Topics in Canonical Gravity Sergio Inglima September 2012 Abstract In this work we review some important topics in canonical theories of General Relativity. In particular we discuss the theory of constrained systems, the Hamiltonian formulation of GR, the difficulty in constructing gauge invariant objects for GR and recent methods involv- ing relational observables to address this problem and finally we consider modern connection formulations of gravity. Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London. 1 Contents 1 Introduction 4 2 Constrained Hamiltonian Systems 8 2.1 Constrained Systems . 8 2.2 Dirac-Bergmann algorithm & Classification of Constraints . 11 2.3 Definitions of gauge symmetry . 15 2.3.1 Dirac’s point gauge transformations . 15 2.3.2 Gauge transformations as a map from solutions to solutions . 17 2.3.3 Geometry of gauge orbits . 19 2.4 Yang Mills as a constrained Hamiltonian system . 21 3 Hamiltonian Formulation of GR 28 3.1 3+1 analysis of Einstein Hilbert action . 28 3.2 Hamiltonian Analysis . 33 3.3 Constraint Algebra analysis . 37 3.4 Matter coupling in the canonical formalism . 41 3.5 Symmetries, diffeomorphisms & the Dirac algebra . 44 3.5.1 Symmetries in GR . 45 3.5.2 Projectability of Noether symmetries to phase space . 46 3.5.3 Finding a representation of LDiff(M) in canonical gravity . 46 3.6 Asymptotically flat case . 55 4 Dirac Observables in GR 60 4.1 Relational Observables for finite dimensional constrained systems . 61 4.2 Important results concerning complete observables . 63 4.2.1 Approximation scheme for complete observables . 63 4.2.2 Poisson Algebra of Complete Observables . 64 4.2.3 Partially complete observables . 65 4.3 Field theories & GR . 65 4.3.1 Reducing the number of constraints (I) . 67 4.3.2 Reducing the number of constraints (II) . 69 4.4 Deparametrisation of constrained systems . 73 4.4.1 Deparametrisation of GR using matter fields . 75 4.5 Discussion . 78 5 Connection Formalism 80 5.1 Tetrad formalism . 81 5.2 Hilbert Palatini Action . 84 5.3 Ashtekar self-dual Action . 93 5.4 Epilogue . 99 6 Discussion 103 2 Acknowledgments I should firstly like to thank Professor Jonathan Halliwell for his supervision during the prepara- tion of this dissertation. In particular for many helpful suggestions regarding references, comments on early drafts and his expertise in answering questions I had on several challenging papers. Also more generally for his advice and help throughout my time at Imperial. Secondly, I’d like to thank all the lecturers on the Quantum Fields & Fundamental Forces MSc. In particular, Professor Magueijo, Dr Rajantie, Professor Stelle, Professor Waldram and Dr Wiseman for answering many questions I had during their lecture courses on general relativity, advanced fields, unification, QFT/ QED and differential geometry. Finally, I should like to thank many of the students who have helped make my time at Imperial a good one. In particular, I’d like to thank Roland Grinis for discussions on maths and especially Rudolph Kalveks for many interesting and helpful discussions on theoretical physics. 3 1 Introduction In this dissertation we review a number of important topics in classical canonical gravity. The aim has been to provide a pedagogical introduction to these subjects, assuming a background in GR and QFT only. The choice of topics has been motivated by an interest in the area of quantum gravity and a desire to cover the classical background to the canonical quantization of GR, so that understanding the material here might make the jump to the quantum theory more manageable. We stress that none of the material is original, it is a review, but we have tried to present it in a logical way and where possible to include explicit computations to illustrate new ideas and to cover missing steps in the literature. We have also cited many references so that a reader can quickly identify some key papers and gain an appreciation of some of the controversies that have and do still exist in the subject. The subject of quantum gravity has a long history and finding such a theory is one of the outstanding problems in theoretical physics today. There are many research programs working on this problem including string/ M-theory, canonical quantum gravity, causal set theory, causal dynamical triangulations (CDT) and several others, a recent review of several approaches can be found in [1]. We will be covering only the classical background to the canonical quantum gravity approach, which has received new impetus over the last two decades following a reformulation of GR into a theory of connections due to Ashtekar, which we shall discuss in section 5. However, we also discuss the earlier metric formulation of canonical gravity both because it is more familiar and because several problems, e.g. the problem of time, the difficulty in constructing gauge invariant observables and the nature of the constraints, are present in both these approaches. One of the first attempts to quantise GR was to employ the standard methods used for classical electrodynamics, i.e. covariant perturbative field theory. This involves separating the metric into a fixed background part ηµν and a fluctuating field to be quantised, see the review [2]. The resulting QFT suffers from the same infinities that occur in QED but unfortunately the methods of renormalization so successful for QED cannot be applied to GR, the theory is perturbatively non-renormalizable, and this fact was one of the original motivations for the development of supergravity and then the superstring theories. Although GR is perturbatively non-renormalizable, there are arguments that can be made to suggest it is still worthwhile to consider a quantization of GR: i.) the divergent terms in the perturbative expansion occur at arbitrarily high energies (short distances) and assume that the background spacetime is Minkowski- this is a questionable assumption we do not even know whether we will have a smooth manifold structure at the Planck scale, and ii.) GR could be non- pertubatively renormalizable, i.e. there may exist viable approaches where one does not separate the metric into a fixed background part before quantizing - the canonical quantization methods we discuss come under this approach as does CDT, which involves a (covariant) discretization of the Einstein Hilbert action. We should now like to discuss the topics we cover in more detail. In section 2 we provide an introduction to the theory of constrained Hamiltonian systems, this theory developed by Dirac and Bergmann provides the framework for the canonical analysis of singular Lagrangian systems, which includes GR and Yang Mills theories. This framework enables one to perform a Legendre transform and to compute the Hamilton-Dirac equations of motion, which will be subject to constraints. These constraints can be classified into first and second class, and enable one to identify where the gauge freedom, present in the Lagrangian, appears at the canonical level. We further show why first class constraints φi are interpreted as the generators of point gauge transformations on phase space. This interpretation due to Dirac is not the only one, an alternative due to Bergmann is to view gauge transformations as maps that act on phase space 4 trajectories, rather than points, and in this way maintain the idea of gauge transformations as a symmetry, i.e. a map from solutions of the equations of motions to other solutions. In this latter interpretation gauge transformations are generated by a function G(t), which is a particular sum of first class constraints φi, [10]. We then discuss some basic geometric results concerning constrained systems, one finds first class constraints generate surfaces in phase space, which represent the gauge equivalence classes of the system- and these surfaces foliate together to fill the entire constraint surface. First class constraints act twice i.) to reduce the effective phase space dimension and ii.) to reduce the effective dimension of the constraint surface by dividing it into equivalence classes. Second class constraints have no interpretation as gauge generators and they simply reduce the effective dimension of the phase space by restricting the dynamics to a surface in phase space. In gauge theories an important issue is the identification of observable quantities, if one wants a consistent physics then such objects need to be gauge invariant, therefore in the current formalism they need to be invariant under the action of the first class constraints φi, (i.e. they Poisson commute with these constraints). In the case of GR constructing such observables has proven very difficult and only relatively recently have approximation methods been developed, which we shall discuss in section 4. We conclude section 2 by applying this formalism to Yang Mills theory on a Minkowski background, we perform the Legendre transform, identify the constraints, prove that the Poisson algebra of constraints is isomorphic to the Lie algebra of the gauge group and show how to recover the correct number of physical degrees of freedom. In section 3 this formalism is applied to GR, in particular the canonical analysis of the Einstein Hilbert action is described. This is more involved than for Yang Mills, on a fixed background, because to express the Lagrangian in terms of spatial objects, evolving in time, one must assume spacetime is of the form Σ × R, where Σ is a 3 dimensional spatial surface of arbitrary but fixed topology. One can then use a couple of geometric identities to reduce the Lagrangian into an appropriate 3 + 1 form. The result, after performing the canonical analysis and reducing the phase space, is the ADM (Arnowitt Deser Misner) phase space, [13] consisting of canonical ab coordinates: a spatial Riemannian metric qab and momentum P , which is closely related to the extrinsic curvature of the embedding of Σ in spacetime.