 In geometrodynamics, all of the constraint equations are difficult to solve (though the super-Hamiltonian constraint, known as the Wheeler-DeWitt equation, is especially difficult), even in the absence of particular boundary conditions. Lacking solutions, one does not have a grip on what the true, physical states of the theory are, and one cannot hope to make much progress in the way of predictions. The difficulties associated with geometric variables are addressed by the program initiated by Ashtekar and developed by his collaborators (for a review and further references see Rovelli 2001b (Other Internet Resources), 2001a). Ashtekar used a different set of variables, a complexified ‘connection’ (rather than a three-metric) and its canonical conjugate, which made it simpler to solve the constraints. This change of variables introduces an additional constraint into the theory (the Gauss law constraint generating SO(3) transformations) on account of the freedom to rotate the vectors without disturbing the metric. The program underwent further refinements with the introduction of the loop transform, and further refinements still when it was understood that equivalence classes of loops could be identified with spin networks. One is able to recover all the standard geometrical features of general relativity from this formulation. (See Smolin (2001, 2004) for a popular introduction; Rovelli, 2004, offers a physically intuitive account; Thiemann, 2008, provides the mathematical underpinnings; Rickles, 2005, offers a philosophically-oriented review.) Note that the problems of time and observables afflict the loop approach just as they did the earlier geometrodynamical approach. The difference is that one has more (mathematical) control over the theory (and its quantization), in terms of a definable inner product, a separable state space, and more. There is still a question mark over the construction of the full physical Hilbert space, since the solution of the Hamiltonian constraint remains a problem. However, some progress is being made in various directions, e.g. Thomas Thiemann’s master constraint programme (see Thiemann, 2006).