539 research outputs found
General relativity as an extended canonical gauge theory
It is widely accepted that the fundamental geometrical law of nature should
follow from an action principle. The particular subset of transformations of a
system's dynamical variables that maintain the form of the action principle
comprises the group of canonical transformations. In the context of canonical
field theory, the adjective "extended" signifies that not only the fields but
also the space-time geometry is subject to transformation. Thus, in order to be
physical, the transition to another, possibly noninertial frame of reference
must necessarily constitute an extended canonical transformation that defines
the general mapping of the connection coefficients, hence the quantities that
determine the space-time curvature and torsion of the respective reference
frame. The canonical transformation formalism defines simultaneously the
transformation rules for the conjugates of the connection coefficients and for
the Hamiltonian. As will be shown, this yields unambiguously a particular
Hamiltonian that is form-invariant under the canonical transformation of the
connection coefficients and thus satisfies the general principle of relativity.
This Hamiltonian turns out to be a quadratic function of the curvature tensor.
Its Legendre-transformed counterpart then establishes a unique Lagrangian
description of the dynamics of space-time that is not postulated but derived
from basic principles, namely the action principle and the general principle of
relativity. Moreover, the resulting theory satisfies the principle of scale
invariance and is renormalizable.Comment: 14 page
Generic Theory of Geometrodynamics from Noether's theorem for the Diff(M) symmetry group
We work out the most general theory for the interaction of spacetime geometry
and matter fields---commonly referred to as geometrodynamics---for spin- and
spin- particles. The minimum set of postulates to be introduced is that (i)
the action principle should apply and that(ii) the total action should by
form-invariant under the (local) diffeomorphism group. The second postulate
thus implements the Principle of General Relativity. According to Noether's
theorem, this physical symmetry gives rise to a conserved Noether current, from
which the complete set of theories compatible with both postulates can be
deduced. This finally results in a new generic Einstein-type equation, which
can be interpreted as an energy-momentum balance equation emerging from the
Lagrangian for the source-free dynamics of gravitation and the
energy-momentum tensor of the source system . Provided that the system
has no other symmetries---such as SU---the canonical energy-momentum
tensor turns out to be the correct source term of gravitation. For the case of
massive spin particles, this entails an increased weighting of the kinetic
energy over the mass in their roles as the source of gravity as compared to the
metric energy momentum tensor, which constitutes the source of gravity in
Einstein's General Relativity. We furthermore confirm that a massive vector
field necessarily acts as a source for torsion of spacetime. Thus, from the
viewpoint of our generic Einstein-type equation, Einstein's General Relativity
constitutes the particular case for spin- and massless spin particle fields,
and the Hilbert Lagrangian as the model for the source-free dynamics
of gravitation.Comment: 33 page
Covariant Canonical Gauge Gravitation and Cosmology
The covariant canonical transformation theory applied to the relativistic
Hamiltonian theory of classical matter fields in dynamical space-time yields a
novel (first order) gauge field theory of gravitation. The emerging field
equations necessarily embrace a quadratic Riemann term added to Einstein's
linear equation. The quadratic term endows space-time with inertia generating a
dynamic response of the space-time geometry to deformations relative to (Anti)
de Sitter geometry. A "deformation parameter" is identified, the inverse
dimensionless coupling constant governing the relative strength of the
quadratic invariant in the Hamiltonian, and directly observable via the
deceleration parameter . The quadratic invariant makes the system
inconsistent with Einstein's constant cosmological term, . In the Friedman model this inconsistency is resolved with the
scaling ansatz of a "cosmological function", , where is the
scale parameter of the FLRW metric. %Moreover, the strain generated by the
quadratic term turns out to act as a geometrical stress. The cosmological
function can be normalized such that with the CDM parameter set the
present-day observables, the Hubble constant and the deceleration parameter,
can be reproduced. %We analyze the asymptotics of the such normalized Friedman
equations with respect to both, the fundamental parameters (coupling constants)
and the scale . With this parameter set we recover the dark energy scenario
in the late epoch. The proof that inflation in the early phase is caused by the
"geometrical fluid" representing the inertia of space-time is yet pending,
though
Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems
We will present a consistent description of Hamiltonian dynamics on the
``symplectic extended phase space'' that is analogous to that of a
time-\underline{in}dependent Hamiltonian system on the conventional symplectic
phase space. The extended Hamiltonian and the pertaining extended
symplectic structure that establish the proper canonical extension of a
conventional Hamiltonian will be derived from a generalized formulation of
Hamilton's variational principle. The extended canonical transformation theory
then naturally permits transformations that also map the time scales of
original and destination system, while preserving the extended Hamiltonian
, and hence the form of the canonical equations derived from .
The Lorentz transformation, as well as time scaling transformations in
celestial mechanics, will be shown to represent particular canonical
transformations in the symplectic extended phase space. Furthermore, the
generalized canonical transformation approach allows to directly map explicitly
time-dependent Hamiltonians into time-independent ones. An ``extended''
generating function that defines transformations of this kind will be presented
for the time-dependent damped harmonic oscillator and for a general class of
explicitly time-dependent potentials. In the appendix, we will reestablish the
proper form of the extended Hamiltonian by means of a Legendre
transformation of the extended Lagrangian .Comment: 24 page
Covariant Canonical Gauge theory of Gravitation resolves the Cosmological Constant Problem
The covariant canonical transformation theory applied to the relativistic
theory of classical matter fields in dynamic space-time yields a new (first
order) gauge field theory of gravitation. The emerging field equations embrace
a quadratic Riemann curvature term added to Einstein's linear equation. The
quadratic term facilitates a momentum field which generates a dynamic response
of space-time to its deformations relative to de Sitter geometry, and adds a
term proportional to the Planck mass squared to the cosmological constant. The
proportionality factor is given by a dimensionless parameter governing the
strength of the quadratic term. In consequence, Dark Energy emerges as a
balanced mix of three contributions, (A)dS curvature plus the residual vacuum
energy of space-time and matter. The Cosmological Constant Problem of the
Einstein-Hilbert theory is resolved as the curvature contribution relieves the
rigid relation between the cosmological constant and the vacuum energy density
of matter
Stochastic effects in real and simulated charged particle beams
The Vlasov equation embodies the smooth field approximation of the
self-consistent equation of motion for charged particle beams. This framework
is fundamentally altered if we include the fluctuating forces that originate
from the actual charge granularity. We thereby perform the transition from a
reversible description to a statistical mechanics' description covering also
the irreversible aspects of beam dynamics. Taking into account contributions
from fluctuating forces is mandatory if we want to describe effects like
intrabeam scattering or temperature balancing within beams. Furthermore, the
appearance of ``discreteness errors'' in computer simulations of beams can be
modeled as ``exact'' beam dynamics that is being modified by fluctuating
``error forces''. It will be shown that the related emittance increase depends
on two distinct quantities: the magnitude of the fluctuating forces embodied in
a friction coefficient , and the correlation time dependent average
temperature anisotropy. These analytical results are verified by various
computer simulations.Comment: 11 pages, 9 figure
Generalized U(N) gauge transformations in the realm of the extended covariant Hamilton formalism of field theory
The Lagrangians and Hamiltonians of classical field theory require to
comprise gauge fields in order to be form-invariant under local gauge
transformations. These gauge fields have turned out to correctly describe
pertaining elementary particle interactions. In this paper, this principle is
extended to require additionly the form-invariance of a classical field theory
Hamiltonian under variations of the space-time curvature emerging from the
gauge fields. This approach is devised on the basis of the extended canonical
transformation formalism of classical field theory which allows for
transformations of the space-time metric in addition to transformations of the
fields. Working out the Hamiltonian that is form-invariant under extended local
gauge transformations, we can dismiss the conventional requirement for gauge
bosons to be massless in order for them to preserve the local gauge
invariance.The emerging equation of motion for the curvature scalar turns out
to be compatible with the Einstein equation in the case of a static gauge
field. The emerging equation of motion for the curvature scalar R turns out to
be compatible with that from a Proca system in the case of a static gauge
field.Comment: 27 page
Improved envelope and emittance description of particle beams using the Fokker-Planck approach
Generating Explanations of Robot Policies in Continuous State Spaces
Transparency in HRI describes the method of making the current state of a robot or intelligent agent understandable to a human user. Applying transparency mechanisms to robots improves the quality of interaction as well as the user experience.
Explanations are an effective way to make a robot’s decision making transparent. We introduce a framework that uses natural language labels attached to a region in the continuous state space of the robot to automatically generate local explanations of a robot’s policy.
We conducted a pilot study and investigated how the generated explanations helped users to understand and reproduce a robot policy in a debugging scenario
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