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Spectral presheaves as quantum state spaces.

Andreas Döring1

  • 1Institute of Theoretical Physics I, Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany andreas.doering@fau.de.

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Summary

We introduce a novel generalized state space, the spectral presheaf, for quantum systems. This framework geometrically unifies quantum variables and time evolution generators, overcoming Kochen-Specker theorem limitations.

Keywords:
categoryflowoperator algebrastate spacetopos

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Area of Science:

  • Quantum mechanics
  • Operator algebras
  • Mathematical physics

Background:

  • Quantum systems are described by operator algebras.
  • The Kochen-Specker theorem poses challenges for defining quantum state spaces.
  • Self-adjoint operators have a dual role as observables and generators of time evolution.

Purpose of the Study:

  • To provide a generalized state space for quantum systems.
  • To formulate quantum time evolution within this new framework.
  • To reveal the geometric mirroring of operator roles in quantum theory.

Main Methods:

  • Construction of the spectral presheaf as a generalized state space.
  • Formulation of time evolution using Hamiltonian flows on the spectral presheaf.
  • Geometric analysis of the spectral presheaf structure.

Main Results:

  • A generalized quantum state space (spectral presheaf) is provided, irrespective of the Kochen-Specker theorem.
  • Quantum time evolution is successfully formulated as Hamiltonian flows on this space.
  • The spectral presheaf structure geometrically reflects the dual nature of self-adjoint operators.

Conclusions:

  • The spectral presheaf offers a unified geometric framework for quantum mechanics.
  • This approach reconciles the Kochen-Specker theorem with a comprehensive state space.
  • It provides new insights into the fundamental roles of operators in quantum dynamics.