arXiv quant-ph/06061232006-06quant-phmath-phmath.MP

相对论性量子力学中隐藏的几何特征

The hidden geometric character of relativistic quantum mechanics

Jose B. Almeida

备注：: 29 pages. Small changes in V3 suggested by referee
发表信息：: J. Math. Phys. 49(1), pp. 012301, (2007)
DOI：: 10.1063/1.2406055

摘要：

该演示使用了几何代数，也称为Clifford代数，在5维时空中。这个空间的选择被赋予了第一原理的特征，仅仅通过从这种选择中得出的结果及其与实验结果的一致性来证明。给定一个任意维的度量空间，可以定义单演函数，这是解析函数向高维的自然扩展;这样的函数具有零向量导数，并且以前已经被其他作者证明在低维空间中起决定性作用。因此，所有单演函数都具有零拉普拉斯算子;在双曲空间中，这一事实不可避免地导致具有类平面解的波动方程。这对5维时空也是如此，我们将探索这些解，建立与狄拉克方程解的平行。为此，我们将调用4x 4矩阵的复代数之间的同构，也称为狄拉克矩阵。这种同构有一个问题，因为狄拉克方程的解通常被称为旋量（列矩阵），不属于4x 4矩阵代数，因此被排除在同构之外。我们将证明用狄拉克旋量表示的解等价于平面波解。正如人们在标准公式中发现的那样，单演函数可以自然地分为正/负能量和左/右能量。这种分裂是由几何投影仪提供的，我们将展示有第二组投影仪提供交替的4倍分裂。这种交替分裂的可能影响尚未完全理解，目前是深入研究的主题。

英文摘要：

The presentation makes use of geometric algebra, also known as Clifford algebra, in 5-dimensional spacetime. The choice of this space is given the character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental results. Given a metric space of any dimension, one can define monogenic functions, the natural extension of analytic functions to higher dimensions; such functions have null vector derivative and have previously been shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in an hyperbolic space this fact leads inevitably to a wave equation with plane-like solutions. This is also true for 5-dimensional spacetime and we will explore those solutions, establishing a parallel with the solutions of the Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of 4x4 matrices, also known as Dirac's matrices. There is one problem with this isomorphism, because the solutions to Dirac's equation are usually known as spinors (column matrices) that don't belong to the 4x4 matrix algebra and as such are excluded from the isomorphism. We will show that a solution in terms of Dirac spinors is equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive/negative energy together with left/right ones. This split is provided by geometric projectors and we will show that there is a second set of projectors providing an alternate 4-fold split. The possible implications of this alternate split are not yet fully understood and are presently the subject of profound research.

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