Some physical models with Minkowski spacetime structure and Lorentz group symmetry

Hong, Hong‐Ki; Liu, Chein‐Shan

doi:10.1016/s0020-7462(00)00072-x

Cited by 11 publications

(4 citation statements)

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“…Then by x = x° + x s and y = y0 + y\ it was shown by Liu [14] that the product rule in Eq. (20) can be represented by xy = x°X s -y s +x° y s + y° X s +x s x y s…”

Section: = (22)mentioning

confidence: 99%

“…(20) as that for the real quaternions, it is not difficult to show that Eqs. (15) ~ (18) still hold when x, y, and z are elements of the complex quaternions and a is a complex number.…”

Section: (23)mentioning

confidence: 99%

“…This fact is reflected in the equation of motion of a charged particle under the electromagnetic field as shown by Eq. (62) in the paper by Hong and Liu [20]. Applying the operator Y on the above two equations from both sides and utilizing Eqs.…”

Section: The Maxwell Equationsmentioning

confidence: 99%

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Theg-Based Jordan Algebra and Lie Algebra Formulations of the Maxwell Equations

Liu

2004

J. mech.

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When it is usually using a bigger algebra system to formulate the Maxwell equations, in this paper we consider a real four-dimensional algebra to express the Maxwell equations without appealing to the imaginary number and higher dimensional algebras. In terms of g-based Jordan algebra formulation the Lorentz gauge condition is found to be a necessary and sufficient condition to render the second pair of Maxwell equations, while the first pair of Maxwell equations is proved to be an intrinsic algebraic property. Then, we transform the g-based Jordan algebra to a Lie algebra of the dilation proper orthochronous Lorentz group, which gives us an incentive to consider a linear matrix operator of the Lie type, rendering more easy to derive the Maxwell equations and the wave equations. The new formulations fully match the requirements for the classical electrodynamic equations and the Lorentz gauge condition. The mathematical advantage of our formulations is that they are irreducible in the sense that, when compared to the formulations which using other bigger algebras (e.g., biquaternions and Clifford algebras), the number of explicit components and operations is minimal. From this aspect, the g-based Jordan algebra and Lie algebra are the most suitable algebraic systems to implement the Maxwell equations into a more compact form.

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“…Then by x = x° + x s and y = y0 + y\ it was shown by Liu [14] that the product rule in Eq. (20) can be represented by xy = x°X s -y s +x° y s + y° X s +x s x y s…”

Section: = (22)mentioning

confidence: 99%

“…(20) as that for the real quaternions, it is not difficult to show that Eqs. (15) ~ (18) still hold when x, y, and z are elements of the complex quaternions and a is a complex number.…”

Section: (23)mentioning

confidence: 99%

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Theg-Based Jordan Algebra and Lie Algebra Formulations of the Maxwell Equations

Liu

2004

J. mech.

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“…In Blanes and Moan, 2001;Zhang and Deng, 2003, some efficient numerical integration schemes are presented for nonlinear dynamic system. In order to solve the nonlinear dynamic system (1), we can convert it into an augmented dynamic system in the Minkowski space of Liu (2001) and Hong and Liu (2001), which results in the system of Lie type _ X ¼ AX, X 2 M nþ1 (M nþ1 is the Minkowski space), A 2 SOðn; 1Þ is a local Lie algebra of the orthochronous Lorentz group SO 0 ðn; 1Þ.…”

Section: Introductionmentioning

confidence: 99%