339 research outputs found
Multi-Lagrangians, Hereditary Operators and Lax Pairs for the Korteweg-de Vries Positive and Negative Hierarchies
We present an approach to the construction of action principles for
differential equations, and apply it to field theory in order to construct
systematically, for integrable equations which are based on a Nijenhuis (or
hereditary) operator, a ladder of action principles which is complementary to
the well-known multi-Hamiltonian formulation. We work out results for the
Korteweg-de Vries (KdV) equation, which is a member of the positive hierarchy
related to a hereditary operator. Three negative hierarchies of (negative)
evolution equations are defined naturally from the hereditary operator as well,
in the context of field theory. The Euler-Lagrange equations arising from the
action principles are equivalent to the original evolution equation +
deformations, which are obtained in terms of the positive and negative
evolution vectors. We recognize the Liouville, Sinh-Gordon, Hunter-Zheng and
Camassa-Holm equations as negative equations. The ladder for KdV is directly
mappable to a ladder for any of these negative equations and other positive
equations (e.g., the Harry-Dym and a special case of the Krichever-Novikov
equations): a new nonlocal action principle for the deformed system Sinh-Gordon
+ spatial translation vector is presented. Several nonequivalent, nonlocal
time-reparametrization invariant action principles for KdV are constructed.
Hamiltonian and Symplectic operators are obtained in factorized form.
Alternative Lax pairs for all negative flows are constructed, using the flows
and the hereditary operator as only input. From this result we prove that all
positive and negative equations in the hierarchies share the same sets of local
and nonlocal constants of the motion for KdV, which are explicitly obtained
using the local and nonlocal action principles for KdV.Comment: Final version, accepted in JMP; RevTeX, 31 page
Consistency of Semiclassical Gravity
We discuss some subtleties which arise in the semiclassical approximation to
quantum gravity. We show that integrability conditions prevent the existence of
Tomonaga-Schwinger time functions on the space of three-metrics but admit them
on superspace. The concept of semiclassical time is carefully examined. We
point out that central charges in the matter sector spoil the consistency of
the semiclassical approximation unless the full quantum theory of gravity and
matter is anomaly-free. We finally discuss consequences of these considerations
for quantum field theory in flat spacetime, but with arbitrary foliations.Comment: 12 pages, LATEX, Report Freiburg THEP-94/2
Test particles behavior in the framework of a lagrangian geometric theory with propagating torsion
Working in the lagrangian framework, we develop a geometric theory in vacuum
with propagating torsion; the antisymmetric and trace parts of the torsion
tensor, considered as derived from local potential fields, are taken and, using
the minimal action principle, their field equations are calculated. Actually
these will show themselves to be just equations for propagating waves giving
torsion a behavior similar to that of metric which, as known, propagates
through gravitational waves. Then we establish a principle of minimal
substitution to derive test particles equation of motion, obtaining, as result,
that they move along autoparallels. We then calculate the analogous of the
geodesic deviation for these trajectories and analyze their behavior in the
nonrelativistic limit, showing that the torsion trace potential has a
phenomenology which is indistinguishable from that of the gravitational
newtonian field; in this way we also give a reason for why there have never
been evidence for it.Comment: 12 pages, no figures, to appear on Int. Journ. Mod. Phys.
Bohmian Mechanics and Quantum Information
Many recent results suggest that quantum theory is about information, and
that quantum theory is best understood as arising from principles concerning
information and information processing. At the same time, by far the simplest
version of quantum mechanics, Bohmian mechanics, is concerned, not with
information but with the behavior of an objective microscopic reality given by
particles and their positions. What I would like to do here is to examine
whether, and to what extent, the importance of information, observation, and
the like in quantum theory can be understood from a Bohmian perspective. I
would like to explore the hypothesis that the idea that information plays a
special role in physics naturally emerges in a Bohmian universe.Comment: 25 pages, 2 figure
The Canonical Approach to Quantum Gravity: General Ideas and Geometrodynamics
We give an introduction to the canonical formalism of Einstein's theory of
general relativity. This then serves as the starting point for one approach to
quantum gravity called quantum geometrodynamics. The main features and
applications of this approach are briefly summarized.Comment: 21 pages, 6 figures. Contribution to E. Seiler and I.-O. Stamatescu
(editors): `Approaches To Fundamental Physics -- An Assessment Of Current
Theoretical Ideas' (Springer Verlag, to appear
Torsion-induced spin precession
We investigate the motion of a spinning test particle in a spatially-flat
FRW-type space-time in the framework of the Einstein-Cartan theory. The
space-time has a torsion arising from a spinning fluid filling the space-time.
We show that for spinning particles with nonzero transverse spin components,
the torsion induces a precession of particle spin around the direction of the
fluid spin. We also show that a charged spinning particle moving in a
torsion-less spatially-flat FRW space-time in the presence of a uniform
magnetic field undergoes a precession of a different character.Comment: latex, 4 eps figure
What is the Geometry of Superspace ?
We investigate certain properties of the Wheeler-DeWitt metric (for constant
lapse) in canonical General Relativity associated with its non-definite nature.
Contribution to the conference on Mach's principle: "From Newtons Bucket to
Quantum Gravity", July 26-30 1993, Tuebingen, GermanyComment: 10 pages, Plain Te
Duality properties of Gorringe-Leach equations
In the category of motions preserving the angular momentum's direction,
Gorringe and Leach exhibited two classes of differential equations having
elliptical orbits. After enlarging slightly these classes, we show that they
are related by a duality correspondence of the Arnold-Vassiliev type. The
specific associated conserved quantities (Laplace-Runge-Lenz vector and
Fradkin-Jauch-Hill tensor) are then dual reflections one of the othe
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