In physics, the topological structure of spinfoam or spin foam[1] consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structures are employed in loop quantum gravity as a version of quantum foam.
See main article: Loop quantum gravity.
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.
See main article: 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.
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 work of Ponzano–Regge model. The idea was introduced by Reisenberger and Rovelli in 1997,[2] 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,[3] 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).
The summary partition function for a spin foam model is
Z:=\sum\Gammaw(\Gamma)\left[
| \sum | |
| jf,ie |
\prodfAf(jf)\prodeAe(jf,ie)\prodvAv(jf,ie)\right]
with:
\Gamma
f
e
v
\Gamma
w(\Gamma)
j
i
Av(jf,ie)
Ae(jf,ie)
Af(jf)
Af(jf)=\dim(jf)