Spin foam
Encyclopedia
In physics
, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral (functional integration
) description of quantum gravity
. It is closely related to loop quantum gravity
.
of general relativity
at the Planck scale
. Any path integral formulation of the theory can be written in the form of a spin foam model, such as the Barrett-Crane model
. A spin network
is defined as a diagram (like the Feynman diagram
) that makes a basis of connections between the elements of a differentiable manifold
for the Hilbert spaces defined over them. Spin networks provide a representation for computations of amplitudes between two different hypersurface
s of the manifold
. Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. A spin foam may be viewed as a quantum history
.
of space. Spin foam does the same job on spacetime. A spin network is a one-dimensional graph
, together with labels on its vertices and edges which encodes aspects of a spatial geometry.
Spacetime is considered as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph we use a higher-dimensional complex. In topology
this sort of space is called a 2-complex. A spin foam is a particular type of 2-complex, together 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.
The concept of a spin foam, although not called that at the time, was introduced in the paper "A Step Toward Pregeometry I: Ponzano-Regge Spin Networks and the Origin of Spacetime Structure in Four Dimensions" by Norman J. LaFave. In this paper, the concept of creating sandwiches of 4-geometry (and local time scale) from spin networks is described, along with the connection of these spin 4-geometry sandwiches to form paths of spin networks connecting given spin-network boundaries (spin foams). Quantization of the structure leads to a generalized Feynman path integral over connected paths of spin networks between spin-network boundaries. This paper goes beyond much of the later work by showing how 4-geometry is already present in the seemingly three dimensional spin networks, how local time scales occur, and how the field equations and conservation laws are generated by simple consistency requirements.
with:
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral (functional integration
Functional integration
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions...
) description of quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
. It is closely related to loop quantum gravity
Loop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
.
Spin foam in loop quantum gravity
In loop quantum gravity there are some results from a possible canonical quantizationCanonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
at the Planck scale
Planck scale
In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...
. Any path integral formulation of the theory can be written in the form of a spin foam model, such as the Barrett-Crane model
Barrett-Crane model
The Barrett–Crane model is a model in quantum gravity which was defined using the Plebanski action.The B field in the action is supposed to be a so-valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term...
. A spin network
Spin network
In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of...
is defined as a diagram (like the Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
) that makes a basis of connections between the elements of a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
for the Hilbert spaces defined over them. Spin networks provide a representation for computations of amplitudes between two different hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
s of the manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. A spin foam may be viewed as a quantum history
Quantum history
In some physics theories, choosing a 3-geometry via spin network and 4-geometry via spin foam, any superposition of spin networks is called a kinematical state and any superposition of spin foams is called a quantum history....
.
The idea
Spin networks provide a language to describe quantum geometryQuantum geometry
In theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales...
of space. Spin foam does the same job on spacetime. A spin network is a one-dimensional graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
, together with labels on its vertices and edges which encodes aspects of a spatial geometry.
Spacetime is considered as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph we use a higher-dimensional complex. In topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
this sort of space is called a 2-complex. A spin foam is a particular type of 2-complex, together with labels for vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
, edges and faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
. 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.
The concept of a spin foam, although not called that at the time, was introduced in the paper "A Step Toward Pregeometry I: Ponzano-Regge Spin Networks and the Origin of Spacetime Structure in Four Dimensions" by Norman J. LaFave. In this paper, the concept of creating sandwiches of 4-geometry (and local time scale) from spin networks is described, along with the connection of these spin 4-geometry sandwiches to form paths of spin networks connecting given spin-network boundaries (spin foams). Quantization of the structure leads to a generalized Feynman path integral over connected paths of spin networks between spin-network boundaries. This paper goes beyond much of the later work by showing how 4-geometry is already present in the seemingly three dimensional spin networks, how local time scales occur, and how the field equations and conservation laws are generated by simple consistency requirements.
Definition
A spin foam model is:with:
- a set of two complexes
each consisting out of faces 
, edges 
and vertices 
. Associated to each two complexes 
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 