Editing Loop quantum gravity (section)

=== 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.