Projective loop quantum gravity. I. State space

Abstract

Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski [14] 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 [24] 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.

A modification of the projective construction of quantum states for field theories

“…In the original paper [1] this idea was applied to linear phase spaces. Further development of the projective construction presented in [3,4,5,6,7] consisted in applying this idea to more and more general phase spaces including finally the one underlying Loop Quantum Gravity (LQG).…”

“…The gauge group of these variables is SL(2, C) and because of its non-compactness no one so far has been able to construct any acceptable quantum state space for this formulation of GR. The Hilbert space used in LQG was obtained in [17] by breaking the SL(2, C) symmetry to the SU (2) one (which amounts to breaking the Lorentz symmetry of GR to that of three-dimensional rotations) and in [7] Lanéry and Thiemann constructed their projective quantum states also for LQG with the SU (2) symmetry. A wish to construct projective quantum states for LQG with the SL(2, C) symmetry was the main motivation for the paper [3].…”

“…The second partial solution to the problem of "too small" spaces of projective quantum states was presented by Lanéry and Thiemann in [7]-they proved that if a projective family {D λ , π λλ ′ } λ∈Λ is constructed on the basis of so called holonomy-flux algebra for a theory of connections with a compact structure group, then its projective limit is "sufficiently large". However, the only theory of this sort among those mentioned above is LQG with the SU (2) symmetry.…”

“…The notion of family of factorized Hilbert spaces was introduced in [6] and the application of the projective construction to LQG with the SU (2) symmetry presented in [7] is based on this notion. However, as we will show in Appendix A, spaces of projective quantum states constructed in the earlier papers [1,4,5] can also be seen as derived from families of factorized Hilbert spaces.…”

Fusion basis for lattice gauge theory and loop quantum gravity

“…Now, one can ask what kind of transformations would preserve the symplectic structure (6.5) hence keeping the interpretation of the variables as holonomies and fluxes intact. Examples for such transformations are constructed in detail in [14,38,78]. We review the construction in [14,38] shortly, as it is closely related to ribbon operators.…”