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The Logic of Bundles

Harding, John ; Yang, Taewon

International Journal of Theoretical Physics, 2015, Vol.54(12), pp.4601-4614 [Periódico revisado por pares]

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  • Título:
    The Logic of Bundles
  • Autor: Harding, John ; Yang, Taewon
  • Assuntos: Orthomodular poset ; Quantum logic ; Vector bundle ; Categorical quantum mechanics ; Decompositions ; Topological orthomodular poset ; State ; Automorphism
  • É parte de: International Journal of Theoretical Physics, 2015, Vol.54(12), pp.4601-4614
  • Descrição: Since the work of Crown (J. Natur. Sci. Math. 15 (1–2), 11–25 1975) in the 1970’s, it has been known that the projections of a finite-dimensional vector bundle E form an orthomodular poset ( omp ) P ( E ) $\mathcal {P}(E)$ . This result lies in the intersection of a number of current topics, including the categorical quantum mechanics of Abramsky and Coecke (2004), and the approach via decompositions of Harding (Trans. Amer. Math. Soc. 348 (5), 1839–1862 1996). Moreover, it provides a source of omp s for the quantum logic program close to the Hilbert space setting, and admitting a version of tensor products, yet having important differences from the standard logics of Hilbert spaces. It is our purpose here to initiate a basic investigation of the quantum logic program in the vector bundle setting. This includes observations on the structure of the omp s obtained as P ( E ) $\mathcal {P}(E)$ for a vector bundle E , methods to obtain states on these omp s, and automorphisms of these omp s. Key theorems of quantum logic in the Hilbert setting, such as Gleason’s theorem and Wigner’s theorem, provide natural and quite challenging problems in the vector bundle setting.
  • Idioma: Inglês

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