Dynamic metaplectic spinor quantization: the projective correspondence for spectral dual pairs

Abstract

From the historical perspective, the technique of X-ray computer tomography featured the predecessor of magnetic spin resonance tomography, which in fact is a non-invasive, high resolution, biomedical diagnostic scanning modality. Based on non-commutative harmonic analysis on the classical (2 + 1)-dimensional real Heisenberg unipotent Lie group $$\mathcal{N}$$ and the gradient controlled inversive and co-inversive chord-contact dynamics, framed by the coadjoint $$\mathcal{N}$$ -orbit model inside the real dual vector space $$\mathfrak {Lie}(\mathcal{N})^*$$ , the paper provides mathematical insight into the intrinsic electromagnetic quantum field and relativistic symmetries associated to the highly resolving clinical modality of magnetic spin resonance tomography by referring to the methodology of the basic control mechanisms of the projective duality correspondence for spectral dual pairs of real Lie groups. In terms of the projective manifold $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) \cong {\mathbb {P}}_3({\mathbb {C}})$$ , which is associated with the (2 + 1)-dimensional dual vector space $$\mathfrak {Lie}(\mathcal{N})^*$$ of the real Heisenberg nilpotent Lie algebra $$\mathfrak {Lie}(\mathcal{N})$$ , the smooth line bundle technique of dynamic metaplectic spinor quantization leads to the twisted action of the metaplectic Lie group $$\mathrm{Mp}(2,{\mathbb {R}}) = \widetilde{\mathrm{Sp}}(2,{\mathbb {R}})$$ . The transitive proper hyperbolic-parabolically ruled gradient action of the projective Lorentz–Mobius Lie group $$\mathrm{PSO}(1,3,{\mathbb {R}})$$ on $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ torque-records dihedrally its spectral incidence projective curve, the plane cubic $$E({\mathbb {C}}) \hookrightarrow {\mathbb {P}}_2({\mathbb {C}}) \cong \mathrm{Sym}^2\left( {\mathbb {P}}_1({\mathbb {C}})\right) $$ , on the two-dimensional pages of the open-book foliation inside the very round sphere $${\mathbb {S}}_3 \cong \mathrm{Spin}(3,{\mathbb {R}}) \cong \mathrm{SU}(2,{\mathbb {C}}) \hookrightarrow {\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ with angular momentum axis of the driving central universal Casimir metaplectic spinor. The projective correspondence for the spectral dual pair $$\left( \mathrm{Mp}(2,{\mathbb {R}}),\mathrm{PSO}(1,3,{\mathbb {R}})\right) $$ provides an efficient mathematical approach to the high resolution imaging modalities of magnetic spin resonance tomography, optical or ocular coherence tomography of neuro-ophthalmology, spin-polarized scanning tunneling microscopy, and the quantum field phenomena of the Hanbury Brown–Twiss experiment for photons and electrons. Due to the Eisenstein meromorphic calculus, the omni-directional interferometric detection of gravitational wavepackets of two polarizations is tomographically performed by the Abel–Jacobi inversion of the relativistic parabolic porism deviation of Kepplerian bifocal periodicity with its metaplectic spinor driven pair $$(\sigma ,{\bar{\sigma }})$$ of projective tangent involutions in the space $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ . The astrophysical emission of gravitational radiation is closely related to the concept of simply connected horned sphere which is homeomorphic to the compact base manifold of the Hopf principal circle bundle $${\mathbb {S}}_1 \hookrightarrow {\mathbb {S}}_3 {\mathop {\longrightarrow }\limits ^{\eta }} {\mathbb {S}}_2$$ . Riemann surface theory provides tomographic insight into the relativistic phenomenon of post-Kepplerian metaplectically driven spinor warping.