The role of phase space geometry in Heisenberg’s uncertainty relation

Abstract

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.