,
160] by following the steps that Feynman took in defining path
integral quantum mechanics starting from the Schrödinger
canonical theory. More precisely, it was proven in [160] that the matrix elements of the operator
U
(T)
obtained exponentiating the (Euclidean) hamiltonian constraint in the proper time gauge (the operator that generates evolution in proper time) can be expanded in a Feynman sum over paths. In conventional QFT each term of a Feynman sum corresponds naturally to a certain Feynman diagram, namely a set of lines in spacetime meeting at vertices (branching points). A similar natural structure of the terms appears in quantum gravity, but surprisingly the diagrams are now given by surfaces is spacetime that branch at vertices. Thus, one has a formulation of quantum gravity as a sum over surfaces in spacetime. Reisenberger [158] and Baez [30] have argued in the past that such a formulation should exist, and Iwasaki has developed a similar construction in 2+1 dimensions. Intuitively, the time evolution of a spin network in spacetime is given by a colored surface. The surfaces capture the gravitational degrees of freedom. The formulation is ``topological'' in the sense that one must sum over topologically inequivalent surfaces only, and the contribution of each surface depends on its topology only. This contribution is given by the product of elementary ``vertices'', namely points where the surface branches.
The transition amplitude between two s-knot states
and
in a proper time
T
is given by summing over all (branching, colored) surfaces
that are bounded by the two s-knots
and
The weight
of the surface
is given by a product over the
n
vertices
v
of
:
The contribution
of each vertex is given by the matrix elements of the
hamiltonian constraint operator between the two s-knots obtained
by slicing
immediately below and immediately above the vertex. They turn
out to depend only on the colors of the surface components
immediately adjacent the vertex
v
. The sum turns out to be finite and explicitly computable order
by order.
As in the usual Feynman diagrams, the vertices describe the elementary interactions of the theory. In particular, here one sees that the complicated structure of the Thiemann hamiltonian, which makes a node split into three nodes, corresponds to a geometrically very simple vertex. Figure 3 is a picture of the elementary vertex. Notice that it represents nothing but the spacetime evolution of the elementary action of the hamiltonian constraint, given in Figure 2.
An example of a surface in the sum is given in Figure 4.
The sum over surfaces version of loop quantum gravity provides
a link with certain topological quantum field theories and in
particular with the the Crane-Yetter model [71,
72,
73,
74,
75], which admit an extremely similar representation. For a
discussion on the precise relation between topological quantum
field theory and diffeomorphism invariant quantum field theory,
see [160] and [171,
124,
83].
The idea of expressing the theory as a sum over surfaces has
been developed by Baez [33], who has studied the general form of generally covariant
quantum field theories formulated in this manner, and by Smolin
and Markopoulou [144], who have studied how to directly capture the Lorentzian causal
structure of general relativity modifying the elementary
vertices. They have also explored the idea that the long range
correlations of the low energy regime of the theory are related
to the existence of a phase transition in the microscopic
dynamics, and have found intriguing connections with the
theoretical description of percolation.
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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 |
