3-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(mu), p(nu)] = delta(munu) + P(mu)P(nu)/m2c2. Position and time then become fully dynamical variables expressible in terms of the canonical conjugate q(i) and p(i) and the time parameter theta as x(i) = q(i) + p(i)(q . p)/m2c2 and t = theta + E(q . p)/m2c4. In the quantized version, the length associated with a particle of mass m is shown to be an integral multiple of the Compton wavelength lambda(c) = HBAR/mc.