Implementing Bell causality in Quantum Sequential Growth

TL;DR

This paper investigates how different implementations of quantum Bell causality affect the algebraic structure of quantum sequential growth models in causal set quantum gravity, revealing conditions for commutativity and challenges in representation.

Contribution

It analyzes the impact of various operator orderings on the algebraic structure of QSG, identifying conditions for commutativity and exploring representation limitations.

Findings

Certain operator orderings lead to commutative algebras.

New commutation relations constrain the algebra but do not ensure commutativity.

Pauli matrix representations are inconsistent, suggesting higher-dimensional representations are needed.

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 non-commuting transition operators. Assuming a non-singular dynamics we show that for the two most natural choices of operator orderings for the QBC, the transition operator algebra 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 the algebra but do not imply commutativity. On the other hand, if any of the generators of the ``antichain subalgebra'' belongs to its center, then this implies commutativity of the full algebra. 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 non-trivial d=2 representation, we find that a Pauli matrix representation of the generators of the antichain subalgebra leads to inconsistencies, implying that if a non-trivial representation exists, it must be higher dimensional. Our work can be viewed as a first step towards finding a non-commutative realisation of QSG.