Research Article  Open Access
Color Confinement and Spatial Dimensions in the ComplexSedenion Space
Abstract
The paper aims to apply the complexsedenions to explore the wave functions and field equations of nonAbelian gauge fields, considering the spatial dimensions of a unit vector as the color degrees of freedom in the complexquaternion 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 complexsedenions 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 YangMills equation. In contrast to the complexnumber wave function, the complexquaternion wave function possesses three new degrees of freedom, that is, three color degrees of freedom. One complexquaternion wave function is equivalent to three complexnumber wave functions. It means that the three spatial dimensions of unit vector in the complexquaternion wave function can be considered as the “three colors”; naturally the color confinement will be effective. In other words, in the complexquaternion 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 strongnuclear 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 complexsedenions, replies partially to this puzzle. The quantum mechanics, described with the complexsedenions, is capable of deducing the wave functions, Dirac wave equations, YangMills equations of the nonAbelian gauge field, field equations of the electroweak field, and so forth. Particularly, their wave functions, described with the complexquaternions, are able to elucidate directly the color degrees of freedom.
C.N. Yang and R. L. Mills found the field equations for a nonAbelian gauge field in 1954. The field equations of nonAbelian 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 YangMill equations can be applied to construct successfully the unified theory of the electromagnetic field and weaknuclear field. The triumph of the electroweak theory edified a part of scholars to extend the YangMill equations into the strongnuclear fields, establishing the quantum chromodynamics (QCD for short).
M. GellMann 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 strongnuclear 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, nonAbelian monopoles, microscopic resonance, hidden local symmetry, color charge, soliton model, and string theory. Using the magnetic symmetry structure of nonAbelian gauge theories of the YangMills type, Chandola et al. [1] discussed the mathematical foundation of dual chromodynamics in fiberbundle form. Nakamura and Saito [2] studied the longrange behavior of the heavy quark potential in Coulomb gauge, using a quenched SU(3) lattice gauge simulation with partiallength 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 nonAbelian 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 nonAbelian color confinement in a gaugeindependent way, by the high precision lattice Monte Carlo simulations in the gluodynamics. The authors [7] studied the mechanism of nonAbelian color confinement in SU(2) lattice gauge theory, in terms of the Abelian fields and monopoles extracted from the nonAbelian 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 quarkgluon plasma during the intermediate stage and after the end of crossover from the hadronic matter to the strongly coupled quarkgluon plasma, by means of the bond percolation model. Braun et al. [10] identified a simple criterion for quark confinement, computing the orderparameter potential from the Landaugauge 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 offshell 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 frameindependent lightfront Hamiltonian theory. The author [16] studied the lightfront 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 strongnuclear 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 complexsedenions, 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, YangMills equation, electroweak field, color confinement, dark matter field, and so forth. When a part of coordinate values are complexnumbers, the quaternion, octonion, and sedenion are called the complexquaternion, complexoctonion, and complexsedenion, 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 spacetime. 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 complexquaternion 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 quantumfield 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 Maxwelllike 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 timedependent 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, YangMills 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. MarquesBonham [35] developed the geometrical properties of a flat tangent spacetime local to the manifold of the EinsteinSchrödinger nonsymmetric theory on an octonionic curved space. De Leo and AbdelKhalek [36] introduced leftright 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 secondorder equation for octonionic wave function can be reformulated in the form of a system of firstorder equations for quantumfields. The authors [40] demonstrated a generalization of relativistic quantum mechanics using the octonions, generating an associative noncommutative spatial algebra. The octonionic secondorder 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 fourindex antisymmetric and nonAbelian field, which satisfies a selfduality relation in eightdimensional curved space. Demir and Tanişli proposed a new formulation for the massive gravielectromagnetism with monopole terms [43]. The author formulated the MaxwellProca 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 4component spinor, which is a combination of two quaternions. In terms of the quaternion in the YangMills 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 DiracMaxwell’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/nonAbelian 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 spacetime 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 complexnumber 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 spacetime algebra and demonstrated a generalization of relativistic quantum mechanics using sedenionic wave functions and sedenionic spacetime 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 spacetime operators and sedenionic wave functions, the authors [59] discussed the gauge invariance of generalized secondorder and firstorder 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 gravitodyons (linear gravity). Chanyal [61] formulated the sedenionic unified potential equations, field equations, and current equations of dyons and gravitodyons and developed the sedenionic unified theory of dyons and gravitodyons, in terms of two eightpotentials 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 ProcaDiracMaxwell equations with respect to the dual octonion form.
The application of the complexsedenions 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 nonAbelian gauge field, electroweak field, and dark matter field. The quantum mechanics described with the complexsedenions 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 complexquaternion space. The four complexquaternion spaces for the four fundamental fields are perpendicular to each other. Consequently the complexoctonion space is able to study two fundamental fields simultaneously, such as, the electromagnetic and gravitational fields. Meanwhile the quantum mechanics in the complexsedenion 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 threedimensional unit vector , in the complexquaternion wave function, is incapable of playing a major role, the unit vector will be degenerated into the imaginary unit or one new threedimensional 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 threedimensional 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 threedimensional unit vector, , embodies the physical properties of “three colors” relevant to the quarks. Consequently one complexquaternion wave function is able to be degenerated into three complexnumber 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 complexquaternions. 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 complexoctonions, 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 complexsedenions, 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 complexsedenions 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 complexquaternions, either complexoctonion or complexsedenion 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 complexsedenions, can be degenerated into the Dirac wave equation or YangMills equation and so forth. Further the unit vector of the complexquaternion 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 complexsedenion space consists of four complexquaternion spaces. On the basis of the basic postulates (see [63]), the complexsedenion 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 strongnuclear field, except for the existing weaknuclear field.
In the electroweak theory, the weaknuclear field and electromagnetic field can be unified into the electroweak field. Either the weaknuclear field or electromagnetic field is merely one constituent of the electroweak field, so the weaknuclear field would not be regarded as one fundamental field any more. Further, in the quantum mechanics described with the complexsedenions, the weaknuclear field can be considered as the adjoint field of electromagnetic field. As one significant component, the weaknuclear field can be merged into the electromagnetic field described with the complexsedenions (in Section 7). And the electromagnetic field, described with the complexsedenions, 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 complexsedenion space, according to the multiplicative closure of sedenions. The four fundamental fields contain the gravitational field, electromagnetic field, strongnuclear field, and one new species of unknown fundamental field. The last is called Wnuclear field temporarily, marking with the initial of weak traditionally. In other words, in the field theory described with the complexsedenions, the four fundamental fields are the gravitational field, electromagnetic field, strongnuclear field, and Wnuclear field. Apparently, the weaknuclear field is replaced by the Wnuclear field, in the paper.
In the complexquaternion space for the gravitational fields, the coordinates are and , the basis vector is , and the complexquaternion radius vector is . Similarly, in the complex 2quaternion (short for the second quaternion) space for the electromagnetic fields, the coordinates are and , the basis vector is , and the complex 2quaternion radius vector is . In the complex 3quaternion (short for the third quaternion) space for the Wnuclear fields, the coordinates are and , the basis vector is , and the complex 3quaternion radius vector is . In the complex 4quaternion (short for the fourth quaternion) space for the strongnuclear fields, the coordinates are and , the basis vector is , and the complex 4quaternion 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, Wnuclear field, and strongnuclear 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 complexquaternion spaces, , , , and , are perpendicular to each other. They can be combined together to become one complexsedenion space . In the complexsedenion space , the basis vectors are , , , and . The complexsedenion radius vector is . Herein, , , and are coefficients.
In the complexquaternion space for the gravitational fields, the complexquaternion operator is , with . In the complex 2quaternion space for the electromagnetic fields, the complex 2quaternion operator is , with . In the complex 3quaternion space for the Wnuclear fields, the complex 3quaternion operator is , with . In the complex 4quaternion space for the strongnuclear fields, the complex 4quaternion operator is , with . As a result, the complexsedenion operator is , in the complexsedenion 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 complexquaternion space to four complexquaternion spaces. Meanwhile the concepts of fields are differentiated also. According to the properties of the complexsedenion 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 complexsedenions, it is able to deduce the field equations relevant to the gravitational fields, electromagnetic fields, strongnuclear fields, and Wnuclear 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 complexsedenion space , the complexsedenion 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 complexsedenion 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 complexsedenion 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 complexquaternion spaces are as follows.
(a) In the complexquaternion space , the component contains the gravitational fundamental field potential , electromagnetic adjoint field potential , Wnuclear adjoint field potential , and strongnuclear adjoint field potential . Herein . . . . . . . . . . , , , , and are all real.
(b) In the complex 2quaternion space , the component consists of the electromagnetic fundamental field potential , gravitational adjoint field potential , Wnuclear adjoint field potential , and strongnuclear adjoint field potential . Herein . . . . . . . . . . , , , , and are all real.
(c) In the complex 3quaternion space , the component includes the Wnuclear fundamental field potential , gravitational adjoint field potential , electromagnetic adjoint field potential , and strongnuclear adjoint field potential . Herein . . . . . . . . . . , , , , and are all real.
(d) In the complex 4quaternion space , the component covers the strongnuclear fundamental field potential , gravitational adjoint field potential , electromagnetic adjoint field potential , and Wnuclear adjoint field potential . Herein . . . . . . . . . . , , , , and are all real.
3.2. Field Strength
From the complexsedenion field potential, it is able to define the complexsedenion field strength aswhere . , , , and are, respectively, the components of the field strength in the spaces, , , , and .
The complexsedenion 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 complexquaternion spaces are as follows.
(a) In the complexquaternion space , the component comprises the gravitational fundamental field strength , electromagnetic adjoint field strength , Wnuclear adjoint field strength , and strongnuclear adjoint field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complexnumbers.
(b) In the complex 2quaternion space , the component contains the electromagnetic fundamental field strength , gravitational adjoint field strength , Wnuclear adjoint field strength , and strongnuclear adjoint field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complexnumbers.
(c) In the complex 3quaternion space , the component covers the Wnuclear fundamental field strength , gravitational adjoint field strength , electromagnetic adjoint field strength , and strongnuclear adjoint field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complexnumbers.
(d) In the complex 4quaternion space , the component includes the strongnuclear fundamental field strength , gravitational adjoint field strength , electromagnetic adjoint field strength , and Wnuclear adjoint field strength . Herein . . . . . . . . . . , , , , and are all real. , , , , and are complexnumbers.
3.3. Field Source
Making use of the physical properties of the field strength, the complexsedenion 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 complexsedenion 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 complexquaternion spaces are as follows.
(a) In the complexquaternion space , the component covers the gravitational fundamental field source , electromagnetic adjoint field source , Wnuclear adjoint field source , and strongnuclear adjoint field source . Herein . . . . . . . . . . , , , and are coefficients. is the conventional gravitational constant, and , in the paper. , , , , and are all real.
(b) In the complex 2quaternion space , the component contains the electromagnetic fundamental field source , gravitational adjoint field source , Wnuclear adjoint field source , and strongnuclear adjoint field source . Herein . . . . . . . . . . , , , and are coefficients. is the conventional electromagnetic constant, and , in the paper. , , , , and are all real.
(c) In the complex 3quaternion space , the component comprises the Wnuclear fundamental field source , gravitational adjoint field source , electromagnetic adjoint field source , and strongnuclear adjoint field source . Herein . . . . . . . . . . , , , and are coefficients. , , , , and are all real.
(d) In the complex 4quaternion space , the component includes the strongnuclear fundamental field source , gravitational adjoint field source , electromagnetic adjoint field source , and Wnuclear adjoint field source . Herein . . . . . . . . . . , , , and are coefficients. , , , , and are all real.
In the complexquaternion 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 2quaternion 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 3quaternion space , from the field equation, , it is able to determine the “charge” (or W charge, ) of the Wnuclear fields. In the complex 4quaternion space , from the field equation, , it is capable of determining the “charge” (or strong charge, ) of the strongnuclear 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 strongnuclear field and Wnuclear field can be neglected, the definitions of field potential, field strength, and field source, described with the complexsedenions, can be reduced, respectively, into that described with the complexoctonions. Further the definitions of field sources, described with the complexoctonions, can be simplified into Newton’s law of universal gravitation and Maxwell equations, described with the complexquaternions. Meanwhile, a species of electromagnetic adjoint field can be chosen as the dark matter field, in the complexoctonion space.
3.4. Angular Momentum
In the complexsedenion space , from the complexsedenion field source, it is able to define the complexsedenion linear momentum aswhere . , , , and are, respectively, the components of the linear momentum in the spaces, , , , and . . . . . . . . . , , , and are all real.
From the complexsedenion radius vector , linear momentum , and integrating function of field potential , it is capable of defining the complexsedenion 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 complexquaternion spaces are as follows:
3.5. Torque
From the complexsedenion angular momentum, it is able to define the complexsedenion torque aswhere . , , , and are, respectively, the components of the torque in the spaces, , , , and . . . . . . . And the ingredients of the torque in the four complexquaternion spaces are as follows:
3.6. Force
From the complexsedenion torque, it is capable of defining the complexsedenion force aswhere . , , , and are, respectively, the components of the force in the spaces, , , , and . And the ingredients of the torque in the four complexquaternion spaces are as follows:
The above shows that the application of complexsedenion 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 complexsedenion space, the complexsedenion angular momentum comprises the orbital angular momentum, magnetic moment, electric moment, and so forth. The complexsedenion torque consists of the conventional torque and energy and so on. In most cases, the complexsedenion 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 strongnuclear field and Wnuclear field can be neglected, the definitions of angular momentum, torque, and force, described with the complexsedenions, can be degenerated into that described with the complexoctonions, 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 complexsedenions, it is necessary to multiply the physical quantity, in the classical mechanics, with the dimensionless complexsedenion auxiliary quantity, transforming it to become the wave function. For instance, the wave function of the complexsedenion 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 complexsedenion quantum physical quantities include the quantum integrating function of field potential, quantumfield potential, quantumfield strength, quantumfield source, quantum linear momentum, quantum angular momentum, quantum torque, and quantum force. In other words, in terms of the concept of function, the quantumfields 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.
