Summary
A quantum theory is derivedvia path integration that possesses a fundamental joint-probability amplitude for position and four-velocity,i.e. the probabilty amplitude that a particle simultaneously has a particular four-velocity at a particular position in space-time. The uncertainity principle is not violated.
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Nash, P.L. Path integral quantization of the dynamics of a classical point particle with intrinsic spin. Nuov Cim B 110, 913–926 (1995). https://doi.org/10.1007/BF02722860
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DOI: https://doi.org/10.1007/BF02722860