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Nuclear Physics B - Proceedings Supplements

A space-time representation of a massive, relativistic, spin zero particle

Richard W. Ziolkowski, Amr Mohamed Shaarawi, Ioannis M. Besieris

Published Mar 1, 1989• 14 citations
Publisher
DOI
10.1016/0920-5632(89)90450-7

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It is shown that to every Galilei-covariant nonrelativistic stochastic phase space representation of a system of massive particles, whose generator is rotationally invariant, corresponds a Poincaré-covariant relativistic representation sharing the same generator. The stochastic phase space probability densities of the two representations overlap in the limit of nonrelativistic velocities in the laboratory frame. The relativistic representations give rise to covariant and conserved probability currents at stochastic space–time points, in complete analogy with their nonrelativistic counterparts. This parallelism extends to the existence of a global representation of the proper Poincaré group in L2(Γ), which is reduced by each subspace of L2(Γ) spanned by the set of phase-space wavefunctions generated by some stochastic phase space representation of all pure states of the system.
Recently, in the context of Relativistic Quantum Information Theory (RQI) of massive spin-1/2 particles, it has been suggested that it is impossible to perform a momentum-independent spin measurement, showing the inadequacy of the spin reduced density matrix as a legitimate information resource. This is because there is an unavoidable ambiguity in the definition of the spin of a moving particle. In this paper, by introducing a vector bundle theoretic way to view the single-particle state space, we try to rule out this ambiguity. The discrete degree of freedom of the resulting representation space contains information about the Pauli–Lubanski four-vector of the particle instead of the ambiguous spin. Comparing this representation with the standard one used in the RQI literature, we show that the discrete degree of freedom of the standard representation space attains the meaning of the Newton–Wigner spin. In addition, using this viewpoint, we give a mathematical proof of why the spin reduced density matrix is meaningless, which is stronger than the previous claims in that it asserts that the matrix is void of any meaning at all, not just in terms of the impossibility of measurement or Lorentz non-covariance. We give a way (which turns out to be the only way) to modify it to obtain the Pauli–Lubanski reduced density matrix, which is covariant under Lorentz transformations.