Inversion Transformation - Transformation On Coordinates

Transformation On Coordinates

In the following we shall use imaginary time so that space-time is Euclidean and the equations are simpler. The Poincaré transformations are given by the coordinate transformation on space-time parametrized by the 4-vectors V

where is an orthogonal matrix and is a 4-vector. Applying this transformation twice on a 4-vector gives a third transformation of the same form. The basic invariant under this transformation is the space-time length given by the distance between two space-time points given by 4-vectors x and y:

These transformations are subgroups of general 1-1 conformal transformations on space-time. It is possible to extend these transformations to include all 1-1 conformal transformations on space-time

$V_\mu ^\prime =\left( A_\tau ^\nu V_\nu +B_\tau \right) \left( C_{\tau \mu }^\nu V_\nu +D_{\tau \mu }\right) ^{-1}.$

We must also have an equivalent condition to the orthogonality condition of the Poincaré transformations:

Because one can divide the top and bottom of the transformation by we lose no generality by setting to the unit matrix. We end up with

$V_\mu ^\prime =\left( O_\mu ^\nu V_\nu +P_\tau \right) \left( \delta _{\tau \mu} + Q_{\tau \mu }^\nu V_\nu \right) ^{-1}. \,$

Applying this transformation twice on a 4-vector gives a transformation of the same form. The new symmetry of 'inversion' is given by the 3-tensor This symmetry becomes Poincaré symmetry if we set When the second condition requires that is an orthogonal matrix. This transformation is 1-1 meaning that each point is mapped to a unique point only if we theoretically include the points at infinity.

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