  One classical theory, as determined by an equation of motion or set of
classical trajectories, can correspond to many unitarily {\em in}equivalent
quantum theories upon canonical quantization. This arises from a remarkable
ambiguity, not previously investigated, in the construction of the classical
(and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is
illustrated for systems with one degree of freedom: An arbitrary function of
the constants of motion can be introduced into this construction. For example,
the nonrelativistic and relativistic free particles follow identical classical
trajectories, but the Hamiltonians or Lagrangians, and the canonically
quantized versions of these descriptions, are inequivalent. Inequivalent
descriptions of other systems, such as the harmonic oscillator, are also
readily obtained.
