**5. Prospects**

As we have seen, our new approach to quantum fields is conceptually related to conventional approaches such as AQFT and TQFT. Elucidating this relationship at a deeper level will be important in the study of quantum fields. In order to carry out such research, we will need to include more detailed structures such as topological or differential structures in addition to the algebraic and noncommutative probabilistic structures that we have discussed in this paper.

Additionally, it should be emphasized that our approach is directly applicable to the lattice gauge theory [27] and other discrete spacetime approaches, as can be seen from the fact that our approach works on general categories. Its applicability extends to the context of unifying general relativity and quantum theory.

Needless to say, the relationship with categorical approaches to quantum theory, such as "categorical quantum mechanics" [37,38] based on the †-category, should also be explored. The categorical structure of the submodules of the category algebra, as a generalized matrix algebra and regarding the computations based on it, will play an important role. It is also interesting to clarify the relationship between our framework and the approach in a recently published article [8], which also investigates the AQFT and Quantum Cellular Automata (QCA) approach [39,40] from a general categorical viewpoint.

The notion of quantum walk (see [41,42] and references therein, for example), which is closely related to the QCA approach, can also be formulated from our standpoint. Based on our framework, we can model a concrete dynamics of quantum fields as a sequence or flow of the states on a category. In general, the dynamics can be irreversible. The typical examples of reversible dynamics are called quantum walks. The notion of quantum walks on general ∗-algebras and quantum walks on †-categories can be defined as follows:

**Definition 39** (quantum walk)**.** *Let A be a* ∗*-algebra. A sequence of states given by*

$$\varphi^t(\mathfrak{a}) = \varphi((\omega^\*)^t \mathfrak{a} \omega^t) \text{ } t = 0, 1, 2, 3, \dots$$

*generated by a unitary element ω* ∈ *<sup>R</sup>*[C]*, i.e., an element satisfying ω*<sup>∗</sup>*ω* = *ωω*<sup>∗</sup> = *, is called a quantum walk on A.*

**Definition 40** (quantum walk on †-category)**.** *Let* C *be a* †*-category and R be a* ∗*-rig. A quantum walk on R*[C] *is said to be a quantum walk on a* †*-category* C*.*

A quantum walk can be considered as a sequence of "state vectors" through the GNS construction. the notion of quantum walk defined on †-category includes the various concrete dynamical models under the name of quantum walks. For example, this includes quantum walks on simple undirected graphs as a certain sequence of states on an indiscrete category. The category algebraic approach will play a fundamental role for the quantum walks on graphs with multiple edges and loops. Quantum walks on graphs have been used in the modeling of the "dressed photon" [43], which cannot be understood without focusing on the off-shell nature of quantum fields [44], i.e., the aspect of quantum fields which cannot be described as the collection of the modes satisfying the on-shell conditions, and quantum walks on categories may become important in quantum field theory in general. They will also connect the QCA approach to quantum fields and other approaches to quantum fields.

One of the most exciting problems is, of course, to construct a model of a non-trivial quantum field with interactions. We believe we can approach such problems. In particular, it seems that the fact that relevant categories have arrows that go through objects in very distant regions, while the local algebras defined on them satisfy commutativity, may be the key to avoiding various no-go theorems. Note also that our approach extends the coefficients to general (commutative or noncommutative) rigs, which greatly expands the possibilities of investigating interactions. Finally, it should be emphasized that our approach is not limited to quantum fields but can be extended to give a very general noncommutative statistical models with causal structures. The author hopes that the present paper will be a small, new step towards these big problems.

**Funding:** This work was partially supported by Research Origin for Dressed Photon, JSPS KAKENHI (grant number 19K03608 and 20H00001), and JST CREST (JPMJCR17N2).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The author is grateful to Hiroshi Ando, Takahiro Hasebe, Soichiro Fujii, Izumi Ojima, Kazuya Okamura, Misa Saigo, and Juzo Nohmi for the fruitful discussions and comments.

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