LQG is then briefly compared to (ii) quantum Regge calculus and (causal) dynamical triangulations.
These are a more straightforward programme for discretizing GR, in that they triangulate spacetime (so all graphs are considered as embedded in spacetime) and assign each link its spatiotemporal length (in dynamical triangulations, all links being required to be of the same length). Then the dynamics is in terms of a sum over histories’ amplitudes. Most of the focus has been on proving phase transitions, and so with an eye on the continuum theory; and so with a less realistic construal of the discrete microscopic degrees of freedom.
This leads to a brief mention of (iii) matrix models; and (iv) causal set theory. As he proceeds, he briefly compares each program’s proposals with aspects of the preceding ones, especially as regards how realistically one tends to take (or should take!) the postulated basic degrees of freedom.
Then he introduces the group field theory approach .The formalism is a field theory over a group manifold (or the corresponding Lie algebra) with the basic variable being a complex field φ (g1, g2, .., gd ) = φ12..d : a function of d group elements, for a d-dimensional spacetime (the most relevant case, therefore, being d = 4). For example, we may have gi ∈ SO (4), for all i.
The field can be represented graphically as a (d-1)-simplex with field arguments associated to the d faces of it, or as a d-valent graph vertex, with field arguments associated to the links. (So for d=4, the field is represented as a 3-simplex ie as a tetrahedron.)
The dynamics is given by an action, which integrates (with the Haar measure) over the group manifold: (of course a multiple integral). The interaction term involves a non-standard (compared with the usual local QFT) convolution of fields defined in terms of their arguments (analogous to the tracing of indices in matrix models). Thus the specific pattern of convolution chosen depends on which requirement replaces the locality of standard QFTs.
For example, in d = 4 and for gi ∈ SO (4), the interaction term might be a fivefold product with a specific identification of certain arguments of the field: say, φ1234 φ4567 φ7389 φ962 10 φ10 851 ; (so that we have a tenfold multiple integration with Haar measure). This combinatorics of convoluted field arguments in the interaction matches the combinatorics of five tetrahedra glued along common triangles to form a 4-simplex. The propagator induces a simple gluing of 4-simplices across one shared tetrahedron, when used in perturbative expansion to form Feynman diagrams. Such Feynman diagrams are representable, by construction, as 4d simplicial complexes.
So in these models, one could say that the usual QFT locality is replaced by simpliciality , i.e. the requirement that interactions are associated to d-simplices.