Quantum States and Measures on the Spectral Presheaf

After a brief introduction to the spectral presheaf, which serves as an analogue of state space in the topos approach to quantum theory, we show that every state of the von Neumann algebra of physical quantities of a quantum system determines a certain measure on the spectral presheaf of the system. The so-called clopen subobjects of the spectral presheaf play the role of measurable sets. Measures on the spectral presheaf can be characterised abstractly, and the main result is that every abstract measure induces a unique state of the von Neumann algebra. Finally, we show how quantum-theoretical expectation values can be calculated from measures associated to quantum states.

[1]  Observables I: Stone Spectra , 2005, math-ph/0509020.

[2]  C. J. Isham,et al.  Topos Perspective on the Kochen–Specker Theorem: IV. Interval Valuations , 2001 .

[3]  I. Moerdijk,et al.  Sheaves in geometry and logic: a first introduction to topos theory , 1992 .

[4]  C. J. Isham,et al.  A Topos Foundation for Theories of Physics: IV. Categories of Systems , 2008 .

[5]  Matthew Tobias Jackson,et al.  A SHEAF THEORETIC APPROACH TO MEASURE THEORY , 2006 .

[6]  C. J. Isham,et al.  A Topos Perspective on the Kochen-Specker Theorem II. Conceptual Aspects and Classical Analogues , 1998 .

[7]  A. Gleason.  Measures on the Closed Subspaces of a Hilbert Space , 1957 .

[8]  C. J. Isham,et al.  Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations , 1998, quant-ph/9803055.

[9]  C. Isham,et al.  The Representation of Physical Quantities With Arrows δ̆ o ( A ) : Σ → , 2007 .

[10]  Thierry Coquand,et al.  Integrals and valuations , 2008, J. Log. Anal..

[11]  Andreas Döring.  Kochen–Specker Theorem for von Neumann Algebras , 2005 .

[12]  C. Isham,et al.  Spacetime and the Philosophical Challenge of Quantum Gravity , 2001 .

[13]  E. Specker,et al.  The Problem of Hidden Variables in Quantum Mechanics , 1967 .

[14]  M. Takesaki,et al.  Analyticity and the Unruh effect: a study of local modular flow , 2024, Journal of High Energy Physics.

[15]  Andreas Doering,et al.  Topos Theory and ‘Neo-Realist’ Quantum Theory , 2009 .

[16]  Bas Spitters,et al.  A Topos for Algebraic Quantum Theory , 2007, 0709.4364.

[17]  Bernhard Banaschewski,et al.  A globalisation of the Gelfand duality theorem , 2006, Ann. Pure Appl. Log..

[18]  J. Neumann,et al.  The Logic of Quantum Mechanics , 1936 .

[19]  R. Kadison,et al.  Fundamentals of the Theory of Operator Algebras , 1983 .

[20]  Andreas Doering,et al.  “What is a Thing?”: Topos Theory in the Foundations of Physics , 2008, 0803.0417.

[21]  C. J. Isham,et al.  A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory , 2008 .

[22]  C. J. Isham,et al.  A topos foundation for theories of physics: I. Formal languages for physics , 2007 .

[23]  Alain Connes,et al.  Noncommutative geometry , 1994 .

[24]  Shûichirô Maeda,et al.  PROBABILITY MEASURES ON PROJECTIONS IN VON NEUMANN ALGEBRAS , 1989 .

[25]  C. Isham,et al.  Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity , 1999, gr-qc/9910005.

[26]  C. J. Isham,et al.  Topos Perspective on the Kochen=nSpeckerTheorem: III. Von Neumann Algebras as theBase Category , 1999 .