Introduction: Defining quantum gravity 1 Why quantum gravity in the twenty-first century? 1 The role of background independence 8 Approaches to quantum gravity 11 Motivation for canonical quantum general relativity Outline of the book I CLASSICAL FOUNDATIONS, INTERPRETATION AND THE CANONICAL QUANTISATION PROGRAMME 1 Classical Hamiltonian formulation of General Relativity 1.1 The ADM action 1.2 Legendre transform and Dirac analysis of constraints 1.3 Geometrical interpretation of the gauge transformations 1.4 Relation between the four-dimensional diffeomorphism group and the transformations generated by the constraints 1.5 Boundary conditions, gauge transformations and symmetries 1.5.1 Boundary conditions 1.5.2 Symmetries and gauge transformations 65 2 The problem of time, locality and the interpretation of quantum mechanics 2.1 The classical problem of time: Dirac observables 2.2 Partial and complete observables for general constrained systems 2.2.1 Partial and weak complete observables 82 2.2.2 Poisson algebra of Dirac observables 2.2.3 Evolving constants 2.2.4 Reduced phase space quantisation of the algebra of Dirac observables and unitary implementation of the multi-fingered time evolution 2.3 Recovery of locality in General Relativity x Contents 2.4 Quantum problem of time: physical inner product and interpretation of quantum mechanics 95 2.4.1 Physical inner product 95 2.4.2 Interpretation of quantum mechanics 98 3 The programme of canonical quantisation 107 3.1 The programme 4 The new canonical variables of Ashtelcar for General Relativity 118 4.1 Historical overview 118 4.2 Derivation of Ashtekar's variables 123 4.2.1 Extension of the ADM phase space 123 4.2.2 Canonical transformation on the extended phase space 126 II FOUNDATIONS OF MODERN CANONICAL QUANTUM GENERAL RELATIVITY

An anomaly-free operator corresponding to the Wheeler-DeWitt constraint of Lorentzian, four-dimensional, canonical, non-perturbative vacuum gravity is constructed in the continuum. This operator is entirely free of factor ordering singularities and can be defined in symmetric and non-symmetric form.We work in the real connection representation and obtain a well-defined quantum theory. We compute the complete solution to the Quantum Einstein Equations for the non-symmetric version of the operator and a physical inner product thereon.The action of the Wheeler-DeWitt constraint on spin-network states is by annihilating, creating and rerouting the quanta of angular momentum associated with the edges of the underlying graph while the ADM-energy is essentially diagonalized by the spin-network states. We argue that the spinnetwork representation is the "non-linear Fock representation" of quantum gravity, thus justifying the term "Quantum Spin Dynamics (QSD)".

Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kuchař model. The main results also pave way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to combined in an appropriate fashion with a coherent state transform to incorporate complex connections.

It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague quantum field theories in a background metric.We demonstrate that this idea is implemented in a precise sense within the framework of four-dimensional canonical Lorentzian quantum gravity in the continuum.Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators.This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory. In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized.

A Wheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric. The technique introduced here can also be used to produce a couple of completely well-defined regulated operators including but not exhausting a) the Euclidean Wheeler-DeWitt operator, b) the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, c) length operators, d) Hamiltonian operators of the matter sector and e) the generators of the asymptotic Poincaré group including the quantum ADM energy.Attempts at defining an operator which corresponds to the Hamiltonian constraint of four-dimensional Lorentzian vacuum canonical gravity [1] have first been made within the framework of the ADM or metric variables (see, for instance, [2]) This formulation of the theory seemed hopelessly difficult because of the complicated algebraic nature of the Hamiltonian (or Wheeler-DeWitt) constraint. It was therefore thought to be mandatory to first cast the Hamiltonian constraint into polynomial form by finding better suited canonical variables. That this is indeed possible was demonstrated by Ashtekar [3]. There are two, a priori, problems with these Ashtekar connection variables for Lorentzian gravity : 1) they are complex valued and are therefore subject to algebraically highly complicated reality conditions, difficult to impose on the quantum level, which make sure that we are still dealing with real general relativity and 2) the Hamiltonian constraint is polynomial only after rescaling it by a non-polynomial function, namely a power of the square root of the determinant of the three-dimensional metric, that is, the original Wheeler-DeWitt constraint has actually been altered. A solution to problem 1) has been suggested in [4] (see also [5]) : namely, one can define real Ashtekar variables [6] which simplify the rescaled Hamiltonian constraint of Euclidean gravity and then construct a Wick rotation transform from the Euclidean to * thiemann@math.harvard.edu 1

Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms.While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract * -algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate *

In this article we outline a rather general construction of diffeomorphism covariant coherent states for quantum gauge theories.By this we mean states ψ (A,E) , labelled by a point (A, E) in the classical phase space, consisting of canonically conjugate pairs of connections A and electric fields E respectively, such that (a) they are eigenstates of a corresponding annihilation operator which is a generalization of A − iE smeared in a suitable way, (b) normal ordered polynomials of generalized annihilation and creation operators have the correct expectation value, (c) they saturate the Heisenberg uncertainty bound for the fluctuations ofÂ,Ê and (d) they do not use any background structure for their definition, that is, they are diffeomorphism covariant. This is the first paper in a series of articles entitled "Gauge Field Theory Coherent States (GCS)" which aim at connecting non-perturbative quantum general relativity with the low energy physics of the standard model. In particular, coherent states enable us for the first time to take into account quantum metrics which are excited everywhere in an asymptotically flat spacetime manifold as is needed for semi-classical considerations.The formalism introduced in this paper is immediately applicable also to lattice gauge theory in the presence of a (Minkowski) background structure on a possibly infinite lattice. * thiemann@aei-potsdam.mpg.de

In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mourão and Thiemann to arbitrary, finite, piecewise analytic graphs.However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space.In this paper we complement these results : First, we explicitly derive the complexification of the configuration space underlying these heat kernel coherent states and, secondly, prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation. ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness.These states therefore comprise a candidate family for the semi-classical analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical calculations and therefore set a new starting point for numerical canonical quantum general relativity and gauge theory.The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed. * thiemann@aei-potsdam.mpg.de † winkler@aei-potsdam.mpg.de of constructions of (interacting) quantum field theories from given classical ones one is almost always forced to regularize and renormalize the operators in that theory and these are operations which have no classical counterpart. Thus, it would be no surprise if it turned out that the classical limit of such quantum field theories is not the classical field theory that one started from. Just to give an example, even if one could rigorously show that the continuum limit of lattice QCD exists, to the best of the knowledge of the authors it is at present unclear whether the classical limit of that continuum quantum field theory would give us back classical SU(3) Yang-Mills theory coupled to quarks. This paper is the second one in a series of papers [45,46,47,48,49,50] entitled "Gauge Field Theory Coherent States" which are geared at shedding light at these questions. Specifically, we are interested in the question whether the non-perturbative quantization of continuum Lorentzian general relativity in four dimensions with and without matter advertized in [21,22,24] has the correct classical limit. In fact we eliminate the criticism stated in [43] and show in [44] that quantum general relativity as presently formulate...