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Symmetry

Towards a Measurement Theory for Off-Shell Quantum Fields

Kazuya Okamura

Published Jun 30, 2021• 4 citations
Publisher

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.

DOI
10.3390/sym13071183

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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.
In this paper, we attempt to establish a systematic treatment of measurement correlations in quantum theory. We define the concept of system of measurement correlations, which is the exact counterpart of instrument in the (generalized) Heisenberg picture. We then redefine the concept of measuring process in order that it becomes consistent with the definition of system of measurement correlations. In quantum mechanical systems, we establish a one-to-one correspondence between systems of measurement correlations and measuring processes up to complete equivalence. This is nothing but a unitary dilation theorem of systems of measurement correlations. Next, from the viewpoint of the statistical approach to quantum measurement theory, we focus on the extendability of CP instruments to systems of measurement correlations. It will be shown that all CP instruments are extended into systems of measurement correlations. Moreover, in the case of injective von Neumann algebras, we show that any CP instruments are approximated by faithful measuring processes within arbitrarily given error limits. Similarly, in the case of von Neumann algebras with separable dual, we show that any CP instruments are pointwisely realized by measuring processes. These new unitary dilation theorems of CP instruments substantially rewrite the results of past investigations.
In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.
In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional (AFD) von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts (DHR-DR) theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.
In previous papers we have explained how a sequence of theorems by John von Neumann on infinite tensor products (ITP) can be understood as providing elements to support both sectorisation of the Hilbert space of large quantum systems, and a mechanism of self decoherence thereof. These two effects may help understanding the articulation of the classical and quantum realms. However, as they involve considering an infinite number of quantum degrees of freedom, legitimate concerns can be raised on their applicability. In this paper, we address explicitly the interface between both realms through the example of a simplified model of a photon polarisation measurement device. Guided by the fact that there is von Neumann sectorisation at infinity, and by the necessity of classical contexts to perform measurements, we show that this limit can be under control, and that although the full force of the sectorisation theorems requires taking the infinite limit, early signs of the macroscopic behaviour appear before infinity. In our example, this shows up in photodiodes through diverging electron avalanches that simultaneously make the system classical, localise it randomly in a macroscopic sector and provide a macroscopic signal. This lays the grounds for justifying the inclusion in quantum physics of the ITP formalism, which involves non-separable Hilbert spaces and potentially type-III von Neumann algebras. Such an approach could make sense of the quantum-classical transition as a primarily algebraic one.
We study universally valid uncertainty relations in general quantum systems described by general σσσ-finite von Neumann algebras to foster developing quantitative analysis in quantum systems with infinite degrees of freedom such as quantum fields. We obtain the most stringent measurement-disturbance relation ever, applicable to systems with infinite degrees of freedom, by refining the proofs given by Branciard and one of the authors (MO) for systems with finite degrees of freedom. In our proof the theory of the standard form of von Neumann algebras plays a crucial role, incorporating with measurement theory for local quantum systems recently developed by the authors.