37 research outputs found
New Supplementary Conditions for a Non-Linear Field Theory: General Relativity
The Einstein theory of general relativity provides a peculiar example of
classical field theory ruled by non-linear partial differential equations. A
number of supplementary conditions (more frequently called gauge conditions)
have also been considered in the literature. In the present paper, starting
from the de Donder gauge, which is not conformally invariant but is the
gravitational counterpart of the Lorenz gauge, we consider, led by geometric
structures on vector bundles, a new family of gauges in general relativity,
which involve fifth-order covariant derivatives of metric perturbations. A
review of recent results by the authors is presented: restrictions on the
general form of the metric on the vector bundle of symmetric rank-two tensor
fields over space-time; admissibility of such gauges in the case of linearized
theory about flat Euclidean space; generalization to a suitable class of curved
Riemannian backgrounds, by solving an integral equation. Eventually, the
applications to Euclidean quantum gravity are discussed.Comment: 8 pages, plain Tex. Talk prepared for the Conference: "Nonlinearity,
integrability and all that (twenty years after needs '79)". Gallipoli, July 1
- July 10, 199
Classical brackets for dissipative systems
We show how to write a set of brackets for the Langevin equation, describing
the dissipative motion of a classical particle, subject to external random
forces. The method does not rely on an action principle, and is based solely on
the phenomenological description of the dissipative dynamics as given by the
Langevin equation. The general expression for the brackets satisfied by the
coordinates, as well as by the external random forces, at different times, is
determined, and it turns out that they all satisfy the Jacobi identity. Upon
quantization, these classical brackets are found to coincide with the
commutation rules for the quantum Langevin equation, that have been obtained in
the past, by appealing to microscopic conservative quantum models for the
friction mechanism.Comment: Latex file, 8 pages, prepared for the Conference Spacetime and
Fundamental Interactions: Quantum Aspects, Vietri sul Mare, Italy, 26-31 May
200