臺灣博碩士論文加值系統

長久以來，物理學家夢想著用一個「終極理論」來統一自然界中所有的交互作用。 過去的十多年來，實驗物理學家不斷地驗證了目前用來描述自然的「標準模型」，也就是SU(3) X SU(2) X U(1)規範場論。 即使標準模型這樣的成功，物理學家仍希望在一個單一群之下描述所有的作用力。這些嘗試包括了 E6, SU(5), SO(10)...等等。在這篇論文裡，我們研究了一般群的結構以及其在物理上的運用。我們也研究了獨特群G2群的相關大統一場論，雖然其結果不竟滿意，但我們仍得到許多有去的結論。

Contents 4
Abstract
For a long time, physicists dream of ”ultimate theory” which unifies all kinds
of interactions in nature. The past years, countless experiments confirm the
elementary particle interactions which are described by SU(3) × SU(2) × U(1) gauge theory so-called ”Standard Model” (SM) nowadays. Even so
the successes of SM, people run for the beauty of gauge theory with single
group which includes SU(3) × SU(2) × U(1) as the unified theory i.e. E6,
SU(5), SO(10),...... . In this thesis, we study the general group structure
and through the way also the applications in physics associates with it. We
investigate the exceptional group G2 unification theory, it turns out that it is
difficult to get the SM spectrum, even so we still get many interesting results.

CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 The standard model . . . . . . . . . . . . . . . . . . . . . . . 5
2. Review of some Grand Unification Models . . . . . . . . . . . . . . 11
2.1 SU(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 SO(10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3. Finding Invariants of Seven-Dimensional Scalar Field of G2 . . . . . 15
3.1 Irreducible representations of G2 . . . . . . . . . . . . . . . . . 15
3.2 Tensor Product and Decomposition of G2 . . . . . . . . . . . . 16
3.3 Branching Rule of SO(7) to G2 . . . . . . . . . . . . . . . . . 17
4. Spontaneous Symmetry Breaking of G2 . . . . . . . . . . . . . . . . 19
4.1 Invariants of Seven Dimensional Scalar Field of G2 . . . . . . 20
4.2 Spontaneous Symmetry Breaking of Seven Dimensional Scalar
Field of G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5. A toy model: G2 × G2 . . . . . . . . . . . . . . . . . . . . . . . . . 24
Appendix 26
A. Irreducible representations of G2 . . . . . . . . . . . . . . . . . . . 27
B. Tensor Products of G2 . . . . . . . . . . . . . . . . . . . . . . . . . 28
C. Branching rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
C.1 SO(7) G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
C.2 G2 SU(3)(R) . . . . . . . . . . . . . . . . . . . . . . . . . . 35
C.3 G2 SU(2) × SU(2)(R) . . . . . . . . . . . . . . . . . . . . . 36
Contents 2
C.4 SU(3) SU(2) × U(1)(R) . . . . . . . . . . . . . . . . . . . 37
C.5 G2 SU(2) × U(1) . . . . . . . . . . . . . . . . . . . . . . . 37
C.6 G2 SU(2)(S) . . . . . . . . . . . . . . . . . . . . . . . . . . 38
D. Symmetry Breaking of SO(N)- Using Roots and Weights . . . . . . 39
D.1 A first experience for SO(3) . . . . . . . . . . . . . . . . . . . 39
D.2 Generalization into SO(2n + 1) . . . . . . . . . . . . . . . . . 41
E. Chiral fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
F. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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