Reduced Phase Space Quantization and Dirac Observables

In her recent work, Dittrich generalized Rovelli's idea of partial observables to construct Dirac observables for constrained systems to the general case of an arbitrary first class constraint algebra with structure functions rather than structure constants. Here we use this framework and propose a new way for how to implement explicitly a reduced phase space quantization of a given system, at least in principle, without the need to compute the gauge equivalence classes. The degree of practicality of this programme depends on the choice of the partial observables involved. The (multi-fingered) time evolution was shown to correspond to an automorphism on the set of Dirac observables so generated and interesting representations of the latter will be those for which a suitable preferred subgroup is realized unitarily. We sketch how such a programme might look like for General Relativity. We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for General Relativity such that they are spatially diffeomorphism invariant. This has the important consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed Master constraint programme.Comment: 18 pages, no figure

Testing the Master Constraint Programme for Loop Quantum Gravity IV. Free Field Theories

This is the fourth paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. We now move on to free field theories with constraints, namely Maxwell theory and linearized gravity. Since the Master constraint involves squares of constraint operator valued distributions, one has to be very careful in doing that and we will see that the full flexibility of the Master Constraint Programme must be exploited in order to arrive at sensible results.Comment: 23 pages, no figure

Testing the Master Constraint Programme for Loop Quantum Gravity II. Finite Dimensional Systems

This is the second paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we begin with the simplest examples: Finite dimensional models with a finite number of first or second class constraints, Abelean or non -- Abelean, with or without structure functions.Comment: 23 pages, no figure

Quantum Spin Dynamics VIII. The Master Constraint

Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the non -- Lie -- algebra structure of the Dirac algebra of Hamiltonian constraints and was successfully tested in various field theory models. For the case of 3+1 gravity itself, so far only a positive quadratic form for the Master Constraint Operator was derived. In this paper we close this gap and prove that the quadratic form is closable and thus stems from a unique self -- adjoint Master Constraint Operator. The proof rests on a simple feature of the general pattern according to which Hamiltonian constraints in LQG are constructed and thus extends to arbitrary matter coupling and holds for any metric signature. With this result the existence of a physical Hilbert space for LQG is established by standard spectral analysis.Comment: 19p, no figure

Algebraic Quantum Gravity (AQG) III. Semiclassical Perturbation Theory

In the two previous papers of this series we defined a new combinatorical approach to quantum gravity, Algebraic Quantum Gravity (AQG). We showed that AQG reproduces the correct infinitesimal dynamics in the semiclassical limit, provided one incorrectly substitutes the non -- Abelean group SU(2) by the Abelean group $U(1)^3$ in the calculations. The mere reason why that substitution was performed at all is that in the non -- Abelean case the volume operator, pivotal for the definition of the dynamics, is not diagonisable by analytical methods. This, in contrast to the Abelean case, so far prohibited semiclassical computations. In this paper we show why this unjustified substitution nevertheless reproduces the correct physical result: Namely, we introduce for the first time semiclassical perturbation theory within AQG (and LQG) which allows to compute expectation values of interesting operators such as the master constraint as a power series in $\hbar$ with error control. That is, in particular matrix elements of fractional powers of the volume operator can be computed with extremely high precision for sufficiently large power of $\hbar$ in the $\hbar$ expansion. With this new tool, the non -- Abelean calculation, although technically more involved, is then exactly analogous to the Abelean calculation, thus justifying the Abelean analysis in retrospect. The results of this paper turn AQG into a calculational discipline

Testing the Master Constraint Programme for Loop Quantum Gravity V. Interacting Field Theories

This is the final fifth paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. Here we consider interacting quantum field theories, specificlly we consider the non -- Abelean Gauss constraints of Einstein -- Yang -- Mills theory and 2+1 gravity. Interestingly, while Yang -- Mills theory in 4D is not yet rigorously defined as an ordinary (Wightman) quantum field theory on Minkowski space, in background independent quantum field theories such as Loop Quantum Gravity (LQG) this might become possible by working in a new, background independent representation.Comment: 20 pages, no figure

Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models

This is the third paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper: These are systems with an SL(2,\Rl) gauge symmetry and the complications arise because non -- compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the Master Constraint does not contain the point zero. However, the minimum of the spectrum is of order $\hbar^2$ which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to $\hbar$ normal ordering constants). The physical Hilbert space can then be be obtained after subtracting this normal ordering correction.Comment: 33 pages, no figure

A Path-integral for the Master Constraint of Loop Quantum Gravity

In the present paper, we start from the canonical theory of loop quantum gravity and the master constraint programme. The physical inner product is expressed by using the group averaging technique for a single self-adjoint master constraint operator. By the standard technique of skeletonization and the coherent state path-integral, we derive a path-integral formula from the group averaging for the master constraint operator. Our derivation in the present paper suggests there exists a direct link connecting the canonical Loop quantum gravity with a path-integral quantization or a spin-foam model of General Relativity.Comment: 19 page

Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework

Recently the Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Wheeler -- DeWitt constraint equations in terms of a single Master Equation. While the proposal has some promising abstract features, it was until now barely tested in known models. In this series of five papers we fill this gap, thereby adding confidence to the proposal. We consider a wide range of models with increasingly more complicated constraint algebras, beginning with a finite dimensional, Abelean algebra of constraint operators which are linear in the momenta and ending with an infinite dimensional, non-Abelean algebra of constraint operators which closes with structure functions only and which are not even polynomial in the momenta. In all these models we apply the Master Constraint Programme successfully, however, the full flexibility of the method must be exploited in order to complete our task. This shows that the Master Constraint Programme has a wide range of applicability but that there are many, physically interesting subtleties that must be taken care of in doing so. In this first paper we prepare the analysis of our test models by outlining the general framework of the Master Constraint Programme. The models themselves will be studied in the remaining four papers. As a side result we develop the Direct Integral Decomposition (DID) for solving quantum constraints as an alternative to Refined Algebraic Quantization (RAQ).Comment: 42 pages, no figure

On (Cosmological) Singularity Avoidance in Loop Quantum Gravity

Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of the phase space of classical General Relativity to spatially homogeneous situations which is then quantized by the methods of LQG. Thus, LQC is a quantum mechanical toy model (finite number of degrees of freedom) for LQG(a genuine QFT with an infinite number of degrees of freedom) which provides important consistency checks. However, it is a non trivial question whether the predictions of LQC are robust after switching on the inhomogeneous fluctuations present in full LQG. Two of the most spectacular findings of LQC are that 1. the inverse scale factor is bounded from above on zero volume eigenstates which hints at the avoidance of the local curvature singularity and 2. that the Quantum Einstein Equations are non -- singular which hints at the avoidance of the global initial singularity. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in full LQG, if curvature singularity avoidance is realized, then not in this simple way. In fact, it turns out that the boundedness of the inverse scale factor is neither necessary nor sufficient for curvature singularity avoidance and that non -- singular evolution equations are neither necessary nor sufficient for initial singularity avoidance because none of these criteria are formulated in terms of observable quantities.After outlining what would be required, we present the results of a calculation for LQG which could be a first indication that our criteria at least for curvature singularity avoidance are satisfied in LQG.Comment: 34 pages, 16 figure