John Baez, UC Riverside
For some time now, the most important outstanding problem in the loop representation of quantum gravity has been to formulate the Wheeler-DeWitt equation in a rigorous way by making the Hamiltonian constraint into a well-defined operator. Thomas Thiemann recently wrote four papers aimed at solving this problem (gr-qc/9606088, 89, 90, 91) which have caused quite a bit of excitement among those working on the loop representation. In this brief introduction to his work and the history leading up to it, I will not attempt to credit the many people to whose work I allude; detailed references can be found in his papers.
An interesting feature of Thiemann's approach is that while it uses the whole battery of new techniques developed in the loop representation of quantum gravity, in some respects it returns to earlier ideas from geometrodynamics. Recall that in geometrodynamics á la Wheeler and DeWitt, the basic canonically conjugate variables were the 3-metric and extrinsic curvature . The idea was to quantize these, making them into operators acting on wavefunctions on the space of 3-metrics, and then to quantize the Hamiltonian and diffeomorphism constraints and seek wavefunctions annihilated by these quantized constraints. In particular, if H denotes the Hamiltonian constraint, a physical state should satisfy the Wheeler-DeWitt equation
However, this program soon became regarded as dauntingly difficult for various reasons, one being that H is not a polynomial in and : it contains a factor of . Experience had taught field theorists that it is difficult to quantize non-polynomial expressions in the canonically conjugate variables.
In the 1980's Ashtekar found a new formulation of general relativity in which the canonically conjugate variables are a densitized complex triad field and a chiral spin connection , where is built from the Levi-Civita connection of the 3-metric and is built from the extrinsic curvature. As their names suggest, and are analogous to the electric field and vector potential in electromagnetism.
At first glance, in terms of and the Hamiltonian constraint appears polynomial in form. This greatly revived optimism in canonical quantum gravity. However, in this new formalism one is really working with the densitized Hamiltonian constraint , which is related to the original Hamiltonian constraint by . Thus in a sense the original problem has been displaced rather than addressed. It took a while, but it was eventually seen that many of the problems with quantizing can be traced to this fact (or technically speaking, the fact that it has density weight 2).
A more immediately evident problem was that because is complex-valued, the corresponding 3-metric is also complex-valued unless one imposes extra `reality conditions'. The reality conditions are easy to deal with in the Riemannian theory, where the signature of spacetime is taken to be ++++. There one can handle them by working with a real densitized triad field and an connection given by . In the physically important Lorentzian theory, however, no such easy remedy is available.
Despite these problems, the enthusiasm generated by the new variables led to a burst of work on canonical quantum gravity. Many new ideas were developed, most prominently the loop representation. In the Riemannian theory, this gives a perfectly rigorous way to construct the Hilbert space on which the Hamiltonian constraint is supposed to be an operator: the Hilbert space of square-integrable wavefunctions on the space of connections. The idea is to work with graphs embedded in space, and for each such graph to define a Hilbert space of wavefunctions depending only on the holonomies of the connection along the edges of the graph. One then forms the union of all these Hilbert spaces and completes it to obtain the desired Hilbert space .
It turns out has a basis of `spin networks', given by graphs with labellings of the edges by representations of --- i.e., spins --- as well as certain labellings of the vertices. One can quantize various interesting observables such as the area of a surface or the volume of a region of space, obtaining operators on . Moreover, the matrix elements of these operators have been explicitly computed in the spin network basis.
Thiemann's approach applies this machinery to Lorentzian gravity by exploiting the interplay between the Riemannian and Lorentzian theories. As in the Riemannian theory, he takes as his canonically conjugate variables a real densitized triad field and an connection . This automatically deals with the reality conditions. He also takes as his Hilbert space the space as defined above, since it turns out that this space is acceptable for the Lorentzian theory as well as the Riemannian theory. Then, modulo some important subtleties we discuss below, he quantizes the Hamiltonian constraint of Lorentzian gravity to obtain an operator on . Interestingly, it is crucial to his approach that he quantizes H rather than the densitized Hamiltonian constraint . This avoids the regularization problems that plagued attempts to quantize .
How does Thiemann quantize the Hamiltonian constraint? First, in the context of classical general relativity he derives a very clever formula for the Hamiltonian constraint in terms of the Poisson brackets of the connection , its curvature --- analogous to the magnetic field in electromagnetism --- and the total volume V of space. (For simplicity, we assume here that space is compact.) Using the trick of replacing Poisson brackets by commutators, this reduces the problem of quantizing the Hamiltonian constraint to the problem of quantizing , , and V. As noted, V has already been successfully quantized, and the resulting `volume operator' is known quite explicitly. This leaves and .
Now, a fundamental fact about the loop representation --- at least as currently formulated --- is that the connection and curvature do not correspond to well-defined operators on , even if one smears them with test functions in the usual way. Instead, one has operators corresponding to parallel transport along paths in space. Classically we can write a formula for in terms of parallel transport along an infinitesimal open path, and a formula for in terms of parallel transport around an infinitesimal loop. However, in loop representation of the quantum theory one cannot take the limit as the path or loop shrinks to zero length. The best one can do when quantizing and is to choose some paths or loops of finite size and use parallel transport along them to define approximate versions of these operators. This introduces a new kind of ambiguity when quantizing polynomial expressions in and : dependence on arbitrary choices of paths or loops.
So, contrary to the conventional wisdom of old, while the factors of in the Hamiltonian constraint are essential in Thiemann's approach, the polynomial expressions in and introduce problematic ambiguities! In short, Thiemann really constructs a large family of different versions of the Hamiltonian constraint operator, depending on how the choices of paths and loops are made. However, by making these choices according to a careful method developed with the help of Jerzy Lewandowski, the ambiguity is such that two different versions acting on a spin network give spin networks differing only by a diffeomorphism of space. Mathematically speaking we may describe this as follows. Let be the space of finite linear combinations of spin networks, and let be the space of finite linear combinations of spin networks modulo diffeomorphisms. Then Thiemann obtains, for any choice of lapse function N, a smeared Hamiltonian constraint operator
independent of the arbitrary choices he needed in his construction.
Since these operators do not map a space to itself we cannot ask whether they satisfy the naively expected commutation relations, the `Dirac algebra'. However, this should come as no surprise, since the Dirac algebra also involves other operators that are ill-defined in the loop representation, such as the 3-metric . Thiemann does check as far as possible that the consequences one would expect from the Dirac algebra really do hold. Thus if one is troubled by how arbitrary choices of paths and loops prevent one from achieving a representation of the Dirac algebra, one is really troubled by the assumption, built into the loop representation, that , , and are not well-defined operator-valued distributions. Ultimately, the validity of this assumption can only be known through its implications for physics.
Thiemann's approach to quantizing the Hamiltonian constraint is certainly not the only one imaginable within the general framework of the loop representation. (Indeed, his papers actually treat two approaches, one yielding a formally Hermitian operator, the other not.) As soon as his work became understood, discussion began on whether it gives the right physics, or perhaps needs some modification, or perhaps exhibits fundamental problems with the loop representation. The quest for a good theory of quantum gravity is far from over. But at the very least, Thiemann's work overturns some established wisdom and opens up exciting new avenues for research.