The Barrett–Crane model is a model in

quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...

, first published in 1998, which was defined using the

Plebanski action General relativity and supergravity in all dimensions meet each other at a common assumption: :''Any configuration space can be coordinatized by gauge fields A^i_a, where the index i is a Lie algebra index and a is a spatial manifold index.'' ...

. The

B

field in the action is supposed to be a

so(3, 1)

-valued

2-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...

, i.e. taking values in the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

of a

special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

. The term :

B^ \wedge B^

in the action has the same symmetries as it does to provide the Einstein–Hilbert action. But the form of :

B^

is not unique and can be posed by the different forms: *

\pm e^i \wedge e^j

\pm \epsilon^ e_k \wedge e_l

where

e^i

is the tetrad and

\epsilon^

is the antisymmetric symbol of the

so(3, 1)

-valued 2-form fields. The Plebanski action can be constrained to produce the

BF model The BF model or BF theory is a topological field, which when quantized, becomes a topological quantum field theory. BF stands for background field B and F, as can be seen below, are also the variables appearing in the Lagrangian of the theory, whic ...

which is a theory of no local

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

John W. Barrett John Barrett or Johnny Barrett may refer to: Clergy * John Barrett (bishop) (1878–1946), British clergyman who held high office in the Roman Catholic Church * John Barrett (dean of Clonmacnoise) (1929–1996), Irish Anglican priest * John Bar ...

and Louis Crane modeled the analogous constraint on the summation over

spin foam In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structur ...

. The Barrett–Crane model on spin foam quantizes the

, but its path integral amplitude corresponds to the degenerate

B

field and not the specific definition :

B^ = e^i \wedge e^j

, which formally satisfies the

Einstein's field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the fo ...

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...

. However, if analysed with the tools of

loop quantum gravity Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...

the Barrett–Crane model gives an incorrect long-distance limi

and so the model is not identical to loop quantum gravity.

References

Loop quantum gravity {{relativity-stub