How is matter modeled in loop quantum gravity?

Question

I have a basic understanding of how gravitational degrees of freedom are modeled in loop quantum gravity, but as far as I know, the main machinery, spin network states and observables, does not encode the matter degrees of freedom. How is matter currently modeled in loop quantum gravity?

Actually, the question applies not only to matter as in fermionic matter, but also the other three forces. The LQG spin networks are based around the SU2 symmetry of Ashtekar's gauge description of gravity. How do they incorporate the other gauge groups?

Answer 1

The original construction of the Ashtekar-Lewandowski space only depends on the fact tha gauge group is compact. As long as you stay at the kinematical level, and don't bother about the dynamics, any gauge group can be implemented if it is compact.

The gauge group of the standard model, in particular, can easily be incorporated by developing spin networks, not for $SU(2)$ but for $SU(2) \times G_\textrm{Standard} \simeq SU(2) \times (SU(3)_C \times SU(2)_L \times U(1)_Y)$. In practice, it means that the edges of the spin network will not carry spins but rather (unitary) representations of the full gauge group, which can be labelled by a spin (for the gravitationnal gauge group) but also labels for the representations of $SU(3)$, $SU(2)$ and $U(1)$ from the Standard Model.

Now, for matter, the subject is less clear to me. It seems that the Hilbert space is well-defined for fermions but, at the dynamical level, we don't know how to do this without issues regarding chirality which translates into fermion doubling problems (http://arxiv.org/abs/1507.01232) though some people think it might be possible to do (http://arxiv.org/abs/1506.08794).

For scalar matter (like the Higgs), for a long time, compactification was the easiest way to deal with the definition of the problem (the scalar field takes value in $U(1)$ for instance) as was done in Thiemann original work (the whole thing is developed here - if you ever find the courage to read it).

More recent developments used Bohr compactification. See for instance : http://arxiv.org/abs/gr-qc/0211012 though the consequence of this doesn't seem clear to me.

All these problems though can be classified into two kinds:

• Problems with non-compact structures (as can be found when doing covariant loop quantum gravity) that leads to problems when defining the projective limit.
• Problems with the dynamics, but which doesn't mean much for the kinematical structure.

At present, I would say there are some pretty convincing ways of incorporating matter into LQG. The main trick though, as always with LQG, is the dynamics.