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Oct17-03, 07:23 PM
P: 97
This thread seems about searching for the way to connect LQG with the low energy world by a correspondence between quantum states as they are expressed in LQG and QFT. The most recent paper I could find which bears on this directly (The specific passage which makes me think this is in bold) and which gives a view that may be more current than previously mentioned papers by Ashtekar et al and Thiemann is another one by Thiemann dated May 1st 2003:

"Seven years ago, for the first time a mathematically well-defined Hamiltonian con-straint operator has been proposed for LQG which is a candidate for the definition of the quantum dynamics of the gravitational field and all known (standard model) matter.

Despite this success, three papers were published which criticized the proposal by doubting the correctness of the classical limit of the Hamiltonian constraint operator. In broad terms, what these papers point out is that while the algebra of commutators among smeared Hamiltonian constraint operators does not not lead to inconsistencies, it does not manifestly reproduce the classical Poisson algebra among the smeared Hamiltonian constraint

While the arguments put forward are inconclusive (e.g. the direct translation of the techniques used in the full theory work extremely well in Loop Quantum Cosmology)

[i]these three papers raised a serious issue and presumably discouraged almost all researchers in the field to work on an improvement of these questions. In fact, except for two papers there has been no publication on possible modifications of the Hamiltonian constraint proposed. Rather, the combination of with path integral techniques and ideas from topological quantum field theory gave rise to the so-called spin foam reformulation of LQG. Most of the activity in LQG over the past five years has focussed on spin foam models, partly because the hope was that spin foam models, which are defined rather independently of the Hamiltonian framework, circumvent the potential problems pointed out. However, the problem reappears as was shown in recent contributions which seem to indicate that the whole virtue of the spin foam formulation, its manifestly covariant
character, does not survive quantization.

One way out could be to look at constraint quantization from an entirely new point of view which proves useful also in discrete formulations of classical GR, that is, numerical GR. While being a fascinating possibility, such a procedure would be a rather drastic step in the sense that it would render most results of LQG obtained so far obsolete.

In this paper we propose a new, more modest, method to cut the Gordic Knot which we will describe in detail in what follows. Namely we introduce the Phoenix Project which aims at reviving interest in the quantization of the Hamiltonian Constraint. However, before the reader proceeds we would like to express a word of warning. So far this is really only a proposal. While there are many promising features as we will see, many mathematical issues, mostly functional analytic in nature, are not yet worked out com-pletely. Moreover, the proposal is, to the best of our knowledge, completely new and thus has been barely tested in solvable models. Hence, there might be possible pitfalls which we are simply unaware of at present and which turn the whole programme obsolete."

Whatever the "recent trends" are in terms of applications like to cosmology etc., to me this indicates that LQG really is at a crisis point. Maybe this is the real reason for the upcoming LQG - m theory conference. What do you guys think?