200 research outputs found
Uniform discretizations: a new approach for the quantization of totally constrained systems
We discuss in detail the uniform discretization approach to the quantization
of totally constrained theories. This approach allows to construct the
continuum theory of interest as a well defined, controlled, limit of well
behaved discrete theories. We work out several finite dimensional examples that
exhibit behaviors expected to be of importance in the quantization of gravity.
We also work out the case of BF theory. At the time of quantization, one can
take two points of view. The technique can be used to define, upon taking the
continuum limit, the space of physical states of the continuum constrained
theory of interest. In particular we show in models that it agrees with the
group averaging procedure when the latter exists. The technique can also be
used to compute, at the discrete level, conditional probabilities and the
introduction of a relational time. Upon taking the continuum limit one can show
that one reproduces results obtained by the use of evolving constants, and
therefore recover all physical predictions of the continuum theory. This second
point of view can also be used as a paradigm to deal with cases where the
continuum limit does not exist. There one would have discrete theories that at
least at certain scales reproduce the semiclassical properties of the theory of
interest. In this way the approach can be viewed as a generalization of the
Dirac quantization procedure that can handle situations where the latter fails.Comment: 17 pages, Revtex, no figures, published versio
Loop quantization of spherically symmetric midi-superspaces
We quantize the exterior of spherically symmetric vacuum space-times using a
midi-superspace reduction within the Ashtekar new variables. Through a partial
gauge fixing we eliminate the diffeomorphism constraint and are left with a
Hamiltonian constraint that is first class. We complete the quantization in the
loop representation. We also use the model to discuss the issues that will
arise in more general contexts in the ``uniform discretization'' approach to
the dynamics.Comment: 18 pages, RevTex, no figures, some typos corrected, published
version, for some reason a series of figures were incorrectly added to the
previous versio
Emergent diffeomorphism invariance in a discrete loop quantum gravity model
Several approaches to the dynamics of loop quantum gravity involve
discretizing the equations of motion. The resulting discrete theories are known
to be problematic since the first class algebra of constraints of the continuum
theory becomes second class upon discretization. If one treats the second class
constraints properly, the resulting theories have very different dynamics and
number of degrees of freedom than those of the continuum theory. It is
therefore questionable how these theories could be considered a starting point
for quantization and the definition of a continuum theory through a continuum
limit. We show explicitly in a model that the {\em uniform discretizations}
approach to the quantization of constrained systems overcomes these
difficulties. We consider here a simple diffeomorphism invariant one
dimensional model and complete the quantization using {\em uniform
discretizations}. The model can be viewed as a spherically symmetric reduction
of the well known Husain--Kucha\v{r} model of diffeomorphism invariant theory.
We show that the correct quantum continuum limit can be satisfactorily
constructed for this model. This opens the possibility of treating 1+1
dimensional dynamical situations of great interest in quantum gravity taking
into account the full dynamics of the theory and preserving the space-time
covariance at a quantum level.Comment: 12 pages, Revte
Gauge Is More Than Mathematical Redundancy
Physical systems may couple to other systems through variables that are not gauge invariant. When we split a gauge system into two subsystems, the gauge-invariant variables of the two subsystems have less information than the gauge-invariant variables of the original system; the missing information regards degrees of freedom that express relations between the subsystems. All this shows that gauge invariance is a formalization of the relational nature of physical degrees of freedom. The recent developments on boundary variables and boundary charges are clarified by this observation
Spherically symmetric Einstein-Maxwell theory and loop quantum gravity corrections
Effects of inverse triad corrections and (point) holonomy corrections,
occuring in loop quantum gravity, are considered on the properties of
Reissner-Nordstr\"om black holes. The version of inverse triad corrections with
unmodified constraint algebra reveals the possibility of occurrence of three
horizons (over a finite range of mass) and also shows a mass threshold beyond
which the inner horizon disappears. For the version with modified constraint
algebra, coordinate transformations are no longer a good symmetry. The
covariance property of spacetime is regained by using a \emph{quantum} notion
of mapping from phase space to spacetime. The resulting quantum effects in both
versions of these corrections can be associated with renormalization of either
mass, charge or wave function. In neither of the versions, Newton's constant is
renormalized. (Point) Holonomy corrections are shown to preclude the undeformed
version of constraint algebra as also a static solution, though
time-independent solutions exist. A possible reason for difficulty in
constructing a covariant metric for these corrections is highlighted.
Furthermore, the deformed algebra with holonomy corrections is shown to imply
signature change.Comment: 38 pages, 9 figures, matches published versio
The volume operator in covariant quantum gravity
A covariant spin-foam formulation of quantum gravity has been recently
developed, characterized by a kinematics which appears to match well the one of
canonical loop quantum gravity. In particular, the geometrical observable
giving the area of a surface has been shown to be the same as the one in loop
quantum gravity. Here we discuss the volume observable. We derive the volume
operator in the covariant theory, and show that it matches the one of loop
quantum gravity, as does the area. We also reconsider the implementation of the
constraints that defines the model: we derive in a simple way the boundary
Hilbert space of the theory from a suitable form of the classical constraints,
and show directly that all constraints vanish weakly on this space.Comment: 10 pages. Version 2: proof extended to gamma > 1
Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory
A covariant spin-foam formulation of quantum gravity has been recently
developed, characterized by a kinematics which appears to match well the one of
canonical loop quantum gravity. In this paper we reconsider the implementation
of the constraints that defines the model. We define in a simple way the
boundary Hilbert space of the theory, introducing a slight modification of the
embedding of the SU(2) representations into the SL(2,C) ones. We then show
directly that all constraints vanish on this space in a weak sense. The
vanishing is exact (and not just in the large quantum number limit.) We also
generalize the definition of the volume operator in the spinfoam model to the
Lorentzian signature, and show that it matches the one of loop quantum gravity,
as does in the Euclidean case.Comment: 11 page
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