**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 yet 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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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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.


**Abstract**

Barrett's esophagus (BE), as a more frequent complication of gastroesophageal reflux disease, is a metaplastic condition in which the normal squamous epithelium of the esophagus is replaced by specialized intestinal metaplastic epithelium, and that, in about 10% of patients with gastroesophageal reflux disease (GERD) and the main condition for dysplasia and adenocarcinoma. The incidence of adenocarcinoma of the cardia is rapidly increasing at a rate that exceeds that of any other cancer. Recently, acid suppression with proton pump inhibitors (PPIs) has become the cornerstone of treatment for patients with BE. Many worldwide investigations showed that PPI is effective in the regression of BE with low-grade dysplasia and especially for the regression of intestinal metaplasia, incomplete or complete, for long-term use of these medicaments. This chapter reviews the specific endpoints of such treatment, included and our results for this dilemma.

**Keywords:** Barrett's esophagus, low-grade dysplasia, proton pump inhibitor, regression, gastroesophageal reflux disease (GERD)


**1.1 Marine riser**

Oceans are quite important fields for us because many resources lurk there which are oil and gas under seabed, mineral resources, water heat energy and so on. Development of submarine oil has been major in the North Sea and in the Gulf of Mexico. Today, submarine oil has been developed at ultra-deep water fields of offshore of Brazil and West Africa, which does deep over 1000m. Deepest field developed is more than 3000m in water depth of Brazilian seas. Riser system is necessary to develop and to production submarine oil. The riser is a tubing structure which is for drilling and production. Diameter of a drilling riser is greater than 50cm and that of a production riser is about 20cm. The riser is thin rope-like tube in oceans. Therefore, the tubing behaves elastically by marine currents and ocean waves and so on.

These motion behaviors are called as Vortex-Induced Vibration (VIV). VIV of the riser is a complex phenomenon, which is dominated by the natural frequency of the riser system and behavior of vortex shedding around the rider. VIV is very important for structural design of the riser system and the platform of the drilling and the production of submarine oil and so on.

There are many studies of VIV of the riser and the drilling or the production system including the riser system in the ocean engineering field with numerical approaches, theoretical approaches and model experimental approaches. Behaviors of time variation of VIV obtained from numerical calculations or model experiments using model risers in a water tank include a complicated mechanism so it is not easy to understand them, because the time variation is not steady but transient and chaos. Therefore, we need to understand VIV phenomenon in not only time characteristics but also frequency characteristics.

For understanding frequency characteristics, we often use a power spectrum with the FFT analysis or others. However, a power spectrum does not inform us time variation of VIV characteristics. Then, the wavelet analysis can be applied to the VIV analysis because we can simultaneously understand the characteristics in time domain and frequency domain.

Application of Wavelet Analysis for the Understanding of Vortex-Induced Vibration 595

Number of degrees of freedom is two in sway motion and roll motion. The sway in this experiment is horizontal displacement of center of gravity in *y* direction and rolling is rotation around *x* axis in a coordinate system of Fig. 1. Freedom of surge corresponding to *x* axis motion is allowed and decided due to forced oscillation by experimental operators.

The cylinder is suspended under a load cell through the flat spring. Most of the cylinder is submerged in still water. Side views of the experimental setup system, which corresponds to the *z*-*x* plane, are illustrated in Fig. 2. The direction of forced oscillations is right and left in

A load cell for measuring the total load in inline direction, which including the inertia force of a cylinder and hydrodynamic forces on a cylinder, is attached under the forced oscillation device. A Doppler current meter is installed at the straightly back of the cylinder, the current meter which moves together with the cylinder due to the forced oscillation. The current meter can measure fluid velocity of three directions in *x*, *y* and *z* axis. We measure the

> Transverse vibration VIV with lock-in

In the experiment, 1) inline displacement of forced oscillation with a potentiometer, 2) the total inline load with the load cell, 3) the bending moment with the flat spring and 4) fluid velocity at the back of the cylinder with the Doppler current meter are measured. The fluid velocity is measured at midship depth of the submerged cylinder. The VIV is evaluated by using the vertical bending moment and cross-flow displacement predicted from the bending

*x*

*o*

Forced oscillation in inline direction

*z*

vertical bending moment with a flat spring on which strain gages are set.

*y*

flat spring for transverse motion

Fig. 1. Idealization of VIV in experiment

moment.

**2.2 Experimental setup system**

Fig. 2.


**Chitin Nanofibers, Preparations and Applications**

Shinsuke Ifuku, Zameer Shervani and Hiroyuki Saimoto

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/57095


Lung Cancer - Modern Multidisciplinary Management

*Edited by Henry S. Park*

Published in London, United Kingdom

## *Supporting open minds since 2005*

Lung Cancer - Modern Multidisciplinary Management http://dx.doi.org/10.5772/intechopen.92053 Edited by Henry S. Park

Contributors

Joyce W.Y. Chan, Rainbow W.H. Lau, Calvin S.H. Ng, Hiroshi Doi, Kozo Kuribayashi, Beatrice Aramini, Valentina Masciale, Federico Banchelli, Roberto D'Amico, Massimo Dominici, Khawaja Husnain Haider, Paul Van Houtte, Charlier Florian, Luigi Moretti, Dirk Van Gestel, Martina Vrankar, Henry S. Park, Milad Rasouli, Nadia Fallah, Kostya (Ken) Ostrikov, Peter Ping Ping Lin, Justin Blasberg, Andrew Li, Guntug Batihan, Ana C. Belzarena, Kenan Can Ceylan

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First published in London, United Kingdom, 2021 by IntechOpen IntechOpen is the global imprint of INTECHOPEN LIMITED, registered in England and Wales, registration number: 11086078, 5 Princes Gate Court, London, SW7 2QJ, United Kingdom Printed in Croatia

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*3.4. In Vivo Implantation*

The Mg-RE sheet with TRC samples was successfully inserted into the subcutaneous tissue of the rat thigh and attached to the femur with an AZ31 screw during surgery, despite the slight mismatches between the Mg-RE sheet and femur owing to manual surgical placement and the fact that there is only one screw to fix the femur and alloy sheet. Slight skin swelling was observed in the hind limbs of the rats during the visual examination of the rats during the study period. The bulge disappeared after a few days.

Figure 10 reveals the surface volume of the degrading Mg-RE alloy sheet and bone growth changes, depicted using MicroCT 3D rendering after various implantation times. As can be seen from Figure 10a,c, two weeks after surgery, no new bone production was found around the Mg-RE alloy sheet and femur, and the screw between them was clearly visible. There are no obvious corrosion pits on the surface of magnesium alloy sheet. Figure 10b,d, six weeks after the surgery, new bone was formed between the Mg-RE alloy sheet and the femur. Figure 10e presents on the cross section of the bone–alloy joint, the new bone is generated and surrounds the original femur. The sheets are attached to the femur by new bone, which appears to be relatively stable in living organisms. In the red area shown in Figure 10, corrosion pits began to appear on the surface of magnesium alloy sheet after six weeks of surgery. This indicates that the films with improved corrosion resistance of Mg-RE alloy sheet began to degrade in vivo after 6 weeks.

Mg-RE sheet degradation and femoral change were assessed by volume quantification in MicroCT 3D rendering separately as shown in Figure 11. Before surgery, the volume of self-made Mg-RE alloy bone sheet was 94.79 mm3. The Mg-RE sheet corroded, resulting in 9.32 ± 0.86 mm3 volume loss after 6 weeks. With regard to the changes in the femur, the volume of the femur did not change significantly during the two weeks following the surgery. In fact, for some time after the surgery, the rat lost significantly more weight, the reason for this being that it took some time for the body to adapt the a foreign body. After about five days, the rats returned to their normal mental state and began to eat normally. By 6 weeks, as new bone had formed between the sheet and the femur, a noticeable

change in bone volume was observed from 2 to 6 weeks. The rat showed no abnormal physical or physiological responses during subsequent breeding.

Uniquely, magnesium alloy application has the potential to enhance bone formation. Several previous studies have shown that the potential performance, presenting increased bone mass, mineral apposition, and bone mineral density around magnesium alloy implants in bone [36–39]. In our present study, new bone appears in the area where the femur contacts the alloy sheet, which confirmed our findings that the Mg-RE alloy with TRC degradation can lead to the promotion of bone formation. The new bone formation is not common in absorbable polymers or permanent metal devices, thus emphasizing the unique advantages of magnesium fixtures. In this way, the degraded fixture will be gradually replaced in the future.

**Figure 10.** Determination of the volume of the degrading Mg-RE alloy sheet and bone growth changes, depicted in MicroCT 3D rendering after implantation times of 2 weeks (**a**,**c**) and 6 weeks (**b**,**d**). A cross section slice of femoral-sheet contact area observed after 6 weeks (**e**) shows that the new bone (Nb is represented in green) was formed between the Mg-RE alloy sheet (S) and the original bone (Ob is represented in blue).

**Figure 11.** Mg-RE sheet degradation after 6 weeks was estimated by the volume of quantification. This shows a sheet volume loss of 9.32 ± 0.86 mm3 after 6 weeks. MicroCT 3D rendering after 2 weeks shows there was no obvious change in bone growth around the degrading Mg-RE sheet. By 6 weeks, as new bone formed between the sheet and the bone, a notable change in bone volume was observed from 2 to 6 weeks.


*4.1.5 Dipeptidyl peptidase-4 inhibitors*

It has been proposed that SARS Cov-2 binds to Dipeptidyl peptidase-4 inhibitors (DPP4), but the clinical implications are not known. Dipeptidyl peptidase-4 inhibitors (DPP4i) are well tolerated in COVID-19 infection [15]. The majority of studies have shown either benefit with DPP4i in PWD and COVID-19, or no harm or benefit [20], Although DPP4 inhibitors appear to be safe in T2D and COVID-19 infection [4], in an observational study of 717 patients, in the diabetes sub-group, patients on DPP4I were at a higher risk of ICU admission [5]. As study that compared GLP-1 RA or DPP4i with SGLT2i did not note associated improved outcomes in patients with COVID-19 infection [28]. As DPP4 upregulation may be an indicator for severity of COVID-19 infection, there is interest regarding the use of DPP4i in COVID-19 infection and available information may form the path to discovering novel therapies [29]. RCTs involving Linagliptin versus placebo and Sitagliptin versus placebo have been registered [30, 31].


**3.2 Artificial vision system**

An algorithm of computational vision is implemented in the welding process to locate spatially the pieces respect to the robot coordinates. The camera is set perpendicular to the working area at a fixed coordinate XYZAB with respect to the center robot position; the CCD makes the capture of the image that contains the pieces to be weld. OpenCV is used for the segmentation process and it is performed applying filters and morphological operation to determine the area of contact to be weld; this area becomes a straight line and the points that will be used for the creation of G code will be loaded in the numerical control.

For the processing of the images, the capture of the pieces to be weld is made, as can be seen in **Figure 6**. The working pieces are located on a surface that

**Figure 6.** *Original image.*

*Implementation of an Artificial Vision System for Welding in the Retrofitting Process… DOI: http://dx.doi.org/10.5772/intechopen.88360*

contrasts with the color of the pieces to facilitate the segmentation and identification of the edges.

Subsequently, the image is converted to a gray scale, and a Sobel filter is applied to create an image emphasizing edges. The resulting image can be observed in **Figure 7**; this process was made with the aim of being able to identify the possible closeness of the working pieces.

Before the pieces are segmented to highlight the edges, a morphological operation of closing is made to eliminate small gaps (filling them) and join components connected nearby. The kernel used is the structuring element of a rectangle of 1 8 pixels. The result is presented in **Figure 8**, where the space that separates the two pieces was totally filled.

In the last process, it is possible to highlight the suitable area for the welding process; the algorithm is able to do it due to a Gaussian filter that is applied to eliminate the rest of the edges of the piece and leave only the highlighted union of both. With the segmentation process done, the line that traces the union of the pieces is transformed into points that will be used to coordinate to which the arm must reach to carry out the welding process.

To verify the precision of the algorithm, several probes are made to check the real coordinates and the coordinates calculated from the computer vision. In this process, the values of the location of the camera and the relationship of the pixels and the location XYZAB change according to the precision of each servomotor.

**3.2 Artificial vision system**

**Figure 5.** *Digital indicator.*

*Digital Imaging*

**Table 3.** *Movements.*

**Figure 6.** *Original image.*

**70**

An algorithm of computational vision is implemented in the welding process to locate spatially the pieces respect to the robot coordinates. The camera is set perpendicular to the working area at a fixed coordinate XYZAB with respect to the center robot position; the CCD makes the capture of the image that contains the pieces to be weld. OpenCV is used for the segmentation process and it is performed applying filters and morphological operation to determine the area of contact to be weld; this area becomes a straight line and the points that will be used for the

**Displacement Axis (X) Axis (Y) Axis (Z)** To 1 (mm) 0.98 0.96 0.99 To 1.5 (mm) 1.48 1.46 1.47 To 2.5 (mm) 2.46 2.45 2.47 To 5 (mm) 4.88 4.84 4.90

For the processing of the images, the capture of the pieces to be weld is made,

as can be seen in **Figure 6**. The working pieces are located on a surface that

creation of G code will be loaded in the numerical control.

**Figure 8.** *Welding trajectory.*