Towards a Measurement Theory for Off-Shell Quantum Fields

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.