规范势分解理论与整体拓扑问题
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摘要
利用段一士提出的规范势可分解和具有内部结构的思想,使用几何代数方法对SO(n)群用单位矢量场进行了分解,给出了一般形式,并讨论这个分解的性质;由此给出了SU(2)群和U(1)群用单位矢量分解的形式,这正是著名物理学家法捷耶夫1999年所给出的结果.使用SO(n)群规范势分解的一般形式讨论了Gauss-Bonnet-Chern密度的局域拓扑结构,其整体拓扑结构正好是Gauss-Bonnet-Chern定理,由拓扑结构很容易得到Euler-Poincar示性数的Morse理论形式.利用SU(2)群规范势分解研究了–1/2 Bose-Einstein凝聚体,得到了一个新的环流条件,也是Mernin-Ho关系的推广.最后,使用段一士发现的三维黎曼几何的Torsion张量与U(1)规范理论的关系,使用U(1)规范势分解研究了位错线与link数的关系.
Using the idea of decomposability of gauge potential and internal structure put forward by Duan Yi-Shi, we decompose gauge potential of SO(n) group in the unit vector field by using geometric algebra method, we get the general decomposition form and discuss the properties of this decomposition. In this paper, we give the form of decomposition of SU(2) group and U(1) group with unit vector field, which is exactly the result given by famous physicist Fadeev in 1999.The local topological structure of Gauss-Bonnet-Chern density is discussed by using the general form of decomposition of SO(n) gauge potential. The global topological structure of the density is Gauss-Bonnet-Chern theorem, and the Morse theoretical form of Euler-Poincar characteristic is easily obtained from the topological structure. A new circulation condition is obtained by using the normal potential decomposition of SU(2) gauge potential to study –1/2 Bose-Einstein condensate, which is also a generalization of the Mernin-Ho relation. Finally, using the relation between Torsion tensor and U(1) gauge theory of three-dimensional Riemannian geometry discovered by Duan Yi-Shi, the relation between dislocation line and link number is studied by using U(1) gauge potential decomposition.
Using the idea of decomposability of gauge potential and internal structure put forward by Duan Yi-Shi, we decompose gauge potential of SO(n) group in the unit vector field by using geometric algebra method, we get the general decomposition form and discuss the properties of this decomposition. In this paper, we give the form of decomposition of SU(2) group and U(1) group with unit vector field, which is exactly the result given by famous physicist Fadeev in 1999.The local topological structure of Gauss-Bonnet-Chern density is discussed by using the general form of decomposition of SO(n) gauge potential. The global topological structure of the density is Gauss-Bonnet-Chern theorem, and the Morse theoretical form of Euler-Poincar characteristic is easily obtained from the topological structure. A new circulation condition is obtained by using the normal potential decomposition of SU(2) gauge potential to study –1/2 Bose-Einstein condensate, which is also a generalization of the Mernin-Ho relation. Finally, using the relation between Torsion tensor and U(1) gauge theory of three-dimensional Riemannian geometry discovered by Duan Yi-Shi, the relation between dislocation line and link number is studied by using U(1) gauge potential decomposition.
引文
1 Yang C N, Mills R L. Conservation of isotopic spin and isotopic gauge invariance. Phys Rev, 1954, 96:191–195
2 Duan Y S. Gauge theory of gravitation. Commu Theor Phys, 1985, 4:661–674
3 Schwarz A. Topology for Physicists. Berlin Heidelberg:Springer-Verlag, 1994; Nash C, Sen S. Topology and Geometry of Physicists. London:Academic Press Inc., 1983
4 Duan Y S. The generalized covariant equation of the field of an arbitrary spin basic particle. J Lanzhou Univ, 1958, 1:29–34
5 Duan Y S, Ge M L. Su(2)Guage Theory and electrodynamics with N magnetic monopoles(in Chinese). Sci China, 1979, 9:1072–1081[段一士,葛墨林. SU(2)规范理论与N个磁单极运动体系的电动力学.中国科学, 1979, 11:1072–1081]
6 Cho Y M. Restricted gauge theory. Phys Rev D, 1980, 21:1080–1088; Cho Y M. Glueball spectrum in extended quantum chromodynamics. Phys Rev Lett, 1981, 46:302–306; Faddeev L, Niemi A J. Partially dual variables in SU(2)Yang-Mills Theory. Phys Rev Lett, 1999, 82:1624–1627
7 Chen X S, LüX F, Sun W M, et al. Spin and orbital angular momentum in gauge theories:Nucleon spin structure and multipole radiation revisited. Phys Rev Lett, 2008, 100:232002, ar Xiv:0806.3166
8 Doran C, Hestenes D, Sommen F, et al. Lie groups as spin groups. J Math Phys, 1993, 34:3642
9 Duan Y S, Lee X G. General decomposition theory of spin connections, topological structure of Gauss-Bonnet-Chern density and Morse theory.Helv Phys Acta, 1995, 68:513–530
10 Duan Y S, Li X G. General decomposition theory of spin connection, topological structure in Gauss-Bonnet-Chern density. Commun Theor Phys,1998, 29:237
11 Li X G, Duan Y S, Song J J. General decomposition theory of SO(n)gauge potential. High Energy Phys Nucl Phys, 2001, 25:296–303
12 Duan Y S, Meng X H. Topological structure of Gauss-Bonnet-Chern density and its topological current. J Math Phys, 1993, 34:1149–1161
13 Chern S S. A simple intrinsic prof of the gauss-bonnet formula for closed rienannian manifolds. Ann Math, 1944, 45:747–752
14 Lee X G. General Decomposition Theory of Gauge potential, Gauss-Bonnet-Chern Therm and Inner Struture of Torsion(in Chinese).Dissertation for Doctoral Degree. Lanzhou:Lanzhou University, 1994[李希国.规范势的一般分解理论, Gauss-Bonnet-Chern定理和绕率得内部结构.博士学位论文.兰州:兰州大学, 1994]
15 Duan Y, Fu L. The general decomposition theory of SU(2)gauge potential, topological structure and bifurcation of SU(2)Chern density. J Math Phys, 1998, 39:4343–4355
16 Lee X G. The decomposable theory of gauge potential and its application(in Chinese). Nucl Phys Rev, 2000, 17:201–209[李希国.规范势的可分解理论及其应用.原子核物理评论, 2000, 17:201–209]
17 Jia D, Duan Y, Lee X. Circulation condition of the two-component Bose-Einstein condensate. Phys Lett A, 2001, 289:245–250
18 Lee X G, Jia D J, Gao Y. Decomposition of the SU(2)gauge field and circulation condition of the Bose-Einstein condensate. High Energy Phys Nucl Phys, 2003, 27:12–14
19 Lee X G. New Torsion structure on Riemann-Cartan manifold and knots. High Energy Phys Nucl Phys, 1999, 23:906–913
20 Duan Y S, Ge M L. On the guage theory(in Chinese). J Lanzhou Univer, 1975, 4:29–37[段一士,葛墨林.关于规范场理论.兰州大学学报,1975, 4:29–37]
21 Lee X, Baldo M, Duan Y. Torsion structure in riemann-cartan manifold and dislocation. General Rel Gravit, 2002, 34:1569–1577
22 Ge M L, Cai R G, Liu Y X. Memorial Volume for Yi-Shi Duan. Singapore:World Scientific, 2018
2 Duan Y S. Gauge theory of gravitation. Commu Theor Phys, 1985, 4:661–674
3 Schwarz A. Topology for Physicists. Berlin Heidelberg:Springer-Verlag, 1994; Nash C, Sen S. Topology and Geometry of Physicists. London:Academic Press Inc., 1983
4 Duan Y S. The generalized covariant equation of the field of an arbitrary spin basic particle. J Lanzhou Univ, 1958, 1:29–34
5 Duan Y S, Ge M L. Su(2)Guage Theory and electrodynamics with N magnetic monopoles(in Chinese). Sci China, 1979, 9:1072–1081[段一士,葛墨林. SU(2)规范理论与N个磁单极运动体系的电动力学.中国科学, 1979, 11:1072–1081]
6 Cho Y M. Restricted gauge theory. Phys Rev D, 1980, 21:1080–1088; Cho Y M. Glueball spectrum in extended quantum chromodynamics. Phys Rev Lett, 1981, 46:302–306; Faddeev L, Niemi A J. Partially dual variables in SU(2)Yang-Mills Theory. Phys Rev Lett, 1999, 82:1624–1627
7 Chen X S, LüX F, Sun W M, et al. Spin and orbital angular momentum in gauge theories:Nucleon spin structure and multipole radiation revisited. Phys Rev Lett, 2008, 100:232002, ar Xiv:0806.3166
8 Doran C, Hestenes D, Sommen F, et al. Lie groups as spin groups. J Math Phys, 1993, 34:3642
9 Duan Y S, Lee X G. General decomposition theory of spin connections, topological structure of Gauss-Bonnet-Chern density and Morse theory.Helv Phys Acta, 1995, 68:513–530
10 Duan Y S, Li X G. General decomposition theory of spin connection, topological structure in Gauss-Bonnet-Chern density. Commun Theor Phys,1998, 29:237
11 Li X G, Duan Y S, Song J J. General decomposition theory of SO(n)gauge potential. High Energy Phys Nucl Phys, 2001, 25:296–303
12 Duan Y S, Meng X H. Topological structure of Gauss-Bonnet-Chern density and its topological current. J Math Phys, 1993, 34:1149–1161
13 Chern S S. A simple intrinsic prof of the gauss-bonnet formula for closed rienannian manifolds. Ann Math, 1944, 45:747–752
14 Lee X G. General Decomposition Theory of Gauge potential, Gauss-Bonnet-Chern Therm and Inner Struture of Torsion(in Chinese).Dissertation for Doctoral Degree. Lanzhou:Lanzhou University, 1994[李希国.规范势的一般分解理论, Gauss-Bonnet-Chern定理和绕率得内部结构.博士学位论文.兰州:兰州大学, 1994]
15 Duan Y, Fu L. The general decomposition theory of SU(2)gauge potential, topological structure and bifurcation of SU(2)Chern density. J Math Phys, 1998, 39:4343–4355
16 Lee X G. The decomposable theory of gauge potential and its application(in Chinese). Nucl Phys Rev, 2000, 17:201–209[李希国.规范势的可分解理论及其应用.原子核物理评论, 2000, 17:201–209]
17 Jia D, Duan Y, Lee X. Circulation condition of the two-component Bose-Einstein condensate. Phys Lett A, 2001, 289:245–250
18 Lee X G, Jia D J, Gao Y. Decomposition of the SU(2)gauge field and circulation condition of the Bose-Einstein condensate. High Energy Phys Nucl Phys, 2003, 27:12–14
19 Lee X G. New Torsion structure on Riemann-Cartan manifold and knots. High Energy Phys Nucl Phys, 1999, 23:906–913
20 Duan Y S, Ge M L. On the guage theory(in Chinese). J Lanzhou Univer, 1975, 4:29–37[段一士,葛墨林.关于规范场理论.兰州大学学报,1975, 4:29–37]
21 Lee X, Baldo M, Duan Y. Torsion structure in riemann-cartan manifold and dislocation. General Rel Gravit, 2002, 34:1569–1577
22 Ge M L, Cai R G, Liu Y X. Memorial Volume for Yi-Shi Duan. Singapore:World Scientific, 2018