Observable Consequences of Fundamental-Length Hypotheses

Abstract

The postulate of the existence of a fundamental length may be expressed in terms of a minimum uncertainty in position measurements, or equivalently as a minimum uncertainty in measurements of the gravitational field. The postulate is expressed mathematically by means of "indeterminate operators," whose properties are discussed. With their aid, it is shown that the postulate of a fundamental length has as a consequence a certain broadening of spectral lines. In the case of a fundamental length of the order of ${10}^{\ensuremath{-}13}$ cm, the predicted broadening is much larger than the widths of nuclear gamma transitions already observed. It is concluded that this fundamental length is already in serious contradiction with experiment. In the case of a fundamental length due to gravitational effects of the order of ${10}^{\ensuremath{-}33}$ cm, as previously suggested by the author, the broadening is too small to have been observed in any experiments done to date. A modified M\"ossbauer experiment is suggested which should be capable of detecting this small broadening, if it is present. The experiment is difficult, but appears to be possible with presently available techniques. The gravitational-field uncertainties calculated by DeWitt, Peres, and Rosen, and others, lead to a still smaller broadening, which our proposed method would be incapable of detecting.