We extend the Poincaré group to the complex Minkowski space-time. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the g...We extend the Poincaré group to the complex Minkowski space-time. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the generalizations of the two Casimir operators.展开更多
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 M...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 $$\eta ^\mu \left[ {\frac{{\partial F_{jk} }}{{\partial x^l }} + C_l F_{jk} - F_{jk} C_l } \right] = 0.$$ .展开更多
文摘We extend the Poincaré group to the complex Minkowski space-time. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the generalizations of the two Casimir operators.
文摘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 $$\eta ^\mu \left[ {\frac{{\partial F_{jk} }}{{\partial x^l }} + C_l F_{jk} - F_{jk} C_l } \right] = 0.$$ .
