Lorentz invariant quantum concurrence for $\textit{SU}(2) \otimes \textit{SU}(2)$ spin–parity states

Abstract

The quantum concurrence of $\textit{SU}(2) \otimes \textit{SU}(2)$ spin–parity states is shown to be invariant under $\textit{SO}(1,3)$ Lorentz boosts and O(3) rotations when the density matrices are constructed in consonance with the covariant probabilistic distribution of Dirac massive particles. Similar invariance properties are obtained for the quantum purity and for the trace of unipotent density matrix operators. The reported invariance features—obtained in the scope of the $\textit{SU}(2) \otimes \textit{SU}(2)$ corresponding to just one of the inequivalent representations enclosed by the $\textit{SL}(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})$ symmetry—set a more universal and kinematical-independent meaning for the quantum entanglement encoded in systems containing not only information about spin polarization but also the correlated information about intrinsic parity. Such a covariant framework is used for computing the Lorentz invariant spin–parity entanglement of spinorial particles coupled to a magnetic field, through which the extensions to more general Poincaré classes of spinor interactions are straightforwardly depicted.

Appendix: The group \(\textit{SU}(2)\otimes \textit{SU}(2)\) as a subgroup of \(\textit{SL}(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})\) and the spin–parity intrinsic structure of Dirac bispinors

The group representations of $sl(2,{\mathbb {C}})\oplus sl(2,{\mathbb {C}})$—the Lie algebra of the $\textit{SL}(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})$ Lie group—are irreducible, since they are tensor products of linear complex representations of $sl(2,{\mathbb {C}})$. Given that $\textit{SU}(2)\otimes \textit{SU}(2) \subset SL(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})$, the unitary irreducible representations (irreps) of the $\textit{SU}(2)\otimes \textit{SU}(2)$ are built through tensor products between unitary representations of $\textit{SU}(2)$. Such an one-to-one correspondence with the $\textit{SL}(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})$ simply connected group imposes a single correspondence with the $sl(2,{\mathbb {C}})\oplus sl(2,{\mathbb {C}})$ algebra. In fact, the existence of inequivalent representations of $\textit{SU}(2) \otimes \textit{SU}(2)$ follows exactly from such one-to-one correspondences. The point is that inequivalent representations (for instance, the one for the chiral basis and another one for the spin–parity basis) do not correspond to the complete set of representations of $\textit{SL}(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})$, and therefore, each of them, independently, does not exhibit a one-to-one correspondence with the homogeneous Lorentz transformations that compose the algebra of the $\textit{SO}(1,3)$ group. The chiral basis described by a double-doublet representation of $\textit{SU}(2)\otimes \textit{SU}(2)$, for instance, maps a subset of transformations of the $\textit{SO}(4) \equiv SO(3)\otimes SO(3)$ group, as for instance, those which include the double-covering rotations.

Turning back to our point, since the $\textit{SU}(2)\otimes \textit{SU}(2)$ transformations can be mapped into a subset of $\textit{SL}(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})$, one may choose two inequivalent subsets of $\textit{SU}(2)$ generators, such that $\textit{SU}(2)\otimes \textit{SU}(2) \subset SL(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})$, with each group transformation generator having its own irrep. In an overall context, despite the effectiveness of the irreps of the Poincaré group, in the Lorentz covariant Hamiltonian formulation of quantum mechanics, one has to pay attention to the inclusion of masses in the relativistic formalism described by the Dirac equation. It requires the inclusion of the parity symmetry and the equalization of its role with the helicity (spin one-half projection, $\hat{{\varvec{e}}}_p\cdot \hat{\varvec{\sigma }} \sim {\hat{\sigma }}_z$) symmetry, as an accomplished $\textit{SU}(2)$ symmetry. Under such conditions, one designates the fundamental representation of the $SU_{\xi }(2)$ as a spinor-like object $\xi$ described by $({\pm },\,0)$, which transforms as a $SU_{\xi }(2)$doublet (2-dim) of parity quantum numbers, ${\pm }$, and as a singlet (1-dim), which is transparent under any $SU_{\chi }(2)$ transformation. Reciprocally, the fundamental object of the $SU_{\chi }(2)$, a typical spinor $\chi$ described by (0, 1/2), transforms as a $SU_{\chi }(2)$doublet of spin quantum numbers, ${\pm } 1/2$, and as a singlet of the $SU_{\xi }$.⁴

The meaning of the above-mentioned quantities related to parity and helicity can be clarified by noticing that the total parity operator ${\hat{P}}$ acts on the direct product $\left| {\pm } \right\rangle \otimes \left| \chi _s{({\varvec{p}})}\right\rangle$ as

$$\begin{aligned} {\hat{P}}\left( \left| {\pm } \right\rangle \otimes \left| \chi _s{({\varvec{p}})}\right\rangle \right) ={\pm } \left( \left| {\pm } \right\rangle \otimes \left| \chi _s(-{\varvec{p}})\right\rangle \right) , \end{aligned}$$

and, for instance, it corresponds to the Kronecker product of two operators, ${\hat{P}}^{(P)}\otimes {\hat{P}}^{(S)}$, where ${\hat{P}}^{(P)}$ is the intrinsic parity (with two eigenvalues, ${\hat{P}}^{(P)}\left| {\pm } \right\rangle ={\pm } \left| {\pm } \right\rangle$) and ${\hat{P}}^{(S)}$ is the spatial parity (with ${\hat{P}}^{(S)}\chi _s \left( {\varvec{p}}\right) =\chi _s \left( -{\varvec{p}}\right)$).

The above construction supports the identification of Dirac Hamiltonian eigenstates with electrons in the double-doublet irrep of the $\textit{SU}(2) \otimes \textit{SU}(2)$. Spatial parity couples quantum states with ${\pm }$ parity and ${\pm } 1/2$ spin quantum numbers given that they are described by irreps of the Poincaré group [46]. To comprehend the covariant behavior of parity, one needs to consider the extended Poincaré group [35, 47] which also accounts for the helicity (spin projection) in a double-doublet $\textit{SL}(2,{\mathbb {C}}) \otimes \textit{SL}(2,{\mathbb {C}})$ representation given by Dirac four-component spinors, the bispinors satisfying the Dirac equation. Reciprocally, the same assertion is applied for the understanding of the factorized covariant behavior of the helicity (spin projection) at such quantum states.