Complete sets of observables

1969 ◽  
Vol 47 (10) ◽  
pp. 1083-1093 ◽  
Author(s):  
Eduard Prugovečki
Keyword(s):  
Hilbert Space ◽  
Cyclic Vector ◽  
Space Formalism ◽  
Complete Set ◽  
Complete Sets

The concept of a complete set of observables is formulated in the Hilbert space formalism in a manner which agrees with Dirac's original intuitive formulation. Theorems are derived which relate the completeness of a commuting set of observables to the existence of a cyclic vector. It is shown that any operator which commutes with a complete set of observables is a function of those observables.

Dense Subalgebras of Left Hilbert Algebras

1982 ◽  
Vol 34 (6) ◽  
pp. 1245-1250 ◽  
Author(s):  
A. van Daele
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Cyclic Vector ◽  
Quantum Field ◽  
Von Neumann ◽  
Hilbert Algebras

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).

scholarly journals Hilbert space multidimensional modelling of continuous measurements

2019 ◽  
Vol 377 (2157) ◽  
pp. 20190142
Author(s):  
J. R. Busemeyer ◽  
Z. Wang
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Data Fusion ◽  
State Vector ◽  
Joint Distribution ◽  
Context Effect ◽  
Theme Issue ◽  
Multiple Contexts ◽  
Complete Set ◽  
Multidimensional Modelling

Data fusion problems arise when a researcher needs to analyse results obtained by measuring empirical variables under different measurement contexts. A context is defined by a subset of variables taken from a complete set of variables under investigation. Multiple contexts can be formed from different subsets, which produce a separate distribution of measurements associated with each context. A context effect occurs when the distributions produced by the different contexts cannot be reproduced by marginalizing over a complete joint distribution formed by all the variables. We propose a Hilbert space multidimensional theory that uses a state vector and measurement operators to account for multiple distributions produced by different contexts. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

A two-parameter eigenvalue problem involving complex potentials

1990 ◽  
Vol 116 (1-2) ◽  
pp. 177-191
Author(s):  
M. Faierman
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Complex Potentials ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Parameter System ◽  
Complete Set ◽  
Necessary And Sufficient ◽  
Two Parameter

SynopsisWe consider a two-parameter system of ordinary differential equations of the second order involving complex potentials and show that, unlike the case of real potentials, the eigenfunctions of the system do not necessarily form a complete set in the usual Hilbert space associated with the problem. We also give a necessary and sufficient condition for the eigenfunctions to be complete. Finally, we establish some results concerning the eigenvalues of the system.

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