Perturbative decoherence of relativistic entanglement

Abstract A consistent theory of quantum entanglement requires that the constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. A covariant theory of relativistic spin and entanglement has been given by Horwitz and Arshansky, in which representations of O (3, 1) are induced with respect to an arbitrarily selected timelike unit vector n µ . In this paper, we construct an induced relativistic representation of spin on an extended phase space {( x µ , p µ ), ( ζ µ , π µ )}, in which we associate the unit vector n µ with the momentum π µ , thus providing a dynamical interpretation for this new quantity. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states. As in non-relativistic quantum theory the magnetic field couples to the rotation generators of SU (2) ⊂ SL (2, C ), however the electric field couples to the non-compact and anti-Hermitian boost generators. We show that to first order in perturbation theory, a constant external magnetic field will produce a perturbative contribution to the total mass (the eigenvalue of the Lorentz scalar Hamiltonian) or a transition between states in the same representation of SL (2, C ), and so will not disrupt a singlet state. However, an electric field normal to π µ can produce a perturbative transition between states in inequivalent representations of SL (2, C ), and therefore potentially cause decoherence of the relativistic entanglement.