The Cumulative Effect of Vacuum Radiation on Particle Coordinates

The action principle can predict the trajectories of a system of particles as determined by the dynamical forces acting on them. However, its predictions do not include the results of quantum fluctuations in the coordinates of the particles. It is proposed that quantum fluctuations shift the particles from one dynamical trajectory to another and that the change in action due to a root mean square shift in an individual coordinate is the same, regardless of which coordinate is shifted. This assumption, together with the uncertainty principle, implies that the cumulative effect of changes in energy and momentum varies as t-1/2, where t is time, so that these quantities tend to be conserved. However, the cumulative effect of changes in spatial coordinate varies as t1/2, so this coordinate shows a Brownian drift over time. An example is given in which this stochastic drift, with its characteristic t1/2 dependence, could be experimentally observed at the beginning of a highly collimated particle beam.