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

The mass quantum and black hole entropy II

In gr-qc/9908036 [Phys. Lett. A 265 (2000) 1] a new method was given which naturally led to a quantum of mass equal to twice the Planck mass. In the present note which, for convenience, we write formally as a continuation of that paper, we show that with spin one of the mass quantum, the physical entropy of a rotating black hole is also given by the Bekenstein-Hawking formula.Comment: 3 plain tex page

Is Barbero's Hamiltonian formulation a Gauge Theory of Lorentzian Gravity?

This letter is a critique of Barbero's constrained Hamiltonian formulation of General Relativity on which current work in Loop Quantum Gravity is based. While we do not dispute the correctness of Barbero's formulation of general relativity, we offer some criticisms of an aesthetic nature. We point out that unlike Ashtekar's complex SU(2) connection, Barbero's real SO(3) connection does not admit an interpretation as a space-time gauge field. We show that if one tries to interpret Barbero's real SO(3) connection as a space-time gauge field, the theory is not diffeomorphism invariant. We conclude that Barbero's formulation is not a gauge theory of gravity in the sense that Ashtekar's Hamiltonian formulation is. The advantages of Barbero's real connection formulation have been bought at the price of giving up the description of gravity as a gauge field.Comment: 12 pages, no figures, revised in the light of referee's comments, accepted for publication in Classical and Quantum Gravit

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 $B\wedge F$ 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

Hamiltonian analysis of SO(4,1) constrained BF theory

In this paper we discuss canonical analysis of SO(4,1) constrained BF theory. The action of this theory contains topological terms appended by a term that breaks the gauge symmetry down to the Lorentz subgroup SO(3,1). The equations of motion of this theory turn out to be the vacuum Einstein equations. By solving the B field equations one finds that the action of this theory contains not only the standard Einstein-Cartan term, but also the Holst term proportional to the inverse of the Immirzi parameter, as well as a combination of topological invariants. We show that the structure of the constraints of a SO(4,1) constrained BF theory is exactly that of gravity in Holst formulation. We also briefly discuss quantization of the theory.Comment: 9 page

Cosmological vector modes and quantum gravity effects

In contrast to scalar and tensor modes, vector modes of linear perturbations around an expanding Friedmann--Robertson--Walker universe decay. This makes them largely irrelevant for late time cosmology, assuming that all modes started out at a similar magnitude at some early stage. By now, however, bouncing models are frequently considered which exhibit a collapsing phase. Before this phase reaches a minimum size and re-expands, vector modes grow. Such modes are thus relevant for the bounce and may even signal the breakdown of perturbation theory if the growth is too strong. Here, a gauge invariant formulation of vector mode perturbations in Hamiltonian cosmology is presented. This lays out a framework for studying possible canonical quantum gravity effects, such as those of loop quantum gravity, at an effective level. As an explicit example, typical quantum corrections, namely those coming from inverse densitized triad components and holonomies, are shown to increase the growth rate of vector perturbations in the contracting phase, but only slightly. Effects at the bounce of the background geometry can, however, be much stronger.Comment: 20 page

Radiation of Quantized Black Hole

The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. The general structure of the horizon spectrum and the value of the Barbero-Immirzi parameter are found. The discrete spectrum of thermal radiation of a black hole fits naturally the Wien profile. The natural widths of the lines are very small as compared to the distances between them. The total intensity of the thermal radiation is calculated.Comment: 11 pages; few comments and a reference added; one more reference and a comment on it added; a note added that the natural widths of the lines are very small as compared to the distances between the

Quantized Black Holes, Their Spectrum and Radiation

Under quite natural general assumptions, the following results are obtained. The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. The general structure of the horizon spectrum is found. The discrete spectrum of thermal radiation of a black hole Under quite natural general assumptions, the following results are obtained. The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. The general structure of the horizon spectrum is found. The discrete spectrum of thermal radiation of a black hole fits the Wien profile. The natural widths of the lines are much smaller than the distances between them. The total intensity of the thermal radiation is estimated. In the special case of loop quantum gravity, the value of the Barbero -- Immirzi parameter is found. Different values for this parameter, obtained under additional assumption that the horizon is described by a U(1) Chern -- Simons theory, are demonstrated to be in conflict with the firmly established holographic bound.Comment: 15 pages, content of few talks given at conferences this summe

Noncommutative Chiral Anomaly and the Dirac-Ginsparg-Wilson Operator

It is shown that the local axial anomaly in $2-$dimensions emerges naturally if one postulates an underlying noncommutative fuzzy structure of spacetime . In particular the Dirac-Ginsparg-Wilson relation on ${\bf S}^2_F$ is shown to contain an edge effect which corresponds precisely to the ``fuzzy'' $U(1)_A$ axial anomaly on the fuzzy sphere . We also derive a novel gauge-covariant expansion of the quark propagator in the form $\frac{1}{{\cal D}_{AF}}=\frac{a\hat{\Gamma}^L}{2}+\frac{1}{{\cal D}_{Aa}}$ where $a=\frac{2}{2l+1}$ is the lattice spacing on ${\bf S}^2_F$, $\hat{\Gamma}^L$ is the covariant noncommutative chirality and ${\cal D}_{Aa}$ is an effective Dirac operator which has essentially the same IR spectrum as ${\cal D}_{AF}$ but differes from it on the UV modes. Most remarkably is the fact that both operators share the same limit and thus the above covariant expansion is not available in the continuum theory . The first bit in this expansion $\frac{a\hat{\Gamma}^L}{2}$ although it vanishes as it stands in the continuum limit, its contribution to the anomaly is exactly the canonical theta term. The contribution of the propagator $\frac{1}{{\cal D}_{Aa}}$ is on the other hand equal to the toplogical Chern-Simons action which in two dimensions vanishes identically .Comment: 26 pages, latex fil