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arXiv:quant-ph/0310132 · Physical Review A

Special relativity and reduced spin density matrices

Cezary Gonera, Piotr Kosiński, Paweł Maślanka

Published Sep 22, 2004• 11 citations
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Abstract

We derive the general formula for Lorentz-transformed spin density matrix. It is shown that an appropriate Lorentz transformation can produce totally unpolarized state out of pure one. Further properties, as depurification by an arbitrary Lorentz boost and its relation to the localization properties are also discussed.
DOI
10.1103/physreva.70.034102

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We have shown that the covariant relativistic spin operator is equivalent to the spin operator commuting with the free Dirac Hamiltonian. This implies that the covariant relativistic spin operator is a good quantum observable. The covariant relativistic spin operator has a pure quantum contribution that does not exist in the classical covariant spin operator. Based on this equivalence, reduced spin states can be clearly defined. We have shown that depending on the relative motion of an observer, the change in the entropy of a reduced spin density matrix sweeps through the whole range.
The spin reduced matrix defined by taking partial trace with the momentum variable has been shown not to transform covariantly under Lorentz transformations. In this paper, I suggest another representation space of the Poincare group, the elements of which are in a sense apparent wave functions. In this new Hilbert space, the naive spin observable 12τ\frac{1}{2}\boldsymbol{\tau}21​τ from nonrelativistic quantum mechanics on the old Hilbert space is represented as the Newton-Wigner spin operator. Also on this space, one can naturally associate to each state a Lorentz covariant spin reduced density matrix, which is then given a new interpretation.