Advertisement

Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 545–584 | Cite as

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

  • Walter J. SchemppEmail author
Original Research
  • 66 Downloads

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–Möbius 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.

Keywords

Dynamic metaplectic spinor quantization Hopf principal circle bundle Metaplectic Lie group \(\mathrm{Mp}(2, {\mathbb {R}})\) Semi-simple Lorentz–Möbius Lie group \({\mathrm{PSO}(1, 3, {\mathbb {R}})}\) Post-Kepplerian metaplectically driven spinor warping Gravitational radiation emission Abel–Jacobi inversion Eisenstein elementary meromorphic functions 

Mathematics Subject Classification

94A40 81S10 81T20 70G65 70G45 57R17 51N35 53C27 30F10 22E46 22E25 14H81 14H55 14H52 14M15 14N05 

References

  1. 1.
    Alexander, J.W.: A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA 9, 93–95 (1923)CrossRefGoogle Scholar
  2. 2.
    Alexander, J.W.: An example of a simply connected surface bounding a region which is not simply connected. Proc. Nat. Acad. Sci. USA 10, 8–10 (1924)CrossRefGoogle Scholar
  3. 3.
    Baez, J.C.: The Octonions. Bull. (New Ser.) Am. Math. Soc. 39, 145–205 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Becker, W. (ed.): Neutron Stars and Pulsars. Springer, Berlin (2009)Google Scholar
  5. 5.
    Donaldson, S.: Riemann Surfaces. Oxford University Press, Oxford (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Eisenstein, F.G.M.: Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenhängenden Doppelreihen (als eine Begründungsweise der Theorie der elliptischen Functionen, mit besonderer Berücksichtigung ihrer Analogie zu den Kreisfunctionen). Crelles J. 35, 153–274 (1847)Google Scholar
  7. 7.
    Eisenstein, F.G.M.: Mathematische Werke, 2nd edn, vol. 1, pp. 357–478. Chelsea, New York (1989)Google Scholar
  8. 8.
    Godement, R.: Analyse Mathématique I et IV. Springer, Berlin (2003)Google Scholar
  9. 9.
    Goldman, W.M.: Complex Hyperbolic Geometry. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  10. 10.
    Goulielmakis, E., Yakovlev, V.S., Cavalieri, A.L., Uiberacker, M., Pervak, V., Apolonski, A., Kienberger, R., Kleineberg, U., Krausz, F.: Attosecond control and measurement: lightwave electronics. Science 317, 769–775 (2007)CrossRefGoogle Scholar
  11. 11.
    Griffiths, P.: Variations on theorem of Abel. Invent. Math. 35, 321–390 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Griffiths, P., Harris, J.: A Poncelet theorem in space. Comment. Math. Helv. 52, 145–160 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Griffiths, P., Harris, J.: On Cayley’s explicit solution to Poncelet’s porism. Enseign. Math. 24, 31–40 (1978)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Howe, R., Tan, E.C.: Non-Abelian Harmonic Analysis: Applications to \({\rm SL}(2,{\mathbb{R}})\). Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  15. 15.
    Husemöller, D.: Elliptic Curves, 2nd edn. Springer, New York (2004)zbMATHGoogle Scholar
  16. 16.
    Johnson, N.W.: Absolute polarities and central inversions. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds.) The Geometric Vein: The Coxeter Festschrift, pp. 443–464. Springer, New York (1981)CrossRefGoogle Scholar
  17. 17.
    Kampschulte, M., Langheinrich, A.C., Sender, J., Litzlbauer, H.D., Althöhn, U., Schwab, J.D., Alejandre-Lafont, E., Martels, G., Krombach, G.A.: Nano-computed tomography: technique and applications. Fortschr. Röntgenstr. 188, 146–154 (2016)CrossRefGoogle Scholar
  18. 18.
    Kapteyn, H., Cohen, O., Christov, I., Murnane, M.: Harnessing attosecond science in the quest for coherent X-rays. Science 317, 775–778 (2007)CrossRefGoogle Scholar
  19. 19.
    Kostant, B.: Quantization and unitary representations. In: Taam, C.T. (ed.) Lectures in Modern Analysis and Applications III, pp. 87–208. Lecture Notes in Mathematics, Volume 170. Springer, Berlin (1970)Google Scholar
  20. 20.
    Kostant, B.: Symplectic spinors. Istituto Nazionale di Alta Matematica Roma. Symposia Mathematica, vol. XIV, pp. 139–152. Academic Press, London (1974)Google Scholar
  21. 21.
    Lion, G., Vergne, M.: The Weil Representation, Maslov Index and Theta Series. Birkhäuser Verlag, Boston (1980)CrossRefzbMATHGoogle Scholar
  22. 22.
    Loth, S., Etzkorn, M., Lutz, C.P., Eigler, D.M., Heinrich, A.J.: Measurement of fast electron spin relaxation times with atomic resolution. Science 329, 1628–1630 (2010)CrossRefGoogle Scholar
  23. 23.
    Narasimhan, R.: Complex Analysis in One Variable. Birkhäuser Verlag, Boston (1985)CrossRefzbMATHGoogle Scholar
  24. 24.
    Nowak, A., Sjögren, P., Szarek, T.Z.: Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions. Adv. Math. 318, 307–354 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pascual-Sánchez, J.-F.: Introducing relativity in global navigation satellite systems. Ann. Phys. (Leipzig) 16, 258–273 (2007)CrossRefzbMATHGoogle Scholar
  26. 26.
    Polishchuk, A.: Analogue of Weil representation for abelian schemes. J. Reine Angew. Math. 543, 1–37 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schempp, W.J.: Magnetic Resonance Imaging: Mathematical Foundations and Applications. Wiley-Liss, New York (1998)zbMATHGoogle Scholar
  28. 28.
    Schempp, W.J.: Non-Abelian harmonic analysis of gravitational radiation emission. In: Proceedings of the 30th International Conference on Systems Research, Informatics and Cybernetics, Baden-Baden, Germany (2018)Google Scholar
  29. 29.
    Schempp, W.J., Non-Abelian harmonic analysis of gravitational radiation emission and terahertz spectroscopy of semiconductor nanostructures. Results Math (to appear)Google Scholar
  30. 30.
    Silverman, J.H., Tate, J.T.: Rational Points on Elliptic Curves, 2nd edn. Springer, Cham (2015)CrossRefzbMATHGoogle Scholar
  31. 31.
    Stephenson, B.: Kepler’s Physical Astronomy. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  32. 32.
    Stejskal, E.O., Tanner, J.E.: Spin diffusion measurements: spin-echo in the presence of a time-dependent field gradient. J. Chem. Phys. 42, 288–292 (1965)CrossRefGoogle Scholar
  33. 33.
    Walker, J.S.: Fast Fourier Transforms. CRC Press, Boca Raton (1992)Google Scholar
  34. 34.
    Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Weil, A.: Elliptic Functions According to Eisenstein and Kronecker. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  36. 36.
    Weil, A.: Collected Papers, vol. III, pp. 1–69, Springer, New York (1979)Google Scholar
  37. 37.
    Wilker, J.B.: Inversive geometry. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds.) The Geometric Vein: The Coxeter Festschrift, pp. 379–442. Springer, New York (1981)CrossRefGoogle Scholar
  38. 38.
    Wiltshire, D.L., Visser, M., Scott, S.M. (eds.): The Kerr Spacetime: Rotating Black Holes in General Relativity. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Lehrstuhl für Mathematik IUniversität SiegenSiegenGermany

Personalised recommendations