John Baez, UC Riverside
baez@math.ucr.edu
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.