Narrow Results

We studied the path integral quantization for the Shape Dynamics formulation of General Relativity in the 2+1 torus universe. We show that the Shape Dynamics path integral on the reduced phase space is equivalent with the previous results obtained for the ADM 2+1 gravity and we found that the Shape Dynamics Hamiltonian allows us to establish a straightforward relation between reduced systems in the (τ, V)-form and the τ-form through the York time gauge fixing.

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.

This short note is an attempt to bring out the geometric structures in the linking theory of shape dynamics. Symplectic induction is applied to give a natural construction of the extended phase space used in the linking theory as a trivial vector bundle over the original phase space for canonical gravity. The geometry of the gauge fixing for shape dynamics is analyzed with the assistance of the Lichnerowicz–York equation lifted to the extended phase space. An alternative description is provided to show how the same geometry simply derives from symplectic induction.

We present a consistent framework for treating the energy and angular-momentum dependence of the nuclear shape evolution in the fission process. It combines microscopically calculated level densities with the Metropolis walk method and it contains no new parameters. The treatment can elucidate the influence on the shape dynamics of warm nuclei of pairing and shell effects as a function of the excitation energy.