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

Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order

In our companion paper we identified a complete set of manifestly gauge-invariant observables for general relativity. This was possible by coupling the system of gravity and matter to pressureless dust which plays the role of a dynamically coupled observer. The evolution of those observables is governed by a physical Hamiltonian and we derived the corresponding equations of motion. Linear perturbation theory of those equations of motion around a general exact solution in terms of manifestly gauge invariant perturbations was then developed. In this paper we specialise our previous results to an FRW background which is also a solution of our modified equations of motion. We then compare the resulting equations with those derived in standard cosmological perturbation theory (SCPT). We exhibit the precise relation between our manifestly gauge-invariant perturbations and the linearly gauge-invariant variables in SCPT. We find that our equations of motion can be cast into SCPT form plus corrections. These corrections are the trace that the dust leaves on the system in terms of a conserved energy momentum current density. It turns out that these corrections decay, in fact, in the late universe they are negligible whatever the value of the conserved current. We conclude that the addition of dust which serves as a test observer medium, while implying modifications of Einstein's equations without dust, leads to acceptable agreement with known results, while having the advantage that one now talks about manifestly gauge-invariant, that is measurable, quantities, which can be used even in perturbation theory at higher orders.Comment: 51 pages, no figure

Eigenvalues of the volume operator in loop quantum gravity

We present a simple method to calculate certain sums of the eigenvalues of the volume operator in loop quantum gravity. We derive the asymptotic distribution of the eigenvalues in the classical limit of very large spins which turns out to be of a very simple form. The results can be useful for example in the statistical approach to quantum gravity.Comment: 12 pages, version accepted in Class. Quantum Gra

LTB spacetimes in terms of Dirac observables

The construction of Dirac observables, that is gauge invariant objects, in General Relativity is technically more complicated than in other gauge theories such as the standard model due to its more complicated gauge group which is closely related to the group of spacetime diffeomorphisms. However, the explicit and usually cumbersome expression of Dirac observables in terms of gauge non invariant quantities is irrelevant if their Poisson algebra is sufficiently simple. Precisely that can be achieved by employing the relational formalism and a specific type of matter proposed originally by Brown and Kucha{\v r}, namely pressureless dust fields. Moreover one is able to derive a compact expression for a physical Hamiltonian that drives their physical time evolution. The resulting gauge invariant Hamiltonian system is obtained by Higgs -- ing the dust scalar fields and has an infinite number of conserved charges which force the Goldstone bosons to decouple from the evolution. In previous publications we have shown that explicitly for cosmological perturbations. In this article we analyse the spherically symmetric sector of the theory and it turns out that the solutions are in one--to--one correspondence with the class of Lemaitre--Tolman--Bondi metrics. Therefore the theory is capable of properly describing the whole class of gravitational experiments that rely on the assumption of spherical symmetry.Comment: 29 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

Born--Oppenheimer decomposition for quantum fields on quantum spacetimes

Quantum Field Theory on Curved Spacetime (QFT on CS) is a well established theoretical framework which intuitively should be a an extremely effective description of the quantum nature of matter when propagating on a given background spacetime. If one wants to take care of backreaction effects, then a theory of quantum gravity is needed. It is now widely believed that such a theory should be formulated in a non-perturbative and therefore background independent fashion. Hence, it is a priori a puzzle how a background dependent QFT on CS should emerge as a semiclassical limit out of a background independent quantum gravity theory. In this article we point out that the Born-Oppenheimer decomposition (BOD) of the Hilbert space is ideally suited in order to establish such a link, provided that the Hilbert space representation of the gravitational field algebra satisfies an important condition. If the condition is satisfied, then the framework of QFT on CS can be, in a certain sense, embedded into a theory of quantum gravity. The unique representation of the holonomy-flux algebra underlying Loop Quantum Gravity (LQG) violates that condition. While it is conceivable that the condition on the representation can be relaxed, for convenience in this article we consider a new classical gravitational field algebra and a Hilbert space representation of its restriction to an algebraic graph for which the condition is satisfied. An important question that remains and for which we have only partial answers is how to construct eigenstates of the full gravity-matter Hamiltonian whose BOD is confined to a small neighbourhood of a physically interesting vacuum spacetime

Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity

We perform a canonical, reduced phase space quantisation of General Relativity by Loop Quantum Gravity (LQG) methods. The explicit construction of the reduced phase space is made possible by the combination of 1. the Brown -- Kuchar mechanism in the presence of pressure free dust fields which allows to deparametrise the theory and 2. Rovelli's relational formalism in the extended version developed by Dittrich to construct the algebra of gauge invariant observables. Since the resulting algebra of observables is very simple, one can quantise it using the methods of LQG. Basically, the kinematical Hilbert space of non reduced LQG now becomes a physical Hilbert space and the kinematical results of LQG such as discreteness of spectra of geometrical operators now have physical meaning. The constraints have disappeared, however, the dynamics of the observables is driven by a physical Hamiltonian which is related to the Hamiltonian of the standard model (without dust) and which we quantise in this paper.Comment: 31 pages, no figure

Manifestly Gauge-Invariant General Relativistic Perturbation Theory: I. Foundations

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 n'th 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 Kuchar as well as the realtional 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 generalisation to higher orders. This will also allow us to settle the stability issue in a future publication.Comment: 77 pages, no figure

On the Relation between Operator Constraint --, Master Constraint --, Reduced Phase Space --, and Path Integral Quantisation

Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. In this article we review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantisation and, more specifically, with the Master constraint quantisation for first class constraints. For first class constraints with non trivial structure functions the equivalence can only be established by passing to Abelian(ised) constraints which is always possible locally in phase space. Generically, the correct configuration space path integral measure deviates from the exponential of the Lagrangian action. The corrections are especially severe if the theory suffers from second class secondary constraints. In a companion paper we compute these corrections for the Holst and Plebanski formulations of GR on which current spin foam models are based.Comment: 43 page

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