arXiv quant-ph/06091072006-09-14quant-ph

尺度相对论中的泡利方程

The Pauli equation in scale relativity

Marie-Noelle Celerier, Laurent Nottale

备注：: 22 pages, 2 figures, accepted for publication in J. Phys. A: Math. & Gen
发表信息：: J.Phys.A39:12565-12585,2006
DOI：: 10.1088/0305-4470/39/40/020

摘要：

在标准量子力学中，不可能直接将薛定谔方程推广到旋量，所以泡利方程必须从狄拉克方程出发，取其非相对论极限。因此，它预言了电子的内禀磁矩的存在，并给出了它的正确值。在尺度相对论框架下，薛定谔方程、克莱因-戈登方程和狄拉克方程都是由第一性原理导出的不可微连续时空的测地线方程。由于这样一个广义几何意味着新的离散对称破缺的发生，这导致我们将狄拉克双旋量写成双四元数（复四元数）的形式。在目前的工作中，我们表明，在标度相对论，正确的泡利方程只能从相对论测地线方程的非相对论极限（积分后，成为狄拉克方程），而不是从非相对论形式主义（涉及对称性破缺分形3-空间）。同样的简并过程，当它被应用到用于推导狄拉克方程的双四元数4-速度时，自然会产生保利型四元数3-速度。因此，它证实了尺度相对论方法对于从量子假设和量子工具的第一原理构建的相关性。这也强化了自旋的相对论性和基本量子性质，我们在标度相对论中将其归因于量子时空几何的不可微性（而不仅仅是量子空间）。最后，我们进行旋量测地线的数值模拟，使人们获得一个物理的自旋性质的几何图片。

英文摘要：

In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and continuous spacetime. Since such a generalized geometry implies the occurence of new discrete symmetry breakings, this has led us to write Dirac bi-spinors in the form of bi-quaternions (complex quaternions). In the present work, we show that, in scale relativity also, the correct Pauli equation can only be obtained from a non-relativistic limit of the relativistic geodesics equation (which, after integration, becomes the Dirac equation) and not from the non-relativistic formalism (that involves symmetry breakings in a fractal 3-space). The same degeneracy procedure, when it is applied to the bi-quaternionic 4-velocity used to derive the Dirac equation, naturally yields a Pauli-type quaternionic 3-velocity. It therefore corroborates the relevance of the scale relativity approach for the building from first principles of the quantum postulates and of the quantum tools. This also reinforces the relativistic and fundamentally quantum nature of spin, which we attribute in scale relativity to the non-differentiability of the quantum spacetime geometry (and not only of the quantum space). We conclude by performing numerical simulations of spinor geodesics, that allow one to gain a physical geometric picture of the nature of spin.

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