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...

In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mourão and Thiemann.In this paper we establish the "Ehrenfest Property" of these states which are labelled by a point (A, E), a connection and an electric field, in the classical phase space. By this we mean that i) The expectation value of all elementary quantum operatorsÔ with respect to the coherent state with label (A, E) is given to zeroth order inh by the value of the corresponding classical function O evaluated at the phase space point (A, E) and ii) The expectation value of the commutator between two elementary quantum operators [Ô 1 ,Ô 2 ]/(ih) divided by ih with respect to the coherent state with label (A, E) is given to zeroth order inh by the value of the Poisson bracket between the corresponding classical functions {O 1 , O 2 } evaluated at the phase space point (A, E).These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order inh indeed given by classical general relativity.

Linear cosmological perturbation theory is pivotal to a theoretical understanding of current cosmological experimental data provided e.g. by cosmic microwave anisotropy probes. A key issue in that theory is to extract the gauge-invariant degrees of freedom which allow unambiguous comparison between theory and experiment. When one goes beyond first (linear) order, the task of writing the Einstein equations expanded to nth order in terms of quantities that are gauge-invariant up to terms of higher orders becomes highly non-trivial and cumbersome. This fact has prevented progress for instance on the issue of the stability of linear perturbation theory and is a subject of current debate in the literature. In this series of papers we circumvent these difficulties by passing to a manifestly gauge-invariant framework. In other words, we only perturb gauge-invariant, i.e. measurable quantities, rather than gauge variant ones. Thus, gauge invariance is preserved non-perturbatively while we construct the perturbation theory for the equations of motion for the gauge-invariant observables to all orders. In this first paper we develop the general framework which is based on a seminal paper due to Brown and Kuchař as well as the relational formalism due to Rovelli. In the second, companion, paper we apply our general theory to FRW cosmologies and derive the deviations from the standard treatment in linear order. As it turns out, these deviations are negligible in the late universe, thus our theory is in agreement with the standard treatment. However, the real strength of our formalism is that it admits a straightforward and unambiguous, gauge-invariant generalization to higher orders. This will also allow us to settle the stability issue in a future publication.

We summarize a recently proposed concrete programme for investigating the (semi)classical limit of canonical, Lorentzian, continuum quantum general relativity in four spacetime dimensions. The analysis is based on a novel set of coherent states labelled by graphs. These fit neatly together with an Infinite Tensor Product (ITP) extension of the currently used Hilbert space. The ITP construction enables us to give rigorous meaning to the infinite volume (thermodynamic) limit of the theory which has been out of reach so far.

In the canonical approach to Lorentzian Quantum General Relativity in four spacetime dimensions an important step forward has been made by Ashtekar, Isham and Lewandowski some eight years ago through the introduction of a Hilbert space structure which was later proved to be a faithful representation of the canonical commutation and adjointness relations of the quantum field algebra of diffeomorphism invariant gauge field theories by Ashtekar, Lewandowski, Marolf, Mourão and Thiemann.This Hilbert space, together with its generalization due to Baez and Sawin, is appropriate for semi-classical quantum general relativity if the spacetime is spatially compact. In the spatially non-compact case, however, an extension of the Hilbert space is needed in order to approximate metrics that are macroscopically nowhere degenerate.For this purpose, in this paper we apply the theory of the Infinite Tensor Product (ITP) of Hilbert Spaces, developed by von Neumann more than sixty years ago, to Quantum General Relativity. The cardinality of the number of tensor product factors can take the value of any possible Cantor aleph, making this mathematical theory well suited to our problem in which a Hilbert space is attached to each edge of an arbitrarily complicated, generally infinite graph.The new framework opens a pandora's box full of techniques, appropriate to pose fascinating physical questions such as quantum topology change, semi-classical quantum gravity, effective low energy physics etc. from the universal point of view of the ITP. In particular, the study of photons and gravitons propagating on fluctuating quantum spacetimes is now in reach, the topic of the next paper in this series.