Inconstant Planck's constant

Motivated by the Dirac idea that fundamental constant are dynamical variables and by conjectures on quantum structure of spacetime at small distances, we consider the possibility that Planck constant $\mathrm{\hslash }$ is a time depending quantity, undergoing random gaussian fluctuations around its measured constant mean value, with variance ${\sigma }^2$ and a typical correlation timescale $\mathrm{\Delta }t$. We consider the case of propagation of a free particle and a one–dimensional harmonic oscillator coherent state, and show that the time evolution in both cases is different from the standard behaviour. Finally, we discuss how interferometric experiments or exploiting coherent electromagnetic fields in a cavity may put effective bounds on the value of $\tau= \sigma^2 \Delta t$.