Geometrical quantum time in the U(1)3 model of Euclidean quantum gravity

Loop quantum gravity faces challenges in constructing a well-defined Hamiltonian constraint and understanding the quantum notion of time. In this paper these issues are studied by quantizing the U(1)3 model, a simplified system exhibiting features similar to general relativity. By isolating a holonomy component within the Hamiltonian constraint, a discrete relative time evolution equation for quantum states is obtained. Then, a Shr & ouml;dinger-like equation is derived in the continuum limit. Thus, the physical states solving this Shr & ouml;dinger-like equation can be written out. The emergence of the time parameter and its corresponding quantum operator are analyzed. It indicates the notion of a geometrical quantum time for quantum gravity.