Validity of Special Relativity at Small Distances and the Velocity Dependence of the Muon Lifetime

Abstract

The velocity dependence of the lifetime of the muon is investigated under the assumption that the Hamiltonian contains a spatial form factor which in the lab frame (the frame at rest with respect to the neighboring macroscopic bodies) vanishes for distances larger than some length $\alpha$. In this model there is a violation of the principles of special relativity at small distances. In particular, space-time is anisotropic at distances smaller than $\alpha$. The lifetime of the muon is calculated to second order in $\alpha$, and it is shown that there will be about 1% deviation from the usual formula $\tau \mathrm{}\left(v\right)={(1-\frac{v^2}{c^2})}^{-\frac{1}{2}}\tau \mathrm{}\left(0\right)\mathrm{}$ (which holds if special relativity is valid at arbitrarily small distances) if, e.g., the muon energy $E_{\mu }={10}^4$ MeV and $\alpha \sim 7\times {10}^{-16\mathrm{}}$ cm. The measurement of the velocity dependence of the muon lifetime at high energies could thus serve as a possible check on the validity of special relativity at small distances.