#### Abstract

The paper aims to apply the complex-sedenions to explore the wave functions and field equations of non-Abelian gauge fields, considering the spatial dimensions of a unit vector as the color degrees of freedom in the complex-quaternion wave functions, exploring the physical properties of the color confinement essentially. J. C. Maxwell was the first to employ the quaternions to study the electromagnetic fields. His method inspires subsequent scholars to introduce the quaternions, octonions, and sedenions to research the electromagnetic field, gravitational field, and nuclear field. The application of complex-sedenions is capable of depicting not only the field equations of classical mechanics, but also the field equations of quantum mechanics. The latter can be degenerated into the Dirac equation and Yang-Mills equation. In contrast to the complex-number wave function, the complex-quaternion wave function possesses three new degrees of freedom, that is, three color degrees of freedom. One complex-quaternion wave function is equivalent to three complex-number wave functions. It means that the three spatial dimensions of unit vector in the complex-quaternion wave function can be considered as the “three colors”; naturally the color confinement will be effective. In other words, in the complex-quaternion space, the “three colors” are only the spatial dimensions, rather than any property of physical substance.

#### 1. Introduction

The essence of the color degrees of freedom in the strong-nuclear fields is attracting increasing attention of many researchers. Since a long time, this conundrum has been intriguing and puzzling numerous scholars. For many years, this puzzle urged some scholars to propose several sorts of hypotheses, attempting to reveal the essence of the color degrees of freedom. Until recently, the appearance of the quantum mechanics on the microscopic scale, described with the complex-sedenions, replies partially to this puzzle. The quantum mechanics, described with the complex-sedenions, is capable of deducing the wave functions, Dirac wave equations, Yang-Mills equations of the non-Abelian gauge field, field equations of the electroweak field, and so forth. Particularly, their wave functions, described with the complex-quaternions, are able to elucidate directly the color degrees of freedom.

C.-N. Yang and R. L. Mills found the field equations for a non-Abelian gauge field in 1954. The field equations of non-Abelian gauge fields were supposed to be applicable to each of four fundamental fields. After the mass problem was finally conquered by means of the spontaneous symmetry breaking and Higgs mechanism, the Yang-Mill equations can be applied to construct successfully the unified theory of the electromagnetic field and weak-nuclear field. The triumph of the electroweak theory edified a part of scholars to extend the Yang-Mill equations into the strong-nuclear fields, establishing the quantum chromodynamics (QCD for short).

M. Gell-Mann and G. Zweig posited independently the model of quarks in 1964. In the same year, O. W. Greenberg introduced a sort of the degree of freedom for the subatomic particles, that is, the color degree of freedom. In the QCD, the scholars assume that a species of “charge” is capable of producing the strong-nuclear field and possesses “three colors.” And it is called the color charge, which is similar to the electric charge in the electromagnetic fields. The color charge is a fundamental and crucial assumption and is the theoretical footstone of the QCD. On the basis of the assumption of color charge, it is able to conduct a variety of theoretical inferences and perform diverse experimental validations, for the QCD. Up to now, the color charge of quarks and gluons has not yet been observed directly in the experiments. As a result, a rule had been summarized to explain correlative physical phenomena. That is, only the “colorless” hadrons can be studied or observed in the physics. This rule was called the quark confinement or color confinement.

However, the QCD is unable to elucidate the physical phenomena of color confinement by itself. Consequently it is necessary to introduce several hypotheses, including the lattice gauge theory, non-Abelian monopoles, microscopic resonance, hidden local symmetry, color charge, soliton model, and string theory. Using the magnetic symmetry structure of non-Abelian gauge theories of the Yang-Mills type, Chandola et al. [1] discussed the mathematical foundation of dual chromodynamics in fiber-bundle form. Nakamura and Saito [2] studied the long-range behavior of the heavy quark potential in Coulomb gauge, using a quenched SU(3) lattice gauge simulation with partial-length Polyakov line correlators. Chaichian and Nishijima [3] adopted that the colored states are unphysical, so the colored particles cannot be observed. And there are two phases in QCD distinguished by different choices of the gauge parameter. Eto et al. [4] argued that the dual transformation of non-Abelian monopoles results in the dual system being in the confinement phase. Wang et al. [5] proposed a new kind microscopic resonance, the color confinement multiquark resonance. Suzuki et al. [6] observed the Abelian mechanism of non-Abelian color confinement in a gauge-independent way, by the high precision lattice Monte Carlo simulations in the gluodynamics. The authors [7] studied the mechanism of non-Abelian color confinement in SU(2) lattice gauge theory, in terms of the Abelian fields and monopoles extracted from the non-Abelian link variables without adopting gauge fixing. Yamamoto and Suganuma [8] proposed a new lattice framework to extract the relevant gluonic energy scale of QCD phenomena, which is based on a “cut” on link variables in momentum space. Meiling et al. [9] studied the liquid properties of the strongly coupled quark-gluon plasma during the intermediate stage and after the end of crossover from the hadronic matter to the strongly coupled quark-gluon plasma, by means of the bond percolation model. Braun et al. [10] identified a simple criterion for quark confinement, computing the order-parameter potential from the Landau-gauge correlation functions. Troshin and Tyurin [11] discussed how the confinement property of QCD results in the rational unitarization scheme and how the unitarity saturation leads to the appearance of a hadron liquid phase at very high temperatures. Kitano [12] identified the gauge theory, with the hidden local symmetry, as the magnetic picture of QCD, enabling a linearly realized version of gauge theory to describe the color confinement and chiral symmetry breaking. Pandey et al. [13] studied an effective theory of QCD, in which the color confinement is realized through the dynamical breaking of magnetic symmetry, leading to the magnetic condensation of QCD vacuum. Gates and Stiffler [14] presented the evidence in some examples that an Adinkra quantum number seems to play a role with regard to off-shell 4D, supersymmetry similar to the role of color in QCD. Brodsky et al. [15] showed that a mass gap and a fundamental color confinement scale will arise, when one extends the formalism of de Alfaro, Fubini, and Furlan to the frame-independent light-front Hamiltonian theory. The author [16] studied the light-front wave functions and the functional form of the QCD running coupling in the nonperturbative domain, connecting the parameter in the QCD running coupling to the mass scale underlying confinement and hadron masses. Kharzeev and Levin [17] modified the gluon propagator to reconcile perturbation theory with the anomalous Ward identities for the topological current in the vacuum, making the connection between confinement and topology of the QCD vacuum explicit. The existence of so many hypotheses, to attempt to elucidate the color confinement, announces that our present understanding with respect to the strong-nuclear field is quite inadequate.

Some scholars proposed the Standard Model of elementary particles, attempting to further unify the QCD and Electroweak theory. This promising unification hypothesis is anticipated to be a huge success, although it does not include the gravitational field. Nevertheless, the Standard Model is unable to unpuzzle the color confinement, color charge, dark matter, and so forth. Consequently, some scholars put the effort towards a few theoretical schemes in recent years, such as “Beyond the Standard Model,” “Superstring theory,” and “Beyond the Relativity.”

Making a detailed comparison and analysis of preceding studies, a few primal problems of these theories are found as follows.

*(a) Four Interactions*. Either of the QCD and Electroweak theories is incapable of containing the gravitational interaction. Even in the theories regarding the “Beyond the Standard Model,” there is not any tangible theoretical scheme, to include the gravitational interaction. In the four fundamental interactions, the gravitational interaction is the first to be discovered in the history. Unfortunately, the gravitational interaction still lies outside the mainstream of current unification theories. It means that the mainstream of current unification theories may be seized of some fatal defects essentially, so that they are unable to cover and describe the four fundamental interactions simultaneously. It is well known that an appropriate unification theory must be able to depict the four fundamental interactions simultaneously, especially the gravitational interaction.

*(b) Color Confinement*. The QCD and other existing theories are incapable of revealing the essence of color degrees of freedom. Either they cannot determine whether the color degrees of freedom belong to the physical substance or spatial dimension and even others. Therefore they are unable to explain effectively the physical properties of color degrees of freedom. Under the circumstance that the essence of color degrees of freedom cannot be comprehended, the QCD assumed that prematurely the color degrees of freedom are induced by one sort of physical substance, that is, the color charge. Undoubtedly this assumption is governed by expediency. As one fundamental assumption of QCD, the color charge may encounter certain setbacks or challenges, bringing negative effects on the subsequent development of theory. However, even though the color charge was deemed as one species of the physical substance, the QCD and other existing theories are still unable to account for the color confinement, laying themselves open to suspicion. The assumption of color charge in the QCD is appealing for more validation experiments. Up to now, the QCD may not be really perfect yet, especially its fundamental assumption.

*(c) Dark Matter*. The existence of the physical phenomena connected with the dark matters was firstly validated in the astronomy and then was accepted generally by the whole academic circle. Nevertheless the Standard Model is blind to the existence of dark matters, and it is unable to elucidate the relevant physical phenomena either. Further the Standard Model and even the “Beyond the Standard Model” are incapable of predicting or inferring the confirmed dark matters. It means that the research scope, in the mainstream of current unification theories, is restricted and insufficient enough. For the unification theory, which is unable to explore the dark matter, there may be still something left to be improved. Apparently an appropriate unification theory must be capable of comprising and exploring the ordinary matter and dark matter simultaneously.

Presenting a striking contrast to the above is that it is able to account for a few problems, derived from the QCD and other existing theories, in the quantum mechanics described with the complex-sedenions, trying to improve the unification theory relevant to the four fundamental interactions to a certain extent. J. C. Maxwell was the first to employ the algebra of quaternions to explore the physical properties of electromagnetic fields. And it inspired the subsequent scholars to apply the quaternions, octonions, and sedenions to investigate the gravitational field, electromagnetic field, nuclear field, Dirac wave equation, Yang-Mills equation, electroweak field, color confinement, dark matter field, and so forth. When a part of coordinate values are complex-numbers, the quaternion, octonion, and sedenion are called the complex-quaternion, complex-octonion, and complex-sedenion, respectively.

In recent years, a part of scholars make use of the algebra of quaternions to research the electromagnetic field equations, quantum mechanics, gravitational fields, dark matters [18], curved space, and so on. Edmonds [19] expressed the wave equation and gravitational theory with the quaternion in the curved space-time. Doria [20] researched the gravitational theory, by means of the quaternions. Winans [21] described some physics quantities by means of the quaternion. Honig [22] and Singh [23], respectively, applied the complex-quaternion to express Maxwell’s equations in the classical electromagnetic theory. Brumby and Joshi [24] studied the possibility that dark matter may arise as a consequence of the underlying quaternionic structure to the universe. And the authors [25] explored some global consequences of a model quaternionic quantum-field theory, enabling the quaternionic structure to induce the cosmic strings and nonbaryonic hot dark matter candidates. Majernik [26] adopted the quaternions to deduce the modified Maxwell-like gravitational field equations. The author [27] studied a model of the universe consisting of a mixture of the ordinary matter with cosmic quaternionic field, researching the interaction of the quaternionic field with the dark matter. Anastassiu et al. [28] explored the physics properties of the electromagnetic field with the quaternions. Grudsky et al. [29] investigated the time-dependent electromagnetic field, by means of the quaternions. Morita [30] studied the quaternion field theory, making use of the quaternions. Rawat and Negi [31] explored the gravitational field, by means of the quaternion terminology. Davies adopted the quaternions to investigate the Dirac equation [32].

Some scholars introduce the algebra of octonions [33] to explore the Dirac wave equation, curved space, electromagnetic field equations, gravitational fields, dark matters, Yang-Mills equation, and so forth. Penney [34] applied the octonions to extract the square root of the classical relativistic Hamiltonian, finding the resulting wave equation to be equivalent to a pair of coupled Dirac equations. Marques-Bonham [35] developed the geometrical properties of a flat tangent space-time local to the manifold of the Einstein-Schrödinger nonsymmetric theory on an octonionic curved space. De Leo and Abdel-Khalek [36] introduced left-right barred operators to obtain a consistent formulation of octonionic quantum mechanics, developing an octonionic relativistic free wave equation. Koplinger [37] provided proof to a statement that the Dirac equation in physics can be found in conic sedenions. As the subalgebra of conic sedenions, the hyperbolic octonions are used to describe the Dirac equation. Gogberashvili [38] applied the octonions to investigate the electromagnetic field equations. V. L. Mironov and S. V. Mironov [39] demonstrated a generalization of relativistic quantum mechanics using the octonionic wave function and octonionic spatial operators. The second-order equation for octonionic wave function can be reformulated in the form of a system of first-order equations for quantum-fields. The authors [40] demonstrated a generalization of relativistic quantum mechanics using the octonions, generating an associative noncommutative spatial algebra. The octonionic second-order equation for the octonionic wave function may describe the particles with spin 1/2. The authors [41] made use of the octonion to describe the electromagnetic field equations and related features. Meanwhile Dündarer [42] defined a four-index antisymmetric and non-Abelian field, which satisfies a self-duality relation in eight-dimensional curved space. Demir and Tanişli proposed a new formulation for the massive gravi-electromagnetism with monopole terms [43]. The author formulated the Maxwell-Proca type field equations of linear gravity [44], in terms of the hyperbolic octonions. The authors [45] made use of the octonion to discuss the gravitational field equations and relevant properties. Castro [46] proposed a nonassociative octonionic ternary gauge field theory, based on a ternary bracket. Furui [47] expressed equivalently a Dirac fermion as a 4-component spinor, which is a combination of two quaternions. In terms of the quaternion in the Yang-Mills Lagrangian, the author [48] discussed the axial anomaly and the triality symmetry of octonion. The author [49] assumed that the dark matter may be able to be interpreted as matter emitting photons in a different triality sector, rather than that of electromagnetic probes in the world. With the help of the algebraic properties of quaternions and octonions, Kalauni and Barata [50] obtained the fully symmetric Dirac-Maxwell’s equations as one single equation. Chanyal et al. [51] applied the octonion to express the consistent form of generalized Maxwell’s equations in presence of electric and magnetic charges (dyons). In terms of the Zorn vector matrix realization, Chanyal et al. [52] described the octonion formulation of Abelian/non-Abelian gauge theory. The authors [53] analyzed the role of octonions in various unified field theories associated with the dyon and dark matter, reconstructing the field equations of hot and cold dark matter by means of split octonions.

A few scholars adopt the sedenions to explore the Dirac wave equation, gravitational field equations, electromagnetic field equations, curved space, dark matters, and so forth. Koplinger [54] applied the conic sedenions to express the Dirac equation in physics through the hyperbolic subalgebra, together with a counterpart on circular geometry that has been proposed for description of gravity. Making use of the conic sedenions, the author [55] proposed the classical Minkowski and a new Euclidean space-time metric, in an alignment program for large body (nonquantum) physics. By transitioning between different geometries through genuine hypernumber rotation, the author [56] demonstrated a method to extend the complex-number algebra using nonreal square roots of +1 to aid the mathematical description of physical laws. V. L. Mironov and S. V. Mironov [57] represented the sedenions to generate the associative noncommutative space-time algebra and demonstrated a generalization of relativistic quantum mechanics using sedenionic wave functions and sedenionic space-time operators. Making use of the sedenions, the authors [58] demonstrated some fundamental aspects of massive field’s theory, including the gauge invariance, charge conservation, Poynting’s theorem, potential of a stationary scalar point source, plane wave solution, and interactions between point sources. On the basis of the sedenionic space-time operators and sedenionic wave functions, the authors [59] discussed the gauge invariance of generalized second-order and first-order wave equations for massive and massless fields. Demir and Tanişli [60] presented the conic sedenionic formulation for the unification of generalized field equations of dyons (electromagnetic theory) and gravito-dyons (linear gravity). Chanyal [61] formulated the sedenionic unified potential equations, field equations, and current equations of dyons and gravito-dyons and developed the sedenionic unified theory of dyons and gravito-dyons, in terms of two eight-potentials leading to the structural symmetry. By means of the dual octonion algebra, the author [62] formulated a new set of electrodynamic equations for massive dyons and made an attempt to obtain the symmetrical form of generalized Proca-Dirac-Maxwell equations with respect to the dual octonion form.

The application of the complex-sedenions is able to express the relevant physical quantities as the wave functions in the paper, describing the quantum mechanics connected with the four fundamental interactions, including the non-Abelian gauge field, electroweak field, and dark matter field. The quantum mechanics described with the complex-sedenions is capable of solving a part of conundrums, derived from the Standard Model and even “Beyond the Standard Model.”

*(a) Four Fundamental Fields*. According to the basic postulates [63], one fundamental field extends its individual space. Each of fundamental fields can be described independently by one complex-quaternion space. The four complex-quaternion spaces for the four fundamental fields are perpendicular to each other. Consequently the complex-octonion space is able to study two fundamental fields simultaneously, such as, the electromagnetic and gravitational fields. Meanwhile the quantum mechanics in the complex-sedenion space is capable of exploring the four fundamental fields, including the electromagnetic field, gravitational field, and nuclear field.

*(b) Color Confinement*. In case the direction of a three-dimensional unit vector , in the complex-quaternion wave function, is incapable of playing a major role, the unit vector will be degenerated into the imaginary unit or one new three-dimensional unit vector , which is independent of the unit vector . The wave functions with the imaginary unit can be applied to study the quantum properties of some particles (such as the electrons). Further the wave functions with the new three-dimensional unit vector, , can be utilized to explore the quantum properties of some other particles (e.g., the quarks), which are in possession of “three colors.” The three-dimensional unit vector, , embodies the physical properties of “three colors” relevant to the quarks. Consequently one complex-quaternion wave function is able to be degenerated into three complex-number wave functions, which are independent of each other. In other words, the color degrees of freedom can be merged into the wave functions, described with the complex-quaternions. And the color degrees of freedom are only the spatial dimensions, rather than any property of physical substance.

*(c) Dark Matter Fields*. In the electromagnetic and gravitational fields described with the complex-octonions, there are two sorts of adjoint fields. One of them can be considered as the dark matter field, because the adjoint field possesses a few major properties of the fundamental fields. In the four fundamental interactions, described with the complex-sedenions, there are four fundamental fields and twelve adjoint fields. Three adjoint fields of them can be regarded as the dark matter fields. Moreover, the field sources of these three adjoint fields can be combined with that of some other adjoint fields or fundamental fields to become certain comparatively complicated particles, producing several physical effects of dark matters.

In the paper, the complex-sedenions can be utilized to investigate simultaneously the four fundamental interactions, deducing the field equations of classical mechanics on the macroscopic scale (in Section 3), including the integrating function of field potential, field potential, field strength, field source, linear momentum, angular momentum, torque, and force. On the basis of the exponential form of complex-quaternions, either complex-octonion or complex-sedenion can be written as the exponential form, expressing the relevant physical quantities as the wave functions. Making use of the composite operator and the wave function of physical quantity, it is able to derive similarly the field equations of quantum mechanics on the microscopic scale (in Section 5), from the preceding field equations of classical mechanics. Under few approximate conditions, the field equations of quantum mechanics, described with the complex-sedenions, can be degenerated into the Dirac wave equation or Yang-Mills equation and so forth. Further the unit vector of the complex-quaternion wave function is seized of three spatial dimensions, which can be interpreted as the color degrees of freedom. In other words, the color degrees of freedom are merely the spatial dimensions, rather than any property of physical substance, accounting for the rule of color confinement naturally.

#### 2. Sedenion Space

By means of the algebraic property of sedenions, it is found that one complex-sedenion space consists of four complex-quaternion spaces. On the basis of the basic postulates (see [63]), the complex-sedenion space is able to describe the four sorts of fundamental fields simultaneously. However, according to the electroweak theory, the four fundamental fields in the paper include the gravitational field, electromagnetic field, and strong-nuclear field, except for the existing weak-nuclear field.

In the electroweak theory, the weak-nuclear field and electromagnetic field can be unified into the electroweak field. Either the weak-nuclear field or electromagnetic field is merely one constituent of the electroweak field, so the weak-nuclear field would not be regarded as one fundamental field any more. Further, in the quantum mechanics described with the complex-sedenions, the weak-nuclear field can be considered as the adjoint field of electromagnetic field. As one significant component, the weak-nuclear field can be merged into the electromagnetic field described with the complex-sedenions (in Section 7). And the electromagnetic field, described with the complex-sedenions, can be degenerated naturally into the existing electroweak field.

However, it is inevitable that there must be simultaneously four sorts of fundamental fields in the complex-sedenion space, according to the multiplicative closure of sedenions. The four fundamental fields contain the gravitational field, electromagnetic field, strong-nuclear field, and one new species of unknown fundamental field. The last is called W-nuclear field temporarily, marking with the initial of weak traditionally. In other words, in the field theory described with the complex-sedenions, the four fundamental fields are the gravitational field, electromagnetic field, strong-nuclear field, and W-nuclear field. Apparently, the weak-nuclear field is replaced by the W-nuclear field, in the paper.

In the complex-quaternion space for the gravitational fields, the coordinates are and , the basis vector is , and the complex-quaternion radius vector is . Similarly, in the complex 2-quaternion (short for the second quaternion) space for the electromagnetic fields, the coordinates are and , the basis vector is , and the complex 2-quaternion radius vector is . In the complex 3-quaternion (short for the third quaternion) space for the W-nuclear fields, the coordinates are and , the basis vector is , and the complex 3-quaternion radius vector is . In the complex 4-quaternion (short for the fourth quaternion) space for the strong-nuclear fields, the coordinates are and , the basis vector is , and the complex 4-quaternion radius vector is . Herein, it is convenient to apply the superscript or subscript, , , , and , to mark, respectively, the physical quantities in the gravitational field, electromagnetic field, W-nuclear field, and strong-nuclear field. , , , and are all real. . is the speed of light, and is the time. . . . . . . . . . . The symbol denotes the multiplication of sedenion. . . is the imaginary unit.

In the above, the four independent complex-quaternion spaces, , , , and , are perpendicular to each other. They can be combined together to become one complex-sedenion space . In the complex-sedenion space , the basis vectors are , , , and . The complex-sedenion radius vector is . Herein, , , and are coefficients.

In the complex-quaternion space for the gravitational fields, the complex-quaternion operator is , with . In the complex 2-quaternion space for the electromagnetic fields, the complex 2-quaternion operator is , with . In the complex 3-quaternion space for the W-nuclear fields, the complex 3-quaternion operator is , with . In the complex 4-quaternion space for the strong-nuclear fields, the complex 4-quaternion operator is , with . As a result, the complex-sedenion operator is , in the complex-sedenion space . Herein . . . . . . . .

#### 3. Classical Field Equations

In contrast to the spaces in the Newtonian mechanics, the multidimensional spaces in the paper are differentiated. On the basis of the basic postulates, the spaces to research the four fundamental interactions have been extended from a single complex-quaternion space to four complex-quaternion spaces. Meanwhile the concepts of fields are differentiated also. According to the properties of the complex-sedenion space, the fields to study the four fundamental interactions have been extended from the four fundamental fields to the four fundamental fields and twelve adjoint fields. The adjoint fields possess merely a few major properties of the fundamental fields (see [18]).

Making use of the properties of the complex-sedenions, it is able to deduce the field equations relevant to the gravitational fields, electromagnetic fields, strong-nuclear fields, and W-nuclear fields on the macroscopic scale, including the field potential, field strength, field source, linear momentum, angular momentum, torque, and force. And some of these physical quantities can be separated and degenerated into the electromagnetic field equations, Newton’s law of universal gravitation, mass continuity equation, current continuity equation, force equilibrium equation, precessional equilibrium equation, and so forth.

##### 3.1. Field Potential

In the complex-sedenion space , the complex-sedenion integrating function of field potential is . Herein , , , and are, respectively, the components of the integrating function of field potential in the spaces, , , , and . . . . . , , , and are all real.

By means of the properties of the integrating function of field potential, the complex-sedenion field potential can be defined aswhere . , , , and are, respectively, the components of the field potential in the spaces, , , , and . . Particularly, the operation of complex conjugate will be applied to some physical quantities, except for the coefficients, , , and . The symbol stands for the complex conjugate.

The complex-sedenion field potential includes the field potentials of four fundamental fields and of twelve adjoint fields obviously. The ingredients of the field potentials in the four complex-quaternion spaces are as follows.

(a) In the complex-quaternion space , the component contains the gravitational fundamental field potential , electromagnetic adjoint field potential , W-nuclear adjoint field potential , and strong-nuclear adjoint field potential . Herein . . . . . . . . . . , , , , and are all real.

(b) In the complex 2-quaternion space , the component consists of the electromagnetic fundamental field potential , gravitational adjoint field potential , W-nuclear adjoint field potential , and strong-nuclear adjoint field potential . Herein . . . . . . . . . . , , , , and are all real.

(c) In the complex 3-quaternion space , the component includes the W-nuclear fundamental field potential , gravitational adjoint field potential , electromagnetic adjoint field potential , and strong-nuclear adjoint field potential . Herein . . . . . . . . . . , , , , and are all real.

(d) In the complex 4-quaternion space , the component covers the strong-nuclear fundamental field potential , gravitational adjoint field potential , electromagnetic adjoint field potential , and W-nuclear adjoint field potential . Herein . . . . . . . . . . , , , , and are all real.

##### 3.2. Field Strength

From the complex-sedenion field potential, it is able to define the complex-sedenion field strength aswhere . , , , and are, respectively, the components of the field strength in the spaces, , , , and .

The complex-sedenion field strength comprises the field strengths of four fundamental fields and of twelve adjoint fields obviously. The ingredients of the field strengths in the four complex-quaternion spaces are as follows.

(a) In the complex-quaternion space , the component comprises the gravitational fundamental field strength , electromagnetic adjoint field strength , W-nuclear adjoint field strength , and strong-nuclear adjoint field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complex-numbers.

(b) In the complex 2-quaternion space , the component contains the electromagnetic fundamental field strength , gravitational adjoint field strength , W-nuclear adjoint field strength , and strong-nuclear adjoint field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complex-numbers.

(c) In the complex 3-quaternion space , the component covers the W-nuclear fundamental field strength , gravitational adjoint field strength , electromagnetic adjoint field strength , and strong-nuclear adjoint field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complex-numbers.

(d) In the complex 4-quaternion space , the component includes the strong-nuclear fundamental field strength , gravitational adjoint field strength , electromagnetic adjoint field strength , and W-nuclear adjoint field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complex-numbers.

##### 3.3. Field Source

Making use of the physical properties of the field strength, the complex-sedenion field source can be defined aswhere . , , , and are, respectively, the components of the field source in the spaces, , , , and . , , , , and are coefficients. The symbol denotes the sedenion conjugate.

The complex-sedenion field source consists of the field sources of four fundamental fields and of twelve adjoint fields obviously. The ingredients of the field sources in the four complex-quaternion spaces are as follows.

(a) In the complex-quaternion space , the component covers the gravitational fundamental field source , electromagnetic adjoint field source , W-nuclear adjoint field source , and strong-nuclear adjoint field source . Herein . . . . . . . . . . , , , and are coefficients. is the conventional gravitational constant, and , in the paper. , , , , and are all real.

(b) In the complex 2-quaternion space , the component contains the electromagnetic fundamental field source , gravitational adjoint field source , W-nuclear adjoint field source , and strong-nuclear adjoint field source . Herein . . . . . . . . . . , , , and are coefficients. is the conventional electromagnetic constant, and , in the paper. , , , , and are all real.

(c) In the complex 3-quaternion space , the component comprises the W-nuclear fundamental field source , gravitational adjoint field source , electromagnetic adjoint field source , and strong-nuclear adjoint field source . Herein . . . . . . . . . . , , , and are coefficients. , , , , and are all real.

(d) In the complex 4-quaternion space , the component includes the strong-nuclear fundamental field source , gravitational adjoint field source , electromagnetic adjoint field source , and W-nuclear adjoint field source . Herein . . . . . . . . . . , , , and are coefficients. , , , , and are all real.

In the complex-quaternion space , the field equation, , is capable of determining the “charge” (or mass, ) of the gravitational fields. Meanwhile the field equation can be expanded into the classical gravitational field equations. And the latter can be degenerated into Newton’s law of universal gravitation. In the complex 2-quaternion space , the field equation, , is able to determine the “charge” (or electric charge, ) of the electromagnetic fields. And the field equation can be expanded into the classical electromagnetic field equations. Similarly, in the complex 3-quaternion space , from the field equation, , it is able to determine the “charge” (or W charge, ) of the W-nuclear fields. In the complex 4-quaternion space , from the field equation, , it is capable of determining the “charge” (or strong charge, ) of the strong-nuclear fields. Further this method can be extended from the fundamental fields into the adjoint fields, defining twelve sorts of “charges” of adjoint fields (Table 1).

In case the strong-nuclear field and W-nuclear field can be neglected, the definitions of field potential, field strength, and field source, described with the complex-sedenions, can be reduced, respectively, into that described with the complex-octonions. Further the definitions of field sources, described with the complex-octonions, can be simplified into Newton’s law of universal gravitation and Maxwell equations, described with the complex-quaternions. Meanwhile, a species of electromagnetic adjoint field can be chosen as the dark matter field, in the complex-octonion space.

##### 3.4. Angular Momentum

In the complex-sedenion space , from the complex-sedenion field source, it is able to define the complex-sedenion linear momentum aswhere . , , , and are, respectively, the components of the linear momentum in the spaces, , , , and . . . . . . . . . , , , and are all real.

From the complex-sedenion radius vector , linear momentum , and integrating function of field potential , it is capable of defining the complex-sedenion angular momentum aswhere . , , , and are, respectively, the components of the angular momentum in the spaces, , , , and . . . . . . . . is a coefficient, to meet the requirement of the dimensional homogeneity (see [63]). And the ingredients of the angular momentum in the four complex-quaternion spaces are as follows:

##### 3.5. Torque

From the complex-sedenion angular momentum, it is able to define the complex-sedenion torque aswhere . , , , and are, respectively, the components of the torque in the spaces, , , , and . . . . . . . And the ingredients of the torque in the four complex-quaternion spaces are as follows:

##### 3.6. Force

From the complex-sedenion torque, it is capable of defining the complex-sedenion force aswhere . , , , and are, respectively, the components of the force in the spaces, , , , and . And the ingredients of the torque in the four complex-quaternion spaces are as follows:

The above shows that the application of complex-sedenion is able to infer some field equations, relevant to fundamental fields and adjoint fields, in the classical mechanics on the macroscopic scale (Table 2). In the complex-sedenion space, the complex-sedenion angular momentum comprises the orbital angular momentum, magnetic moment, electric moment, and so forth. The complex-sedenion torque consists of the conventional torque and energy and so on. In most cases, the complex-sedenion force will be equal to zero. From the equation, it is able to deduce the force equilibrium equation, precessional equilibrium equation, power equation, mass continuity equation, current continuity equation, and so forth. The force equilibrium equation includes the inertial force, gravity, electromagnetic force, and energy gradient. As a force term, the energy gradient can be applied to explore the astrophysical jets (see [33]), condensed dark matters, new principle of the accelerator, and so forth. Meanwhile the precessional equilibrium equation can be utilized to account for certain precessional motions, deducing the angular velocity of Larmor precession and so forth.

In case the strong-nuclear field and W-nuclear field can be neglected, the definitions of angular momentum, torque, and force, described with the complex-sedenions, can be degenerated into that described with the complex-octonions, respectively.

#### 4. Wave Function

In the quantum mechanics, what plays an important role is the wave function, connected with the physical quantities in the classical mechanics, rather than any pure physical quantity in the classical mechanics. Therefore, in the classical mechanics described with the complex-sedenions, it is necessary to multiply the physical quantity, in the classical mechanics, with the dimensionless complex-sedenion auxiliary quantity, transforming it to become the wave function. For instance, the wave function of the complex-sedenion angular momentum is . Herein is a dimensionless auxiliary quantity, and is the Planck constant. Apparently, the wave functions or quantum physical quantities in the quantum mechanics are some functions of the physical quantities in the classical mechanics (Table 3). In Table 3, the complex-sedenion quantum physical quantities include the quantum integrating function of field potential, quantum-field potential, quantum-field strength, quantum-field source, quantum linear momentum, quantum angular momentum, quantum torque, and quantum force. In other words, in terms of the concept of function, the quantum-fields of the quantum mechanics on the microscopic scale are just the functions of the classical fields of the classical mechanics on the macroscopic scale. There are some similarities as well as differences between the quantum physical quantity and classical physical quantity.

In the subspace of the complex-octonion space , the complex-quaternion wave function is defined as , in terms of the complex-quaternion angular momentum . The major ingredient of is . By means of certain appropriate transformations (Appendix A), it is able to achieve two new wave functions, and . Herein and are all real. is a three-dimensional unit vector, and is a vector.

In the complex-octonion space , which consists of two complex-quaternion spaces, and , any complex-quaternion wave function can be degenerated into two correlative wave functions. (a) Imaginary unit: when the direction of unit vector is incapable of playing a major role in the complex-quaternion wave function, the unit vector will be degenerated into the imaginary unit . If the unit vector can be replaced by the imaginary unit , the above wave functions, and , will be degraded into the wave function, described with the complex-number, in the conventional quantum mechanics. (b) Unit vector: when the vector property of the unit vector can be neglected partially in the complex-quaternion wave function, the unit vector will be degenerated into a new three-dimensional unit vector , which is independent of the unit vector . In case the unit vector can be replaced by the unit vector , the three-dimensional unit vector will possess three new degrees of freedom, in contrast to the imaginary unit . That is, one wave function with the unit vector is equivalent to three conventional wave functions with the complex-numbers. In a general way, a complex-quaternion wave function may be degraded into three complex-number wave functions, which are independent of each other. If we mistake the three-dimensional unit vector, , for the imaginary unit, either of two wave functions, and , must be considered equivalently as the complex-number wave functions, which are in possession of three new degrees of freedom. Herein , , and .

Similarly, in the complex-quaternion space , in terms of some other physical quantities, also it is able to define their corresponding complex-quaternion wave functions. What is more interesting is that it is capable of defining further some complex-quaternion wave functions, connected with the physical quantities, in the rest of complex-quaternion spaces, , , and . Moreover, under certain different circumstances, one should choose different approximate results (imaginary unit or unit vector , etc.) for the complex-quaternion wave functions, in order to facilitate these complex-quaternion wave functions to be degenerated into a few wave functions, described with complex-numbers, in the conventional quantum mechanics.

#### 5. Quantum-Field Equations

In the complex-sedenion space , making use of the exponential forms and wave functions of the complex-sedenions, it is able to deduce the quantum-field equations, on the microscopic scale, for the gravitational fields, electromagnetic fields, W-nuclear fields, and strong-nuclear fields, including the quantum-field potential, quantum-field strength, quantum-field source, quantum linear momentum, angular momentum, quantum torque, and quantum force.

The quantum-fields of the quantum mechanics on the microscopic scale are just the functions or transformations of the classical fields of the classical mechanics on the macroscopic scale, in the complex-sedenion space . In a similar way to the method of the mathematical inference, for the field equations of classical mechanics on the macroscopic scale, it is capable of deducing the field equations of the quantum mechanics on the microscopic scale. Starting from the integrating function of field potential , of the field equations in the classical mechanics, one can infer gradually the field equations in the quantum mechanics (Table 4). Obviously, a majority of physical quantities and field equations in the Table 4 are different from that in the Table 2. For instance, the quantum-field source in the Table 4 is independent of the classical field source in the Table 2.

##### 5.1. Quantum-Field Potential

In the complex-sedenion space , the complex-sedenion quantum integrating function of field potential is . , , , and are, respectively, the components of the quantum integrating function of field potential in the spaces, , , , and . Herein , and is one auxiliary quantity. . . . . , , , and are all real.

In the quantum mechanics, we may encounter a new composite operator, . Herein . . . . . . Meanwhile, . . . . Herein , , , and are, respectively, the components of the auxiliary quantity, , in the complex-quaternion spaces, , , , and . And the ingredients of this new composite operator in the four complex-quaternion spaces are as follows:

In the complex-sedenion space , by means of the physical quantities of the complex-sedenion quantum integrating function of field potential , the complex-sedenion quantum-field potential is defined aswhere and is one auxiliary quantity. . , , , and are, respectively, the components of the quantum-field potential in the complex-quaternion spaces, , , , and . .

According to the multiplication rule of the sedenions, the complex-sedenion quantum-field potential includes the quantum-field potentials of the four fundamental quantum-fields (i.e., gravitational fundamental quantum-field, electromagnetic fundamental quantum-field, W-nuclear fundamental quantum-field, and strong-nuclear fundamental quantum-field) and twelve adjoint quantum-fields. And the ingredients of the quantum-field potential in the four complex-quaternion spaces are as follows.

(a) In the complex-quaternion space , the component contains the gravitational fundamental quantum-field potential , electromagnetic adjoint quantum-field potential , W-nuclear adjoint quantum-field potential , and strong-nuclear adjoint quantum-field potential . Herein . . . . . . . . . . , , , , and are all real.

(b) In the complex 2-quaternion space , the component consists of the electromagnetic fundamental quantum-field potential , gravitational adjoint quantum-field potential , W-nuclear adjoint quantum-field potential , and strong-nuclear adjoint quantum-field potential . Herein . . . . . . . . . . , , , , and are all real.

(c) In the complex 3-quaternion space , the component includes the W-nuclear fundamental quantum-field potential , gravitational adjoint quantum-field potential , electromagnetic adjoint quantum-field potential , and strong-nuclear adjoint quantum-field potential . Herein . . . . . . . . . . , , , , and are all real.

(d) In the complex 4-quaternion space , the component covers the strong-nuclear fundamental quantum-field potential , gravitational adjoint quantum-field potential , electromagnetic adjoint quantum-field potential , and W-nuclear adjoint quantum-field potential . Herein . . . . . . . . . . , , , , and are all real.

##### 5.2. Quantum-Field Strength

From the complex-sedenion quantum-field potential, it is able to define the complex-sedenion quantum-field strength aswhere and is one auxiliary quantity. . , , , and are, respectively, the components of the quantum-field strength in the spaces, , , , and .

The complex-sedenion quantum-field strength comprises the quantum-field strengths of four fundamental quantum-fields and of twelve adjoint quantum-fields obviously. The ingredients of the quantum-field strengths in the four complex-quaternion spaces are as follows.

(a) In the complex-quaternion space , the component comprises the gravitational fundamental quantum-field strength , electromagnetic adjoint quantum-field strength , W-nuclear adjoint quantum-field strength , and strong-nuclear adjoint quantum-field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complex-numbers.

(b) In the complex 2-quaternion space , the component contains the electromagnetic fundamental quantum-field strength , gravitational adjoint quantum-field strength , W-nuclear adjoint quantum-field strength , and strong-nuclear adjoint quantum-field strength . Herein . . . . . . .