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Science in China Series A: Mathematics

, Volume 41, Issue 10, pp 1061–1067 | Cite as

The Yang-Mills fields on the Minkowski space

  • Qikeng Lu
Article

Abstract

Let the coordinatex=(x 0,x 1,x 2,x 3) of the Minkowski spaceM 4 be arranged into a matrix
$$H_x = \left( {\begin{array}{*{20}c} {x^0 + x^1 x^2 + ix^3 } \\ {x^2 - ix^3 x^0 - x^1 } \\ \end{array} } \right).$$
Then the Minkowski metric can be written as
$$ds^2 = \eta _{jk} dx^j dx^k = det dH_x $$
. Imbed the space of 2 × 2 Hermitian matrices into the complex Grassmann manifoldF(2,2), the space of complex 4-planes passing through the origin ofC 2×4. The closure\(\bar M^4 \) ofM 4 inF(2,2) is the compactification ofM 4. It is known that the conformal group acts on\(\bar M^4 \). It has already been proved that onF(2,2) there is anSu(2)-connection
$$B(Z, dZ) = \Gamma (Z, dZ) - \Gamma (Z, dZ)^ + - \frac{{tr[\Gamma (Z, dZ) - \Gamma (Z, dZ^ + ]}}{2}I.$$
whereZ is a 2 × 2 complex matrix andZ the complex conjugate and transposed matrix ofZ. Restrict this connection to\(\bar M^4 \)
$$C(H_x ,dH_x ) = [B(Z, dZ)]_{z = H_x } = C_j (x)dx^j ,$$
which is anSu(2)-connection on\(\bar M^4 \). It is proved that its curvature form
$$F: = dC + C \Lambda C = \frac{1}{2}\left[ {\frac{{\partial C_k }}{{\partial x^j }} - \frac{{\partial C_j }}{{\partial x^k }} + C_j C_k - C_k C_j } \right]dx^j \Lambda dx^k = :F_{jk} dx^j \Lambda dx^k $$
satisfies the Yang-Mills equation
$$\eta ^\mu \left[ {\frac{{\partial F_{jk} }}{{\partial x^l }} + C_l F_{jk} - F_{jk} C_l } \right] = 0.$$
.

Keywords

Yang-Mills fields Minkowski space Lorentz manifold 

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References

  1. 1.
    Yang, C. N., Mills, P. L., Conservation of isotropic spin and isotropic gauge transformation,Phys. Review, 1954, 96: 91.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Lu Qikeng, On self-dual gauge field,Advances in Math., 1993, 22: 92.MATHGoogle Scholar
  3. 3.
    Lawson, B.,The Theory of Gauge Fields in Four Dimensions, CBMS, 58, 1983.Google Scholar
  4. 4.
    Lu Qikeng, A study of Maxwell equation by spinor analysis, Proc. Symp. in Honor of Prof. Pei Yuan Zhou’s 90th Birthday (eds. Lin, C.C., Wu Ning), Beijing: Peking Univ. Press, 1993.Google Scholar
  5. 5.
    Wells. R.O., Complex manifolds and mathematical physics,Bull. AMS (NS), 1978, 1: 296.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Lu Qikeng,The Classical Manifolds and Classical Domains (in Chinese), Beijing: Sci. Press, 1963.Google Scholar
  7. 7.
    Lu Qikeng, The unitary connections on the complex Grassmann manifold, preprint.Google Scholar
  8. 8.
    Lu Qikeng,The Differential Geometry and its Application to Physics (in Chinese), Shanghai: Sci. and Tech. Press, 1982.Google Scholar

Copyright information

© Science in China Press 1998

Authors and Affiliations

  • Qikeng Lu
    • 1
    • 2
  1. 1.Institute of MathematicsShantou UniversityShantouChina
  2. 2.Institute of MathematicsChinese Academy of SciencesBeijingChina

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