In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, 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 integrationFunctional 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 gravityQuantum gravity is the field of theoretical physics attempting to unify quantum mechanics with general relativity in a self-consistent manner, or more precisely, to formulate a self-consistent theory which reduces to ordinary quantum mechanics in the limit of weak gravity and which reduces to...
. It is closely related to
loop quantum gravityLoop 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...
.
In
loop quantum gravityLoop 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...
there are some results from a possible
canonical quantizationIn physics, canonical quantization is one of many procedures for quantizing a classical theory. Historically, this was the earliest method to be used to build quantum mechanics. When applied to a classical field theory it is also called second quantization...
of
general relativityGeneral 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. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
at the
Planck scaleIn particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1028 eV 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 modelThe Barrett-Crane model is a model in quantum gravity which was defined using the Plebanski action. The field in the action is supposed to be a -valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term...
.
In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, 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 integrationFunctional 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 gravityQuantum gravity is the field of theoretical physics attempting to unify quantum mechanics with general relativity in a self-consistent manner, or more precisely, to formulate a self-consistent theory which reduces to ordinary quantum mechanics in the limit of weak gravity and which reduces to...
. It is closely related to
loop quantum gravityLoop 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 gravityLoop 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...
there are some results from a possible
canonical quantizationIn physics, canonical quantization is one of many procedures for quantizing a classical theory. Historically, this was the earliest method to be used to build quantum mechanics. When applied to a classical field theory it is also called second quantization...
of
general relativityGeneral 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. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
at the
Planck scaleIn particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1028 eV 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 modelThe Barrett-Crane model is a model in quantum gravity which was defined using the Plebanski action. The field in the action is supposed to be a -valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term...
. A
spin networkIn 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
Feynman diagramIn quantum field theory a Feynman diagram is an intuitive graphical representation of a contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory....
) which make a basis of connections between the elements of a
differentiable manifoldA differentiable manifold is a type of manifold that is locally similar enough to Euclidean 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 networkIn 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...
s provide a representation for computations of amplitudes between two different
hypersurfaceIn 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
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
. Any evolution of
spin networkIn 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...
provides a spin foam over a
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
of one dimension higher than the dimensions of the corresponding spin network. A spin foam may be viewed as a
quantum historyIn 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 networkIn 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...
s provide a language to describe
quantum geometryIn 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
graphIn 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 diagramIn quantum field theory a Feynman diagram is an intuitive graphical representation of a contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory....
where instead of a graph we use a higher-dimensional complex. In
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example 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
verticesIn geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3D models, where each such point is given as a vector.-Of an angle:The vertex of an angle is the...
, edges and
facesIn 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 suffix -hedron is derived from the Greek word hedra which means face....
. The boundary of a spin foam is 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 (gr-qc/9310036) (1993). 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.