**1. Introduction**

Quantum fields are the most fundamental entities in modern physics. Intuitively, the notion of quantum field is the unification of relativity theory and quantum theory. However, the existence of a non-trivial interacting quantum field model defined on a four-dimensional Minkowski spacetime, which is a covariant with respect to the Poincaré group, has not ye<sup>t</sup> been proven. In axiomatic approaches to the quantum field theory, there have been shown many fundamental theorems including no-go theorems such as Haag's theorem [1,2], which implies that the "interaction picture exists only if there is no interaction" [3] through the clarification of the concept of the quantum field (see [3,4] and references therein). To put it roughly, we cannot go beyond the free fields if we remain at the axioms that we take for granted in conventional quantum field theories.

In this paper, we propose a new approach to quantum fields: The core idea is to investigate quantum fields in terms of category algebra, which is noncommutative, in general, over a rig ("ring without negatives"), i.e., an algebraic system equipped with addition and multiplication, in which the category and rig correspond to the "relativity" aspect and the "quantum" aspect of nature, respectively. By utilizing category algebra and states on categories instead of simply considering categories, we can directly integrate relativity as a category theoretic structure and quantum nature as a noncommutative probabilistic structure. The cases in which the rig is an algebra over C, the field of complex numbers become especially important for our approach to quantum fields. For other regions of physics, such as classical variational contexts, the tropical semiring (originally introduced in [5]), i.e., a rig with "min" and "plus" as addition and multiplication, will be useful. The author believes that it is quite interesting to see the quantum–classical correspondence from the unified viewpoint of the category algebras over rigs.

As is well known, the essence of the relativity is nothing but the structure of the possible relationships between possible events. If we assume the structure of the relationships between events, we can essentially reconstruct the relativity structure. More concretely, in [6], it is shown that two future-and-past-distinguishing Lorentzian manifolds are conformally equivalent if and only if the associated posets are isomorphic, where the poset consists of events and of the order relation defined by the existence of future-directed causal curves, based on [7]; what really matters are the causal relationships (for details, see [8] and reference therein). This viewpoint is quite essential and there is an interesting

**Citation:** Saigo, H. Quantum Fields as Category Algebras. *Symmetry* **2021**, *13*, 1727. https://doi.org/10.3390/ sym13091727

Academic Editor: Alexei Kanel-Belov

Received: 30 August 2021 Accepted: 14 September 2021 Published: 17 September 2021

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order-theoretic approach to spacetime (for example, the "causal set" approach [9]). However, to investigate the off-shell nature of quantum fields, which seems to be essential in modeling interacting fields on the spacetime, we need to take not only causal relationships but also more general relationships between spacelike events into consideration. Then, the question arises: how should we generalize a framework of previous approaches?

The strategy we propose is to think categories, which are generalizations of both of ordered sets (causality structures) and groups (symmetry structures), "as" relativity in a generalized sense. More concretely, we identify the notion of causal category equipped with partial involution structure, introduced in Section 2, as the generalized relativity structure. To combine this relativity structure with quantum theory, which can be modeled by noncommutative rigs, especially effectively by noncommutative algebras over C as history has shown, we need noncommutative algebras that reflect the structures of categories. Category algebras are just such algebras. As categories are generalized groups, category algebras are generalized group algebras.

The above discussion intuitively explains why we use category algebras to model quantum fields. For simplicity, in this paper, we focus on a category algebra which satisfies a suitable finiteness condition. Importantly, the category algebras can be considered as generalized matrix algebras over R as well as generalized polynomial algebras [10], which provides a platform for concrete and flexible studies as well as calculations. The extension to larger algebras is, of course, of interest but the category algebras we focus on already have rich structures as covariance and local structures of subalgebras reflect the causal and partial involution structure of the category, as we will see in Section 3. By focusing on these structures, we can also see the conceptual relationship between our approach and the preceding approaches such as Algebraic Quantum Field Theory (AQFT) [4,11] and Topological Quantum Field Theory (TQFT) [12,13].

Identifying a quantum field to be a category algebra over a rig, the next problem, which is treated in Section 4, concerns how to define a state of it. In general, the notion of state on ∗-algebra over C is defined as positive normalized linear functional. We can naturally extend the notion in the context of algebras with involution over rigs ([10] for details). We call the states on category algebras as states on categories. If the number of objects in the category is finite, states can be characterized by functions on arrows satisfying certain conditions [10], which is a generalization of the result in [14] for groupoids with finite numbers of objects. More generally, to define a state on a category whose support is contained in a subcategory with finite numbers of objects is equivalent to defining the corresponding function which assigns the weight to each arrow. By considering such states, we can see a quantum mechanical system as an aspect of the quantum field. This viewpoint will shed light on the foundation of quantum theory.

For the study of quantum fields, a localized notion of state, or a "local state" [15,16], is important. We can define the counterpart of the notion, originally studied in the AQFT approach, as the system of states on certain subalgebras of category algebras called local algebras, introduced in Section 3. These matters will be explained in Section 4 with more clarification of the conceptual relationship with AQFT and TQFT. The discussion in Section 4 will provide a new basis for generalizing the DHR (Doplicher–Haag–Roberts)–DR (Doplicher– Roberts) sector theory [17–23] as well as for developing Ojima's micro–macro duality [24,25] and quadrality scheme [26] from the viewpoint of category algebras and states on categories.

In the last section, we will discuss the prospect of research directions based on our framework. In addition to the importance of mathematical research, such as taking topological or differential structures into account, there is the challenge of integrating various approaches to quantum fields and of conducting research on quantum foundations based on our framework. These are where new concepts such as quantum walks on categories will be useful. One of the most exciting problems is, of course, to construct a model of a non-trivial quantum field with interactions. The author hopes that the present paper will be a small, new step towards these big problems.

### **2. Structure of Dynamics as Category**

In this section, the "relativistic structure" as the basic structure of dynamics, consisting of possible events and relations (or "processes") between them, is formulated in terms of category theory.
