Classification of Lorentz Transformations

For a further classification of the set of all Lorentz transformations we need some elementary properties of these transformations. From Eq. (3.17) and det g = —1 it is immediately clear that Lorentz transformations feature a determinant det A = 1. And according to Eq. (3.17) with p = v = 0 it follows that 1. We can thus define the subgroup of proper Lorentz transformations by the condition 

Proper Lorentz transformations can always be continuously converted into the identical transformation given by A = 1 = id. Improper Lorentz transformations involve either space inversion (A q 1, det A = —1) or time reversal (A g — 1, det A = —1) or both. These improper transformations do not play any role in quantum chemistry and will thus not be considered in this presentation. 

Pure rotations of the coordinate axes of IS with reference to those of IS also belong to the homogeneous Lorentz transformations and are given by 

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