﻿ 在平面二元树族上的量子电动力学的H-Hopf模 The H-Hopf Module of Quantum Electrodynamics on Planar Binary Tree

Vol.04 No.02(2015), Article ID:15307,9 pages
10.12677/AAM.2015.42022

The Module of Quantum Electrodynamics on Planar Binary Tree

Miaohao Jiang, Beishang Ren*, Ruju Zhao, Junwei Liu

School of Mathematical and Statistical Sciences, Guangxi Teachers Education University, Nanning Guangxi

*通讯作者。

Received: May 5th, 2015; accepted: May 22nd, 2015; published: May 27th, 2015

ABSTRACT

Using the combination of Feynman diagrams and the planar binary tree, the renormalization group of QED was defined and the module of semidirect product of the group, the semidirect coproduct of the module, the spread of the module on the planar binary tree and the module with charge were pointed out. Finally it concluded the module of quantum electrodynamics and the renormalization coaction of electron and photon.

Keywords:Module, Quantum Electrodynamics, Semidirect Coproduct, Renormalization Coaction

1. 引言

2. 重正化群和模的半直余积

2.1. 费曼图

(a) (b) (c)

Figure 1. Feynman diagram

2.2. QED的重正化群

1) 通过嵌入

2) 投射到上，且

2.3. 群上半直积的

，对偶于的群作用，即有，其中，即，也可记为，它的余作用对偶于的作用，即

2.4.模的半直余积

， (1)

， (2)

(3)

3. 平面二元树上的模的传播

3.1. 修剪余模(The Pruning Comodules)

3.2. 电子和量子传播的

4. 树上带电的

4.1. 一个带电的模

4.2. 一个带电的

1)

2)

4.3. 一个非交换的带电的

5. 平面二元树上的QED的模和余作用

5.1. 电子和光子的余作用

, ,

,

.

2) 分别用代替，即满足

.

,

5.2. QED的

，这样，是一个模。

The H-Hopf Module of Quantum Electrodynamics on Planar Binary Tree. 应用数学进展,02,172-181. doi: 10.12677/AAM.2015.42022

1. 1. Beck, I. (1988) Coloring of commutative rings. Journal of Algebra, 116, 208-226.

2. 2. Schmitt, W.R. (1994) Incidence Hopf algebras. Journal of Pure and Applied Algebra, 96, 299-330.

3. 3. Schmitt, W.R. (1995) Hopf algebra methods in graph theory. Journal of Pure and Applied Algebra, 101, 77-90.

4. 4. Anderson, D.F. and Livingston, P.S. (1999) The zero-divisor graph of a commutative ring. Algebra, 217, 434-447.

5. 5. 邓汉元, 胡国权, 何国梁 (1998) 二元树族的H-Hopf代数结构. 湖南大学学报(自然科学学报), 3, 1-3.

6. 6. 赵燕 (2005) 完全图与完全二部图上的H-Hopf代数结构. 曲阜师范大学学报(自然科学版), 3, 25-29.

7. 7. 江妙浩, 任北上, 赵汝菊 (2015) 平面二元树族上的H-Hopf模结构. 广西师范院学报(自然科学学报), 3, 21-23.

8. 8. Dascalescu, S., Nastasescu, C. and Raianu, S. (2000) Hopf algebra: An introduction. CRC Press, Boca Raton.

9. 9. Brouder, Ch. (2000) On the trees of quantum fields. The European Physical Journal C, 12, 535-549.

10. 10. Grossman, R. and Larson, R.G. (1989) Hopf algebraic structure of families of trees. Journal of Algebra, 126, 184-210.

11. 11. Brouder, Ch. and Frabetti, A. (2001) Renormalization of QED with planar binary trees. European Physical Journal C, 19, 715-741.

12. 12. Brouder, Ch. and Frabetti, A. (2003) QED Hopf algebra on planar binary trees. Journal of Algebra, 267, 298-322.

13. 13. Connes, A. and Kreimer, D. (2000) Renormalization in quantum field theory and the Riemann-Hilbert problem I: The Hopf algebra structure of graphs and the main theorem. Communications in Mathematical Physics, 210, 249-273.

14. 14. Molnar, R.K. (1977) Semi-direct products of Hopf algebras. Journal of Algebra, 47, 29-51.

15. 15. Itzykson, C. and Zuber, J.B. (1980) Quantum field theory. McGraw-Hill, New York.

16. 16. Loday, J.L. (2002) Arithmetree. Journal of Algebra, 258, 275-309.

17. 17. Kreimer, D. (1998) On the Hopf algebra structure of perturbative quantum field theories. Advances in Theoretical and Mathematical Physics, 2, 303-334.

18. 18. Loday, J.L. and Ronco, M.O. (1998) Hopf algebra of the planar binary trees. Advances in Mathematics, 139, 293-309.

19. 19. Foissy, L. (2002) Les algèbres de Hopf des arbres enracinés décorés I. Bulletin des Sciences Mathématiques, 126, 193- 239.

20. 20. Holtkamp, R. (2003) Comparison of Hopf algebras on trees. Archiv der Mathematik, 80, 368-383.

21. 21. Loday, J.L. and Ronco, M.O. Order structure on the algebra of permutations and planar binary trees. Journal of Algebraic Combinatorics, to Appear.

22. 22. Peskin, M.E. and Schroeder, D.V. (1995) An introduction to quantum field theory. Perseus Books Pub. L.L.C., New York.