Editing Loop quantum gravity (section)

== Spin foams ==
{{main| spin network|spin foam|BF model|Barrett–Crane model}}
In loop quantum gravity (LQG), a spin network represents a "quantum state" of the gravitational field on a 3-dimensional [[hypersurface]]. The set of all possible spin networks (or, more accurately, "s-knots" – that is, equivalence classes of spin networks under diffeomorphisms) is countable; it constitutes a basis of LQG [[Hilbert space]].

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.

=== Spin foam derived from the Hamiltonian constraint operator ===

The Hamiltonian constraint generates 'time' evolution. Solving the Hamiltonian constraint should tell us how quantum states evolve in 'time' from an initial spin network state to a final spin network state. One approach to solving the Hamiltonian constraint starts with what is called the [[Dirac delta function]]. The summation of which over different sequences of actions can be visualized as a summation over different histories of 'interaction vertices' in the 'time' evolution sending the initial spin network to the final spin network. Each time a Hamiltonian operator acts it does so by adding a new edge at the vertex.{{sfn|Thiemann|2008|pp=458–462}}

This then naturally gives rise to the two-complex (a combinatorial set of faces that join along edges, which in turn join on vertices) underlying the spin foam description; we evolve forward an initial spin network sweeping out a surface, the action of the Hamiltonian constraint operator is to produce a new planar surface starting at the vertex. We are able to use the action of the Hamiltonian constraint on the vertex of a spin network state to associate an amplitude to each "interaction" (in analogy to [[Feynman diagrams]]). See figure below. This opens a way of trying to directly link canonical LQG to a path integral description. Just as a spin networks describe quantum space, each configuration contributing to these path integrals, or sums over history, describe 'quantum spacetime'. Because of their resemblance to soap foams and the way they are labeled [[John Baez]] gave these 'quantum spacetimes' the name 'spin foams'.

[[File:Spin foam from Hamiltonian constraint.jpg|right|thumb|upright=2.2|The action of the Hamiltonian constraint translated to the [[Functional integration|path integral]] or so-called spin foam description. A single node splits into three nodes, creating a spin foam vertex. <math>N (x_n)</math> is the value of <math>N</math> at the vertex and <math>H_{nop}</math> are the matrix elements of the Hamiltonian constraint <math>\hat{H}</math>.]]

There are however severe difficulties with this particular approach, for example the Hamiltonian operator is not self-adjoint, in fact it is not even a [[normal operator]] (i.e. the operator does not commute with its adjoint) and so the [[spectral theorem]] cannot be used to define the exponential in general. The most serious problem is that the <math>\hat{H} (x)</math>'s are not mutually commuting, it can then be shown the formal quantity <math display="inline">\int [d N] e^{i \int d^3 x N (x) \hat{H} (x)}</math> cannot even define a (generalized) projector. The master constraint (see below) does not suffer from these problems and as such offers a way of connecting the canonical theory to the path integral formulation.

=== 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.

=== Modern formulation of spin foams ===
Just as imposing the classical simplicity constraint recovers general relativity from BF theory, it is expected that an appropriate quantum simplicity constraint will recover quantum gravity from quantum BF theory.

Progress has been made with regard to this issue by Engle, Pereira, and Rovelli,{{sfn|Engle|Pereira|Rovelli|2009|p=161301}} Freidel and Krasnov{{sfn|Freidel|Krasnov|2008|p=125018}} and Livine and Speziale{{sfn|Livine|Speziale|2008|p=50004}} in defining spin foam interaction amplitudes with better behaviour.

An attempt to make contact between EPRL-FK spin foam and the canonical formulation of LQG has been made.{{sfn|Alesci|Thiemann|Zipfel|2011|p=024017}}

=== Spin foam derived from the master constraint operator ===
See below.<!--

=== Spin foams from consistent discretisations ===-->