臺灣博碩士論文加值系統

「幾何代數」(Geometric Algebra，簡稱GA)，為1960年代，由David Hestenes所推廣的一套新興的數學語言。GA的前身：Clifford代數，為主要用來描述量子旋量的工具，當時人們只注意到它的代數性質，而Hestenes卻發現了它許多的幾何性質，並推廣了它的數學寬廣度，並為它命名為「幾何代數」這個較為貼切的名字。從事幾何代數研究的團隊相信，GA是可以統一物理上所有數學工具的數學語言。截至目前為止，GA也已經在理論物理、應用物理和實際的工程之上，獲得了很廣泛的運用。
GA可以有效地用來描述彎曲的時空，目前已有好幾套等效的推導方法，本論文將著重於其中一個稱為GTG (Gauge Theory of Gravity) 的重力規範理論。以GTG理論來得到幾何量的GA表示式，與其他方法推導所得的結論是相同的。
有了幾何量的GA表示式，接下來可以討論GA與其他主要的幾何數學語言之間的轉換。在這裡，我們找出了GA與Clifforms之間的轉換方法，加上已知的張量、微分形式間之轉換，以及林永康之論文 [12] 所提及之GA、微分形式之轉換，至此，便有了這四套重要的幾何數學語言相互間的轉換方法。
有了轉換方法，我們就可以更進一步探討，在對於從前以一些其他的數學工具，就能夠處理得很好的物理問題上，GA是否能夠同樣地有效和方便，這也等於是在測試GA在此方面，是否是一個成熟的統一語言。我們的研究方法，是以重力哈密頓函數和準局域量做為例子，來加以探究。我們的結論是，以統一數學語言這方面來說，以微分形式和Clifforms來描述的物理問題，是完全可以用GA來重新闡釋的；然而使用GA來做為描述的方法，在物理意義的解讀、以及在計算上，就不一定是同樣的有效和方便了。

Mathematical languages which describe geometric structures are used to expresses physical ideas and concepts. In particular the quaternions, matrix algebra, vectors, tensors, differential forms and spinors each have their advantages in certain applications, but none of them gives a sufficient algebraic structure for all purposes. Physicists look for a unified theory for physics and hence need an unified mathematical language. Geometric algebra may be a good choice for unifying the geometric tools of mathematics; no fundamental limitations of the technique has yet been identified.
Geometric (Clifford) algebra and the associated calculus make a promising tool for theoretical physics [1-10]. Multivectors of various grades are used to represent linear subspaces of space (or spacetime). The foundation is the geometric product of two vectors, which splits into a symmetric part giving the usual scalar dot product and an antisymmetric bivector part representing an oriented plane element. In this algebra one can divide by vectors! There is a single simple universal formula for rotations of all objects. The vector derivative operator has a unique integral inverse. With it one forms a single theorem which includes the fundamental theorem of calculus, Stokes theorem, the Gauss divergence theorem plus an infinite number of higher order generalizations. The vector derivative operator allows compact and remarkably similar formulations and easy manipulations of various fundamental equations of mathematical physics including the Cauchy-Riemann equations, the Dirac equation and Maxwell's equations. Recently a group of researchers at Cambridge have developed much tutorial material and many applications [1].
Geometric algebra and calculus are very useful for theoretical physics work in flat spacetime. There have been several schemes developed to extend these techniques to also include gravity and curved spacetime. The original approach of Hestenes can be used [10]. The “Clifform” technique [11] has already been proven to be quite effective, but it is somewhat complicated, not at all elementary, and thus not so well suited for beginning students. The “vector manifold” approach developed later by Hestenes [2] has promise but has not actually been used much yet. More recently the Cambridge research group have developed a fairly simple “flat space gauge theory of gravity” (GTG) [7].
People are still trying to know how good are these schemes, especially, if the GTG approach is too simple. In this thesis we look in detail at GTG. It models gravitational interactions in terms of (gauge) fields defined in the flat spacetime, and thus is very different from general relativity. It is also quite different from other versions of gauge theory of gravity like the Poincaré gauge theory.
Although in this thesis we just discuss the GTG approach, basically, the difference schemes of dealing with curved spacetime, which were mentioned in the above paragraph, can deduce the same formulae for geometric objects. This means that GA is at least a possible tool for dealing with geometric problems. One aim of our research group is to investigate the efficacy of GA. In particular we want to look at several applications, which certain other techniques manage neatly, and see if GA handles them well.
The beginning of the job is to find out the methods of translation between various mathematic languages. In our research area, we need especially the relations between the GA with differential forms and clifforms, which are widely used in geometrical physics. Lin [12] has shown us the relation between GA and differential forms and an application: the energy-momentum pseudotensors. Liang [16] give us two other applications: the Nester-Witten positive energy proof; Tung and Nester’s quadratic spinor Lagrangian. Here in this thesis we will show the relation between GA and the clifform approach, and then make a whole picture for translating between these three languages; finally we give another application: the gravitational Hamiltonian and quasilocal quantities.

Acknowledgements..............ii
Contents..............iii
Introduction..............1
Chapter 1 Geometric Algebra and Geometric Calculus..............3
1.1Geometric Algebra (GA)..............3
1.2Geometric Calculus..............6
Chapter 2 Some Physics Applications..............10
2.1Spacetime Algebra..............10
2.2Electrodynamics..............12
2.3The Dirac Equation..............12
2.4Lagrangian Field Theory..............13
Chapter 3 Gauge Theory of Gravity (GTG)..............15
3.1Gauge Principles..............15
3.2The Gravitational Gauge Fields..............15
3.3The Gravitation Field Strength..............17
3.4The Gravitational Field Equations..............17
3.5The Translation of Tensor Calculus..............19
3.6Spherically Symmetric Sources..............20
Chapter 4 Relating GA with Differential Forms and Clifforms..............22
4.1Convention..............22
4.2GA and Differential Forms..............23
4.3Clifforms..............26
4.4Conclusion..............31
Chapter 5 Gravitational Hamiltonian and Quasilocal Quantities..............33
5.1Differential Forms Version..............33
5.2Clifforms Version..............34
5.3GA Version..............36
5.4Conclusion..............40
Appendix A Differential Forms and Clifforms..............41
Appendix B Hamiltonian and Quasilocal Quantities in Differential Forms Style..............46
Appendix C Formulae sheet..............50
Bibliography..............57

[1] The Geometric Algebra Research Group, http://www.mrao.cam.ac.uk/~clifford/
[2] D. Hestenes and G. Sobczyk, "Cliford Algebra to Geometric Calculus", D. Reidel Publishing, 1984.
[3] William E. Baylis, Editor, "Clifford (Geometric) Algebras", Birkhauser Publishing, 1996.
[4] A. N. Lasenby and C. J. L. Doran, "A Lecture Course in Geometric Algebra", http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/course99/
[5] C. J. L. Doran and A. N. Lasenby, "Physical Applications of Geometric Algebra", http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/
[6] C. J. L. Doran, "Geometric Algebra and its Application to Mathematical Physics", PhD thesis, Cambridge University, 1994.
[7] A. N. Lasenby, C. J. L. Doran and S. F. Gull, "Gravity, gauge theories and geometric algebra", Phil. Trans. R. Soc. Lond. A356, 487-582 (1998).
[8] A. N. Lasenby, C. J. L. Doran and S. F. Gull, "A Multivector Derivative Approach to Lagrangian Field Theory", Found. Phys. 23(10), 1295-1327 (1993).
[9] A. Lewis, C. Doran and A.N. Lasenby, "Quadratic Lagrangians and Topology in Gauge Theory Gravity", General Rel. Grav. 32(1), 161 (2000).
[10] D. Hestenes, "Space Time Algebra", Gordon & Breach, New York,1966.
[11] A. Dmakis and F. Müller-Hoissen, "Clifform calculus with applications to classical field theories", Class. Quantum Grav. 8 (1991) 2093-2132.
[12] Yung-Kang Lin, "Geometric Algebra and Differential Forms: Translation and Gravitational Application", MSc Thesis, National Central University, 2003.
[13] Chiang-Mei Chen and James M. Nester, "Quasilocal quantities for GR and other gravity theories", Class. Quantum Grav. 16 (1999) 1279-1304.
[14] Chiang-Mei Chen, "Quasilocal Quantities for Gravity Theories", MSc Thesis, National Central University, 1994.
[15] 孟繁蕃, "Quasilocal center-of-mass moment in general relativity", MSc Thesis, National Central University, 2002.
[16] Shiuan-Ni Liang, "Geometric Algebra: Spinors and the Positivity of Gravitational Energy", MSc Thesis, National Central University, 2003