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# DYK-06 Position Operator in Relativistic Quantum Mechanics

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... In a famous paper, Newton and Wigner [2] showed that the required behaviour of a position operator under space translations and rotations almost uniquely determines this operator. However the resulting operator q is noncovariant and, due to its energy being positive, has the ugly  property that a state that is an eigenstate of it at a given time ( a localized state) will spread out over all of space n infinitesimal time later....

For a Dirac spin -1/2 particle, the Newton Wigner position operator turns out to be identical to the Foldy Wouthuysen mean position operator...

This case is particularly interesting because when Dirac equation was conceived  in 1928, the space part of $$\bf x$$ of the four vector $$x=\vec{x}, ct)$$  appearing as an argument of Dirac's four spinor wave function  $$\psi(x)$$  was identified with the position of the electron. This identification had the embarrassing consequence that the corresponding velocity  of the electron would always found to be the velocity of light. It took twenty years before this problem was solved and ....

2. T. D. Newton and E.P. Wigner, Localized states for elementary  system'',  Revs. Mod Phys 21 (1949) 400-406.