Maps for Lorentz transformations of spin

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.

Figure 1

(Color online) $\mid ⟨\mathbf{\Sigma }{⟩}^{\Lambda }\mid$ as a function of the velocity $v$ of the Lorentz transformation, for ${\mathbf{p}}_1=-{\mathbf{p}}_2$ perpendicular to the direction of the Lorentz transformation and $\mid {\mathbf{p}}_1\mid ∕m=10$, with ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ perpendicular to the axes of the Wigner rotations, $r_1$ and $r_2$ both $1∕2$, and ${\mathbf{r}}_1$ in the same direction as ${\mathbf{r}}_2$. On the right are the vectors $W_1\left({\mathbf{r}}_1\right)$ (blue dotted arrows), $W_2\left({\mathbf{r}}_2\right)$ (red dashed arrows), and $⟨\mathbf{\Sigma }{⟩}^{\Lambda }$ (black solid arrows) for different values of $v$.

Figure 2

(Color online) The same as in Fig. 1 except that ${\mathbf{r}}_1$ is perpendicular to ${\mathbf{r}}_2$.

Figure 3

(Color online) The size of the $⟨\mathbf{\Sigma }{⟩}^{\Lambda }$ described by Eq. (4.8), that a Lorentz transformation can completely remove, or produce starting from zero, as a function of the velocity $v$ of the Lorentz transformation, for different values of ${\mathbf{p}}_1\bullet {\mathbf{p}}_2∕\mid {\mathbf{p}}_1{\mid }^2$ with ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$ along the same line perpendicular to the direction of the Lorentz transformation and ${\mathbf{p}}_1∕m=10$, with ${\mathbf{r}}_1$ perpendicular to the axis of the Wigner rotation and $r_1=1∕2$.