For practical purposes, to enable calculation, string theories are constructed on a fixed geometric background. Like a given multidim. spacetime manifold with a specific distance function (aka "metric" function) defined on it.
The strings have to have some "target space" in which to vibrate.
So I would not call string "backgroundless" or "background independent". You need already a fairly elaborate geometric setup to start with.
When you hears someone say "spin network" and "spin foam" it suggests they are talking about LQG---those terms have technical meanings in LQG, and are the basic objects dealt with. They go together. a foam is the 4D picture of a network evolving, like its trajectory. A spin network is a quantum state of 3D geometry, and a foam is how it changes over time.
A network does not live in space, it is space (restricted to a finite number of degrees of freedom---dumbed down, so to speak). It is not located anywhere, it is where.
I think the answer to your question would depend on what you mean by a lattice. Probably the answer is no because you probably mean a regular lattice with a fixed size, like a cubic lattice with a some definite edge length.
The spin networks are not like that. There is no definite length of any of the links, and there is no definite number of links that have to meet at any given node. You use different mathematical rules to define and use them, from what you might expect with a regular lattice.
So you wouldn't get similar results. Unless you changed the lattice rules so that your lattice was really a spin network going by a different name. It might be good to read something about LQG.
The key prereq. is familiarity with the Lie groups SU(2) and SL(2,C). Everything is defined on cartesian products of these basic symmetry groups. I think that would be the primary barrier to understanding papers like December's#4719 survey.
http://arxiv.org/abs/1011.3667
Coarse graining theories with gauge symmetries
Benjamin Bahr, Bianca Dittrich, Song He
(Submitted on 16 Nov 2010)
Discretizations of continuum theories often do not preserve the gauge symmetry content. This occurs in particular for diffeomorphism symmetry in general relativity, which leads to severe difficulties both in canonical and covariant quantization approaches. We discuss here the method of perfect actions, which attempts to restore gauge symmetries by mirroring exactly continuum physics on a lattice via a coarse graining process. Analytical results can only be obtained via a perturbative approach, for which we consider the first steps, namely the coarse graining of the linearized theory. The linearized gauge symmetries are exact also in the discretized theory, hence we develop a formalism to deal with gauge systems. Finally we provide a discretization of linearized gravity as well as a coarse graining map and show that with this choice the 3D linearized gravity action is invariant under coarse graining.
http://arxiv.org/abs/1012.4719 See footnote 1 on page 2 for a comment on the possible non-role of Nielsen-Ninomiya theorem in the context of LQG.
Last edited:
#4
Yes I am think of a nice simple cubic lattice with a single spacing interval. Thanks for making clear that strings already assume a background.
I am surprise that a coarse grained might be better than a fine grained.
I was hoping that things could be done a lattice rather than a semi-random network. I still hold out hope that a lattice is a close enough approximation for many calculations and hope to see someone show the physics differences between a regular lattice and a natural network/foam. Maybe some of the differences will be experimentally observable.
LQG is not completely background independent. The derivation of the theory uses that there is a spacetime manifold with topology R*M³. Somehwo this background dependence seems to be "washed away" when deriving the spin networks as these do not rely on a manifold structure a on a certain dimension. Nevertheless one cannot be sure that not some relicts of the initial starting point survive.
I think your question cannot be answered generically. It is of course possible (but usually unlikely) that a certain approach breaks a symmetry but restores it in a certain limit.