Congruence and metaplectic covariance: Rational biquadratic reciprocity and quantum entanglement

Yıl 2021, Cilt: 4 Sayı: 1, 61 - 80, 01.03.2021

Walter J. Schempp

https://doi.org/10.33205/cma.804852

Cited By: 1

Öz

The purpose of the paper is to elucidate the cyclotomographic applications of the coadjoint orbit methodology to the Legendre-Hilbert-Artin symbolic tower of class field theory in the sense of the theories of Chevalley, Hasse, Weil and Witt. The Witt arithmetics concludes with the law of rational biquadratic reciprocity and quantum entanglement.
The purpose of the paper is to elucidate the cyclotomographic applications of the coadjoint orbit methodology to the Legendre-Hilbert-Artin symbolic tower of class field theory in the sense of the theories of Chevalley, Hasse, Weil and Witt. The Witt arithmetics concludes with the law of rational biquadratic reciprocity and quantum entanglement.

Anahtar Kelimeler

Third order principle of spinor triality, spaces of even and odd half-spinor, metaplectic Lie group Mp(2, R), Hopf principal circle bundle, the metaplectic coadjoint orbit model, half-spinor Maslov index, Witt quartic groups, Legendre-Hilbert-Artin symbolic tower of class field theory, valuation and module function, rational quantum entanglement, adeles and ideles

Kaynakça

• D.M.S. Bagguley, Editor, Pulsed Magnetic Resonance: NMR, ESR, and Optics. A recognition of E.L. Hahn. Oxford University Press, Oxford, New York, Toronto 1992
• O. Baumgart, The Quadratic Reciprocity Law: A Collection of Classical Proofs. Birkhäuser and Springer, Cham, Heidelberg, New York, Dordrecht, London 2015
• K. Burde, Ein rationales biquadratisches Reziprozitätsgesetz. J. Reine und Angew. Math. 235, 175-184 (1969)
• J.C. Carr, T.J. Carroll, Magnetic Resonance Angiography, Principles and Applications. Springer New York, Dordrecht, Heidelberg, London 2012
• C. Chevalley, The Algebraic Theory of Spinors and Clifford Algebras. Collected Works, Volume 2, SpringerVerlag, Berlin, Heidelberg, New York 1997
• N. Childress, Class Field Theory. Springer-Verlag, Berlin, Heidelberg, New York 2009
• C.W. Curtis, Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer. American Mathematical Society, Providence, Rhode Island 1999
• J.A. Dieudonné, Review of The Algebraic Theory of Spinors by C. Chevalley. Bull. Amer. Math. Soc. 60, 408-413 (1954)
• J. Franchi, Y. Le Jan, Hyperbolic Dynamics and Brownian Motion: An Introduction. Oxford University Press, Oxford, New York 2012
• L.J. Goldstein, Analytic Number Theory. Prentice Hall, Englewood Cliffs, New Jersey 1971
• G. Gras, Class Field Theory: From Theory to Practice. Springer-Verlag, Berlin, Heidelberg, NewYork 2005
• V.W. Guillemin, Symplectic spinors and partial differential equations. In: Géomeétrie Symplectique et Physique Mathématique, pp. 217-252, Centre National de la Recherche Scientifique, Paris 1975
• V.W. Guillemin, S. Sternberg, Supersymmetry and Equivariant de Rham Theory. Springer-Verlag, Berlin, Heidelberg, New York 1999
• D. Harari, Galois Cohomology and Class Field Theory. Springer Nature, Cham, Switzerland 2020
• R. Howe, On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. (New Series), Vol. 3, 821-843 (1980)
• R. Howe, Quantum mechanics and partial differential equations. J. Funct. Anal. 38, 188-254 (1980)
• R. Howe, The oscillator semigroup. In: The Mathematical Heritage of Hermann Weyl. R.O. Wells, Jr., Editor, Proceedings of Symposia in Pure Mathematics, Vol. 48, pp. 61-132, American Mathematical Society, Providence, Rhode Island 1988
• H. Johansen-Berg, T.E.J. Behrens, Editors, Diffusion MRI: From Quantitative Measurement to In vivo Neuroanatomy. Elsevier, Academic Press, Amsterdam, Boston, Heidelberg 2009
• D.K. Jones, Editor, Diffusion MRI: Theory, Methods, and Applications. Oxford University Press, Oxford, New York 2010
• B. Kostant, Symplectic spinors. In: Istituto Nazionale di Alta Matematica Roma, Symposia Mathematica, Volume XIV, pp. 139–152, Academic Press, London, New York 1974
• S. Lang, Introduction to Algebraic and Abelian Functions. Second edition, Springer-Verlag, New York, Heidelberg, Berlin 1987
• G. Lion, M. Vergne, The Weil Representation, Maslov Index and Theta Series. Birkhäuser Verlag, Boston, Basel, Stuttgart 1980
• A. Merkurjev, On the norm residue symbol of degree 2. Sovjet Math. Doklady 24, 546-551 (1981)
• P. Morandi, Field and Galois Theory. Springer-Verlag, New York, Berlin, Heidelberg 1996
• S. Mori, Introduction to Diffusion Tensor Imaging. Elsevier, Amsterdam, Boston, Heidelberg 2007
• J. Neukirch, Class Field Theory. Springer-Verlag, Berlin, Heidelberg, New York 1986
• O.T. O’Meara, Introduction to Quadratic Forms. Springer-Verlag, Berlin, Heidelberg, NewYork 2000
• A.M. Ozorio de Almeida, Entanglement in phase space. In: Entanglement and Decoherence: Foundations and Modern Trends, A. Buchleitner, C. Viviescas, M. Tiersch, Editors, Lecture Notes in Physics 768, pp. 157-219, Springer-Verlag, Berlin, Heidelberg, New York 2010
• R.S. Pierce, Associative Algebras. Springer-Verlag, New York, Heidelberg, Berlin 1982
• C.R. Riehm, Introduction to Orthogonal, Symplectic and Unitary Representations of Finite Groups. American Mathematical Society, Providence, Rhode Island 2011
• H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel, Compact Projective Planes, With an Introduction to Octonion Geometry. Walter de Gruyter, Berlin, New York 1995
• W. Scharlau, Quadratic and Hermitian Forms. Springer-Verlag, Berlin, Heidelberg, New York 1985
• W.J. Schempp, Magnetic Resonance Imaging: Mathematical Foundations and Applications. Wiley-Liss, New York, Chichester, Weinheim 1998
• W.J. Schempp, Dynamic metaplectic spinor quantization: The projective correspondence for spectral dual pairs. Journal of Applied Mathematics and Computing 59, 545-584 (2019)
• W.J. Schempp, Applications of metaplectic cohomology and global-local contact holonomy. Journal of Applied Mathematics and Computing 67 (2), 2021
• E.O. Stejskal, Use of spin echoes in a pulsed magnetic-field gradient to study non-isotropic, restricted diffusion and flow. J. Chem. Phys. 43, 3597-3603 (1965)
• E.O. Stejskal, J.E. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient. J. Chem. Phys. 42, 288-292 (1965)
• A. Weil, Basic Number Theory. Third edition, Springer-Verlag, Berlin, Heidelberg, New York 1974
• A. Weil, Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964); Collected Papers, Vol. III, pp. 1-69, Springer-Verlag, New York, Heidelberg, Berlin 1979
• A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math. 113, 1-87 (1965); Collected Papers, Vol. III, pp. 71-157, Springer-Verlag, New York, Heidelberg, Berlin 1979