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An order for quantum observables

In: Mathematica Slovaca, vol. 56, no. 5
Stanley Gudder

Details:

Year, pages: 2006, 573 - 589
About article:
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators $S(H)$ on a complex Hilbert space $H$. The usual order $A≤ B$ on $S(H)$ is determined by assuming that the expectation of $A$ is not greater than the expectation of $B$ for every state of the system. We may think of $≤$ as a numerical order on $S(H)$. In this article we introduce a new order $\preceq$ on $S(H)$ that may be interpreted as a logical order. This new order is determined by assuming that $A\preceq B$ if the proposition that $A$ has a value in $Δ$ implies the proposition that $B$ has a value in $Δ$ for every Borel set $Δ$ not containing $0$. We give various characterizations of this order and show that it is generated by an orthosum $\oplus$ that endows $S(H)$ with the structure of a generalized orthoalgebra. We also show that the usual order $≤$ cannot be generated by an orthosum. We demonstrate that if we restrict $\oplus$ to an interval $[0,A]\subseteqS (H)$, then we obtain a structure that is isomorphic to an orthomodular lattice of projections on $H$. The lattice structure of $S (H)$ is investigated and unlike $(S(H),{≤})$ it is shown that $(S(H),{\preceq})$ is a near-lattice in the sense that if $A,B\preceq C$, then $A\wedge B$ and $A\vee B$ exist. Moreover, we show that if $\dim (H)<∞$, then $A\wedge B$ always exists. We also consider the commutative case in which observables are represented by fuzzy random variables.
How to cite:
ISO 690:
Gudder, S. 2006. An order for quantum observables. In Mathematica Slovaca, vol. 56, no.5, pp. 573-589. 0139-9918.

APA:
Gudder, S. (2006). An order for quantum observables. Mathematica Slovaca, 56(5), 573-589. 0139-9918.