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

The congruential rule advanced by Graves for polarization basis transformation of the radar backscatter matrix is now often misinterpreted as an example of consimilarity transformation. However, consimilarity transformations imply a physically unrealistic antilinear time-reversal operation. This is just one of the approaches found in literature to the description of transformations where the role of conjugation has been misunderstood. In this paper, the different approaches are examined in particular in respect to the role of conjugation. In order to justify and correctly derive the congruential rule for polarization basis transformation and properly place the role of conjugation, the origin of the problem is traced back to the derivation of the antenna height from the transmitted field. In fact, careful consideration of the role played by the Green’s dyadic operator relating the antenna height to the transmitted field shows that, under general unitary basis transformation, it is not justified to assume a scalar relationship between them. Invariance of the voltage equation shows that antenna states and wave states must in fact lie in dual spaces, a distinction not captured in conventional Jones vector formalism. Introducing spinor formalism, and with the use of an alternate spin frame for the transmitted field a mathematically consistent implementation of the directional wave formalism is obtained. Examples are given comparing the wider generality of the congruential rule in both active and passive transformations with the consimilarity rule.

The “pain matrix” in pain-free individuals

Human functional imaging provides a correlative picture of brain activity during pain. A particular set of central nervous system structures (eg, the anterior cingulate cortex, thalamus, and insula) consistently respond to transient nociceptive stimuli causing pain. Activation of this so-called pain matrix or pain signature has been related to perceived pain intensity, both within and between individuals,1,2 and is now considered a candidate biomarker for pain in medicolegal settings and a tool for drug discovery. The pain-specific interpretation of such functional magnetic resonance imaging (fMRI) responses, although logically flawed,3,4 remains pervasive. For example, a 2015 review states that “the most likely interpretation of activity in the pain matrix seems to be pain.”4 Demonstrating the nonspecificity of the pain matrix requires ruling out the presence of pain when highly salient sensory stimuli are presented. In this study, we administered noxious mechanical stimuli to individuals with congenital insensitivity to pain and sampled their brain activity with fMRI. Loss-of-function SCN9A mutations in these individuals abolishes sensory neuron sodium channel Nav1.7 activity, resulting in pain insensitivity through an impaired peripheral drive that leaves tactile percepts fully intact.5 This allows complete experimental disambiguation of sensory responses and painful sensations