用变分法证明磁单极子及双荷子的存在性
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摘要
本文首先分别应用直接变分法和不定变分法证明了任意(4p-1)维Yang-Mills理论磁单极子的存在性和SO(3)规范Skyrme模型双荷子的存在性.前者推广了Maison和杨在低维情况下的结论,后者对Y.Brihaye, B.Hartmann and D.H.Tchrakian提出的数值解给出了理论证明.另外,应用分析的技巧给出了解的渐近估计.
In this paper, we obtain the existence , uniqueness and equivalence theorems formagnetic monopoles in general (4p?1)-dimensional Yang-Mills theory by variationalmethod. It generalized the theorems of Maison and Yang in low dimensions. On theother hand, we get the existence and asymptotic of dyons in SO(3) gauge Skyrmemodel by indefinite variational method. It present the analytic prove of the solutionconstrued numerically by Y.Brihaye, B.Hartmann and D.H.Tchrakian.
In this paper, we obtain the existence , uniqueness and equivalence theorems formagnetic monopoles in general (4p?1)-dimensional Yang-Mills theory by variationalmethod. It generalized the theorems of Maison and Yang in low dimensions. On theother hand, we get the existence and asymptotic of dyons in SO(3) gauge Skyrmemodel by indefinite variational method. It present the analytic prove of the solutionconstrued numerically by Y.Brihaye, B.Hartmann and D.H.Tchrakian.
引文
[1] A. Actor, Classical solutions of SU(2) Yang–Mills theories, Rev. Modern Phys. 51(1979) 461–525.
[2] K.Arthur and D.H.tchrakian,Solitons in gauged Sigma models:2 dimensions, Phys.Lett.B 352 (1995) 327.
[3] A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Pseudoparticlesolutions of the Yang–Mills equations, Phys. Lett. B 59 (1975) 85–87.
[4] E. B. Bogomol’nyi, The stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976)449–454.
[5] Y.Brihaye and D.H.tchrakian,Solitons/instantons in d-dimensional SO (d) gauged O(d+ 1) Skyrme models,Nolinearity 11 (1998) 891.
[6] Y.Brihaye, B.Hrtmann and D.H.Tchrakian, Monopoles and Dyons in SO(3) gaugedSkyrme models,Jour. Math. Phys.42.8(2001)
[7] L. Ca?arelli, B. Gidas, and J. Spruck, On multimeron solutions of the Yang–Millsequation, Commun. Math. Phys. 87 (1983) 485–495.
[8] P. A. M. Dirac, Quantized singularities in the electromagnetic field, Proc. Roy. Soc.133 (1931) 60–72.
[9] G.t Hooft,Nucl. Phys.B77 (1974).461.B75 (1974).461.
[10] R. Jackiw and C. Rebbi, Degrees of freedom in pseudoparticle systems, Phys. Lett.B 67 (1977) 189–192.
[11] A. Ja?e and C. H.Taubes, Vortices and Monopoles, Birkh¨auser, Boston, 1980.
[12] B. Julia and A. Zee, Poles with both electric and electric charges in non-Abeliangauge theory, Phys. Rev. D 11 (1975) 2227–1232.
[13] D. Maison, Uniqueness of the Prasad–Sommerfeld monopole solution, Nucl. Phys. B182 (1981) 144–150.
[14] T. H. Parker, Nonminimal Yang–Mills fields and dynamics, Invent. Math. 107 (1992)397–420.
[15] M. K. Prasad and C. M. Sommerfeld, Exact classical solutions for the’t Hooftmonopole and the Julia–Zee dyon, Phys. Rev. Lett. 35 (1975) 760–762.
[16] E. Radu and D. H. Tchrakian, Static BPS‘monopoles’in all even spacetime dimen-sions. Phys. Rev. D 71 (2005) 125013.
[17] R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982.
[18] L. Sadun and J. Segert, Non-self-dual Yang–Mills connections with quadrupole sym-metry, Commun. Math. Phys. 145 (1992) 362–391.
[19] J. Schwinger, A magnetic model of matter, Science 165 (1969) 757–761.
[20] L. Sibner, R. Sibner, and K. Uhlenbeck, Solutions to Yang–Mills equations that arenot self-dual, Proc. Nat. Acad. Sci. USA 86 (1989) 8610–8613.
[21] T.H.R.Skyrme, Static properties of nucleons in the Skyrme model ,Proc.Roy.Soc.A260 (1961) 127
[22] T.H.R.Skyrme, Baby skyrme on the two-sphere, Nucl. Phys.31 (1962). 556.
[23] C. H. Taubes, The existence of a non-minimal solution to the SU(2) Yang–Mills–Higgs equations on R3, Parts I and II, Commun. Math. Phys. 86 (1982) 257–320.
[24] G.’t Hooft, A property of electric and magnetic ?ux in nonabelian gauge theories,Nucl. Phys. B 153 (1979) 141–160.
[25] X. Wang and Y. Yang, Existence of static BPS monopoles and dyons in arbitrary(4p-1)-dimensional space, Lett. Math. Phys. 77 (2006) 249-263
[26] E. Witten, Some exact multipseudoparticle solutions of classical Yang–Mills theory,Phys. Rev. Lett. 38 (1977) 121–124.
[27] Y. Yang, Existence of solutions for a generalized Yang–Mills theory, Lett. Math. Phys.19 (1990) 257–267.
[28] Y. Yang, Self-dual monopoles in a seven-dimensional gauge theory, Lett. Math. Phys.20 (1990) 285–290.
[29] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer, New York, 2001.
[2] K.Arthur and D.H.tchrakian,Solitons in gauged Sigma models:2 dimensions, Phys.Lett.B 352 (1995) 327.
[3] A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Pseudoparticlesolutions of the Yang–Mills equations, Phys. Lett. B 59 (1975) 85–87.
[4] E. B. Bogomol’nyi, The stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976)449–454.
[5] Y.Brihaye and D.H.tchrakian,Solitons/instantons in d-dimensional SO (d) gauged O(d+ 1) Skyrme models,Nolinearity 11 (1998) 891.
[6] Y.Brihaye, B.Hrtmann and D.H.Tchrakian, Monopoles and Dyons in SO(3) gaugedSkyrme models,Jour. Math. Phys.42.8(2001)
[7] L. Ca?arelli, B. Gidas, and J. Spruck, On multimeron solutions of the Yang–Millsequation, Commun. Math. Phys. 87 (1983) 485–495.
[8] P. A. M. Dirac, Quantized singularities in the electromagnetic field, Proc. Roy. Soc.133 (1931) 60–72.
[9] G.t Hooft,Nucl. Phys.B77 (1974).461.B75 (1974).461.
[10] R. Jackiw and C. Rebbi, Degrees of freedom in pseudoparticle systems, Phys. Lett.B 67 (1977) 189–192.
[11] A. Ja?e and C. H.Taubes, Vortices and Monopoles, Birkh¨auser, Boston, 1980.
[12] B. Julia and A. Zee, Poles with both electric and electric charges in non-Abeliangauge theory, Phys. Rev. D 11 (1975) 2227–1232.
[13] D. Maison, Uniqueness of the Prasad–Sommerfeld monopole solution, Nucl. Phys. B182 (1981) 144–150.
[14] T. H. Parker, Nonminimal Yang–Mills fields and dynamics, Invent. Math. 107 (1992)397–420.
[15] M. K. Prasad and C. M. Sommerfeld, Exact classical solutions for the’t Hooftmonopole and the Julia–Zee dyon, Phys. Rev. Lett. 35 (1975) 760–762.
[16] E. Radu and D. H. Tchrakian, Static BPS‘monopoles’in all even spacetime dimen-sions. Phys. Rev. D 71 (2005) 125013.
[17] R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982.
[18] L. Sadun and J. Segert, Non-self-dual Yang–Mills connections with quadrupole sym-metry, Commun. Math. Phys. 145 (1992) 362–391.
[19] J. Schwinger, A magnetic model of matter, Science 165 (1969) 757–761.
[20] L. Sibner, R. Sibner, and K. Uhlenbeck, Solutions to Yang–Mills equations that arenot self-dual, Proc. Nat. Acad. Sci. USA 86 (1989) 8610–8613.
[21] T.H.R.Skyrme, Static properties of nucleons in the Skyrme model ,Proc.Roy.Soc.A260 (1961) 127
[22] T.H.R.Skyrme, Baby skyrme on the two-sphere, Nucl. Phys.31 (1962). 556.
[23] C. H. Taubes, The existence of a non-minimal solution to the SU(2) Yang–Mills–Higgs equations on R3, Parts I and II, Commun. Math. Phys. 86 (1982) 257–320.
[24] G.’t Hooft, A property of electric and magnetic ?ux in nonabelian gauge theories,Nucl. Phys. B 153 (1979) 141–160.
[25] X. Wang and Y. Yang, Existence of static BPS monopoles and dyons in arbitrary(4p-1)-dimensional space, Lett. Math. Phys. 77 (2006) 249-263
[26] E. Witten, Some exact multipseudoparticle solutions of classical Yang–Mills theory,Phys. Rev. Lett. 38 (1977) 121–124.
[27] Y. Yang, Existence of solutions for a generalized Yang–Mills theory, Lett. Math. Phys.19 (1990) 257–267.
[28] Y. Yang, Self-dual monopoles in a seven-dimensional gauge theory, Lett. Math. Phys.20 (1990) 285–290.
[29] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer, New York, 2001.