| 摘要(英) |
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. |
| 參考文獻 |
1. http://modelingnts.la.asu.edu/GC_R&D.html.
2. D. Hestenes, "Space-Time Algebra", Gordon and Breach, New York (1966).
3. D. Hestenes and G. Sobczyk, "Clifford Algebra to Geometric Calculus", D. Reidel Publishing (1984).
4. D. Hestenes, "New Foundations for Classical Mechanics", Kluwe, Dordrecht (1990).
5. D. Hestenes, "Oersted Medal Lecture 2002 : Reforming the
Mathematical Language of Physics", (2002).
6. D. Hestenes, "Spacetime Physics with Geometric Algebra", (2002).
7. P. Lounesto, "Clifford Algebras and Spinors 2nd Ed.", Cambridge University Press (2001).
8. http://www.mrao.cam.ac.uk/~clifford/.
9. A. Lasenby and C. Doran, "A Lecture Course in Geometric Algebra"
10. C. Doran and A. Lasenby, "Physical Applications of Geometric Algebra".
11. A. Lasenby, C. Doran and S. Gull, "Gravity, Gauge Theories and Geometric Algebra", Phil. Trans. R. Soc. Lond. A, 356:487-582 (1999).
12. C. Doran, A. Lasenby, S. Gull, S. Somaroo and A. Challinor, "Spacetime Algebra and Electron Physics", In P. W. Hawkes, editor, "Advances in Imaging and Electron Physics", 95, page 271-386 (Academic Press) (1996).
13. A. Lasenby, C. Doran and S. Gull, "2-spinors, Twistors and Supersymmetry in the Spacetime Algebra", In Z. Oziewicz, A. Borowiec, and B. Jancewicz, editors, "Spinors, Twistors and Clifford Algebras", page 233. Kluwer (1993).
14. C. Doran, A. Lasenby and S. Gull, "States and Operators in the Spacetime Algebra", Found. Phys. 23, 1239, (1993).
15. E. Witten, Comm. Math. Phys. 80 (1981) 381.
16. J. M. Nester, "The Gravitational Hamiltonian", Springer Lect. Notes in Physics 202 (1984) 155.
17. J. M. Nester, "A New Gravitational Energy Expression with a Simple Positivity Proof", Phys. Lett. 83A, 6, (1981) 241-242.
18. C. M. Chen, J. M. Nester and R. S. Tung, "Spinor Formulations for Gravitational Energy-Momentum", arXiv: gr-qc/0209100 to appear in
the Proceeding of the 6th Conference on Clifford Algebras and
their Applications in Mathematical Physics (Cookeville, Tennessee,
20-25 May, 2002).
19. J. M. Nester and R. S. Tung, "A Quadratic Spinor Lagrangian for General Relativity", General Rel. Grav. 27 (1995) 115.
20. J. M. Nester, R. S. Tung and V. V. Zhytnikov, "Some Spinor Curvature Identities", Class. Quat. Grav. 11 (1994) 983.
21. A. Dimakis and F. Muller-Hoissen, "Clifform calculus with applications to classical field theories", Class. Quant. Grav. 8 (1991) 2093.
22. C. W. Misner, K. S. Thorne and J. A. Wheeler, "Gravitation", Freeman San Franscesco (1973).
23. J. Y. Lin, "Application of geometric algebra to gravity
theory", M. Sc. thesis, National Central University (1997).
24. Y. K. Lin, "Geometric Algebra and Differential Forms: Translation and Gravitational Application", M. Sc. thesis, National Central University (2003). |
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