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Scientific Report 2007-2009
Theoretical physics
T11. Quantum Cosmology
The necessity for a quantum theory of gravity arises
from fundamental considerations, and, in particular,
from the space-time singularity problem. In fact, the
classical theory of gravity implies the well known singularity
theorems, among which the cosmological one.
Several difficulties in implementing a quantum theory
for the gravitational field can be overcome in minisuperspace
models, for which some degrees of freedom are
frozen out in view of the adopted symmetries. These
models are still highly meaningful, since the most relevant
case is a cosmological space-time.
The study performed within our group improves a research
line centered in the investigation of cosmological
models with a minimal scale. The introduction of a cutoff
can be implemented by inequivalent approaches to
quantum mechanics, which are expected to mimic some
features of the final Quantum Gravity theory.
The polymer representation of quantum mechanics for
a particular homogeneous cosmological space-time (the
Taub Universe) was analyzed in [1] . This approach is
based on a non-standard representation of the canonical
commutation relations and it is relevant in treating
the quantum-mechanical properties of a backgroundindependent
canonical quantum theory of gravity. The
modifications induced by the cut-off scale on ordinary
trajectories were studied from a classical point of view.
Furthermore, the quantum regime was explored in detail
by the investigation of the evolution of wave packets, unveiling
an interference phenomenon between such wave
packets and the potential wall. Nevertheless, the wave
function of the Universe is not peaked far away from the
singularity and falls into it following a classical trajectory;
thus we have to conclude that the singularity is not
removed on a probabilistic level.
A different intuitive approach to introduce a cut-off
is based on deforming the canonical uncertainty relations
leading to the so-called Generalized Uncertainty
Principle (GUP). Such a modification appeared in perturbative
string theory. In the work [2], the Bianchi IX
cosmological model (the Mixmaster Universe) was studied
within the GUP framework. To perform the analysis,
two necessary steps, i.e. the study of the Bianchi I and II
cosmological models, were necessary. The main results
are: (i) The Bianchi I dynamics is still Kasner-like but
is deeply modified since the GUP effects allow for the
existence of two negative Kasner exponents. (ii) The
Bianchi II model is no longer analytically integrable and
therefore no BKL map can be obtained. (iii) The potential
walls of Bianchi IX become stationary with respect
to the point-Universe when the momentum of the latter
is of the same order of the cut-off. We conclude that
the deformed evolution of the Mixmaster Universe is still
chaotic.
The comparison between the polymer- and the GUP-
Taub model illustrates that the interference phenomena
are produced in a complementary way. This feature appears
both at classical level and in the quantum regime,
as the behavior of the wave packets is investigated.
A further research line within our group deals with
the definition of a background independent quantization
of the gravitational field in a generic local Lorentz
frame. This investigation is motivated by the standard
requirement of Loop Quantum Gravity to restrict the local
Lorentz frame by the so-called time gauge condition.
The Hamiltonian formulation without such a gauge fixing
is performed in [3]. The main technical issue is the
emergence of a second-class system of constraints, which
is reduced to a first-class one without fixing the local
Lorentz frame but restricting to a suitable hypersurface
in the full phase space. A privileged set of variables is
selected out and is constituted by non-dynamical boost
parameters and SU(2) connections. Hence, the standard
loop quantization in terms of holonomies and fluxes of
the SU(2) group is still well-grounded. Furthermore,
boost invariance on a quantum level is reproduced by
wave-functionals which do not exhibit any dependence
on boost parameters. The results of this analysis outline
the invariant nature of the discrete space structure
proper of Loop Quantum Gravity and elucidates the fundamental
role that the SU(2) symmetry plays in the
phase-space of gravity.
Finally, wide attention is devoted to study of generalized
formulations of differential geometry in order
to incorporate physical features of fundamental fields
into a unified picture. In particular, in [4], a generalized
connection, including Christoffel coefficients, torsion,
non-metricity tensor and metric-asymmetricity objects,
is analyzed according to the Schouten classification.
The inverse structure matrix is obtained in the
linearized regime, autoparallel trajectories are defined,
and the contribution of the connection components are
clarified at first-order approximation. The restricted sector
in which is retained only a torsion field, is currently
under investigation towards its implementation in the
framework of a Lorentz gauge theory.
References
1. M. V. Battisti et al., Phys. Rev. D78, 103514 (2008).
2. M. V. Battisti et al., Phys. Lett. B681, 179 (2009).
3. F. Cianfrani et al., Phys. Rev. Lett. 102, 091301 (2009).
4. S. Casanova et al., Mod. Phys. Lett. A23, 17 (2008).
Authors
M. V. Battisti 6 , R. Benini 6 , F. Cianfrani 6 , O. M. Lecian 6 , G.
Montani 68 , R. Ruffini
Sapienza Università di Roma 34 Dipartimento di Fisica