The
covariant formulation of loop quantum gravity provides the best formulation of the dynamics of the theory of
quantum gravity – a
quantum field theory where the invariance under
diffeomorphisms of
general relativity applies. The resulting path integral represents a sum over all the possible configurations of spin foam.
Spin network A spin network is a two-dimensional
graph, together with labels on its vertices and edges which encode aspects of a spatial geometry. A spin network is defined as a diagram like the
Feynman diagram which makes a basis of
connections between the elements of a
differentiable manifold for the
Hilbert spaces defined over them, and for computations of amplitudes between two different
hypersurfaces of the
manifold. Any evolution of the spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. A spin foam is analogous to
quantum history.
Spacetime Spin networks provide a language to describe the
quantum geometry of space. Spin foam does the same job for spacetime. Spacetime can be defined as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph, a higher-dimensional complex is used. In
topology this sort of space is called a 2-
complex. A spin foam is a particular type of 2-complex, with labels for
vertices, edges and
faces. The boundary of a spin foam is a spin network, just as in the theory of manifolds, where the boundary of an n-manifold is an (n-1)-manifold. In loop quantum gravity, the present spin foam theory has been inspired by the
Ponzano–Regge model. The idea was introduced by Reisenberger and
Rovelli in 1997, and later developed into the
Barrett–Crane model. The formulation that is used nowadays is commonly called EPRL after the names of the authors of a series of seminal papers, but the theory has also seen fundamental contributions from the work of many others, such as
Laurent Freidel (FK model) and
Jerzy Lewandowski (KKL model). ==Definition==