The paper leads from Keppler to Heisenberg in order to demonstrate the tight control which Lie group theory exercises over the non-invasive MRI modality via the geometric quantization strategy. For the first time, it allows for a visualization of the coadjoint orbit fibration associated to the coadjoint \(G\)-action in the dual vector space Lie\((G)^*\) of the Heisenberg Lie algebra Lie\((G)\) as a stack of energetic strata forming tomographic slices, and the associated bundle of \({\mathcal C}^*\)-algebras. In addition, the paper offers the three-dimensional compact super-encoding projective space \({\mathbf P}({\mathbf R}\times \text{Lie}(G)^*)\) with homogeneous transverse direction line \({\mathbf R}\) as the natural frame of the solvable affine Lauterbur subband encoding technique of spatial localization by directional derivatives, or linear magnetic field gradients. The three-dimensional real projective space \({\mathbf P}({\mathbf R}\times \text{Lie}(G)^*)\) represents the frame for the line geometric implementation of the Larmor equation for the magnetic spin precession. The planar frame is performed by the de Rham cohomology associated to the \(L^2\)-sections of the homogeneous line bundle of transvections over the projectively immersed, natural symplectic affine structure of the planar coadjoint orbits \({\mathcal O}_\nu (\nu\neq 0)\) of \(G\), and its rotational curvature form.
Tomographic slice selection, readout procedure, and phase encoding linear gradients operating on the \(L^2\)-sections of a homogeneous hologram line bundle perform the basic MRI functions and are refocused in every repetition period. The refocusing procedures by spatial rewinder gradients and phase conjugation flip depend on Fourier analysis of the metaplectic group pointwise attached to the one-dimensional center \(C\) of \(G\) as a group of dynamical symmetries specific for the selected energetic stratum. Each point \(\nu\) of the line \(\check C-\{0\}\) with the origin removed corresponds to the distinguished closed exterior differential 2-form \(\omega_\nu\) associated to the curvature form of the planar coadjoint orbit \({\mathcal O}_\nu \hookrightarrow\text{Lie}(G)^* (\nu\neq 0)\) of \(G\) which is immersed as a linear symplectic affine variety into the three-dimensional compact super-encoding projective space \({\mathbf P}({\mathbf R}\times \text{Lie} (G)^*)\). Similar to the refocusing procedures by spatial rewinder gradients, the rephasing procedure of spin echoes by phase conjugation flip depends upon the fact that the normal subgroup \({\mathbf G}{\mathbf O}^+(2,{\mathbf R})\) of direct linear similitudes has index two in the non-abelian group \({\mathbf G} {\mathbf O}(2,{\mathbf R})\) of all linear similitudes of the symplectic affine plane \({\mathbf R}\oplus{\mathbf R}\).
The fascinating aspects of electronic engineering concerned with the implementation of the dynamical symmetries inherent to the semi-classical approach to clinical magnetic resonance tomography by large scale integrated (LSI) microcircuit technology are also indicated. Once greater understanding has been gained in the quantized calculus foundations of MRI, and the dynamical system approach to the solvable affine Lauterbur subband encoding technique of spatial localization by linear magnetic field gradients operating on the \(L^2\)-sections of a homogeneous hologram line bundle, it will be possible to design new wavelet packets of spin excitation profiles with the desired contrast resolution capability.