In the loop quantum gravity approach, space-time is quantized by a procedure that encodes it in a discretized structure, consisting of spin networks and spin foams.
A spin network consists of an oriented embedded graph in a 3-dimensional manifold with edges labelled by SU(2) representations and edges labelled by intertwiners between the representations attached to incoming and outgoing vertices. These representations relate to gravity in terms of holonomies of connections, and the formulation of Einstein’s equations in terms of vierbein, or tetrads, and dual co-tetrads.
Thus, to a spin networks, or the 1-skeleton of a triangulation by tetrahedra, one assigns operators of quantized area and volume, coming from counting intersection points of a surface, or 3-dimensional regions, with the edges or vertices of the spin network with a multiplicity given in terms of the spin representation attached to the edges and the intertwiners attached to the vertices.
SOURCE: Listening to Golem
Loop quantum gravity is one of a few general frameworks that may eventually form the basis of how physicists think of the smallest scales of time and distance.
(These frameworks are sometimes called Theories of Everything but they’re really just thoughts-on-the-way-to-theories of brief-and-tiny matter.)
- SU(2) is the group of 2×2 unitary matrices with determinant 1. They have the form
SU(2) is just like the unit quaternions, which represent rotations in 3-D. Here are the pieces that make up SU(2) .
- 3-dimensional manifold — any shape that can be made with dough. Including that if you stretch the dough, the outside and the inside stretch.
- oriented graph — things like this:
- embedded oriented graph — oriented graphs sculpted in 3-D
- edge — arrows in the above pictures
- spin network — a graph like above and each circle has value +½ or −½
- representation — every group can be represented as a matrix
- holonomy — how to move things in parallel within the dough (curved 3-manifold)
- connection — parallel transport
- tetrad or vierbein — a 4-D spacetime frame of reference
I love randomly throwing out dense mathematical statements from theoretical physicists like this. Sometimes math people sound like wizards canting magical runes.