Background Independence and Quantum Causal Structure

Abstract

One of the key ways in which quantum mechanics differs from general 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 '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 imposing that the probabilities arising in the formalism-which we ascribe to measurement outcomes across the points of a discrete spacetime-be invariant under permutations of spacetime points. 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.