Owing to the non-differentiable nature of the theory of Scale Relativity, the emergence of complex wave functions, then of spinors and bi-spinors occurs naturally in its framework. The wave function is here a manifestation of the velocity field of geodesics of a continuous and non-differentiable (therefore fractal) space-time. In a first paper (Paper I), we have presented the general argument which leads to this result using an elaborate and more detailed derivation than previously displayed. We have therefore been able to show how the complex wave function emerges naturally from the doubling of the velocity field and to revisit the derivation of the non relativistic Schrödinger equation of motion. In the present paper (Paper II) we deal with relativistic motion and detail the natural emergence of the bi-spinors from such first principles of the theory. Moreover, while Lorentz invariance has been up to now inferred from mathematical results obtained in stochastic mechanics, we display here a new and detailed derivation of the way one can obtain a Lorentz invariant expression for the expectation value of the product of two independent fractal fluctuation fields in the sole framework of the theory of Scale Relativity. These new results allow us to enhance the robustness of our derivation of the two main equations of motion of relativistic quantum mechanics (the Klein-Gordon and Dirac equations) which we revisit here at length.
Ternary relative velocity; astonishing conflict of the Lorentz group with relativity
我们证明了洛仑兹推进需要相对速度是三元的：三元相对速度是一个物体相对于内部观察者的速度，就像一个优选的外部观察者看到的那样。洛伦兹增强意味着三元相对速度的非结合相加。在爱因斯坦的狭义相对论中，每一个首选的观测者（恒星、以太等）决定了每对大质量天体之间唯一的相对速度。因此，建立在闵可夫斯基公理上的狭义相对论，即每对参考系必须通过洛伦兹等距线相关联，需要一个优选的参考系，以便在每对大质量物体之间具有唯一的爱因斯坦相对速度。这种相对速度的选择依赖性违反了所有参考系必须等效的相对性原理。洛伦兹相对论群与相对性原理之间的这种惊人冲突，可以用两种不同的方式来解决。或者，放弃相对性原理，而采用一个更好的参考系[de Abreu 2004; de Abreu and Guerra 2005，2006]。或者，在相对性原理中，用相对性广群代替洛伦兹相对性群，用闵可夫斯基在1908年引入的无选择的二元相对速度。二进制-意味着相对速度是一对参考系的函数。遵循闵可夫斯基[1908]，我们认为二元相对速度是闵可夫斯基类空向量（而不是像Hestenes理论[Hestenes 1974]中那样的闵可夫斯基双向量）。
