Quantum mechanics with a minimal length and time scale

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Abstract
Many theories of quantum gravity suggest that there should exist a minimal scale with which it is possible to measure distances and time. We present a formalism of quantum mechanics exchibiting a minimal length and time scale. The work is based on the Page-Wootters formalism which gives a possible solution to the problem of time in quantum mechanics. This formalism is an example of a relational approach to quantum dynamics. It is based on the idea that reference frames with respect to which motion is described are themselves physical systems interacting with the degrees of freedom they wish to describe. Quantum dynamics is described in terms of quantum correlations between a clock and a system. In our work we modify the commutation relations between the time and frequency operators leading to a minimal uncertainty of time measurement. This results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We further investigate the e ects of such modi cation on a couple simple quantum systems.