24,078 research outputs found
Field Equations in the Complex Quaternion Spaces
The paper aims to adopt the complex quaternion and octonion to formulate the
field equations for electromagnetic and gravitational fields. Applying the
octonionic representation enables one single definition to combine some physics
contents of two fields, which were considered to be independent of each other
in the past. J. C. Maxwell applied simultaneously the vector terminology and
the quaternion analysis to depict the electromagnetic theory. This method
edified the paper to introduce the quaternion and octonion spaces into the
field theory, in order to describe the physical feature of electromagnetic and
gravitational fields, while their coordinates are able to be the complex
number. The octonion space can be separated into two subspaces, the quaternion
space and the S-quaternion space. In the quaternion space, it is able to infer
the field potential, field strength, field source, field equations, and so
forth, in the gravitational field. In the S-quaternion space, it is able to
deduce the field potential, field strength, field source, and so forth, in the
electromagnetic field. The results reveal that the quaternion space is
appropriate to describe the gravitational features; meanwhile the S-quaternion
space is proper to depict the electromagnetic features.Comment: arXiv admin note: substantial text overlap with arXiv:1503.0609
Color Confinement and Spatial Dimensions in the Complex-sedenion Space
The paper aims to apply the complex-sedenions to explore the wavefunctions
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 wavefunctions, exploring the physical properties of the
color confinement essentially. J. C. Maxwell was the first to employ the
quaternions to study the physical properties of electromagnetic fields. His
method inspires some subsequent scholars to introduce the quaternions,
octonions, and sedenions to research the electromagnetic field, gravitational
field, nuclear field, quantum mechanics, and gauge field. The application of
complex-sedenions is capable of depicting not only the field equations of the
classical mechanics on the macroscopic scale, but also the field equations of
the quantum mechanics on the microscopic scale. The latter can be degenerated
into the Dirac equation and Yang-Mills equation. In contrast to the
complex-number wavefunction, the wavefunction in the complex-quaternion space
possesses three new degrees of freedom, that is, three color degrees of
freedom. One complex-quaternion wavefunction is equivalent to three
conventional wavefunctions with the complex-numbers. It means that the three
spatial dimensions of unit vector in the complex-quaternion wavefunction 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. The existing `three colors' can be merged into the wavefunction,
described with the complex-quaternions
Evans v. Michigan: The Impact of Judicial Error on Double Jeopardy Protection
This commentary previews an upcoming Supreme Court case, Evans v. Michigan, in which the Court has an opportunity to clarify the bounds of the prohibition on double jeopardy. More specifically, the Court will determine what, if any, impact judicial error has on double jeopardy protection under the Fifth Amendment
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