超越相对论性点粒子：一个互反不变系统及其推广
Beyond the Relativistic Point Particle: A Reciprocally Invariant System and its Generalisation
摘要：
我们研究了Low和Govaerts等人提出的一个非线性不变系统，其作用量包含正交形式和辛形式，并且在全局$O(2,4)\cap Sp(2,4)$变换下是不变的。我们发现经典运动方程的通解在演化参数$τ$中没有线性项，而只有振荡项，因此不能代表粒子在时空中的传播。作为补救措施，我们考虑通过采用类似于Bars等人的程序来概括该行动，他引入了$τ$导数的概念，该导数在相空间变量$x^μ(τ)$和$p^μ(τ)$之间的局部Sp（2）变换下是协变的。这个系统特别类似于一个刚性粒子，它的作用包含了世界线的外在曲率，而世界线在时空中是螺旋形的。另一种可能的推广是引进了Montesinos提出的辛势。我们展示了后一种方法是如何与卡鲁扎-克莱因理论和克利福德空间的概念相关的，克利福德空间是一个流形，其切空间在任何一点上都是克利福德代数Cl（8），这是一个有前途的统一粒子和力的框架。
英文摘要：
We investigate a reciprocally invariant system proposed by Low and Govaerts et al., whose action contains both the orthogonal and the symplectic forms and is invariant under global $O(2,4)\cap Sp(2,4)$ transformations. We find that the general solution to the classical equations of motion has no linear term in the evolution parameter, $τ$, but only the oscillatory terms, and therefore cannot represent a particle propagating in spacetime. As a remedy, we consider a generalisation of the action by adopting a procedure similar to that of Bars et al., who introduced the concept of a $τ$ derivative that is covariant under local Sp(2) transformations between the phase space variables $x^μ(τ)$ and $p^μ(τ)$. This system, in particular, is similar to a rigid particle whose action contains the extrinsic curvature of the world line, which turns out to be helical in spacetime. Another possible generalisation is the introduction of a symplectic potential proposed by Montesinos. We show how the latter approach is related to Kaluza-Klein theories and to the concept of Clifford space, a manifold whose tangent space at any point is Clifford algebra Cl(8), a promising framework for the unification of particles and forces.
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