An essential ambiguity of quantum theory

It is shown that there is a substantial family of Lagrangians for one and the same ground problem of Newtonian mechanics where linear momentum is conserved. These alternative Lagrangians involve an arbitrary function of conserved momentum, which reduces to a one-parameter family when it is required that the Galilean group be canonically represented. Then the canonical momentum becomes parameter-dependent: and this leads quantally to ambiguous commutation rules and Heisenberg uncertainty principles—for example the coordinate of one particle may be complementary not to its own mechanical momentum (as in standard treatments) but to that of a different particle. One possible way of resolving the ambiguity is by reformulating many-particle dynamics as higher-order one-particle (HOOP) dynamics by decoupling the standard lower-order equations of motion. This also admits relativistic generalization in which the Poincaré group is canonically represented.