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Implementing Bell causality in Quantum Sequential Growth

Ritesh Srivastava, Sumati Surya

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.

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 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.

Paper Structure

This paper contains 15 sections, 12 theorems, 147 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The Sequential Growth Paradigm
  3. Preliminaries
  4. Sequential growth dynamics(SGD)
  5. Atomisation and Decimation
  6. Classical Sequential Growth (CSG) Models
  7. Quantum Sequential growth
  8. Definition
  9. Quantum sequential growth dynamics
  10. Implementing Quantum Bell Causality
  11. Time Ordered Bell Causality
  12. Non-Time Ordered Bell Causality (NTOBC)
  13. Causal Past Ordered (CPOBC)
  14. Discussion
  15. Acknowledgements

Key Result

Lemma 2.1

Every transition probability $A_n^i:c_n \rightarrow c_{n+1}^i$ at stage $n$ can be expressed in terms of the gregarious transition probability $G_n:c_n \rightarrow c_{n+1}^g$ as well as lower stage probabilities. Here $i \in \mathcal{J}(c_n)$ the index set of children of $c_n$.

Figures (13)

  • Figure 1: Labelled Poscau $\mathcal{P}$.
  • Figure 2: Poscau $\tilde{\mathcal{P}}$.
  • Figure 3: Ordering of transitions
  • Figure 4: Bell related pairs of transitions
  • Figure 5: Atomisation takes $c_n$ to $a_n$, along an atomisation path $S(c_n)$ while decimation takes $c_n$ back to $a_k$, along the related decimation path $\Gamma(c_n)$ where $k$ is the number of minimal elements in $a_n$. While $S(c_n)$ is not unique for a given $c_n$, it determines $\Gamma(c_n)$.
  • ...and 8 more figures

Theorems & Definitions (23)

  • Definition
  • Definition
  • Lemma 2.1: Lemma 1 of csgone
  • proof
  • Lemma 2.2: Lemma 2 of csgone
  • proof
  • Definition
  • Definition
  • Lemma 3.1
  • Lemma 3.2
  • ...and 13 more