复数的本质是什么？复数是真实的吗？ 一个数学家在脑海里凭空产生的数学概念何以与现实的自然如此息息相关，以至于没了复数，我们根本无法准确地描述我们的时空结构，特别是描述微观粒子的量子力学，复数更是不可或缺。复数的引入，使得量子力学出现了无数神奇的特性。 费曼（ R.P.Feynman ）曾说“ I think I can safely say that nobody understands quantum mechanics ”，其中一个很关键的问题就是，量子力学中的复数是怎么回事。。。 几何代数这门学科，将给这些问题提供一种新思路！ 这门学科可以看做是复数、四元数、八元数等等的推广，深刻地揭示了时空结构的数学基础，而且基于此重新表述了经典力学、电磁理论、狭义相对论、量子力学，甚至给出了一种平直时空下基于规范变换的重力理论，与爱因斯坦的弯曲时空重力等价 这种重新表述有诸多优越之处，比如，重新表述的电磁理论把麦克斯韦方程组高度精炼地合为了一个方程。 我相信，上帝能用一个方程写清楚的理论，他不会用四个，所以，这种整合，将是指向更进一步统一的正确方向！ 奇文共欣赏： 最后，这个是一种抽象的数学与物理的统一的美，要领略这种美，不是冥想得出的，是得花点力气学点数学的，各位根据自己的爱好，请随意。 Imaginary Numbers are not Real - the Geometric Algebra of Spacetime Stephen Gull (a), Anthony Lasenby (a) and Chris Doran (b) (a) MRAO, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK (b) DAMTP, Silver Street, Cambridge, CB3 9EW, UK February 9, 1993 Abstract: This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a `geometric product' of vectors in 2- and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analysed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics). Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained - results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics. Introduction An Outline of Geometric Algebra How to Multiply Vectors A Little Un-Learning The Geometric Product Geometric Algebra of the Plane The Algebra of 3-Space Interlude Rotations and Geometric algebra Analytic and Monogenic Functions The Algebra of Spacetime Concluding Remarks References http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html
标签: 几何代数
