Bianca Dittrich

Partial and Complete Observables for Canonical General Relativity

In this work we will consider the concepts of partial and complete observables for canonical general relativity. These concepts provide a method to calculate Dirac observables. The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint. For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints. It will turn out that these can be made to be Abelian. Furthermore the methods outlined here provide a connection between observables in the spacetime picture, i.e. quantities invariant under spacetime diffeomorphisms, and Dirac observables in the canonical picture.

Partial and complete observables for Hamiltonian constrained systems

We will pick up the concepts of partial and complete observables introduced by Rovelli in Conceptional Problems in Quantum Gravity, Birkhäuser, Boston (1991); Class Quant Grav, 8:1895 (1991); Phys Rev, D65:124013 (2002); Quantum Gravity, Cambridge University Press, Cambridge (2007) in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kuchař’s Bubble-Time Formalism (J Math Phys, 13:768, 1972). Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.

Phase space descriptions for simplicial 4D geometries

Starting from the canonical phase space for discretised (4d) BF–theory, we implement a canonical version of the simplicity constraints and construct phase spaces for simplicial geometries. Our construction allows us to study the connection between different versions of Regge calculus and approaches using connection variables, such as loop quantum gravity. We find that on a fixed triangulation the (gauge invariant) phase space associated to loop quantum gravity is genuinely larger than the one for length and even area Regge calculus. Rather, it corresponds to the phase space of area–angle Regge calculus, as defined in [1] (prior to the imposition of gluing constraints, which ensure the metricity of the triangulation). Finally, we show that for a subclass of triangulations one can construct first class Hamiltonian and Diffeomorphism constraints leading to flat 4d space–times.

Area–angle variables for general relativity

We introduce a modified Regge calculus for general relativity on a triangulated four-dimensional Riemannian manifold where the fundamental variables are areas and a certain class of angles. These variables satisfy constraints which are local in the triangulation. We expect the formulation to have applications to classical discrete gravity and non-perturbative approaches to quantum gravity.

Improved and Perfect Actions in Discrete Gravity

We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they - although being defined on a triangulation - reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of General Relativity. We construct the perfect action in three dimensions with cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge Calculus with curved simplices, which arises naturally in this context.

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, Abelian algebra of constraint operators which are linear in the momenta and ending with an infinite-dimensional, non-Abelian 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 particular, as we will see, that we can possibly construct a master constraint operator for a nonlinear, that is, interacting quantum field theory underlines the strength of the background-independent formulation of LQG. 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) programme for solving quantum constraints as an alternative to refined algebraic quantization (RAQ).

Broken) Gauge Symmetries and Constraints in Regge Calculus

We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore, we derive a canonical formulation that exactly matches the dynamics and hence symmetries of the covariant picture. In this canonical formulation broken symmetries lead to the replacements of constraints by so-called pseudo constraints. These considerations should be taken into account in attempts to connect spin foam models, based on the Regge action, with canonical loop quantum gravity, which aims at implementing proper constraints. We will argue that the long-standing problem of finding a consistent constraint algebra for discretized gravity theories is equivalent to the problem of finding an action with exact diffeomorphism symmetries. Finally, we will analyze different limits in which the pseudo constraints might turn into proper constraints. This could be helpful to infer alternative discretization schemes in which the symmetries are not broken.

Are the spectra of geometrical operators in Loop Quantum Gravity really discrete

One of the celebrated results of Loop Quantum Gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area and volume operators. This is an indication that Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present, because the afore mentioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli’s partial and complete observables provides a possible mechanism for turning a non gauge invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that “fundamental discreteness at Planck scale in LQG” is far from established. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.

Coarse graining methods for spin net and spin foam models

We undertake first steps in making a class of discrete models of quantum gravity, spin foams, accessible to a large-scale analysis by numerical and computational methods. In particular, we apply the Migdal?Kadanoff and tensor network renormalization (TNR) schemes to spin net and spin foam models based on finite Abelian groups and introduce ?cutoff models? to probe the fate of gauge symmetries under various such approximated renormalization group flows. For the TNR analysis, a new Gau??constraint preserving algorithm is introduced to improve numerical stability and aid physical interpretation. We also describe the fixed point structure and establish the equivalence of certain models.

From the discrete to the continuous - towards a cylindrically consistent dynamics

Discrete models usually represent approximations to continuum physics. Cylindrical consistency provides a framework in which discretizations exactly mirror the continuum limit. As a standard tool for the kinematics of loop quantum gravity, we propose a coarse-graining procedure that aims at constructing a cylindrically consistent dynamics in the form of transition amplitudes and Hamiltons principal functions. The coarse-graining procedure, which is motivated by tensor network renormalization methods, provides a systematic approximation scheme for this purpose. A crucial role in this coarse-graining scheme is played by the embedding maps that allow interpretation of discrete boundary data as continuum configurations. These embedding maps should be selected according to the dynamics of the system, as the choice of embedding maps will determine the truncation of the renormalization flow.