**6. Discussion and Perspectives**

In the study, we have defined instruments by using central subspaces of the dual of a C∗-algebra. We have checked its consistency with the definition in the von Neumann algebraic setting. This result means that the extension of the measurement theory to C∗-algebra in the paper is valid. Furthermore, we have proposed a unification of the measurement theory and the sector theory: we have defined and characterized the centrality of instruments. In addition, we have discussed the operational characterization and macroscopic nature of quantum measurement. In the context, we have actively used the disjointness of states to distinguish different output values of the meter. Our results are, of course, applicable to systems described by C\*-algebras generated from field operators, and the macroscopic aspects of quantum fields can now be discussed in terms of measurement theory.

In the setting of AQFT, we use a local net {A(O)}O∈R1 on a space *M*1 in order to describe the DP phenomena. In describing the measurement of DPs, only the use of the local net first adopted is not enough. In fact, to detect (the effect of) DPs, we need an operation wherein some probe is brought closer to the spatial scale at which DPs are generated. We introduced an extension of a local net to mathematically describe the operation at the level of observable algebras.

**Definition 12.** *Let* {A(O)}O∈R1 *and* {B(O)}O∈R2 *be local nets on M*1 *and M*2*, respectively.* {B(O)}O∈R2 *is an extension of* {A(O)}O∈R1 *if it satisfies the following three conditions:* (*i*) *M*1 ⊂ *M*2*.* (*ii*) R1 ⊂ R2*.* (*iii*)*Forevery*O∈R<sup>1</sup>*,*A(O)⊂B(O)*.*

We use the extensions of a local net because the construction of the composite system of the system of interest and a measuring apparatus is not so simple. In particular, the construction of the composite system by the tensor product is not always applicable to quantum fields.

Let {B(O)}O∈R2 be a local net on *M*2 and an extension of a local net {A(O)}O∈R1 on *M*1. We suppose that *M*1 is bounded. The composite system of the original system and a probe, which is close to the original system on the spatial scale where DPs are generated, is described by {B(O)}O∈R2 as a quantum field. Furthermore, the material system, which is a part of the composite system, is assumed to be localized in the neighborhood of *M*1. In the composite system, the generation and annihilation of DPs constantly occur near non-uniform materials in the unstable situation where light continues to incident constantly. By measuring the emitted light at regions far from *M*1, we check (or estimate) the effect of DPs generated in *M*1.

Constructing a concrete model of DPs as a quantum field in order to correlate experiments of DPs with the theory is a future task. We hope to describe the DP phenomena as open systems at the next stage. In the future, clarification of the relationship between this study and the recent trends in DP research [33] is required. Moreover, the mathematical theory of quantum measurement for quantum systems described by C∗-algebras should be further developed.

**Funding:** This research received no external funding.

**Acknowledgments:** The author thanks anonymous reviewers for their comments to improve the quality of this paper.

**Conflicts of Interest:** The author declares no conflict of interest.
