Complexifier Coherent States for Quantum General Relativity
Thomas Thiemann
TL;DR
This work ties Varadarajan’s polymer-like, background-dependent states for Maxwell theory to the complexifier coherent-state program in quantum general relativity, showing that Varadarajan states can be derived from a complexifier on the kinematical Hilbert space ${\cal H}_0$ and hence belong to the broad family of complexifier coherent states. It then analyzes the prospects and obstacles of extending these states to non-Abelian gauge groups ${\rm SU}(2)$, finding that naively constructed SU(2) complexifiers either fail to be densely defined or do not yield the correct classical limit, and that shadows or area-based variants do not circumvent fundamental issues with electric flux fluctuations. The authors argue that, within the current kinematical framework, semiclassical analysis favors normalizable, graph-dependent coherent states, and that Dirichlet-Voronoi averaging does not restore semiclassicality for coordinate-dependent holonomies, although it can improve certain geometric observables. They propose a shift in focus toward diffeomorphism-invariant observables (potentially linked to matter) whose semiclassical behavior is well captured by these graph-dependent states, offering a practical path to test the classical limit of QGR under the present formalism. Overall, the work clarifies the relationship between Varadarajan’s constructions and the complexifier approach, highlights the indispensability of graph dependence for semiclassical analysis, and suggests that diffeomorphism-invariant observables provide the most robust arena for semiclassical consistency checks in canonical QGR.
Abstract
Recently, substantial amount of activity in Quantum General Relativity (QGR) has focussed on the semiclassical analysis of the theory. In this paper we want to comment on two such developments: 1) Polymer-like states for Maxwell theory and linearized gravity constructed by Varadarajan which use much of the Hilbert space machinery that has proved useful in QGR and 2) coherent states for QGR, based on the general complexifier method, with built-in semiclassical properties. We show the following: A) Varadarajan's states {\it are} complexifier coherent states. This unifies all states constructed so far under the general complexifier principle. B) Ashtekar and Lewandowski suggested a non-Abelean generalization of Varadarajan's states to QGR which, however, are no longer of the complexifier type. We construct a new class of non-Abelean complexifiers which come close to the one underlying Varadarajan's construction. C) Non-Abelean complexifiers close to Varadarajan's induce new types of Hilbert spaces which do not support the operator algebra of QGR. The analysis suggests that if one sticks to the present kinematical framework of QGR and if kinematical coherent states are at all useful, then normalizable, graph dependent states must be used which are produced by the complexifier method as well. D) Present proposals for states with mildened graph dependence, obtained by performing a graph average, do not approximate well coordinate dependent observables. However, graph dependent states, whether averaged or not, seem to be well suited for the semiclassical analysis of QGR with respect to coordinate independent operators.