Editing Loop quantum gravity (section)

=== Spin foams from BF theory ===
It turns out there are alternative routes to formulating the path integral, however their connection to the Hamiltonian formalism is less clear. One way is to start with the [[BF theory]]. This is a simpler theory than general relativity, it has no local degrees of freedom and as such depends only on topological aspects of the fields. BF theory is what is known as a [[topological field theory]]. Surprisingly, it turns out that general relativity can be obtained from BF theory by imposing a constraint,{{sfn|Bojowald|Perez|2009|p=877}} BF theory involves a field <math>B_{ab}^{IJ}</math> and if one chooses the field <math>B</math> to be the (anti-symmetric) product of two tetrads
<math display="block">B_{ab}^{IJ} = {1 \over 2} \left(E^I_a E^J_b - E^I_b E^J_a\right)</math>
(tetrads are like triads but in four spacetime dimensions), one recovers general relativity. The condition that the <math>B</math> field be given by the product of two tetrads is called the simplicity constraint. The spin foam dynamics of the topological field theory is well understood. Given the spin foam 'interaction' amplitudes for this simple theory, one then tries to implement the simplicity conditions to obtain a path integral for general relativity. The non-trivial task of constructing a spin foam model is then reduced to the question of how this simplicity constraint should be imposed in the quantum theory. The first attempt at this was the famous [[Barrett–Crane model]].{{sfn|Barrett|Crane|2000|pp=3101–3118}} However this model was shown to be problematic, for example there did not seem to be enough degrees of freedom to ensure the correct classical limit.{{sfn|Rovelli|Alesci|2007|p=104012}} It has been argued that the simplicity constraint was imposed too strongly at the quantum level and should only be imposed in the sense of expectation values just as with the [[Lorenz gauge condition]] <math>\partial_\mu \hat{A}^\mu</math> in the [[Gupta–Bleuler formalism]] of [[quantum electrodynamics]]. New models have now been put forward, sometimes motivated by imposing the simplicity conditions in a weaker sense.

Another difficulty here is that spin foams are defined on a discretization of spacetime. While this presents no problems for a topological field theory as it has no local degrees of freedom, it presents problems for GR. This is known as the problem triangularization dependence.