The Bertlmann–Martin Inequality and Spin Degrees of Freedom

Abstract

The Bertlmann–Martin inequality based on the dipole sum rule is revisited taking into account the spin degrees of freedom. We consider 1 and 2 particles of spin 1/2 in a mean field, adding a spin dependent interaction. The derivation of the inequality relies on the closure relation. We discuss the effect of the Pauli principle, and the restrictions it imposes on the use of the closure relation. The problem is exemplified by a simple model based on harmonic forces. Moreover, in the 2 particle case, the model we use is separable in the relative and centre of mass coordinates. In this case, we show that for operators connecting only singlet states, their sum rule can be calculated in the usual way, i.e. via the double commutator of this operator with the Hamiltonian. An upper bound can also be obtained by using the Bertlmann–Martin technique. This is not possible for operators involving a transition between singlet and triplet states.