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

Complexifier Coherent States for Quantum General Relativity

Recently, substantial amount of activity in Quantum General Relativity (QGR) has focussed on the semiclassical analysis of the theory. In this paper we want to comment on two such developments: 1) Polymer-like states for Maxwell theory and linearized gravity constructed by Varadarajan which use much of the Hilbert space machinery that has proved useful in QGR and 2) coherent states for QGR, based on the general complexifier method, with built-in semiclassical properties. We show the following: A) Varadarajan's states {\it are} complexifier coherent states. This unifies all states constructed so far under the general complexifier principle. B) Ashtekar and Lewandowski suggested a non-Abelean generalization of Varadarajan's states to QGR which, however, are no longer of the complexifier type. We construct a new class of non-Abelean complexifiers which come close to the one underlying Varadarajan's construction. C) Non-Abelean complexifiers close to Varadarajan's induce new types of Hilbert spaces which do not support the operator algebra of QGR. The analysis suggests that if one sticks to the present kinematical framework of QGR and if kinematical coherent states are at all useful, then normalizable, graph dependent states must be used which are produced by the complexifier method as well. D) Present proposals for states with mildened graph dependence, obtained by performing a graph average, do not approximate well coordinate dependent observables. However, graph dependent states, whether averaged or not, seem to be well suited for the semiclassical analysis of QGR with respect to coordinate independent operators.Comment: Latex, 54 p., no figure

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 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

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

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

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

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

The Proca-field in Loop Quantum Gravity

In this paper we investigate the Proca-field in the framework of Loop Quantum Gravity. It turns out that the methods developed there can be applied to the symplectically embedded Proca-field, giving a rigorous, consistent, non-perturbative quantization of the theory. This can be achieved by introducing a scalar field, which has completely different properties than the one used in spontaneous symmetry breaking. The analysis of the kernel of the Hamiltonian suggests that the mass term in the quantum theory has a different role than in the classical theory.Comment: 15 pages. v2: 19 pages, amended sections 2 and 6, references added v3: 20 pages, amended section 6 and minor correction

The large cosmological constant approximation to classical and quantum gravity: model examples

We have recently introduced an approach for studying perturbatively classical and quantum canonical general relativity. The perturbative technique appears to preserve many of the attractive features of the non-perturbative quantization approach based on Ashtekar's new variables and spin networks. With this approach one can find perturbatively classical observables (quantities that have vanishing Poisson brackets with the constraints) and quantum states (states that are annihilated by the quantum constraints). The relative ease with which the technique appears to deal with these traditionally hard problems opens several questions about how relevant the results produced can possibly be. Among the questions is the issue of how useful are results for large values of the cosmological constant and how the approach can deal with several pathologies that are expected to be present in the canonical approach to quantum gravity. With the aim of clarifying these points, and to make our construction as explicit as possible, we study its application in several simple models. We consider Bianchi cosmologies, the asymmetric top, the coupled harmonic oscillators with constant energy density and a simple quantum mechanical system with two Hamiltonian constraints. We find that the technique satisfactorily deals with the pathologies of these models and offers promise for finding (at least some) results even for small values of the cosmological constant. Finally, we briefly sketch how the method would operate in the full four dimensional quantum general relativity case.Comment: 21 pages, RevTex, 2 figures with epsfi