**1. Introduction**

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. The C∗-algebraic approach to quantum fields is not unrelated to the usual approach by field operators. It is a powerful way to remove the difficulty of unbounded operators by making them bounded operators. For example, in a free real Bose field, the exponential *eiφ*(*f*) (or resolvent) of the field operator *φ*(*f*), where *f* is a real function, is a bounded operator, and the collection of them generates a C<sup>∗</sup>- algebra. This study is inspired by the measurement of the quantum field generated by the interaction between the electromagnetic field and electrons at the nanoscale, which is called the dressed photon (DP) phenomenon [1]. It is known to behave completely differently from electromagnetic waves propagating in free space or electromagnetic fields in a uniform medium, and has long been studied as near-field optics. The measurement theory for such systems is still unexplored, and we believe that a framework extending the current theory is necessary. For this reason, we adopt an approach based on both algebraic quantum field theory (AQFT) and quantum measurement theory and their mathematics. There are many examples of the contribution of mathematics to the progress of physical theories, and the introduction of new mathematics contributes greatly to the implementation of new physical concepts. In the study, we will actively use the mathematical framework for conceptual advancement.

In the algebraic formulation of quantum theory, the observable algebra of a quantum system is described by a ∗-algebra X , and a state is described by an expectation functional *ω*on X . From an algebraic point of view, Hilbert space is treated as a secondary one

**Citation:** Okamura, K. Towards a Measurement Theory for Off-Shell Quantum Fields. *Symmetry* **2021**, *13*, 1183. https://doi.org/10.3390/ sym13071183

Academic Editor: Motoichi Ohtsu

Received: 1 June 2021 Accepted: 28 June 2021 Published: 30 June 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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to be used in analysis as needed. For each state *ω*, a Hilbert space is given by the GNS representation (*πω*, H*<sup>ω</sup>*, <sup>Ω</sup>*ω*):

$$
\omega(X) = \langle \Omega\_{\omega} | \pi\_{\omega}(X) \Omega\_{\omega} \rangle \tag{1}
$$

for all *X* ∈ X . C∗-algebras, a special case of ∗-algebra, are used in AQFT [2–4]. Various Hilbert spaces can be given by the GNS representation, and the fact that the representation has a physical meaning as well as the Hilbert space itself primarily promotes the conceptual understanding of the algebraic formulation. The contribution of Haag and Kastler [2] to this progress has been significant. Although there are studies on the algebraic formulation prior to their study, Ref. [2] is probably the first to successfully confront the fact that there are many different representations (depending on the choice of state). In [2], the "physical equivalence" of representations (also called weak equivalence) was used to give a clear meaning to the replacement between equivalent representations. In [5–8], a physical meaning was given to the situation in (A)QFT where different representations chosen by the DHR selection criterion coexist. It is a criterion that selects representations equivalent (through unitary transformations) to the vacuum representation (obtained from the GNS representation from the vacuum state) of the observable algebra on the domain which is spatial to some bounded domain. A representation satisfying this criterion describes a situation in which localized excitations of the quantum field exist. It was shown in [9] that a class (collection) of representations satisfying certain conditions corresponds to a situation where topological charges exist, and that, by using these representations, field algebra F and global gauge group *G* are reconstructed from observable algebra A. This result is known as an iconic result in AQFT. Representations with different charges form their own sectors (with unitary equivalence), which are not only unitarily inequivalent but also mutually "disjoint", giving rise to the so-called "superselection rule". This result is closely related to the representation theory of field operators including the algebra of canonical commutation relations, where unitarily inequivalent representations arise (see [10–13] and references therein). Global gauge group *G* here is an unbroken symmetry, and the results of [9] are not valid for broken symmetries [14]. The extension of Ref. [9]'s results to broken symmetry situations was done in [14,15], and Ojima [16] defined the generalized sector as a "quasi-equivalence class of factor states", allowing for a unified treatment of macroscopic aspects in quantum systems in various contexts, including measurement.

To date, the instrument introduced by Davies and Lewis [17] has contributed greatly to the development of quantum measurement theory. They introduced instruments from a statistical viewpoint, and specified probability distributions and states after the measurement obtained by measuring a system using the measurement apparatus. However, because the relationship between the instrument and the usual quantum mechanical description was not clear at first, the analysis using the instrument did not progress until the investigation by Ozawa [18]. He introduced a completely positive instrument and a measuring process, the latter being used for quantum mechanical modeling of measurement. Every measuring process defines a completely positive instrument. The main result of [18] is the converse in a quantum system with finite degrees of freedom, i.e., every completely positive instrument in such a system is defined by a measuring process. This is a standard fact in quantum measurement theory now. Furthermore, 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]. C∗-algebras and von Neumann algebras can be viewed as non-commutative versions of topological and measurable spaces, respectively. The latter is a special case of the former, but their analysis methods are very different. In the current measurement theory, focusing on probability distributions and states after the measurement has led to the selection of components to be macroscopic by the measurement andthesuccessfulinvestigationoftherelationshipwithquantummechanicalmodeling.

In order to formulate the measurement theory for quantum systems described by C<sup>∗</sup>- 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

of the macroscopic aspect of the quantum system. The reason for this is that, because the concept of state is statistically characterized, we consider that the difference of values output by the measurement should be macroscopically distinguished by the disjointness of states of the composite system of the system and the measuring apparatus. In other words, 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 of course important historically and theoretically, but it should not necessarily be the first consideration in establishing the physical meaning and description of the measurement. On the other hand, this study is advantageous in that the identification of sectors by the measurement is justified by the measurement-theoretic description. We are convinced that the establishment of the measurement theory in quantum systems described by C∗-algebras will open up new perspectives for the understanding of macroscopic aspects of quantum systems. Herein, we reexamine the result of [21]. While [21] focused on the use of measuring processes, we make thorough use of the instrument in this study.

In Section 2, the local net and open system are discussed and the description of dynamics as an open system in AQFT is stated. In Section 3, we review the sector theory and its mathematics. In Section 4, the central subspaces of the dual of a C∗-algebra are defined. In the C∗-algebraic setting, we define instruments in terms of central subspaces. Furthermore, we define and characterize central instruments in order to examine the differences between the C∗-algebraic setting and the von Neumann algebraic setting. In Section 5, we summarize the results of the study and present the perspective.

### **2. Systems of Interest: Local Nets and Open System**
