Modern Canonical Quantum General Relativity

Abstract

Introduction: Defining quantum gravity 1 Why quantum gravity in the twenty-first century? 1 The role of background independence 8 Approaches to quantum gravity 11 Motivation for canonical quantum general relativity Outline of the book I CLASSICAL FOUNDATIONS, INTERPRETATION AND THE CANONICAL QUANTISATION PROGRAMME 1 Classical Hamiltonian formulation of General Relativity 1.1 The ADM action 1.2 Legendre transform and Dirac analysis of constraints 1.3 Geometrical interpretation of the gauge transformations 1.4 Relation between the four-dimensional diffeomorphism group and the transformations generated by the constraints 1.5 Boundary conditions, gauge transformations and symmetries 1.5.1 Boundary conditions 1.5.2 Symmetries and gauge transformations 65 2 The problem of time, locality and the interpretation of quantum mechanics 2.1 The classical problem of time: Dirac observables 2.2 Partial and complete observables for general constrained systems 2.2.1 Partial and weak complete observables 82 2.2.2 Poisson algebra of Dirac observables 2.2.3 Evolving constants 2.2.4 Reduced phase space quantisation of the algebra of Dirac observables and unitary implementation of the multi-fingered time evolution 2.3 Recovery of locality in General Relativity x Contents 2.4 Quantum problem of time: physical inner product and interpretation of quantum mechanics 95 2.4.1 Physical inner product 95 2.4.2 Interpretation of quantum mechanics 98 3 The programme of canonical quantisation 107 3.1 The programme 4 The new canonical variables of Ashtelcar for General Relativity 118 4.1 Historical overview 118 4.2 Derivation of Ashtekar's variables 123 4.2.1 Extension of the ADM phase space 123 4.2.2 Canonical transformation on the extended phase space 126 II FOUNDATIONS OF MODERN CANONICAL QUANTUM GENERAL RELATIVITY

Semiclassical regime of Regge calculus and spin foams

“…For this we first specify a set of values for the boundary lengths, L i = L 0 i . When used as Dirichlet boundary conditions in the equations of motion (10), there may be for the internal lengths either (a) no solution, (b) a unique solution, (c) a discrete set of solutions, or (d) a continuum of solutions. We are not interested in the case (a), because our goal is to write a state that peaks us on a classical solution; let us therefore assume that we choose {L 0 i } so there is at least one solution.…”

Black hole entropy from loop quantum gravity in higher dimensions

“…In order to be able to quantize them, we will select a maximally commuting subset adapted to the puncturing spin network in the next section, that is, we perform gauge unfixing [28]. This subset does however not form a closing algebra with the Gauß constraints, since the smearing functions Λ IJ are not constant on H. We thus further restrict to constant Λ IJ on H and note that also in the U(1) framework in 3 + 1 dimensions [15,24], the restriction to constant Λ IJ on H becomes necessary, however for different reasons [7].…”

Loop quantum modified gravity and its cosmological application

“…It is worth noting that the spatial geometric operator, such as the area [32] , the volume [33] and the length [34,35] operators, are still valid in H gr kin of quantum STT. As in LQG, it is natural to promote the Gaussian constraint G(Λ) as a well-defined operator [2,4]. Then it's kernel is the internal gauge invariant Hilbert space H G with gauge invariant spin-scalar-network basis.…”

“…As a background independent approach to quantize general relativity (GR), Loop quantum gravity(LQG) has been widely investigated in past 25 years [1][2][3][4]. Recently, this non-perturbative loop quantization procedure has been generalized to the metric f (R) theories, [5,6] Brsns-Dicke theory [7] and scalar-tensor theories [8].…”