From time operator to chronons

Abstract

A time operator, which incorporates the idea of time as a dynamical variable, was first introduced in the context of a theory of irreversible evolution. The existence of a time operator has interesting implications in several areas of physics. Here we demonstrate a close link between the existence of the time operator for relativistic particles and the existence of an indivisible time interval or chronons for dynamical evolution. More explicitly, we consider a Klein-Gordon particle and require the existence of a time operator for its evolution. We also make a natural choice of the form of the time operator which expresses it in terms of the generators of the Poincaré group. These then imply that the physical time evolution group must be the discrete subgroup U_nτ (n integers) of the originally given evolution group U_t of the Klein-Gordon particle and the constant τ is given by τ=h/2mc². This means that the requirement of the existence of a time operator implies that the time evolution cannot be followed to time intervals smaller than τ and, as such, τ emerges as a chronon for the dynamical evolution. Expecting that the same results hold for a Dirac particle also, we conclude that the so-called Zitterbewegungdoes not occur in reality. Thus, possible confirmation of the existence of chronons would result if no observableconsequence of Zitterbewegungis actually realized in nature. This calls for a search of observable consequences of the Zitterbewegungand a re-examination of their agreement (if any) with experiments. A possible consequence of Zitterbewegung,the so-called Darwin term present in the Dirac Hamiltonian in an electric field, is briefly considered.