（2+1）-空间中的分裂四元数与粒子
Split Quaternions and Particles in (2+1)-Space
摘要：
已知四元数表示3D欧几里得和闵可夫斯基空间中的旋转。然而，四元数的乘积同时在两个独立的平面上旋转，为了获得单平面旋转，必须从左边和右边（及其逆）应用半角四元数两次。这种“双覆盖”性质在分裂四元数的几何应用中是一个潜在的问题，因为它们的范数的（2+2）-签名对于每个乘积都不应该改变。如果分裂四元数形成微观物理学的适当代数结构，则在（2+1）空间中表示增强导致将四元数的标量部分解释为粒子的波长。时空间隔的不变性和一些量子行为，如非对易性和基本旋量表示，可能也是代数性质。在我们的方法中，狄拉克方程代表柯西-黎曼解析性条件和两个基本的物理参数（光速和普朗克常数）出现从四元数规范的正定性的要求。
英文摘要：
It is known that quaternions represent rotations in 3D Euclidean and Minkowski spaces. However, product by a quaternion gives rotation in two independent planes at once and to obtain single-plane rotations one has to apply by half-angle quaternions twice from the left and on the right (with its inverse). This 'double cover' property is potential problem in geometrical application of split quaternions, since (2+2)-signature of their norms should not be changed for each product. If split quaternions form proper algebraic structure for microphysics, representation of boosts in (2+1)-space leads to the interpretation of the scalar part of quaternions as wavelength of particles. Invariance of space-time intervals and some quantum behavior, like noncommutativity and fundamental spinor representation, probably also are algebraic properties. In our approach the Dirac equation represents the Cauchy-Riemann analyticity condition and the two fundamental physical parameters (speed of light and Planck's constant) appear from the requirement of positive definiteness of quaternionic norms.
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