On Group Averaging for SO(n,1)

TL;DR

This work probes the practical implementation of refined algebraic quantization via group averaging for the gauge group $SO_c(n,1)$, focusing on when standard averaging diverges and how a renormalized procedure can still yield a meaningful physical inner product. By decomposing the auxiliary space into inside- and outside-light-cone sectors, the authors construct a renormalized rigging map $\eta_2$ on the exterior sector and demonstrate its compatibility with observables, while the interior sector enjoys convergent group averaging through $\eta_1$. A key result is that a linear combination $a_1\eta_1\oplus a_2\eta_2$ satisfies the rigging-map conditions, thereby enforcing a superselection rule between sectors. The findings illuminate how divergence properties of the group-averaging integral relate to the emergence of superselection sectors and point toward nuanced renormalization schemes in refined algebraic quantization, with implications for gauge theories and quantum gravity.

Abstract

The technique known as group averaging provides powerful machinery for the study of constrained systems. However, it is likely to be well defined only in a limited set of cases. Here, we investigate the possibility of using a `renormalized' group averaging in certain models. The results of our study may indicate a general connection between superselection sectors and the rate of divergence of the group averaging integral.