# The Yang-Mills fields on the Minkowski space

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## Abstract

Let the coordinate Then the Minkowski metric can be written as. 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 of where which is anSu(2)-connection on\(\bar M^4 \). It is proved that its curvature form satisfies the Yang-Mills equation.

*x*=(*x*^{0},*x*^{1},*x*^{2},*x*^{3}) of the Minkowski space*M*^{4}be arranged into a matrix$$H_x = \left( {\begin{array}{*{20}c} {x^0 + x^1 x^2 + ix^3 } \\ {x^2 - ix^3 x^0 - x^1 } \\ \end{array} } \right).$$

$$ds^2 = \eta _{jk} dx^j dx^k = det dH_x $$

*C*^{2×4}. The closure\(\bar M^4 \) of*M*^{4}inF(2,2) is the compactification of*M*^{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$$B(Z, dZ) = \Gamma (Z, dZ) - \Gamma (Z, dZ)^ + - \frac{{tr[\Gamma (Z, dZ) - \Gamma (Z, dZ^ + ]}}{2}I.$$

*Z*is a 2 × 2 complex matrix and*Z*^{†}the complex conjugate and transposed matrix of*Z*. Restrict this connection to\(\bar M^4 \)$$C(H_x ,dH_x ) = [B(Z, dZ)]_{z = H_x } = C_j (x)dx^j ,$$

$$F: = dC + C \Lambda C = \frac{1}{2}\left[ {\frac{{\partial C_k }}{{\partial x^j }} - \frac{{\partial C_j }}{{\partial x^k }} + C_j C_k - C_k C_j } \right]dx^j \Lambda dx^k = :F_{jk} dx^j \Lambda dx^k $$

$$\eta ^\mu \left[ {\frac{{\partial F_{jk} }}{{\partial x^l }} + C_l F_{jk} - F_{jk} C_l } \right] = 0.$$

## Keywords

Yang-Mills fields Minkowski space Lorentz manifold## Preview

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## References

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© Science in China Press 1998