195 research outputs found
Real and complex connections for canonical gravity
Both real and complex connections have been used for canonical gravity: the
complex connection has SL(2,C) as gauge group, while the real connection has
SU(2) as gauge group. We show that there is an arbitrary parameter
which enters in the definition of the real connection, in the Poisson brackets,
and therefore in the scale of the discrete spectra one finds for areas and
volumes in the corresponding quantum theory. A value for could be could
be singled out in the quantum theory by the Hamiltonian constraint, or by the
rotation to the complex Ashtekar connection.Comment: 8 pages, RevTeX, no figure
Fuzzy Nambu-Goldstone Physics
In spacetime dimensions larger than 2, whenever a global symmetry G is
spontaneously broken to a subgroup H, and G and H are Lie groups, there are
Nambu-Goldstone modes described by fields with values in G/H. In
two-dimensional spacetimes as well, models where fields take values in G/H are
of considerable interest even though in that case there is no spontaneous
breaking of continuous symmetries. We consider such models when the world sheet
is a two-sphere and describe their fuzzy analogues for G=SU(N+1),
H=S(U(N-1)xU(1)) ~ U(N) and G/H=CP^N. More generally our methods give fuzzy
versions of continuum models on S^2 when the target spaces are Grassmannians
and flag manifolds described by (N+1)x(N+1) projectors of rank =< (N+1)/2.
These fuzzy models are finite-dimensional matrix models which nevertheless
retain all the essential continuum topological features like solitonic sectors.
They seem well-suited for numerical work.Comment: Latex, 18 pages; references added, typos correcte
Duality in Fuzzy Sigma Models
Nonlinear `sigma' models in two dimensions have BPS solitons which are
solutions of self- and anti-self-duality constraints. In this paper, we find
their analogues for fuzzy sigma models on fuzzy spheres which were treated in
detail by us in earlier work. We show that fuzzy BPS solitons are quantized
versions of `Bott projectors', and construct them explicitly. Their
supersymmetric versions follow from the work of S. Kurkcuoglu.Comment: Latex, 9 pages; misprints correcte
Quantum Computation toward Quantum Gravity
The aim of this paper is to enlight the emerging relevance of Quantum
Information Theory in the field of Quantum Gravity. As it was suggested by J.
A. Wheeler, information theory must play a relevant role in understanding the
foundations of Quantum Mechanics (the "It from bit" proposal). Here we suggest
that quantum information must play a relevant role in Quantum Gravity (the "It
from qubit" proposal). The conjecture is that Quantum Gravity, the theory which
will reconcile Quantum Mechanics with General Relativity, can be formulated in
terms of quantum bits of information (qubits) stored in space at the Planck
scale. This conjecture is based on the following arguments: a) The holographic
principle, b) The loop quantum gravity approach and spin networks, c) Quantum
geometry and black hole entropy. Here we present the quantum version of the
holographic principle by considering each pixel of area of an event horizon as
a qubit. This is possible if the horizon is pierced by spin networks' edges of
spin 1\2, in the superposed state of spin "up" and spin "down".Comment: 11 pages. Contributed to XIII International Congress on Mathematical
Physics (ICMP 2000), London, England, 17-22 Jul 2000. Typos corrected.
Accepted for publication in General Relativity and Gravitatio
Reality conditions for Ashtekar gravity from Lorentz-covariant formulation
We show the equivalence of the Lorentz-covariant canonical formulation
considered for the Immirzi parameter to the selfdual Ashtekar
gravity. We also propose to deal with the reality conditions in terms of Dirac
brackets derived from the covariant formulation and defined on an extended
phase space which involves, besides the selfdual variables, also their
anti-selfdual counterparts.Comment: 14 page
Regge calculus and Ashtekar variables
Spacetime discretized in simplexes, as proposed in the pioneer work of Regge,
is described in terms of selfdual variables. In particular, we elucidate the
"kinematic" structure of the initial value problem, in which 3--space is
divided into flat tetrahedra, paying particular attention to the role played by
the reality condition for the Ashtekar variables. An attempt is made to write
down the vector and scalar constraints of the theory in a simple and
potentially useful way.Comment: 10 pages, uses harvmac. DFUPG 83/9
Quasinormal Modes, the Area Spectrum, and Black Hole Entropy
The results of canonical quantum gravity concerning geometric operators and
black hole entropy are beset by an ambiguity labelled by the Immirzi parameter.
We use a result from classical gravity concerning the quasinormal mode spectrum
of a black hole to fix this parameter in a new way. As a result we arrive at
the Bekenstein - Hawking expression of for the entropy of a black
hole and in addition see an indication that the appropriate gauge group of
quantum gravity is SO(3) and not its covering group SU(2).Comment: 4 pages, 2 figure
Topological Lattice Gravity Using Self-Dual Variables
Topological gravity is the reduction of general relativity to flat
space-times. A lattice model describing topological gravity is developed
starting from a Hamiltonian lattice version of B\w F theory. The extra
symmetries not present in gravity that kill the local degrees of freedom in
theory are removed. The remaining symmetries preserve the
geometrical character of the lattice. Using self-dual variables, the conditions
that guarantee the geometricity of the lattice become reality conditions. The
local part of the remaining symmetry generators, that respect the
geometricity-reality conditions, has the form of Ashtekar's constraints for GR.
Only after constraining the initial data to flat lattices and considering the
non-local (plus local) part of the constraints does the algebra of the symmetry
generators close. A strategy to extend the model for non-flat connections and
quantization are discussed.Comment: 22 pages, revtex, no figure
Dirac Operators on Coset Spaces
The Dirac operator for a manifold Q, and its chirality operator when Q is
even dimensional, have a central role in noncommutative geometry. We
systematically develop the theory of this operator when Q=G/H, where G and H
are compact connected Lie groups and G is simple. An elementary discussion of
the differential geometric and bundle theoretic aspects of G/H, including its
projective modules and complex, Kaehler and Riemannian structures, is presented
for this purpose. An attractive feature of our approach is that it
transparently shows obstructions to spin- and spin_c-structures. When a
manifold is spin_c and not spin, U(1) gauge fields have to be introduced in a
particular way to define spinors. Likewise, for manifolds like SU(3)/SO(3),
which are not even spin_c, we show that SU(2) and higher rank gauge fields have
to be introduced to define spinors. This result has potential consequences for
string theories if such manifolds occur as D-branes. The spectra and
eigenstates of the Dirac operator on spheres S^n=SO(n+1)/SO(n), invariant under
SO(n+1), are explicitly found. Aspects of our work overlap with the earlier
research of Cahen et al..Comment: section on Riemannian structure improved, references adde
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