Three-dimensional relativistic-covariant poisson brackets

Using the algebraic properties of Poisson brackets, we extend the three-dimensional brackets (for a single free particle) to conform to the demands of special relativity. This yields, in an essentially unique way, the manifestly covariant extension x ,p _v]=+p p _v/m ² c ². Position and time then become fully dynamical variables expressible in terms of the canonical conjugateq _i andp _i and the time parameter asx _i =q _i +p _i(q ·p)/m ²c² andt = +E(q ·p)/m ²c⁴. In the quantized version, the length associated with a particle of massm is shown to be an integral multiple of the Compton wavelength _C =/mc.