Projective Loop Quantum Gravity I. State Space

TL;DR

This paper extends the projective approach to quantum state spaces in loop quantum gravity to arbitrary gauge groups, providing explicit formulas and a more balanced treatment of variables, potentially improving state construction.

Contribution

It generalizes the projective state space construction from Abelian to arbitrary gauge groups, including explicit formulas and a connection to the Ashtekar-Lewandowski space.

Findings

Extended the construction to non-Abelian, non-compact gauge groups.

Derived explicit formulas relating Hilbert spaces for different labels.

Showed the projective space as an extension of the standard Hilbert space.

Abstract

Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski to describe quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces. Beside the physical motivations for this approach, it could help designing a quantum state space holding the states we need. In [Oko{\l}\'ow 2013, arXiv:1304.6330] the description of a theory of Abelian connections within this framework was developed, an important insight being to use building blocks labeled by combinations of edges and surfaces. The present work generalizes this construction to an arbitrary gauge group G (in particular, G is neither assumed to be Abelian nor compact). This involves refining the definition of the label set, as well as deriving explicit formulas to relate the Hilbert spaces attached to different labels. If the gauge group happens to be compact, we also have at our disposal the well-established Ashtekar-Lewandowski Hilbert space, which is defined as an inductive limit using building blocks labeled by edges only. We then show that the quantum state space presented here can be thought as a natural extension of the space of density matrices over this Hilbert space. In addition, it is manifest from the classical counterparts of both formalisms that the projective approach allows for a more balanced treatment of the holonomy and flux variables, so it might pave the way for the development of more satisfactory coherent states.