4,391 research outputs found
Canonical formalism for simplicial gravity
We summarise a recently introduced general canonical formulation of discrete
systems which is fully equivalent to the covariant formalism. This framework
can handle varying phase space dimensions and is applied to simplicial gravity
in particular.Comment: 4 pages, 5 figures, based on a talk given at Loops '11 in Madrid, to
appear in Journal of Physics: Conference Series (JPCS
(Broken) Gauge Symmetries and Constraints in Regge Calculus
We will examine the issue of diffeomorphism symmetry in simplicial models of
(quantum) gravity, in particular for Regge calculus. We find that for a
solution with curvature there do not exist exact gauge symmetries on the
discrete level. Furthermore we derive a canonical formulation that exactly
matches the dynamics and hence symmetries of the covariant picture. In this
canonical formulation broken symmetries lead to the replacements of constraints
by so--called pseudo constraints. These considerations should be taken into
account in attempts to connect spin foam models, based on the Regge action,
with canonical loop quantum gravity, which aims at implementing proper
constraints. We will argue that the long standing problem of finding a
consistent constraint algebra for discretized gravity theories is equivalent to
the problem of finding an action with exact diffeomorphism symmetries. Finally
we will analyze different limits in which the pseudo constraints might turn
into proper constraints. This could be helpful to infer alternative
discretization schemes in which the symmetries are not broken.Comment: 32 pages, 15 figure
Time-domain scars: resolving the spectral form factor in phase space
We study the relationship of the spectral form factor with quantum as well as
classical probabilities to return. Defining a quantum return probability in
phase space as a trace over the propagator of the Wigner function allows us to
identify and resolve manifolds in phase space that contribute to the form
factor. They can be associated to classical invariant manifolds such as
periodic orbits, but also to non-classical structures like sets of midpoints
between periodic points. By contrast to scars in wavefunctions, these features
are not subject to the uncertainty relation and therefore need not show any
smearing. They constitute important exceptions from a continuous convergence in
the classical limit of the Wigner towards the Liouville propagator. We support
our theory with numerical results for the quantum cat map and the harmonically
driven quartic oscillator.Comment: 10 pages, 4 figure
Non-adiabatic pumping in an oscillating-piston model
We consider the prototypical "piston pump" operating on a ring, where a
circulating current is induced by means of an AC driving. This can be regarded
as a generalized Fermi-Ulam model, incorporating a finite-height moving wall
(piston) and non trivial topology (ring). The amount of particles transported
per cycle is determined by a layered structure of phase-space. Each layer is
characterized by a different drift velocity. We discuss the differences
compared with the adiabatic and Boltzmann pictures, and highlight the
significance of the "diabatic" contribution that might lead to a
counter-stirring effect.Comment: 6 pages, 4 figures, improved versio
A perturbative approach to Dirac observables and their space-time algebra
We introduce a general approximation scheme in order to calculate gauge
invariant observables in the canonical formulation of general relativity. Using
this scheme we will show how the observables and the dynamics of field theories
on a fixed background or equivalently the observables of the linearized theory
can be understood as an approximation to the observables in full general
relativity. Gauge invariant corrections can be calculated up to an arbitrary
high order and we will explicitly calculate the first non--trivial correction.
Furthermore we will make a first investigation into the Poisson algebra between
observables corresponding to fields at different space--time points and
consider the locality properties of the observables.Comment: 23 page
From covariant to canonical formulations of discrete gravity
Starting from an action for discretized gravity we derive a canonical
formalism that exactly reproduces the dynamics and (broken) symmetries of the
covariant formalism. For linearized Regge calculus on a flat background --
which exhibits exact gauge symmetries -- we derive local and first class
constraints for arbitrary triangulated Cauchy surfaces. These constraints have
a clear geometric interpretation and are a first step towards obtaining
anomaly--free constraint algebras for canonical lattice gravity. Taking higher
order dynamics into account the symmetries of the action are broken. This
results in consistency conditions on the background gauge parameters arising
from the lowest non--linear equations of motion. In the canonical framework the
constraints to quadratic order turn out to depend on the background gauge
parameters and are therefore pseudo constraints. These considerations are
important for connecting path integral and canonical quantizations of gravity,
in particular if one attempts a perturbative expansion.Comment: 37 pages, 5 figures (minor modifications, matches published version +
updated references
From the discrete to the continuous - towards a cylindrically consistent dynamics
Discrete models usually represent approximations to continuum physics.
Cylindrical consistency provides a framework in which discretizations mirror
exactly the continuum limit. Being a standard tool for the kinematics of loop
quantum gravity we propose a coarse graining procedure that aims at
constructing a cylindrically consistent dynamics in the form of transition
amplitudes and Hamilton's principal functions. The coarse graining procedure,
which is motivated by tensor network renormalization methods, provides a
systematic approximation scheme towards this end. A crucial role in this coarse
graining scheme is played by embedding maps that allow the interpretation of
discrete boundary data as continuum configurations. These embedding maps should
be selected according to the dynamics of the system, as a choice of embedding
maps will determine a truncation of the renormalization flow.Comment: 22 page
Can chaos be observed in quantum gravity?
Full general relativity is almost certainly 'chaotic'. We argue that this
entails a notion of nonintegrability: a generic general relativistic model, at
least when coupled to cosmologically interesting matter, likely possesses
neither differentiable Dirac observables nor a reduced phase space. It follows
that the standard notion of observable has to be extended to include
non-differentiable or even discontinuous generalized observables. These cannot
carry Poisson-algebraic structures and do not admit a standard quantization;
one thus faces a quantum representation problem of gravitational observables.
This has deep consequences for a quantum theory of gravity, which we
investigate in a simple model for a system with Hamiltonian constraint that
fails to be completely integrable. We show that basing the quantization on
standard topology precludes a semiclassical limit and can even prohibit any
solutions to the quantum constraints. Our proposed solution to this problem is
to refine topology such that a complete set of Dirac observables becomes
continuous. In the toy model, it turns out that a refinement to a polymer-type
topology, as e.g. used in loop gravity, is sufficient. Basing quantization of
the toy model on this finer topology, we find a complete set of quantum Dirac
observables and a suitable semiclassical limit. This strategy is applicable to
realistic candidate theories of quantum gravity and thereby suggests a solution
to a long-standing problem which implies ramifications for the very concept of
quantization. Our work reveals a qualitatively novel facet of chaos in physics
and opens up a new avenue of research on chaos in gravity which hints at deep
insights into the structure of quantum gravity.Comment: 6 pages + references -- matches published version (clarifications
added for why GR with cosmologically interesting matter likely fails our
notion of weak-integrability
Gauge invariant perturbations around symmetry reduced sectors of general relativity: applications to cosmology
We develop a gauge invariant canonical perturbation scheme for perturbations
around symmetry reduced sectors in generally covariant theories, such as
general relativity. The central objects of investigation are gauge invariant
observables which encode the dynamics of the system. We apply this scheme to
perturbations around a homogeneous and isotropic sector (cosmology) of general
relativity. The background variables of this homogeneous and isotropic sector
are treated fully dynamically which allows us to approximate the observables to
arbitrary high order in a self--consistent and fully gauge invariant manner.
Methods to compute these observables are given. The question of backreaction
effects of inhomogeneities onto a homogeneous and isotropic background can be
addressed in this framework. We illustrate the latter by considering
homogeneous but anisotropic Bianchi--I cosmologies as perturbations around a
homogeneous and isotropic sector.Comment: 39 pages, 1 figur
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