Gauge field theory coherent states (GCS): II. Peakedness properties

Abstract

In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mourão and Thiemann to arbitrary, finite, piecewise analytic graphs.However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space.In this paper we complement these results : First, we explicitly derive the complexification of the configuration space underlying these heat kernel coherent states and, secondly, prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation. ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness.These states therefore comprise a candidate family for the semi-classical analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical calculations and therefore set a new starting point for numerical canonical quantum general relativity and gauge theory.The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed. * thiemann@aei-potsdam.mpg.de † winkler@aei-potsdam.mpg.de of constructions of (interacting) quantum field theories from given classical ones one is almost always forced to regularize and renormalize the operators in that theory and these are operations which have no classical counterpart. Thus, it would be no surprise if it turned out that the classical limit of such quantum field theories is not the classical field theory that one started from. Just to give an example, even if one could rigorously show that the continuum limit of lattice QCD exists, to the best of the knowledge of the authors it is at present unclear whether the classical limit of that continuum quantum field theory would give us back classical SU(3) Yang-Mills theory coupled to quarks. This paper is the second one in a series of papers [45,46,47,48,49,50] entitled "Gauge Field Theory Coherent States" which are geared at shedding light at these questions. Specifically, we are interested in the question whether the non-perturbative quantization of continuum Lorentzian general relativity in four dimensions with and without matter advertized in [21,22,24] has the correct classical limit. In fact we eliminate the criticism stated in [43] and show in [44] that quantum general relativity as presently formulate...

Gauge field theory coherent states (GCS): III. Ehrenfest theorems

“…It is closely connected with the companion paper [41]. In [41] we established peakedness properties of the coherent states of the heat kernel family introduced by Hall [46] for arbitrary compact gauge groups which were later applied to gauge field theories in [47].…”

“…To settle this issue we have launched an attack based on coherent states which has culminated in a series of papers called "Gauge Field Theory Coherent States" [40,41,42,43,44,45] and this paper is the third one of this collection (to be continued). It is closely connected with the companion paper [41].…”

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

“…In simple toy examples, such as the harmonic oscillator, it has been shown that the LQG Hilbert space indeed admits states (complicated linear superpositions) whose properties are close to those of the usual Fock space coherent states [19]. In full (3+1)-dimensional LQG, the classical limit is, however, far from understood (so far only kinematical coherent states are known [20][21][22][23][24][25], i.e. states which do not satisfy the quantum constraints).…”

“…This 'weak closure' should hold for all states |X in a restricted habitat of states that are 'naturally' expected to be annihilated by the r.h.s. of (21), and that are subject to the further requirement that the Hamiltonian can be applied twice without leaving the 'habitat'. The latter condition is, for instance, met by the 'vertex smooth' states of [33].…”

“…The same conclusion was already arrived at in an earlier computation of the constraint algebra in [30,37], which was done from a different perspective (no 'habitats'), and makes essential use of the space of diffeomorphism invariant states H diff , the 'natural' kernel of the r.h.s. of (21). Here the idea is to verify that [30,37] 6 For reasons of space, we here restrict attention to the bracket between two Hamiltonian constraints, because the remainder of the algebra involving the kinematical constraints is relatively straightforward to implement.…”