Spin foam

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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.

In loop quantum gravity[edit]

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.[how?]

Spin network[edit]

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.[clarification needed] A spin foam is analogous to quantum history.[why?]


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


  • a set of 2-complexes each consisting out of faces , edges and vertices . Associated to each 2-complex is a weight
  • a set of irreducible representations which label the faces and intertwiners which label the edges.
  • a vertex amplitude and an edge amplitude
  • a face amplitude , for which we almost always have

See also[edit]


  1. ^ Perez, Alejandro (2004). "[gr-qc/0409061] Introduction to Loop Quantum Gravity and Spin Foams". arXiv:gr-qc/0409061.
  2. ^ Michael Reisenberger; Carlo Rovelli (1997). "'Sum over surfaces' form of loop quantum gravity". arXiv:gr-qc/9612035.
  3. ^ Jonathan Engle; Roberto Pereira; Carlo Rovelli; Etera Livine (2008). "LQG vertex with finite Immirzi parameter". arXiv:0711.0146 [gr-qc].

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