A positive and local formalism for quantum theory

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The general boundary formulation (GBF) is an atemporal, but spacetime local formulation of quantum theory. Usually it is presented in terms of the amplitude formalism, which, in the presence of a background time, recovers the pure state formalism of the standard formulation of quantum theory. After reviewing the essentials of the amplitude formalism I will introduce a new "positive formalism", which recovers instead a mixed state formalism. This allows to define general quantum operations within the GBF and opens it to quantum information theory. Moreover, the transition to the positive formalism eliminates operationally irrelevant structure, making the extraction of measurement probabilities more direct. As a consequence, the probability interpretation takes on a remarkably simple and compelling form. I shall describe implications of the positive formalism, both for our understanding of quantum theory and for the practical formulation of quantum theories. I also observe a certain convergence with Lucien Hardy's operator tensor formulation of quantum theory, on which I hope to comment