arXiv 1101.36062011-01gr-qcquant-ph

四元数Weyl-Cartan几何中标量曲率的内蕴Dirac行为

Intrinsic Dirac Behavior of Scalar Curvature in a Quaternionic Weyl-Cartan Geometry

J. E. Rankin

备注：: 37 pages (of which, 24 are pages in the appendix). Major additions to the appendix, and minor changes to the body of the paper. Copy at http://www.researchgate.net/profile/James_Rankin/publications may be updated more often than arXiv copy

摘要：

狄拉克理论的二阶手征形式的“上旋”和“下旋”投影被证明符合早期经典复标量规范理论（1992年4月，第10卷）中预测的形式的叠加。Quantum Grav.）。在某种意义上，似乎可以将两分量狄拉克旋量视为一个单分量、四元数的时空标量。“自旋空间”变换可以被认为是内部四元数基的变换。从本质上讲，四元数狄拉克理论巧妙地投射到复平面上，在那里自旋与度规的自对偶反对称部分相关。正确的狄拉克本征值和行为良好的本征函数完整地投影到早期理论的Weyl-Cartan型几何中标量曲率的一对复解中。一些估计作出预测，有趣的原子和亚原子尺度的现象。一种电磁量子出现了。一个概括的复杂的几何结构，然后勾勒在附录中，允许四元数规范和曲率，并有一些外尔非度量混合扭转。它似乎是一个定义明确的结构，并导致完整的，二阶，四元数狄拉克方程的形式，和一个密切相关的一阶方程，辅助波函数。一个家庭的“自由粒子”的解决方案检查的对称部分的度量的洛伦兹极限。更一般地说，当被限制为两个四元数维度（只有两个分量）时，合理相似的解可以线性叠加成新的解，并分成两个具有不同对易特性的族。辅助波函数方程的可积性条件对原始波函数施加了六个条件，满足了所研究的“自由粒子”解。检查协方差。考察了氢原子的达尔文解。

英文摘要：

The "spin-up" and "spin-down" projections of the second order, chiral form of Dirac Theory are shown to fit a superposition of forms predicted in an earlier classical, complex scalar gauge theory (April, 1992 Class. Quantum Grav.). In some sense, it appears to be possible to view the two component Dirac spinor as a single component, quaternionic, spacetime scalar. "Spin space" transformations can be considered transformations of the internal quaternion basis. Essentially, quaternionic Dirac Theory projects into the complex plane neatly, where spin becomes related to the self-dual antisymmetric part of the metric. The correct Dirac eigenvalues and well-behaved eigenfunctions project intact into a pair of complex solutions for the scalar curvature in the earlier theory's Weyl-Cartan type geometry. Some estimates are made for predicted, interesting atomic and subatomic scale phenomena. A form of electromagnetic quanta appears. A generalization of the complex geometric structure is then sketched in an appendix that allows quaternionic gauges and curvatures, and has some Weyl nonmetricity mixed with torsion. It appears to be a well defined structure, and leads to the full, second order, quaternionic Dirac Equation form, and a first order equation for a closely related, auxiliary wavefunction. A family of "free particle" solutions is examined in the Lorentzian limit of the symmetric part of the metric. More generally, when limited to two quaternion dimensions (just two components), reasonably similar solutions can be superposed linearly into new solutions, and separate into two families with different commutation characteristics. The integrability conditions for the equation for the auxiliary wavefunction impose six conditions on the original wavefunction, satisfied for the "free particle" solutions examined. Covariance is examined. The Darwin solution for the hydrogen atom is examined.

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摘要：

英文摘要：

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