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The congruential rule advanced by Graves for polarization basistransformation of the radar backscatter matrix is now often misinterpreted asan example of consimilarity transformation. However, consimilaritytransformations imply a physically unrealistic antilinear time-reversaloperation. This is just one of the approaches found in literature to thedescription of transformations where the role of conjugation has beenmisunderstood. In this paper, the different approaches are examined inparticular in respect to the role of conjugation. In order to justify andcorrectly derive the congruential rule for polarization basis transformationand properly place the role of conjugation, the origin of the problem is tracedback to the derivation of the antenna hight from the transmitted field. Infact, careful consideration of the role played by the Green's dyadic operatorrelating the antenna height to the transmitted field shows that, under generalunitary basis transformation, it is not justified to assume a scalarrelationship between them. Invariance of the voltage equation shows thatantenna states and wave states must in fact lie in dual spaces, a distinctionnot captured in conventional Jones vector formalism. Introducing spinorformalism, and with the use of an alternate spin frame for the transmittedfield a mathematically consistent implementation of the directional waveformalism is obtained. Examples are given comparing the wider generality of thecongruential rule in both active and passive transformations with theconsimilarity rule.

On mathematical and physical principles of transformations of the coherent radar backscatter matrix