Science in China Series A: Mathematics

, Volume 41, Issue 10, pp 1061–1067

# 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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