Working Group VI:

Classical Limit

Members: K. Mecke, K.-H. Neeb, T. Thiemann

The analysis of the classical limit of Quantum Geometry has several aspects to it. If one chooses an approach based on minimal uncertainty vectors, then one first has to decide for which degrees of freedom this property should hold. Secondly, one has to provide an explicit construction for these minimal uncertainty vectors. Finally, one may wish to sample over coherent vectors for different choices of degrees of freedom in order to achieve the minimal uncertainty property for the maximal number of degrees of freedom and to improve on the degree of symmetry of the resulting object which then becomes a ``coherent density matrix''. Similar techniques are also employed in the representation theory of infinite dimensional Lie groups [32].

A concrete proposal addressing the first two aspects within the context of Loop Quantum Gravity already exists [33]. Here the degrees of freedom for which the fluctuations are controlled are based on a polyhedral complex on the spatial slices together with an associated dual graph respectively on whose faces and edges respectively the electric and magnetic degrees of freedom are probed. That one cannot directly control all degrees of freedom in the representation underlying Loop Quantum Gravity is again due to the non-separability of the Hilbert space. Next one employs a projective limit generalisation of the heat kernel coherent states on cotangent bundles over compact Lie groups invented in [34]. The semiclassical properties of these coherent vectors have been confirmed.

There is much room for improvement. First of all, if we manage to find a representation with improved separability properties then one can control the fluctuations of all degrees of freedom simultaneously with coherent vectors. Next, the method of [34] allows for a natural generalisation outlined in [33] which touches on largely unexplored mathematical disciplines such as complex (Kaehler) geometry on projective limits and Segal-Bargmann type representations [35] based thereon. This freedom could be used in order to encode better dynamical stability properties of the resulting coherent vectors and to also treat the case of non compact Lie groups. Moreover, one should take an integral or sum of the associated pure coherent states in order to obtain a coherent density matrix. An idea [36] for doing this while implementing the symmetry of the geometry to be approximated employs methods from Stochastic Geometry, specifically the Dirichlet-Voronoi construction [4] and Integral Geometry [37], [38] which are widely used in Statistical Physics [39]. Roughly speaking, the sampling over coherent states is based on a random process that generates the corresponding polyhedral complexes as well as a weight that depends on the geometry to be approximated. It is of ultimate interest to implement this idea with all mathematical details and to study the semiclassical properties of this construction as well as of possible variations thereof. Finally, the available constructions in the literature are kinematical in nature as they do not yield semiclassical states satisfying the constraints in the strong or weak* operator topology, only in the sense of expectation values. In order to restrict the freedom in the choice of the semiclassical states and thus to improve on their predictability, the missing dynamical input should enter the construction.

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