimes.
We discuss the properties of this bi-tensor with the method of
point-separation, and derive a regularized expression of the noise-kernel for a
scalar field in general curved spacetimes. One collorary of our finding is that
for a massless conformal field the trace of the noise kernel identically
vanishes. We outline how the general framework and results derived here can be
used for the calculation of noise kernels for Robertson-Walker and
Schwarzschild spacetimes.Comment: 22 Pages, RevTeX; version accepted for publication in PR
Influence Action and decoherence of hydrodynamic modes
We derive an influence action for the heat diffusion equation and from its
spectral dependence show that long wavelength hydrodynamic modes are most
readily decohered. The result is independent of the details of the microscopic
dynamics, and follows from general principles alone.Comment: 5 pages, no figure
Non-equilibrium dynamics of a thermal plasma in a gravitational field
We introduce functional methods to study the non-equilibrium dynamics of a
quantum massless scalar field at finite temperature in a gravitational field.
We calculate the Close Time Path (CTP) effective action and, using its formal
equivalence with the influence functional, derive the noise and dissipation
kernels of the quantum open system in terms of quantities in thermodynamical
equilibrium. Using this fact, we formally prove the existence of a
Fluctuation-Dissipation Relation (FDR) at all temperatures between the quantum
fluctuations of the plasma in thermal equilibrium and the energy dissipated by
the external gravitational field. What is new is the identification of a
stochastic source (noise) term arising from the quantum and thermal
fluctuations in the plasma field, and the derivation of a Langevin-type
equation which describes the non-equilibrium dynamics of the gravitational
field influenced by the plasma. The back reaction of the plasma on the
gravitational field is embodied in the FDR. From the CTP effective action the
contribution of the quantum scalar field to the thermal graviton polarization
tensor can also be derived and it is shown to agree with other techniques, most
notably, Linear Response Theory (LRT). We show the connection between the LRT,
which is applicable for near-equilibrium conditions and the functional methods
used in this work which are useful for fully non-equilibrium conditions.Comment: Final version published in Phys. Rev.
A Generalized Fluctuation-Dissipation Theorem for Nonlinear Response Functions
A nonlinear generalization of the Fluctuation-Dissipation Theorem (FDT) for
the n-point Green functions and the amputated 1PI vertex functions at finite
temperature is derived in the framework of the Closed Time Path formalism. We
verify that this generalized FDT coincides with known results for n=2 and 3.
New explicit relations among the 4-point nonlinear response and correlation
(fluctuation) functions are presented.Comment: 34 pages, Revte
Quantum Fluctuations, Decoherence of the Mean Field, and Structure Formation in the Early Universe
We examine from first principles one of the basic assumptions of modern
quantum theories of structure formation in the early universe, i.e., the
conditions upon which fluctuations of a quantum field may transmute into
classical stochastic perturbations, which grew into galaxies. Our earlier works
have discussed the quantum origin of noise in stochastic inflation and quantum
fluctuations as measured by particle creation in semiclassical gravity. Here we
focus on decoherence and the relation of quantum and classical fluctuations.
Instead of using the rather ad hoc splitting of a quantum field into long and
short wavelength parts, the latter providing the noise which decoheres the
former, we treat a nonlinear theory and examine the decoherence of a quantum
mean field by its own quantum fluctuations, or that of other fields it
interacts with. This is an example of `dynamical decoherence' where an
effective open quantum system decoheres through its own dynamics. The model we
use to discuss fluctuation generation has the inflation field coupled to the
graviton field. We show that when the quantum to classical transition is
properly treated, with due consideration of the relation of decoherence, noise,
fluctuation and dissipation, the amplitude of density contrast predicted falls
in the acceptable range without requiring a fine tuning of the coupling
constant of the inflation field (). The conventional treatment which
requires an unnaturally small stems from a basic
flaw in naively identifying classical perturbations with quantum fluctuations.Comment: 35 pages, latex, 0 figure
Entropy and Uncertainty of Squeezed Quantum Open Systems
We define the entropy S and uncertainty function of a squeezed system
interacting with a thermal bath, and study how they change in time by following
the evolution of the reduced density matrix in the influence functional
formalism. As examples, we calculate the entropy of two exactly solvable
squeezed systems: an inverted harmonic oscillator and a scalar field mode
evolving in an inflationary universe. For the inverted oscillator with weak
coupling to the bath, at both high and low temperatures, , where r is
the squeeze parameter. In the de Sitter case, at high temperatures, where , being the coupling to the bath and H
the Hubble constant. These three cases confirm previous results based on more
ad hoc prescriptions for calculating entropy. But at low temperatures, the de
Sitter entropy is noticeably different. This result, obtained
from a more rigorous approach, shows that factors usually ignored by the
conventional approaches, i.e., the nature of the environment and the coupling
strength betwen the system and the environment, are important.Comment: 36 pages, epsfig, 2 in-text figures include
Stochastic Gravity: A Primer with Applications
Stochastic semiclassical gravity of the 90's is a theory naturally evolved
from semiclassical gravity of the 70's and 80's. It improves on the
semiclassical Einstein equation with source given by the expectation value of
the stress-energy tensor of quantum matter fields in curved spacetimes by
incorporating an additional source due to their fluctuations. In stochastic
semiclassical gravity the main object of interest is the noise kernel, the
vacuum expectation value of the (operator-valued) stress-energy bi-tensor, and
the centerpiece is the (stochastic) Einstein-Langevin equation. We describe
this new theory via two approaches: the axiomatic and the functional. The
axiomatic approach is useful to see the structure of the theory from the
framework of semiclassical gravity. The functional approach uses the
Feynman-Vernon influence functional and the Schwinger-Keldysh close-time-path
effective action methods which are convenient for computations. It also brings
out the open systems concepts and the statistical and stochastic contents of
the theory such as dissipation, fluctuations, noise and decoherence. We then
describe the application of stochastic gravity to the backreaction problems in
cosmology and black hole physics. Intended as a first introduction to this
subject, this article places more emphasis on pedagogy than completeness.Comment: 46 pages Latex. Intended as a review in {\it Classical and Quantum
Gravity
Stochastic dynamics of correlations in quantum field theory: From Schwinger-Dyson to Boltzmann-Langevin equation
The aim of this paper is two-fold: in probing the statistical mechanical
properties of interacting quantum fields, and in providing a field theoretical
justification for a stochastic source term in the Boltzmann equation. We start
with the formulation of quantum field theory in terms of the Schwinger - Dyson
equations for the correlation functions, which we describe by a
closed-time-path master () effective action. When the hierarchy
is truncated, one obtains the ordinary closed-system of correlation functions
up to a certain order, and from the nPI effective action, a set of
time-reversal invariant equations of motion. But when the effect of the higher
order correlation functions is included (through e.g., causal factorization--
molecular chaos -- conditions, which we call 'slaving'), in the form of a
correlation noise, the dynamics of the lower order correlations shows
dissipative features, as familiar in the field-theory version of Boltzmann
equation. We show that fluctuation-dissipation relations exist for such
effectively open systems, and use them to show that such a stochastic term,
which explicitly introduces quantum fluctuations on the lower order correlation
functions, necessarily accompanies the dissipative term, thus leading to a
Boltzmann-Langevin equation which depicts both the dissipative and stochastic
dynamics of correlation functions in quantum field theory.Comment: LATEX, 30 pages, no figure
Non-Equilibrium Quantum Fields in the Large N Expansion
An effective action technique for the time evolution of a closed system
consisting of one or more mean fields interacting with their quantum
fluctuations is presented. By marrying large expansion methods to the
Schwinger-Keldysh closed time path (CTP) formulation of the quantum effective
action, causality of the resulting equations of motion is ensured and a
systematic, energy conserving and gauge invariant expansion about the
quasi-classical mean field(s) in powers of developed. The general method
is exposed in two specific examples, symmetric scalar \l\F^4 theory
and Quantum Electrodynamics (QED) with fermion fields. The \l\F^4 case is
well suited to the numerical study of the real time dynamics of phase
transitions characterized by a scalar order parameter. In QED the technique may
be used to study the quantum non-equilibrium effects of pair creation in strong
electric fields and the scattering and transport processes in a relativistic
plasma. A simple renormalization scheme that makes practical the
numerical solution of the equations of motion of these and other field theories
is described.Comment: 43 pages, LA-UR-94-783 (PRD, in press), uuencoded PostScrip
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