| 摘要(英) |
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. |
| 參考文獻 |
[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 |
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