Complexifier coherent states for quantum general relativity

By T. Thiemann. 2002

163 citations recorded by the source corpus.

Recently, substantial amount of activity in quantum general relativity (QGR) has focused 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 are complexifier coherent states. This unifies all states constructed so far under the general complexifier principle. (B) Ashtekar and Lewandowski suggested a non-Abelian generalization of Varadarajan's states to QGR which, however, are no longer of the complexifier type. We construct a new class of non-Abelian complexifiers which come close to that underlying Varadarajan's construction. (C) Non-Abelian 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.

Subfields: Astronomy and Astrophysics, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9fa1913bdda8bd31ae248853f419f244b2953ab4

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Citations in the corpus

The Spin-Foam Approach to Quantum Gravity (2012, background): “1 Coherent states Coherent states associated with the representation theory of a compact group have been studied by Thiemann and collaborators [357, 355, 342, 360, 361, 362, 358, 50, 51, 177]; see also [76].”
FUNDAMENTAL STRUCTURE OF LOOP QUANTUM GRAVITY (2005, background): “One may find comparisons of the two approaches from both sides [145][21].”
Towards the QFT on curved spacetime limit of QGR: II. A concrete implementation (2002, methodology): “Other states derived by the complexifier method [20] could be used as well, but for the exploratory purposes of this series of papers it is sufficient to stick to those simplest ones.”
Zakopane lectures on loop gravity (2011, background): “Here I describe the “holomorphic" coherent states, developed by a number of people [84, 85, 52, 86, 87] and recently discussed in detail by Bianchi-Magliaro-Perini [55], as well as the “semi coherent" states of Livine-Speziale (LS) [17].”
Twisted geometries: A geometric parametrisation of SU(2) phase space (2010, unknown): “After an early attempt with the weave states [1], an extensive approach to this question has been developed by Thiemann and collaborators [2, 3, 4].”
Quantum gravity, shadow states and quantum mechanics (2002, background): “They will now be discussed in detail and significantly extended in a series of papers, of which this is the first (see also [3, 4]).”
Polyhedra in loop quantum gravity (2010, methodology): “What is the relevance of this geometric construction to the quantum theory? Coherent states for loop quantum gravity have been introduced and extensively studied by Thiemann and collaborators [35, 36, 34].”