Coarse Graining with SU(1,1)

This is the second post in a series about SU(1,1) techniques for coarse graining loop quantum cosmology. While the previous post was about the general idea of group quantisation techniques, this one will focus on coarse graining.


Coarse graining implements the idea that coarse physical systems, such as the universe at large, are built up from smaller systems. The physics at large scales is determined by "integrating out" physics at small scales, i.e the description at large scales is reduced to a few parameters that take into account how the details of the system behave on average.

Obtaining such an effective description at a coarse scale may be very difficult, in particular via an analytic computation. Therefore, explicit examples, even in simplified models, are of great interest.

In a recent paper with a former Bachelor student, I was able to provide such an example using systems quantised with group quantisation techniques. The result is quite general and and requires only that one quantises a classical Poisson algebra isomorphic to su(1,1) in which the 3 generators scale like extensive quantities with the system size and interactions between neighbouring subsystems can be neglected. This is for example the case in the application to quantum cosmology discussed before.

It turns out that the representation label $j$ of certain SU(1,1) representations behaves as the scale of the system if one uses certain coherent states. $N$ systems with representation label (scale) $j_0$ then jointly behave like a single system with representation label $j = N j_0$.

The agreement of the $N$ fine and the single coarse systems is correct for all moments of the 3 generators and the eigenvalues with associated probabilities. The derivation necessary to arrive at this result is the core part of the cited paper. The coarse graining procedure is even dynamically stable if dynamics is generated by one of the group generators. Again, this can be achieved in the quantum cosmology application.

In a future post, another paper with a former Master student will be discussed which lifts these results directly onto the Hilbert space of loop quantum cosmology. This application is interesting because it gives an example of a renormalisation group flow of Hamiltonian operators from small to large spins.