Abstract
We present a compositional algebraic framework to describe the evolution of quantum fields in discretised spacetimes. We show how familiar notions from Relativity and quantum causality can be recovered in a purely order-theoretic way from the causal order of events in spacetime, with no direct mention of analysis or topology. We formulate theory-independent notions of fields over causal orders in a compositional, functorial way. We draw a strong connection to Algebraic Quantum Field Theory (AQFT), using a sheaf-theoretical approach in our definition of spaces of states over regions of spacetime. We introduce notions of symmetry and cellular automata, which we show to subsume existing definitions of Quantum Cellular Automata (QCA) from previous literature. Given the extreme flexibility of our constructions, we propose that our framework be used as the starting point for new developments in AQFT, QCA and more generally Quantum Field Theory.
Highlights
Like much of classical physics, the study of Relativity and quantum field theory has deep roots in topology and geometry
When quantum cellular automata are considered in a relativistic context—e.g. as discrete models of quantum field theories—the requirement of locality is meant to capture the idea that the action of the automaton should respect the causal structure of spacetime
In an effort to connect to Algebraic Quantum Field Theory (AQFT), we have constructed complex spaces of states over regions of spacetime and discussed how the associated information redundancy can be reduced in selected cases
Summary
Like much of classical physics, the study of Relativity and quantum field theory has deep roots in topology and geometry. While the result by Malament guarantees that future-and-past–distinguishing manifolds (up to conformal equivalence) can be identified with their causal orders, it does not provide a characterization of which partial orders arise as causal orders on manifolds (or restrictions thereof to manifold subsets) This lack of exact correspondence between topology and order is the motivation behind many past and current lines of enquiry. The question whether a causal set can always be (suitably) embedded as a discrete subset of a Lorentzian manifold is central to the programme and—as far as we are aware—one which is still to be completely answered [7] When it comes to incorporating quantum fields into the spacetimes, efforts have mostly been focused in three directions: algebraic approaches, topological approaches and quantum cellular automata.
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