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Journal of Lie Theory 30 (2020), No. 2, 445--459
Copyright Heldermann Verlag 2020

Fidelity and Metrics on Lorentz Boosts

Sejong Kim
Department of Mathematics, Chungbuk National University, Cheongju 28644, Korea

We see in this article that the extended version of a real counterpart of qubit density matrices introduced by Abraham Ungar, called a Moebius matrix, is indeed a normalized Lorentz boost by finding its spectral decomposition. Using the gyrogroup isomorphism between the set of all Lorentz boosts and the Einstein gyrogroup on the open unit ball of the n-dimensional Euclidean space, we give a gyrogroup structure on the set of Lorentz boosts and compute various metric formulas of Lorentz boosts such as the Riemannian trace metric, Hilbert projective metric, fidelity and Wasserstein distance in terms of Lorentz gamma factors.

Keywords: Lorentz boost, Einstein gyrogroup, rapidity metric, Hilbert projective metric, fidelity, Wasserstein distance.

MSC: 22E43, 20N05, 81P15.

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