We analyze two branches of five-dimensional theories from a methodological point of view: Kaluza’s theory and the Klein-Fock-Rumer (KFR) 5-optics, aimed at a geometric description of electromagnetism and masses of elementary particles, respectively. These two theories have a number of problematic points such as the Planckian masses in Kaluza’s theory, or appearing of a configuration space in the KFR scheme. We propose a simple six-dimensional toy model of Kaluza-Klein type with compactification on a two-dimensional torus \(\mathbb{T}^2\), which demonstrates a possible way to overcome these difficulties in quite a simple manner. The key features of our approach are: merging of the above two 5D theories into a unique 6D theory; using the signature (+-) of extra space allowing one to renormalize the mass spectrum; and a special kind of truncation of the full isometry group to the electromagnetic gauge group U(1). The latter feature allows a geometrical interpretation in terms of an effective 5D theory on a hypersurface \(\Sigma\subset\mathbb{T}^2\) being a leaf of a linear foliation of the 2-torus. Possible consequences and open questions arising in such a scheme are briefly discussed.

中文翻译：

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关于Kaluza-Klein型6D理论中的质荷描述

我们从方法论的角度分析五维理论的两个分支：Kaluza理论和Klein-Fock-Rumer（KFR）5光学器件，分别针对电磁学和基本粒子质量进行几何描述。这两种理论都有许多问题点，例如卡卢扎理论中的普朗克质量，或者在KFR方案中出现了配置空间。我们提出了一个Kaluza-Klein类型的简单六维玩具模型，并在二维圆环\（\ mathbb {T} ^ 2 \）上进行了压缩，它展示了一种以非常简单的方式克服这些困难的可能方法。我们方法的主要特征是：将上述两个5D理论合并为一个独特的6D理论；使用多余空间的签名（+-）来使质谱图重新归一化；完整等轴测图组的截断是电磁量规组U（1）的一种特殊截断法。后一个特征允许根据超曲面\（\ Sigma \ subset \ mathbb {T} ^ 2 \）上的有效5D理论进行几何解释，该曲面是2瓣的线性叶的叶。简要讨论了这种方案可能产生的后果和未解决的问题。