Background Independence and Quantum Causal Structure

TL;DR

This paper develops a background-independent quantum formalism that allows for indefinite causal structures, addressing a key conceptual challenge in unifying quantum mechanics and relativity while maintaining operational measurement consistency.

Contribution

It introduces a background-independent process matrix framework that preserves indefinite causal structures and explores symmetry constraints without relying on fixed spacetime backgrounds.

Findings

Nontrivial indefinite causal structures are compatible with background independence.

Local operations can be recovered by encoding reference frames into system states.

Permutation invariance leads to symmetry constraints akin to superselection rules.

Abstract

One of the key ways in which quantum mechanics differs from relativity is that it requires a fixed background reference frame for spacetime. In fact, this appears to be one of the main conceptual obstacles to uniting the two theories. Additionally, a combination of the two theories is expected to yield non-classical, or "indefinite", causal structures. In this paper, we present a background-independent formulation of the process matrix formalism - a form of quantum mechanics that allows for indefinite causal structure - while retaining operationally well-defined measurement statistics. We do this by postulating an arbitrary probability distribution of measurement outcomes across discrete "chunks" of spacetime, which we think of as physical laboratories, and then requiring that this distribution be invariant under any permutation of laboratories. We find (a) that one still obtains nontrivial, indefinite causal structures with background independence, (b) that we lose the idea of local operations in distinct laboratories, but can recover it by encoding a reference frame into the physical states of our system, and (c) that permutation invariance imposes surprising symmetry constraints that, although formally similar to a superselection rule, cannot be interpreted as such.