Projection evolution and quantum spacetime

Abstract: We discuss the problem of time in quantum mechanics. In the traditional formulation time enters the model as a~parameter, not an observable. In our model time is a quantum observable as any other quantum quantity and it is also a component of the spacetime position operator. In this case, instead of the unitary time evolution, other operators, usually projection or POVM operators which map the space of initial states into the space of final states at each step of the evolution can be used. The quantum evolution itself is a stochastic process. This allows to treat time as a quantum observable in a consistent, observer independent way, which is a very important feature to resolve some quantum paradoxes and the time problem in cosmology. An idea of construction of a quantum spacetime as a special set of the allowed states is presented. An example of a structureless quantum Minkowski-like spacetime is also considered. We present the projection evolution model and show how the traditional Schroedinger evolution and relativistic equations can be obtained from it, in the flat structureless spacetime. We propose the form of the time operator which satisfies the energy-time uncertainty relation based on the same inequality as the space position and spatial momenta observables. The sign of the temporal component of the four-momentum operator defines the basic arrow of time in spacetime.