I present some general ideas about quantum entanglement in relativistic quantum field theory, especially entanglement in the physical vacuum. Here, entanglement is defined between different single particle states (or modes), parametrized either by energy-momentum together with internal degrees of freedom, or by spacetime coordinate together with the component index in the case of a vector or spinor field. In this approach, the notion of entanglement between different spacetime points can be established. Some entanglement properties are obtained as constraints from symmetries, e.g., under Lorentz transformation, space inversion, time reverse, and charge conjugation.
Abstract In this paper we study the single particle entanglement induced by a Lorentz boost. We described the particle as a solution of the free Dirac equation, a Dirac bispinor, and compare the induced momentum-spin entanglement with the result obtained in the widely considered framework of relativistic spin 1/2 states. The spin linear entropy for both approaches agree in the ultra relativistic limit. We also verify that the spin-momentum entanglement differers from the spin entropy for the bispinorial case, indicating a true multipartite entanglement involving the degrees of freedom of a Dirac bispinor state: momentum, spin and intrinsic parity. The fact that Dirac bispinors belongs to an irreducible representation of the complete Lorentz group, which also included parity as a symmetry, is the ultimate reason behind such non-trivial structure.
We consider two spacelike separated Dirac particles and construct five invariants under the spinor representations of the local proper orthochronous Lorentz groups. All of the constructed Lorentz invariants are identically zero for product states. The behavior of the Lorentz invariants under local unitary evolutions that act unitarily on any subspace with fixed particle momenta is studied. All of the Lorentz invariants have invariant absolute values on such subspaces if the evolutions are generated by local zero-mass Dirac Hamiltonians. Some of them also for the case of nonzero mass. Therefore, they are considered potential candidates for describing spinor entanglement of two Dirac particles, with either zero or arbitrary mass. Furthermore, their relations to the Wootters concurrence is investigated and their representations in the Foldy-Wouthuysen picture is given.
In several recent papers on entanglement in relativistic quantum systems and relativistic Bell's inequalities, relativistic Bell-type two-particle states have been constructed in analogy to nonrelativistic states. These constructions do not have the form suggested by relativistic invariance of the dynamics. Two relativistic formulations of Bell-type states are shown for massive particles, one using the standard Wigner spin basis and one using the helicity basis. The construction hinges on the use of Clebsch-Gordan coefficients of the Poincar\'e group to reduce the direct product of two unitary irreducible representations (UIRs) into a direct sum of UIRs.
Analysis of the generation of spin entanglement from nonrelativistic QED is presented. The results of entanglement are obtained with relativistic correction to the leading order of
(v/c)2. It is shown that to this order, the degree of entanglement of a singlet state does not change under time evolution, whereas the triplet state can change.
Motivated by the difficulties which persist in the construction of a consistent description of interacting relativistic particles with spin greater than 1, we examine the analogous problem within the framework of nonrelativistic quantum mechanics and find that all of the difficulties vanish in the (Galilei-invariant) limit. It is found that a unique, first-order, Galilei-covariant wave equation describing massive, spin-
s particles follows from general invariance assumptions and a minimality condition on the number of components of the wave function. The minimal theory has
6s+1 components (
2s+1 of which are independent) and admits a consistent quantum-mechanical interpretation. An external electromagnetic field interaction is introduced via the minimal-coupling replacement, and, in contrast with the relativistic case, a consistent theory emerges for arbitrary spin. The Galilean spin-
s particles so described are found to possess only an electric charge and a magnetic dipole with a
g factor of
s1. The Galilei-invariant addition of arbitrary moment terms is also described. The extension of the formalism to second-quantized spin-
s fields is discussed, and it is found that in that case, too, the difficulties are peculiar to the relativistic case. Reasons for the simplicity of the Galilei case are presented. Finally, for the sake of completeness, the first-order form of the Schr\"odinger theory is presented, as well as examples of theories which violate the minimality condition.