We describe the basic assumptions and key results of loop quantum gravity, which is a background independent approach to quantum gravity. The emphasis is on the basic physical principles and how one deduces predictions from them, at a level suitable for physicists in other areas such as string theory, cosmology, particle physics, astrophysics and condensed matter physics. No details are given, but references are provided to guide the interested reader to the literature. The present state of knowledge is summarized in a list of 35 key results on topics including the hamiltonian and path integral quantizations, coupling to matter, extensions to supergravity and higher dimensional theories, as well as applications to black holes, cosmology and Plank scale phenomenology. We describe the near term prospects for observational tests of quantum theories of gravity and the expectations that loop quantum gravity may provide predictions for their outcomes. Finally, we provide answers to frequently asked questions and a list of key open problems.

Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models ” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm. 1.

smolin@phys.psu.edu y permanent address I. Introduction In the last years significant progress has been made towards the construction of a quantum theory of gravity in several different directions. Three of these directions, in particular, have involved the use of new ideas and mathematical structures that seem, in different ways, well suited to the problem of describing the geometry of spacetime quantum mechanically. These are string theory[1], topological quantum field theory[2, 3, 4, 5, 6, 7], and non-perturbative quantum gravity, based on the loop representation [8, 9, 10, 11, 12, 13, 14]. Furthermore, despite genuine differences, there are a number of concepts shared by these approaches, which suggests the possibility of a deeper relation between them[15, 54]. These include the common use of one dimensional rather than pointlike excitations, as well as the appearance of structures associated with knot theory, spin networks and duality. There are also senses in which each deve...

1. Introduction. The Witten-Reshetikhin-Turaev (WRT) invariant of a compact connected oriented 3-manifold M may be formally defined by [16]

Abstract: We provide, with proofs, a complete description of the authors ’ construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of Rep(Uq(sl2)) (q a principal 4r th root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. 1 1

this paper a new approach to the problem of constructing a quantum theory of gravity in the cosmological context is proposed. It is founded on results from four separate directions of investigation, which are: 1) A new point of view towards the interpretation problem in quantum cosmology[1, 2, 3, 4], which rejects the idea that a single quantum state, or a single Hilbert space, can provide a complete description of a closed system like the universe. Instead, the idea is to accept Bohr's original proposal that the quantum state requires for its interpretation a context in which we distinguish two subsystems of the universe-the quantum system and observer. However, we seek to relativize this split, so that the boundary between the part of the universe that is considered the system and that which might be considered the observer may be chosen arbitrarily. The idea is then that a quantum theory of cosmology is specified by giving an assignment of a Hilbert space and algebra of observables to every possible boundary that can be considered to split the universe into two such subsystems. A quantum state of the universe is then an assignment of a statistical state to every one of these Hilbert spaces, subject to certain conditions of consistency. Each of these states is interpreted to contain the information that an observer on one side of each boundary might have about the system of the other side. This formulation then accepts the idea that each observer can only have incomplete information about the universe, so that the most complete description possible of the universe is given by the whole collection of incomplete, but mutually compatible quantum state descriptions of all the possible observers. At the same time, the information of different observers is, to some extent, ...

There is exactly one straight line passing through any two given distinct points; there is exactly one quadratic curve on the complex projective plane

The formulations of Classical Mechanics by Lagrange and Hamilton are the modern foundation of classical physics [Ar]. Not only do these theories describe the motion of systems of particles, but Maxwell’s theory of electromagnetism, as well as other field theories, can also be formulated in Lagrangian and Hamiltonian terms. A Lagrangian field theory is defined by a local functional of the fields, called the lagrangian, and its integral over spacetime, 1 called the action. The classical solutions of the field theory are the critical points of the action. In particular, the minima satisfy the “least action principle ” of Maupertius. 2 The Hamiltonian theory is defined by a function, the hamiltonian, on phase space, or more generally on a symplectic manifold. The classical motion of the system is then described by Hamilton’s equations, whose solutions are integral curves of the symplectic gradient vector field of the hamiltonian. For many mechanical systems of particles, which should be regarded as 0 + 1 dimensional field theories, there is both a Lagrangian and Hamiltonian formulation. Then the relationship between them is expressed by the Legendre transform, if the lagrangian is nondegenerate. A typical example

We describe a deformation of the observable algebra of quantum gravity in which the loop algebra is extended to framed loops. This allows an alternative nonperturbative quantization which is suitable for describing a phase of quantum gravity characterized by states which are normalizable in the measure of Chern-Simons theory. The deformation parameter, q, is e i¯h 2 G 2 =6 , where is the cosmological constant. The Mandelstam identities are extended to a set of relations which are governed by the Kauffman bracket so that the spin network basis is deformed to a basis of SU(2) q spin networks. Corrections to the actions of operators in non-perturbative quantum gravity may be readily computed using recoupling theory; the example of the area observable is treated here. Finally, eigenstates of the q-deformed Wilson loops are constructed, which may make possible the construction of a q-deformed connection representation through an inverse transform. internet addresses: seth@phys.psu....

: The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques gives the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any Riemann surface with arbitrary coloured punctures. * Supported by DFG, SFB 288 "Differentialgeometrie und Quantenphysik" 1 e-mail: karowski@vax1.physik.fu-berlin.dbp.de 2 e-mail: schrader@vax1.physik.fu-berlin.dbp.de 1 1. Introduction Since the early days of topological quantum field theories there was the question whether such field theories have a lattice formulation analogous to lattice gauge theory. The reason is that one would like to work in a context with mathematically well defined quantities instead of more or less formal functional integrals. This qu...