The Yang-Mills fields on the Minkowski space

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

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

.