is a ``huge'' non-separable space. It is far ``too large'' for a
quantum field theory. However, most of this redundancy is gauge,
and disappears when one solves the diffeomorphism constraint,
defining the diff-invariant Hilbert space
. This is the reason for which the loop representation, as
defined here, is of great value in diffeomorphism invariant
theories only.
The second reason is that
turns out to have a natural basis labeled by knots. More
precisely by ``s-knots''. An s-knot
s
is an equivalence class of spin networks
S
under diffeomorphisms. An s-knot is characterized by its
``abstract'' graph (defined only by the adjacency relations
between links and nodes), by the coloring, and by its knotting
and linking properties, as in knot-theory. Thus, the physical
quantum states of the gravitational field turn out to be
essentially classified by knot theory.
There are various equivalent ways of obtaining
from
. One can use regularization techniques for defining the quantum
operator corresponding to the classical diffeomorphism constraint
in terms of elementary loop operators, and then find the kernel
of such operator. Equivalently, one can factor
by the natural action of the diffeomorphism group that it
carries. Namely
There are several rigorous ways for defining the quotient of a Hilbert space by the unitary action of a group. See in particular the construction in [18], which follows the ideas of Marolf and Higuchi [145, 147, 148, 108].
In the quantum gravity literature, a big deal has been made of
the problem that a scalar product is not defined on the space of
solutions of a constraint
, defined on a Hilbert space
. This, however, is a false problem. It is true that if zero is
in the continuum spectrum of
, then the corresponding eigenstates are generalized states and
the
scalar product is not defined between them. But the generalized
eigenspaces of
, including the kernel, inherit
nevertheless
a scalar product from
. This can be seen in a variety of equivalent ways. For
instance, it can be seen from the following theorem. If
is self adjoint, then there exist a measure
on its spectrum and a family of
Hilbert
spaces
such that
where the integral is the continuous sum of Hilbert spaces
described, for instance, in [101]. Clearly
is the kernel of
equipped with a scalar product.
This is discussed, for instance, in [162].
There are two distinct possibilities for factoring away the diffeomorphisms in the quantum theory, yielding two distinct version of the theory. The first possibility is to factor away smooth transformations of the manifold. In doing so, finite dimensional moduli spaces associated with high valence nodes appear [98], so that the resulting Hilbert space is still non-separable. The physical relevance of these moduli parameters is unclear at this stage, since they do not seem to play any role in the quantum theory. Alternatively, one can consistently factor away continuous transformations of the manifold. This possibility has been explored by Zapata in [215, 216], and seems to lead to a consistent theory free of the residual non separability.
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Loop Quantum Gravity
Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |