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Projective Limits of State Spaces IV. Fractal Label Sets

Abstract : Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski (1977) to represent quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces (see Lanéry (2016) [1] for a concise introduction to this formalism). One can thus bypass the need to select a vacuum state for the theory, and still be provided with an explicit and constructive description of the quantum state space, at least as long as the label set indexing the projective structure is countable. Because uncountable label sets are much less practical in this context, we develop in the present article a general procedure to trim an originally uncountable label set down to countable cardinality. In particular, we investigate how to perform this tightening of the label set in a way that preserves both the physical content of the algebra of observables and its symmetries.
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Submitted on : Wednesday, October 31, 2018 - 10:14:39 AM
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Suzanne Lanéry, Thomas Thiemann. Projective Limits of State Spaces IV. Fractal Label Sets. J.Geom.Phys., 2018, 123, pp.127-155. ⟨10.1016/j.geomphys.2017.08.008⟩. ⟨hal-01909403⟩



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