Projective State Spaces for Theories of Connections
Suzanne Lanéry · OPUS FAU (Kooperativer Bibliotheksverbund Berlin-Brandenburg (KOBV), on behalf of the Universitätsbibliothek Erlangen-Nürnberg) · 2015
Quantum gravity aims at combining the insights of Quantum Field Theory (QFT) and General Relativity (GR) into a consistent fundamental theory. In a non-perturbative, canonical approach like Loop Quantum Gravity (LQG), one attempts to directly quantize the spacetime geometry, while resolutely avoiding the introduction of any supporting background metric. The price for this background independence is that the technologies commonly employed in QFT to extract physics in a computationally tractable way are not readily available. Thus, new tools need to be developed, for example to check the semi-classical limit of the quantum theory or to rigorously derive its cosmological and astrophysical implications. In particular, the vacuum states of Fock type used when doing QFT on Minkowski background have no equivalent when spacetime itself is to be quantized. While the non-standard properties of the Ashtekar-Lewandowski vacuum used in LQG lead to a compelling picture at Planck scale, revealing notably a fundamental discreteness of quantum geometry, they also contribute to long-standing issues regarding the design of semi-classical states and the implementation of the dynamics. This motivates the search for an extension of the LQG Hilbert space, that could accommodate more general quantum states while retaining the key insights of the original construction. This is achieved in the present work using a projective framework, which allows to dispense altogether from the selection of a vacuum state: instead of defining states as vectors in one `big' Hilbert space, or more generally as density matrices thereon, one constructs them as projective families of partial density matrices over a system of `small' Hilbert spaces, each of which extracts specific degrees of freedom from the full quantum theory. This approach is physically motivated by interpreting each small Hilbert space as the arena to describe a given experiment, while the projections binding these partial descriptions together ensure the overall consistency of the theory. We will set up projective state spaces of this kind for general theories of connections: this includes the reformulation of GR using Ashtekar variables that constitutes the starting point of LQG, and could have applications to other quantum gauge theories as well. To this intend, we will develop the projective framework beyond the context of linear configuration spaces in which it was originally formulated, laying down fairly generic prescriptions to turn classical projective system into quantum ones. To ascertain that the thus obtained quantum state spaces indeed extend existing ones, we will investigate in detail their relations with various Hilbert spaces. Finally, we will explore how this approach could help making progress on the aforementioned issues, in particular by paving the way for the development of more satisfactory semi-classical states.