Abstract
In [20, 21] the authors show how it is possible to interpret self-adjoint operators affiliated with a von Neumann algebra \(\mathcal {N}\), as real-valued functions on the projection lattice \(P(\mathcal {N})\) of the algebra. These functions are called q-observable functions . The method of utilising real-valued function on \(P(\mathcal {N})\) to define self-adjoint operators was first introduced in [12] and, independently, in [8]. However, the novelty of the approach defined [20, 21] consists in the fact that these real valued functions are related to both the daseinisation map, central to topos quantum theory, and to quantum probabilities.
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Notes
- 1.
Given two posets (P, ≤ p ) and (Q, ≤ q ), then a map f : P → Q is called monotone (order-preserving) if for all a, b ∈ P, when a ≤ p b then f(a) ≤ q f(b).
- 2.
A conditionally complete lattice is a lattice in which every non-empty bounded subset has a least upper bound and a greatest lower bound. As an example of a conditionally-complete lattice one may take the set of all real numbers with the usual order.
- 3.
A lattice is complemented if every element a has a complement a ⊥. It is orthocomplemented if it is equipped with an involution that sends each element to a complement. An orthomodular lattice is an orthocomplemented lattice such that a ≤ c implies that a ∨ (a ⊥∧ c) = c.
- 4.
Here the notation A −1 is only symbolic since there may not exist any function A whose inverse is A −1. We used this notation to resemble the Definition in 5.5.2.
References
H. Comman, Upper regularization for extended self-adjoint operators. J. Oper. Theory 55(1), 91–116 (2006)
H.F. de Groote, Observables IV: the presheaf perspective (2007). arXiv:0708.0677 [math-ph]
A. Doering, B. Dewitt, Self-adjoint operators as functions I: lattices, Galois connections, and the spectral order (2012). arXiv:1208.4724 [math-ph]
A. Doering, B. Dewitt, Self-adjoint operators as functions II: quantum probability. arXiv:1210.5747 [math-ph]
C. Flori, A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol. 868 (Springer, Heidelberg, 2013)
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Flori, C. (2018). Interpreting Self-Adjoint Operators as q-Functions. In: A Second Course in Topos Quantum Theory. Lecture Notes in Physics, vol 944. Springer, Cham. https://doi.org/10.1007/978-3-319-71108-9_5
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