  Aiming towards a geometric description of quantum theory, we study the
coherent states-induced metric on the phase space, which provides a geometric
formulation of the Heisenberg uncertainty relations (both the position-momentum
and the time-energy ones). The metric also distinguishes the original
uncertainty relations of Heisenberg from the ones that are obtained from
non-commutativity of operators. Conversely, the uncertainty relations can be
written in terms of this metric only, hence they can be formulated for any
physical system, including ones with non-trivial phase space. Moreover, the
metric is a key ingredient of the probability structure of continuous-time
histories on phase space. This fact allows a simple new proof the impossibility
of the physical manifestation of the quantum Zeno and anti-Zeno paradoxes.
Finally, we construct the coherent states for a spinless relativistic particle,
as a non-trivial example by which we demonstrate our results.
