ABSTRACT
There are unfortunately a large variety of possibilities for writing down a rotation; also there are two different interpretations of the word ‘rotation’. Obviously, if one rotates first the coordinate frame and then, by the same rotation, the space, all points will obtain again the same coordinate as before these rotations; the same is true in the reverse order. Rotational invariance means then that the result of this rotation, seen by an observer who did not participate in the rotation, is again a possible state of the unrotated system. One says the rotations in the physical three-dimensional space induce unitary transformations in the Hilbert space, whose matrices give rise to (infinitely many independent) representations of the rotation group. The active interpretation means carrying out the sequence of rotations using a fixed system of rotation axes; the passive interpretation means carrying out the rotations using a system of rotation axes which participates in the rotation.
