Abstract
One of the most fundamental open problems in physics is the unification of general relativity and quantum theory to a theory of quantum gravity. An aspect that might become relevant in such a theory is that the dynamical nature of causal structure present in general relativity displays quantum uncertainty. This may lead to a phenomenon known as indefinite or quantum causal structure, as captured by the process matrix formalism. Due to the generality of that framework, however, for many process matrices there is no clear physical interpretation. A popular approach towards a quantum theory of gravity is the Page-Wootters formalism, which associates to time a Hilbert space structure similar to spatial position. By explicitly introducing a quantum clock, it allows to describe time-evolution of systems via correlations between this clock and said systems encoded in history states. In this paper we combine the process matrix framework with a generalization of the Page-Wootters formalism in which one considers several agents, each with their own discrete quantum clock. We describe how to extract process matrices from scenarios involving such agents with quantum clocks, and analyze their properties. The description via a history state with multiple clocks imposes constraints on the implementation of process matrices and on the perspectives of the agents as described via causal reference frames. While it allows for scenarios where different definite causal orders are coherently controlled, we explain why certain noncausal processes might not be implementable within this setting.
Highlights
Indefinite causal structure is an extension of the usual notion of causal structure that is expected to become relevant in quantum gravity: In general relativity, causal structure is dynamical instead of fixed and attributing quantum properties [1–8] would imply the existence of exotic causal structures, like superpositions of space-times and superpositions of the order of events
Can the formalism account for more general causal structures such as coherently controlled causal orders or even those that lead to the violation of causal inequalities? To approach this question, in the present paper, we present a general definition of what it means for a history state to implement a pure process matrix [10], for the case of finite dimensional systems and many clocks
In this paper we showed how the Page-Wootters approach and the process matrix formalism may be combined to give a history state description of noncausal processes
Summary
Indefinite causal structure is an extension of the usual notion of causal structure that is expected to become relevant in quantum gravity: In general relativity, causal structure is dynamical instead of fixed and attributing quantum properties [1–8] would imply the existence of exotic causal structures, like superpositions of space-times and superpositions of the order of events. They showed how the Page-Wootters formalism can recover the so-called gravitational quantum switch [5] Their approach works for important examples, but it is not clear in general which process (if any) is implemented by a given history state, or what is the set of noncausal processes that can be implemented within such a framework. The observations in [35] motivate us to investigate the relation between causal order and several quantum clocks in further detail, and develop a full framework that systematically combines the process matrix and the Page-Wootters formalism. While the Page-Wootters formalism with many clocks can enable the extraction of processes with definite causal order and quantum controlled causal order, it provides insights into why some processes might not be realizable within the framework.
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