Maps for Lorentz transformations of spin (6 pages)

Elcuno A. Perelstein, Maxim E. Perelstein

Thomas F. Jordan, Anil Shaji, E. C. G. Sudarshan

Lorentz transformations of spin density matrices for a particle with positive mass and spin $1∕2$ are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are not independent, the maps generally are not completely positive and act in limited domains. States with two momentum values are considered, so the maps are for the spin qubit correlated with the qubit made from the two momentum values, and results from the open quantum dynamics of two coupled qubits can be applied. Inverses are used to show that every Lorentz transformation completely removes the spin polarization, and so completely removes the information, from a number of spin density matrices. The size of the spin polarization that is removed is calculated for particular cases.

Hai-Dong Li, Jiangfeng Du

Badriram Rajagopalan

This paper extends the derivation of the Lorentz transformation from count-based observables to the quantum level. First, each ingredient of the Bondi derivation is mapped onto the Page-Wootters (PaW) framework, and the k-factor is extracted as a ratio of PaW clock POVM eigenvalues — completing the derivational chain from Ĥ|Ψ⟩ = 0 to t′ = γ(t − vx/c²). Second, the k-factor is promoted to an operator k̂ = (Ĥ_cm + p̂c)/(mc²), revealing a two-sector decomposition of the Lorentz transformation: a time dilation sector (k̂ + k̂⁻¹ = 2E/(mc²), nonlinear in momentum) and a simultaneity sector (k̂ − k̂⁻¹ = 2pc/(mc²), linear in momentum). Third, the Bondi decomposition is applied to the proper time ω₀τ = ω₀Lm/p of a clock traversing baseline L, showing that the standard result separates into a time dilation contribution ω₀(Lm/p + Lp/(mc²)) and a simultaneity contribution −ω₀Lp/(mc²), which sum to the proper time. The two sectors have different behavior under quantum averaging: the simultaneity sector, being linear in p, is protected — its mean receives no corrections from momentum spread σ_p for any symmetric distribution. The time dilation sector, being nonlinear, produces mean corrections at order σ_p²/(m²c²). This asymmetry clarifies the structure of the Pikovski decoherence mechanism: momentum-dependent dephasing of internal clocks originates entirely from the time dilation sector's nonlinearity, while the simultaneity sector contributes a deterministic offset protected by linearity. All corrections vanish in the classical limit σ_p → 0, recovering the companion paper's Bondi derivation exactly.

Fabián H. Gaioli, Edgardo T. Garcia Alvarez

Thomas F. Jordan, Anil Shaji, E. C. G. Sudarshan

Simple examples are presented of Lorentz transformations that entangle the spins and momenta of two particles with positive mass and spin $1∕2$. They apply to indistinguishable particles, produce maximal entanglement from finite Lorentz transformations of states for finite momenta, and describe entanglement of spins produced together with entanglement of momenta. From the entanglements considered, no sum of entanglements is found to be unchanged.