Quantum cosmology with SU(1,1)

This is the first post in an upcoming series covering some recent papers that deal with quantum cosmology models that are strongly relying on an SU(1,1), or equivalently SL(2,R), structure. In brief, one identifies classical phase space functions whose Poisson algebra is isomorphic to the Lie algebra su(1,1) and then quantises the cosmological model by promoting those functions to the generators of su(1,1) in some representation. The main advantage of this quantisation method is that the representation theory of the group under consideration is well known, so that the crucial "find a representation" step in constructing the quantum theory is essentially trivial.


The exercise of finding such phase space functions can for example be successfully completed for classical spatially flat homogeneous and isotropic cosmology. There, the phase space is described by the canonical pair $\{v,b\}=1$, where $v$ is the spatial volume and $b$ the mean curvature. Instead of writing the details for the case of classical cosmology, we will in the following give a flavour of a similar analysis in the context of a modification of classical cosmology inspired by effective loop quantum cosmology. For a brief recap, see this post. In short, the transition from classical cosmology to the effective loop quantum cosmology model is performed by substituting $b$ by functions such as $\sin(b)$, which are capturing some key quantum effects like a limiting curvature at the Planck scale.

The first main work along these lines was this paper. It turns out that the identification $j_z=v$, $k_x = v \cos(b)$, $k_y = v \sin(b)$ reproduces the su(1,1) algebra with the standard identifications of the generators. Quantisation can then proceed as said before by promoting them to the su(1,1) generators in a certain representation space.

This is only one identification of su(1,1) generators from the cosmology phase space and others are possible. Several examples are given in this follow-up work. The details of the identification are relevant for the correct choice of the representation, as they determine the classical value of the Casimir operators as either positive, zero, or negative. Corresponding SU(1,1) representation are available in all cases, but differ in their properties.

There are several avenues that can be followed from here:

• On the SU(1,1) representation spaces, one can define standard Perelomov coherent states. They have the nice property that one can transfer the action of su(1,1) generators on them directly to a spinor label in the defining representation, thus allowing an explicit computation of the quantum dynamics via exponentiating su(1,1) generators. This also allows to easily compare evolution in different representations. 
• Expectation values of the su(1,1) generators feature a splitting of the dependence on the spinor label and representation label (for the discrete representation with $j=1/2, 3/2, \ldots$) such that the extensive state of the system (its size) decouples from its intensive properties. This suggests to study coarse graining based on such states. 
• An alternative way of quantising is to not start with classical phase space functions forming the su(1,1) algebra, but to start with a given quantum cosmological system and to find compound operators built from the elementary operators, e.g. $\hat v$ and $\hat b$, that have the su(1,1) algebra. This may reduce quantisation ambiguities if interesting operators, such as the Hamiltonian, are linear combinations of the generators.
• It would be interesting to check whether there is a geometric interpretation of the action of the conformal group SL(2,R) on spacetime. 

Some results have already appeared in the literature and we will comment on them in future posts.