Quantization of Diffeomorphism-Invariant Theories with Fermions

We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P -> Sigma be a principal G-bundle over space and let F be a vector bundle associated to P whose fiber is a sum of continuous unitary irreducible representations of the compact connected gauge group G, each representation appearing together with its dual. We consider theories whose classical configuration space is A x F, where A is the space of connections on P and F is the space of sections of F, regarded as a collection of Grassmann-valued fermionic fields. We construct the `quantum configuration space a x f as a completion of A x F. Using this we construct a Hilbert space L^2(a x f) for the quantum theory on which all automorphisms of P act as unitary operators, and determine an explicit `spin network basis' of the subspace L^2((a x f)/G) consisting of gauge-invariant states. We represent observables constructed from holonomies of the connection along paths together with fermionic fields and their conjugate momenta as operators on L^2((a x f)/G). We also construct a Hilbert space H_diff of diffeomorphism-invariant states using the group averaging procedure of Ashtekar, Lewandowski, Marolf, Mourao and Thiemann.Comment: 28 pages, latex, 7 ps-files (included) are needed to process the source fil

Loop Quantization of Maxwell Theory and Electric Charge Quantization

We consider the loop quantization of Maxwell theory. A quantization of this type leads to a quantum theory in which the fundamental excitations are loop-like rather than particle-like. Each such loop plays the role of a quantized Faraday's line of electric flux. We find that the quantization depends on an arbitrary choice of a parameter e that carries the dimension of electric charge. For each value of e an electric charge that can be contained inside a bounded spatial region is automatically quantized in units of hbar/4*pi*e. The requirement of consistency with the quantization of electric charge observed in our Universe fixes a value of the, so far arbitrary, parameter e of the theory. Finally, we compare the ambiguity in the choice of parameter e with the beta-ambiguity that, as pointed by Immirzi, arises in the loop quantization of general relativity, and comment on a possible way this ambiguity can be fixed.Comment: 7 pages, Revtex, no figures, typos corrected and one reference adde

Cosmological perturbations in a family of deformations of general relativity

We study linear cosmological perturbations in a previously introduced family of deformations of general relativity characterized by the absence of new degrees of freedom. The homogeneous and isotropic background in this class of theories is unmodified and is described by the usual Friedmann equations. The theory of cosmological perturbations is modified and the relevant deformation parameter has the dimension of length. Gravitational perturbations of the scalar type can be described by a certain relativistic potential related to the matter perturbations just as in general relativity. A system of differential equations describing the evolution of this potential and of the stress-energy density perturbations is obtained. We find that the evolution of scalar perturbations proceeds with a modified effective time-dependent speed of sound, which, contrary to the case of general relativity, does not vanish even at the matter-dominated stage. In a broad range of values of the length parameter controlling the deformation, a specific transition from the regime of modified gravity to the regime of general relativity in the evolution of scalar perturbations takes place during the radiation domination. In this case, the resulting power spectrum of perturbations in radiation and dark matter is suppressed on the comoving spatial scales that enter the Hubble radius before this transition. We estimate the bounds on the deformation parameter for which this suppression does not lead to observable consequences. Evolution of scalar perturbations at the inflationary stage is modified but very slightly and the primordial spectrum generated during inflation is not noticeably different from the one obtained in general relativity.Comment: 45 pages, version published in JCAP; minor changes, one section moved to the appendi

Counting surface states in the loop quantum gravity

We adopt the point of view that (Riemannian) classical and (loop-based) quantum descriptions of geometry are macro- and micro-descriptions in the usual statistical mechanical sense. This gives rise to the notion of geometrical entropy, which is defined as the logarithm of the number of different quantum states which correspond to one and the same classical geometry configuration (macro-state). We apply this idea to gravitational degrees of freedom induced on an arbitrarily chosen in space 2-dimensional surface. Considering an `ensemble' of particularly simple quantum states, we show that the geometrical entropy $S(A)$ corresponding to a macro-state specified by a total area $A$ of the surface is proportional to the area $S(A)=\alpha A$, with $\alpha$ being approximately equal to $1/16\pi l_p^2$. The result holds both for case of open and closed surfaces. We discuss briefly physical motivations for our choice of the ensemble of quantum states.Comment: This paper is a substantially modified version of the paper `The Bekenstein bound and non-perturbative quantum gravity'. Although the main result (i.e. the result of calculation of the number of quantum states that correspond to one and the same area of 2-d surface) remains unchanged, it is presented now from a different point of view. The new version contains a discussion both of the case of open and closed surfaces, and a discussion of a possibility to generalize the result obtained considering arbitrary surface quantum states. LaTeX, 21 pages, 6 figures adde

Quantum Geometry and Thermal Radiation from Black Holes

A quantum mechanical description of black hole states proposed recently within non-perturbative quantum gravity is used to study the emission and absorption spectra of quantum black holes. We assume that the probability distribution of states of the quantum black hole is given by the ``area'' canonical ensemble, in which the horizon area is used instead of energy, and use Fermi's golden rule to find the line intensities. For a non-rotating black hole, we study the absorption and emission of s-waves considering a special set of emission lines. To find the line intensities we use an analogy between a microscopic state of the black hole and a state of the gas of atoms.Comment: 19 pages, 4 figures, modified version to appear in Class. Quant. Gra

SO_0(1,d+1) Racah coefficients: Type I representations

We use AdS/CFT inspired methods to study the Racah coefficients for type I representations of the Lorentz group SO_0(1,d+1) with d>1. For such representations (a multiple of) the Racah coefficient can be represented as an integral of a product of 6 bulk-to-bulk propagators over 4 copies of the hyperbolic space H_{d+1}. To compute the integrals we represent the bulk-to-bulk propagators in terms of bulk-to-boundary ones. The bulk integrals can be computed explicitly, and the boundary integrations are carried out by introducing Feynman parameters. The final result is an integral representation of the Racah coefficient given by 4 Barnes-Mellin type integrals.Comment: 20 pages, 1 figure. v2: Case d=1 corrected, case d>1 clarifie

The Universal Phase Space of AdS3 Gravity

We describe what can be called the "universal" phase space of AdS3 gravity, in which the moduli spaces of globally hyperbolic AdS spacetimes with compact spatial sections, as well as the moduli spaces of multi-black-hole spacetimes are realized as submanifolds. The universal phase space is parametrized by two copies of the Universal Teichm\"uller space T(1) and is obtained from the correspondence between maximal surfaces in AdS3 and quasisymmetric homeomorphisms of the unit circle. We also relate our parametrization to the Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the holographic (Fefferman-Graham) description. In particular, we obtain a relation between the generators of quasiconformal deformations in each T(1) sector and the chiral Brown-Henneaux vector fields. We also relate the charges arising in the holographic description (such as the mass and angular momentum of an AdS3 spacetime) to the periods of the quadratic differentials arising via the Bers embedding of T(1)xT(1). Our construction also yields a symplectic map from T*T(1) to T(1)xT(1) generalizing the well-known Mess map in the compact spatial surface setting.Comment: 41 pages, 2 figures, revised version accepted for publication in Commun.Math.Phy

Non-Metric Gravity I: Field Equations

We describe and study a certain class of modified gravity theories. Our starting point is Plebanski formulation of gravity in terms of a triple B^i of 2-forms, a connection A^i and a ``Lagrange multiplier'' field Psi^ij. The generalization we consider stems from presence in the action of an extra term proportional to a scalar function of Psi^ij. As in the usual Plebanski general relativity (GR) case, a certain metric can be constructed from B^i. However, unlike in GR, the connection A^i no longer coincides with the self-dual part of the metric-compatible spin-connection. Field equations of the theory are shown to be relations between derivatives of the metric and components of field Psi, as well as its derivatives, the later being in contrast to the GR case. The equations are of second order in derivatives. An analog of the Bianchi identity is still present in the theory, as well as its contracted version tantamount to energy conservation equation.Comment: 21 pages, no figures (v2) energy conservation equation simplified, note on reality conditions added (v3) minor change

Minimal surfaces and particles in 3-manifolds

We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these metrics has a simple description in terms of Teichm\"uller theory. In the hyperbolic settings both questions have positive answers for a certain subset of the quasi-Fuchsian manifolds: those containing a closed surface with principal curvatures at most 1. We show that this subset is parameterized by an open domain of the cotangent bundle of Teichm\"uller space. These results are extended to ``quasi-Fuchsian'' manifolds with conical singularities along infinite lines, known in the physics literature as ``massive, spin-less particles''. Things work better for globally hyperbolic anti-de Sitter manifolds: the parameterization by the cotangent of Teichm\"uller space works for all manifolds. There is another description of this moduli space as the product two copies of Teichm\"uller space due to Mess. Using the maximal surface description, we propose a new parameterization by two copies of Teichm\"uller space, alternative to that of Mess, and extend all the results to manifolds with conical singularities along time-like lines. Similar results are obtained for de Sitter or Minkowski manifolds. Finally, for all four settings, we show that the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichm\"uller space is the same as the 3-dimensional gravity one.Comment: 53 pages, no figure. v2: typos corrected and refs adde

Quantum Loop Representation for Fermions coupled to Einstein-Maxwell field

Quantization of the system comprising gravitational, fermionic and electromagnetic fields is developed in the loop representation. As a result we obtain a natural unified quantum theory. Gravitational field is treated in the framework of Ashtekar formalism; fermions are described by two Grassmann-valued fields. We define a $C^{*}$-algebra of configurational variables whose generators are associated with oriented loops and curves; ``open'' states -- curves -- are necessary to embrace the fermionic degrees of freedom. Quantum representation space is constructed as a space of cylindrical functionals on the spectrum of this $C^{*}$-algebra. Choosing the basis of ``loop'' states we describe the representation space as the space of oriented loops and curves; then configurational and momentum loop variables become in this basis the operators of creation and annihilation of loops and curves. The important difference of the representation constructed from the loop representation of pure gravity is that the momentum loop operators act in our case simply by joining loops in the only compatible with their orientaiton way, while in the case of pure gravity this action is more complicated.Comment: 28 pages, REVTeX 3.0, 15 uuencoded ps-figures. The construction of the representation has been changed so that the representation space became irreducible. One part is removed because it developed into a separate paper; some corrections adde