What Are the Differences Between Riemannian and Lorentzian Spin Networks?

[meteor]

Can someone tell me what's the difference between a Riemannian spin network and a Lorentzian spin network?

 

[selfAdjoint]

It's the underlying manifold they are implemented on. Although spin networks look forward to replacing metric geometry, they must as yet be built on an underlying manifold. That manifold can be either Riemannian, with a Euclidean tangent space, or Lorentzian, with a Minkowskian tangent space. The spacetime of GR is Lorentzian.

Generally speaking a Lorentzian network is harder to work with because you have to deal with links in timelike directions. For this reason you will see basic work on networks being carried out in a spacelike slice - a three dimensional Riemannian manifold embedded in the higher dimensional Lorentzian one.

 

[Ambitwistor]

Originally posted by meteor
Can someone tell me what's the difference between a Riemannian spin network and a Lorentzian spin network?

Usually Riemannian spin networks are used to describe connections defined on space, and Lorenztian spin networks are used to describe connections defined on spacetime. You could say that the difference is in the signature of the metric, e.g. +++ vs. -+++, but since the metric is not fundamental, what really matters is the gauge group. Spatial spin networks have gauge groups like SU(2) or SO(3), whereas spacetime spin networks have gauge groups like SO(1,3). (These connections do induce a fidicial flat Euclidean/Mikowskian metric to transform frames.) Lorentzian spin networks are harder to work with because, unlike SU(2) or SO(3), SO(1,3) is non-compact. So instead of sums over a discrete spectrum of representations, you end up with integrals over a continuous spectrum of representations.