  The Immirzi parameter of loop quantum gravity is a one parameter ambiguity of
the theory whose precise interpretation is not universally agreed upon. It is
an inherent characteristic of the quantum theory as it appears in the spectra
of geometric operators, despite being irrelevant at the classical level. The
parameter's appearance in the area and volume spectra to the same power as the
Planck area suggest that it plays a role in determining the fundamental length
scale of space. In fact, a consistent interpretation is that it represents a
constant rescaling of the kinematical spatial geometry. An interesting
realization is that promoting the Immirzi parameter to be a general conformal
transformation leads to a system which can be identified as analogous to the
linking theory of shape dynamics. A three-dimensional gravitational gauge
connection is then constructed within the linking theory in a manner analogous
to loop quantum gravity, thereby facilitating the application of the
established procedure of loop quantization.
