Implementing Bell causality in quantum sequential growth

Abstract

We explore different implementations of the quantum Bell causality (QBC) condition in the quantum sequential growth (QSG) dynamics of causal set quantum gravity, for noncommuting transition operators. Assuming a nonsingular dynamics we show that for the two most natural choices of operator orderings for the QBC, the transition operator algebra 𝒜 reduces to a commutative one. As a third choice, we take the operator ordering to depend on the size of the precursor set. We find several new commutation relations which further constrain 𝒜 but do not imply commutativity. On the other hand, if any of the generators of the “antichain subalgebra” 𝒬 belongs to its center, then this implies commutativity of 𝒜. The complexity of the algebra prevents us from obtaining a general form for the transition operators, which hinders computability. In an attempt to construct the simplest nontrivial 𝑑 =2 representation of 𝒜, we find that a Pauli matrix representation of the generators of 𝒬 leads to inconsistencies. Our work can be viewed as a first step toward finding a noncommutative realization of QSG.