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 $\beta$ 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 $\beta$ 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

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

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

Reality conditions for Ashtekar gravity from Lorentz-covariant formulation

We show the equivalence of the Lorentz-covariant canonical formulation considered for the Immirzi parameter $\beta=i$ 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

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

Generalized Lagrangian of N = 1 supergravity and its canonical constraints with the real Ashtekar variable

We generalize the Lagrangian of N = 1 supergravity (SUGRA) by using an arbitrary parameter $\xi$, which corresponds to the inverse of Barbero's parameter $\beta$. This generalized Lagrangian involves the chiral one as a special case of the value $\xi = \pm i$. We show that the generalized Lagrangian gives the canonical formulation of N = 1 SUGRA with the real Ashtekar variable after the 3+1 decomposition of spacetime. This canonical formulation is also derived from those of the usual N = 1 SUGRA by performing Barbero's type canonical transformation with an arbitrary parameter $\beta (=\xi^{-1})$. We give some comments on the canonical formulation of the theory.Comment: 17 pages, LATE

Constraints and Reality Conditions in the Ashtekar Formulation of General Relativity

We show how to treat the constraints and reality conditions in the $SO(3)$-ADM (Ashtekar) formulation of general relativity, for the case of a vacuum spacetime with a cosmological constant. We clarify the difference between the reality conditions on the metric and on the triad. Assuming the triad reality condition, we find a new variable, allowing us to solve the gauge constraint equations and the reality conditions simultaneously.Comment: LaTeX file, 12 pages, no figures; to appear in Classical and Quantum Gravit

Quantum Geometry and Black Hole Entropy

A `black hole sector' of non-perturbative canonical quantum gravity is introduced. The quantum black hole degrees of freedom are shown to be described by a Chern-Simons field theory on the horizon. It is shown that the entropy of a large non-rotating black hole is proportional to its horizon area. The constant of proportionality depends upon the Immirzi parameter, which fixes the spectrum of the area operator in loop quantum gravity; an appropriate choice of this parameter gives the Bekenstein-Hawking formula S = A/4*l_p^2. With the same choice of the Immirzi parameter, this result also holds for black holes carrying electric or dilatonic charge, which are not necessarily near extremal.Comment: Revtex, 8 pages, 1 figur

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