are of much importance. Let us write.
(70) [function]F =| S_{11} - L S_{12} S_{13} S_{14} |
| S_{21} S_{22} - L S_{23} S_{24} |
| S_{31} S_{32} S_{33} - L S_{34} |
| S_{41} S_{42} S_{43} S_{44} - L |
Then (71) S_{11} + S_{22} + S_{33} + S_{44} = 0.
Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by
(72) L = 1/2([function]_{23} F_{23} + [function]_{31}F_{31} + [function]_{12} + F_{12} + [function]_{14} F_{14}
+ [function]_{24} F_{24} + [function]_{34} F_{34})
Then we shall have
(73) S_{11} = 1/2([function]_{23} F_{23} + [function]_{34} F_{34} + [function]_{42} F_{42} - [function]_{12} F_{12}
- [function]_{13} F_{13} [function]_{14} F_{14})
S_{12} = [function]_{13} F_{32} + [function]_{14} F_{42} etc. . . .
In order to express in a real form, we write
(74) S = | S_{11} S_{12} S_{13} S_{14} |
| S_{21} S_{22} S_{23} S_{24} |
| S_{31} S_{32} S_{33} S_{34} |
| S_{41} S_{42} S_{43} S_{44} |
= | X_{x} Y_{x} Z_{x} -iT_{x} |
| X_{y} Y_{y} Z_{y} -iT_{y} |
| X_{z} Y_{z} Z_{z} -iT_{z} |
| -iX_{t} -iY_{t} -iZ_{t} T_{t} |
Now X_{x} = 1/2[m_{x}M_{x} - m_{y}M_{y} - m_{z}M_{z} + e_{x}E_{x} - e_{y}E_{y} - e_{z}E_{z}]
