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      Causal categories: relativistically interacting processes

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          Abstract

          A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a `causal category'. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.

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          Quantum Mechanics and Hilbert Space

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            Quantum causal histories

            Quantum causal histories are defined to be causal sets with Hilbert spaces attached to each event and local unitary evolution operators. The reflexivity, antisymmetry, and transitivity properties of a causal set are preserved in the quantum history as conditions on the evolution operators. A quantum causal history in which transitivity holds can be treated as ``directed'' topological quantum field theory. Two examples of such histories are described.
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              Author and article information

              Journal
              29 July 2011
              Article
              10.1007/s10701-012-9646-8
              1107.6019
              2849fb25-bbdd-4952-bd0b-a1cca616e829

              http://arxiv.org/licenses/nonexclusive-distrib/1.0/

              History
              Custom metadata
              Found. Phys. 43(4), 458-501 (2012)
              43 pages, lots of figures
              gr-qc math.CT quant-ph

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