摘要: | 有部份學者(其中有David Hestenes, DGL等)認為幾何代數 (GA, 亦稱為 Clifford代數)是一套強而有力, 可以解決所有數學和物理問題的工具. 這篇論文的目的在於測試幾何代數的極限.我們將GA應用在Nester-Witten重力場正能量的証明和QSL, 希望能藉由GA將式子簡化, 或從中得到更多新的成果. Geometric Algebra (GA, also called Clifford Algebra) is a mathematical system and language introduced by two of the late 19th century's greatest mathematicians, Grassmann (1877) and Clifford (1878). It was not given serious treatment and development at that time because of the introduction of another mathematical language, the vector algebra of Gibbs, which people saw as a more generally applicable and a more straightforward algebra. Although some special cases were rediscovered over the years it was only much later in the mid-1960's, that an American physicist and mathematician, David Hestenes, pioneered and promoted the general mathematical language. He claims that GA is no less than the universal language for physics and mathematics. Now, throughout the world, there are an increasing number of research groups, especially the Cambridge research group (Doran, Gull and Lasenby (DGL)), applying GA to many scientific problems. In this thesis, we will first introduce the basic ideas of GA and some of its elementary applications developed by Hestenes and DGL in chapter 2. We will illustrate the generality and portability of this powerful mathematical language when it is applied to quantum mechanics, relativity and electromagnetism. We will also emphasize the development of GA on spinors and spacetime algebra (STA) which plays an important role in the application in the next two chapters. Our main job is to apply the Gauge Theory Gravity (GTG, introduced by the Cambridge research group (DGL)) to two positive energy proofs: (i) the Nester-Witten Positive Energy Proof and (ii) Tung and Nester's Quadratic Spinor Lagrangian. It is important to look into these proofs because from thermodynamics and stability, an essential fundamental theoretical requirement for isolated gravitating systems is that the energy of gravitating systems should be positive. Otherwise, systems could emit an unlimited amount of energy while decaying deeper into ever more negative energy states. Meaning, gravity acts like a purely attractive force. Thus in chapter 3 we will re-express the Nester-Witten positive energy proof in terms of GA. This positive energy proof was originally presented in tensor index form, then later re-expressed in terms of differential forms and also in Clifforms. GA is claimed to be a powerful and universal language for physics and mathematics; our principle goal is to test it - by seeing if it works efficiently in this advanced application. In chapter 4 we consider a second application: Tung and Nester's Quadratic Spinor Lagrangian (QSL). This alternative is fundamentally different from the above approach. The spinor field in this approach enters into the Lagrangian as a dynamic physical field. Again our main goal is to re-express the new Spinor Curvature Identity and the QSL in terms of GA. We hope these GA expressions not only will give simple and neat formulae but also provide new insight of the positive energy proofs. Finally, we will draw our conclusion by comparing the relative efficiency between Clifforms and GA proof. |