On the equivalence of nonrelativistic quantum mechanics based upon sharp and fuzzy measurements

Abstract

Starting from the idea that physical measurements may have residual imprecisions, the possibility of replacing the nonrelativistic, three‐dimensional configuration space by a so‐called fuzzy configuration space, having an isomorphic Borel structure, is discussed. A quantization procedure with respect to such a space is developed, and the invariance of nonrelativistic quantum mechanics under such Borel isomorphisms is exploited to prove the equivalence of this quantization procedure to the usual quantization procedure on a fuzzy‐free configuration space. Further, any Galilean invariant dynamics is shown to be insensitive to such imprecisions in the measurements of position and momentum.