Abstract
In a gravitational context, canonical methods offer an intuitive picture of the dynamics and simplify an identification of the degrees of freedom. Nevertheless, extracting dynamical information from background independent approaches to quantum gravity is a highly non-trivial challenge. In this thesis, the conundrum of (quantum) gravitational dynamics is approached from
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two different directions by means of new canonical tools. This thesis is accordingly divided into two parts: In the first part, a general canonical formalism for discrete systems featuring a variational action principle is developed which is equivalent to the covariant formulation following directly from the action. This formalism can handle evolving phase spaces and is thus appropriate for describing evolving lattices. Attention will be devoted to a characterization of the constraints, symmetries and degrees of freedom appearing in such discrete systems which, in the case of evolving phase spaces, is time step dependent. The advantage of this formalism is that it does not depend on the particular discretization and, hence, is suitable for coarse graining procedures. This formalism is applicable to discrete mechanics, lattice field theories and discrete gravity models---underlying some approaches to quantum gravity---and, furthermore, may prove useful for numerical implementations. For concreteness, these new tools are employed to formulate Regge Calculus canonically as a theory of the dynamics of discrete hypersurfaces in discrete spacetimes, thereby removing a longstanding obstacle to connecting covariant simplicial gravity models with canonical frameworks. This result is interesting in view of several background independent approaches to quantum gravity. In addition, perturbative expansions around symmetric background solutions of Regge Calculus are studied up to second order. Background gauge modes generically become propagating at second order as a consequence of a symmetry breaking. In the second part of this thesis, the paradigm of relational dynamics is considered. Dynamical observables in gravity are relational. Unfortunately, their construction and evaluation is notoriously difficult, especially in the quantum theory. An effective canonical framework is devised which permits to evaluate the semiclassical relational dynamics of constrained quantum systems by sidestepping technical problems associated with explicit constructions of physical Hilbert spaces. This effective approach is well-geared for addressing the concept of relational evolution in general quantum cosmological models since it (i) allows to depart from idealized relational `clock references’ and, instead, to employ generic degrees of freedom as imperfect relational `clocks’, (ii) enables one to systematically switch between different such `clocks’ and (iii) yields a consistent (temporally) local time evolution with transient observables so long as semiclassicality holds. These techniques are illustrated by toy models and, finally, are applied to a non-integrable cosmological model. It is argued that relational evolution is generically only a transient and semiclassical phenomenon
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