Abstract
In this study, we develop quantum measurement theory for quantum systems described by C∗-algebras. This is the first step to establish measurement theory for interacting quantum fields with off-shell momenta. Unlike quantum mechanics (i.e., quantum systems with finite degrees of freedom), measurement theory for quantum fields is still in development because of the difficulty of quantum fields that are typical quantum systems with infinite degrees of freedom. Furthermore, the mathematical theory of quantum measurement is formulated in the von Neumann algebraic setting in previous studies. In the paper, we aim to extend the applicable area of quantum measurement theory to quantum systems described by C∗-algebras from a mathematical viewpoint, referring to the sector theory that is related to symmetry and based on the theory of integral decomposition of states. In particular, we define central subspaces of the dual space of a C∗-algebra and use them to define instruments. This attempt makes the connection between measurement theory and sector theory explicit and enables us to understand the macroscopic nature and the physical meaning of measurement.
Highlights
In this study, we develop a measurement theory for quantum systems described by C∗-algebras
Interacting quantum fields assumed in this study are quantum systems with infinite degrees of freedom and with off-shell momenta, whose observables are given by self-adjoint elements of C∗-algebras
In order to formulate the measurement theory for quantum systems described by C∗algebras, the more general case compared to von Neumann algebras, we believe that it is necessary to integrate a completely positive instrument and the sector theoretical treatment
Summary
We develop a measurement theory for quantum systems described by C∗-algebras. The theory of completely positive instruments in quantum systems with infinite degrees of freedom described by the general von Neumann algebra has recently been developed in [19,20]. In order to formulate the measurement theory for quantum systems described by C∗algebras, the more general case compared to von Neumann algebras, we believe that it is necessary to integrate a completely positive instrument and the sector theoretical treatment. A measurement is a physical process that leads to the situation wherein different output values of the measuring apparatus correspond to mutually disjoint states of the composite system From this viewpoint, a measuring process, a quantum mechanical modeling of the measurement, is important historically and theoretically, but it should not necessarily be the first consideration in establishing the physical meaning and description of the measurement.
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