1,212 research outputs found
Quantum Mechanics and Discrete Time from "Timeless" Classical Dynamics
We study classical Hamiltonian systems in which the intrinsic proper time
evolution parameter is related through a probability distribution to the
physical time, which is assumed to be discrete. - This is motivated by the
``timeless'' reparametrization invariant model of a relativistic particle with
two compactified extradimensions. In this example, discrete physical time is
constructed based on quasi-local observables. - Generally, employing the
path-integral formulation of classical mechanics developed by Gozzi et al., we
show that these deterministic classical systems can be naturally described as
unitary quantum mechanical models. The emergent quantum Hamiltonian is derived
from the underlying classical one. It is closely related to the Liouville
operator. We demonstrate in several examples the necessity of regularization,
in order to arrive at quantum models with bounded spectrum and stable
groundstate.Comment: 24 pages, 1 figure. Lecture given at DICE 2002. To be published in:
Decoherence and Entropy in Complex Systems, Lecture Notes in Physics
(Springer-Verlag, Berlin 2003). - Comprises quant-ph/0306096 and
gr-qc/0301109, additional reference
The issue of time in generally covariant theories and the Komar-Bergmann approach to observables in general relativity
Diffeomorphism-induced symmetry transformations and time evolution are
distinct operations in generally covariant theories formulated in phase space.
Time is not frozen. Diffeomorphism invariants are consequently not necessarily
constants of the motion. Time-dependent invariants arise through the choice of
an intrinsic time, or equivalently through the imposition of time-dependent
gauge fixation conditions. One example of such a time-dependent gauge fixing is
the Komar-Bergmann use of Weyl curvature scalars in general relativity. An
analogous gauge fixing is also imposed for the relativistic free particle and
the resulting complete set time-dependent invariants for this exactly solvable
model are displayed. In contrast with the free particle case, we show that
gauge invariants that are simultaneously constants of motion cannot exist in
general relativity. They vary with intrinsic time
The spectral data for Hamiltonian stationary Lagrangian tori in R^4
This article determines the spectral data, in the integrable systems sense,
for all weakly conformally immersed Hamiltonian stationary Lagrangian in
. This enables us to describe their moduli space and the locus of branch
points of such an immersion. This is also an informative example in integrable
systems geometry, since the group of ambient isometries acts non-trivially on
the spectral data and the relevant energy functional (the area) need not be
constant under deformations by higher flows.Comment: 30 pages. Version 3: a complete rewrite of version 2 with new results
and two significant correction
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Progress in Surface Theory
Over the last 30 years global surface theory has become pivotal in the understanding of low dimensional global phenomena. At the same time surface geometry became a platform on which seemingly different areas of mathematics – such as geometric and topological analysis, integrable systems, algebraic geometry of curves, and mathematical physics – coalesce to produce far reaching ideas, conjectures, methods and results. The workshop hosted talks on the resolutions of famous conjectures in surface geometry, including the Willmore conjecture, and on exciting new progress in the understanding of moduli spaces of special surface classes
Recent mathematical developments in the Skyrme model
In this review we present a pedagogical introduction to recent, more
mathematical developments in the Skyrme model. Our aim is to render these
advances accessible to mainstream nuclear and particle physicists. We start
with the static sector and elaborate on geometrical aspects of the definition
of the model. Then we review the instanton method which yields an analytical
approximation to the minimum energy configuration in any sector of fixed baryon
number, as well as an approximation to the surfaces which join together all the
low energy critical points. We present some explicit results for B=2. We then
describe the work done on the multibaryon minima using rational maps, on the
topology of the configuration space and the possible implications of Morse
theory. Next we turn to recent work on the dynamics of Skyrmions. We focus
exclusively on the low energy interaction, specifically the gradient flow
method put forward by Manton. We illustrate the method with some expository toy
models. We end this review with a presentation of our own work on the
semi-classical quantization of nucleon states and low energy nucleon-nucleon
scattering.Comment: 129 pages, about 30 figures, original manuscript of published Physics
Report
Gauge Theories of Gravitation
During the last five decades, gravity, as one of the fundamental forces of
nature, has been formulated as a gauge theory of the Weyl-Cartan-Yang-Mills
type. The present text offers commentaries on the articles from the most
prominent proponents of the theory. In the early 1960s, the gauge idea was
successfully applied to the Poincar\'e group of spacetime symmetries and to the
related conserved energy-momentum and angular momentum currents. The resulting
theory, the Poincar\'e gauge theory, encompasses Einstein's general relativity
as well as the teleparallel theory of gravity as subcases. The spacetime
structure is enriched by Cartan's torsion, and the new theory can accommodate
fermionic matter and its spin in a perfectly natural way. This guided tour
starts from special relativity and leads, in its first part, to general
relativity and its gauge type extensions \`a la Weyl and Cartan. Subsequent
stopping points are the theories of Yang-Mills and Utiyama and, as a particular
vantage point, the theory of Sciama and Kibble. Later, the Poincar\'e gauge
theory and its generalizations are explored and special topics, such as its
Hamiltonian formulation and exact solutions, are studied. This guide to the
literature on classical gauge theories of gravity is intended to be a
stimulating introduction to the subject.Comment: 169 pages, pdf file, v3: extended to a guide to the literature on
classical gauge theories of gravit
Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics
We review the Lagrangian formulation of Noether symmetries (as well as
"generalized Noether symmetries") in the framework of Calculus of Variations in
Jet Bundles, with a special attention to so-called "Natural Theories" and
"Gauge-Natural Theories", that include all relevant Field Theories and physical
applications (from Mechanics to General Relativity, to Gauge Theories,
Supersymmetric Theories, Spinors and so on). It is discussed how the use of
Poincare'-Cartan forms and decompositions of natural (or gauge-natural)
variational operators give rise to notions such as "generators of Noether
symmetries", energy and reduced energy flow, Bianchi identities, weak and
strong conservation laws, covariant conservation laws, Hamiltonian-like
conservation laws (such as, e.g., so-called ADM laws in General Relativity)
with emphasis on the physical interpretation of the quantities calculated in
specific cases (energy, angular momentum, entropy, etc.). A few substantially
new and very recent applications/examples are presented to better show the
power of the methods introduced: one in Classical Mechanics (definition of
strong conservation laws in a frame-independent setting and a discussion on the
way in which conserved quantities depend on the choice of an observer); one in
Classical Field Theories (energy and entropy in General Relativity, in its
standard formulation, in its spin-frame formulation, in its first order
formulation "`a la Palatini" and in its extensions to Non-Linear Gravity
Theories); one in Quantum Field Theories (applications to conservation laws in
Loop Quantum Gravity via spin connections and Barbero-Immirzi connections).Comment: 27 page
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