Internal clocks in timeless universe

The Hamiltonian formalism of general relativity involves a Hamiltonian constraint. Attempts at quantisation of the Hamiltonian constraint formalism face an obstacle associated with the lack of predetermined time parameter whose existence is assumed in usual quantisation prescriptions. A way to deal with it is to employ an arbitrary internal degree of freedom as the internal clock to describe the evolution of a gravitational system. We use the so called reduced phase space approach in which the choice of internal clock is made prior to quantisation. We discuss the construction of reduced phase spaces based on the essential role played by internal clocks. Then we introduce the so called extended transformations which extend the well-known notion of canonical transformations. The extended transformations elucidate the relation between the canonical structure of the reduced phase space and the internal clock. Quantisation of reduced phase spaces and respective Hamiltonians reveals the relation between quantum dynamics and the choice of internal clock. Finally, we discuss concrete examples of canonical formalisms of the Friedmann-Lemaitre universe and Bianchi type I universe. We find that many physical features of the quantum dynamics depend on the choice of internal clock. In the Conclusions we speculate about the possible physical meaning of the result.