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arXiv:2608.00468

What is electric charge?: Charge as a local observable in relativistic quantum field theory

Hideyasu Yamashita

quant-phSubmitted Aug 1, 2026• Announced Aug 4, 2026• 0 citations
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Abstract

It is rather surprising that modern quantum physics does not appear to have provided any clear answer to the simple question ``what is electric charge?''. Even when the total charge operator $Q$ is well-defined, the non-locality of $Q$ implies that it is not an observable in the usual sense, which can be measured by a (local) experimental apparatus. A candidate for the ``local version'' of charge operator is the 4-current operator $j=(j^μ)_{μ=0,1,2,3}$. However, it is known that the rigorous definition of $j$ is difficult in $(3+1)$-dimensional Minkowski space. Although it was found that the current can be defined in $(1+1)$-dimensions (Carey et al.), I argue that even when $j$ can be suitably defined, the interpretability of $j$ as the ``local charge operator'' is dubious. Instead I return to Araki and Wyss (1964), and propose the concept of ``scope-local charge'' $Q_ζ(P)$ for a ``scope'' $P$, expressed by a finite-dimensional projection. I work in an abstract $C^{*}$-algebraic setting.

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16 pages

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It is rather surprising that modern quantum physics does not appear to have provided any clear answer to the simple question what is electric charge?''. Even when the total charge operator $Q$ is well-defined, the non-locality of $Q$ implies that it is not an observable in the usual sense, which can be measured by a (local) experimental apparatus. A candidate for the local version'' of charge operator is the 4-current operator j=(jμ)μ=0,1,2,3j=(j^μ)_{μ=0,1,2,3}j=(jμ)μ=0,1,2,3​. However, it is known that the rigorous definition of jjj is difficult in (3+1)(3+1)(3+1)-dimensional Minkowski space. Although it was found that the current can be defined in (1+1)(1+1)(1+1)-dimensions (Carey et al.), I argue that even when jjj can be suitably defined, the interpretability of jjj as the local charge operator'' is dubious. Instead I return to Araki and Wyss (1964), and propose the concept of scope-local charge'' Qζ(P)Q_ζ(P)Qζ​(P) for a ``scope'' PPP, expressed by a finite-dimensional projection. I work in an abstract C∗C^{*}C∗-algebraic setting.
In relativistic quantum field theory the notion of a local operation is regarded as basic: each open space-time region is associated with an algebra of observables representing possible measurements performed within this region. It is much more difficult to accommodate the notions of events taking place in such regions or of localized objects. But how can the notion of a local operation be basic in the theory if this same theory would not be able to represent localized measuring devices and localized events? After briefly reviewing these difficulties we discuss a strategy for eliminating the tension, namely by interpreting quantum theory in a realist way. To implement this strategy we use the ideas of the modal interpretation of quantum mechanics. We then consider the question of whether the resulting scheme can be made Lorentz invariant.
The problem of measuring the charge contained in a finite volume for the theory of free fermions is discussed. The expectation values of Q/sub V/ (an operator for total charge contained in a volume V) are calculated for localized one-particle solutions of the Dirac equation in both 2 and 4 dimensional space time. This measurement procedure is well-defined and leads to physically reasonable results. Two-particle states are considered to determine whether or not the measurement of the charge of one of the particles can be carried out. This measurement procedure is found to be well-defined and leads to physically reasonable results. (GHT)
Abstract: The concept of fractionability or decomposability in parts of a physical system has its mathematical counterpart in the lattice–theoretic concept of orthomodularity. Systems with a finite number of degrees of freedom can be decomposed in different ways, corresponding to different groupings of the degrees of freedom. The orthomodular structure of these simple systems is trivially manifest. The problem then arises as to whether the same property is shared by physical systems with an infinite number of degrees of freedom, in particular by the quantum relativistic ones. The latter case was approached several years ago by Haag and Schroer (1962; Haag, 1992) who started from noting that the causally complete sets of Minkowski spacetime form an orthomodular lattice and posed the question of whether the subalgebras of local observables, with topological supports on such subsets, form themselves a corresponding orthomodular lattice. Were it so, the way would be paved to interpreting spacetime as an intrinsic property of a local quantum field algebra. Surprisingly enough, however, the hoped property does not hold for local algebras of free fields with superselection rules. The possibility seems to be instead open if the local currents that govern the superselection rules are driven by gauge fields. Thus, in the framework of local quantum physics, the request for algebraic orthomodularity seems to imply physical interactions! Despite its charm, however, such a request appears plagued by ambiguities and criticities that make of it an ill–posed problem. The proposers themselves, indeed, concluded that the orthomodular correspondence hypothesis is too strong for having a chance of being practicable. Thus, neither the idea was taken seriously by the proposers nor further investigated
We investigate whether commutativity is necessary to represent relativistic locality for localization observables of relativistic quantum systems in Minkowski spacetime. A well known no-go theorem by Halvorson and Clifton shows that commutativity of localization effects for causally separated regions is incompatible with other seemingly natural assumptions about spatial localization. Since commutativity is taken to represent locality in the Araki-Haag-Kastler framework of QFT, this prompts the question whether it follows from more elementary locality principles of quantum theory. Using Busch's operational analysis in terms of no-signaling and relativistic consistency, we argue that for particle-like systems commutativity is not implied by these principles. Assuming a natural local detectability principle, elementary localization observables are not localized in arbitrarily small spacetime neighborhoods of the relevant spatial regions, but rather in regions containing the entire rest space (a Cauchy surface) on which the measurement is performed. This reflects the particle picture itself, where localization occurs at a unique place on a rest space filled with ideal detectors, and therefore does not directly conflict with the Araki-Haag-Kastler notion of locality. We also show that commutativity and localization can coexist for less idealized localization procedures. To this end, we introduce conditional localization POVMs associated with bounded spatial regions interpreted as laboratories. By the gentle measurement lemma, these observables describe conditional localization probabilities and can, in principle, satisfy commutativity for causally separated laboratories. They may therefore be represented by local observables in the Araki-Haag-Kastler sense. Explicit examples are presented in a second work within local QFT.