Some coordinate transformations relevant to refractive indices

Weng, Zihua

doi:10.1063/5.0068969

Cited by 5 publications

(4 citation statements)

References 43 publications

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“…In the electromagnetic and gravitational theories described by the octonions, the quaternion operator ◊ and the field strength of two fundamental fields (electromagnetic field and gravitational field) can be combined together to form a composite operator, ◊( , ) . This has been discussed in some papers 17,18 . Subsequently, this viewpoint can be further promoted.…”

Section: Electromagnetic and Gravitational Fieldsmentioning

confidence: 96%

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Gauge fields and four interactions in the trigintaduonion spaces

Weng

2024

Preprint

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The paper aims to apply the trigintaduonion spaces to explore the physical properties of four interactions simultaneously, including the electromagnetic fields, gravitational fields, weak nuclear fields, and strong nuclear fields. J. C. Maxwell first applied the algebra of quaternions to study the physical properties of electromagnetic fields. It inspired some subsequent scholars to introduce the quaternions, octonions, sedenions, and trigintaduonions to research the electromagnetic fields, gravitational fields, weak nuclear fields, strong nuclear fields, quantum mechanics, gauge fields, and curved spaces and so forth. The algebra of trigintaduonions is able to discuss the physical quantities of four interactions, including the field potential, field strong, field source, linear momentum, angular momentum, torque, and force. In the field theories described with the algebra of trigintaduonions, the weak nuclear field is composed of three types of fundamental fields. These three fundamental fields, related to weak nuclear fields, can describe the physical properties of weak nuclear fields collectively. This is consistent with the conclusion of the electroweak theory. Meanwhile the strong nuclear field consists of three types of fundamental fields. These three fundamental fields relevant to strong nuclear fields may investigate the physical properties of strong nuclear fields mutually. It is coincident with the deduction of quark theory. According to the properties of trigintaduonions, one can deduce the Yang-Mills equation related to the gauge fields. It means that the electromagnetic field occupies a quaternion space. The gravitational field owns one different quaternion space. The weak nuclear fields occupy three mutually independent quaternion spaces. The properties of weak nuclear fields are different from those of electromagnetic fields or gravitational fields. According to the multiplicative closure, the strong nuclear fields also own three quaternion spaces independent of each other. These explorations further deepen the understanding of the physical properties of weak and strong nuclear fields.

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Section: Electromagnetic and Gravitational Fieldsmentioning

confidence: 96%

“…[17]), second-torque continuity equation (see Ref. [18]), second-force equilibrium equation (see Ref. [19]), precession equilibrium equation 20 , and second-precession equilibrium equation 21 .…”

Section: Gauge Fieldsmentioning

confidence: 99%

Gauge fields and four interactions in the trigintaduonion spaces

Weng

2024

Preprint

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“…From the two basic postulates, it is capable of deducing the Galilean transformation with eight coordinate values. In case the octonion space O is degenerated into the quaternion space H g , the Galilean transformation with eight coordinate values [20] will be reduced into the Galilean transformation in the classical field theory. 3) Lorentz transformation.…”

Section: Coordinate Transformationsmentioning

confidence: 99%

“…From the two basic postulates, it is capable of deducing the Lorentz transformation with eight coordinate values (see Ref. [20]). When the scalar part of octonion radius vector plays a major role, two basic postulates, Eq.…”

Section: Coordinate Transformationsmentioning

confidence: 99%

Two incompatible types of invariants in the octonion spaces

Weng

2022

Preprint

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The paper aims to study some invariants and conservation laws relevant to electromagnetic and gravitational fields, by means of the rotational transformations of octonion coordinate systems. The scholars utilize the octonions to analyze the electromagnetic and gravitational fields simultaneously, including the octonion field potential, field strength, field source, linear momentum, angular momentum, torque and force. When the octonion coordinate system transforms rotationally, the vector part of one octonion may alter, while the scalar part of the octonion will remain unchanged. This property allows one to deduce a few invariants, such as the scalar part of octonion radius vector, speed of light, and norm of octonion radius vector. These invariants are the basic postulates for the Galilean transformation and Lorentz transformation. Further, from the rotation transform of octonion coordinate systems, it is capable of deducing several invariants, including the mass, energy and power relevant to gravitational fields, in one octonion space O. And the term relevant to the electric charge will transform with the rotation of coordinate systems. In another octonion space Ou, it is capable of inferring a few invariants, including the electric charge related to electromagnetic fields. And the terms relevant to the mass and energy will vary with the rotation of octonion coordinate systems. So the invariants are divided into two different groups. In particular, the mass conservation law and energy conservation law can be effective simultaneously. But the charge conservation law and mass conservation law are unable to be valid simultaneously, in the strict sense. It is beneficial to further understand the laws of conservation.

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Basic postulates of some coordinate transformations within material media

Weng

2022

AIP Advances

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This paper aims to explore the physical quantities of several invariants, including the basic postulates of some types of crucial coordinate transformations, conservation laws, and continuity equations, in the electromagnetic and gravitational fields. Maxwell first utilized quaternions to describe the electromagnetic theory. Subsequent scholars make use of octonions to study the physical properties of electromagnetic and gravitational fields simultaneously, including the octonion field strength, field source, angular momentum, torque, and force. When an octonion coordinate system transforms rotationally, the scalar part of one octonion will remain unchanged, although the vector part of the octonion may alter. In the octonion space [Formula: see text], some invariants can be derived from this octonion property. A part of these invariants can be selected as the basic postulates of Galilean transformation or Lorentz transformation. Similarly, several invariants can be derived from the octonion property in the transformed octonion space [Formula: see text], and the invariants can be chosen as the basic postulates of a few new types of coordinate transformations. Furthermore, the combination of invariants in the octonion spaces can be applied as the basic postulates of some new coordinate transformations, relevant to the norm of physical quantities. Through the analysis, it is easy to find that each conserved quantity has its preconditions from the perspective of octonion spaces. This is helpful to deepen the further understanding of the physical properties of conservation laws and other invariants.

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Some coordinate transformations relevant to refractive indices

Cited by 5 publications

References 43 publications

Gauge fields and four interactions in the trigintaduonion spaces

Gauge fields and four interactions in the trigintaduonion spaces

Two incompatible types of invariants in the octonion spaces

Basic postulates of some coordinate transformations within material media

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