Lagrangian analysis and quantum mechanics: a mathematical structure related to asymptotic expansions and the Maslov index
This work might have been entitled The Introduction of Planck's Constant into Mathematics, in that it introduces quantum conditions in a purely mathematical way in order to remove the singularities that arise in obtaining approximations to solutions of complex differential equations. The book's first chapter develops the necessary mathematical apparatus: Fourier transforms, metaplectic and symplectic groups, the Maslov index, and lagrangian varieties. The second chapter orders Maslov's conceptions in a manner that avoids contraditions and creates step by step an essentially new structure-the lagrangian ayalysis. Unexpectedly and strangely the last step requires the datum of a constant, which in applications to quantum mechanics is identified with Planck's constant. The final two chapters apply lagrangian analysis directly to the Schrodinger, the Klein-Gordon, and the Dirac equations. Magnetic field effects and even the Paschen-Back effect are taken into account. Jean Leray-who has been professor at the College de France for the past thirty years-has made fundamental contributions to theoretical hydrodynamics, to the study of elliptic, hyperbolic, and analytic linear and nonlinear equations, and to algebraic topology and its applications to analysis. His motivations always had their origin in physical problems, except during World War II: As a prisoner of war in Germany for five years, he concealed his interest in mathematical applications by making fruitful investigations in the field of pure algebraic topology.
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The Fourier Transform and Symplectic Group
Differential Operators with Polynomial Coefficients
Indices of Inertia
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adjoint affine functions apparent contour arctan Assume assumption asymptotic automorphism characteristic system characteristic vector choose co2 a co3 coefficients commute compact lagrangian manifold const coordinates covering space defined mod defined mod(1/v definition denoted differential operators Dirac equation electron elements energy levels equivalent exists expression following theorem formal function formal number formula frame given hamiltonian hence hermitian structure homeomorphism hypersurface implies independent infinitely differentiable integers invariant measure involution isomorphism Klein-Gordon equation lagrangian amplitude lagrangian functions lagrangian operators associated lagrangian solution lagrangian system lemma m(SA mapping Maslov index Maslov's quantum condition matrix morphism neighborhood Notation partition of unity pfaffian Poisson bracket polynomial Proof prove q-oriented quadratic form relations Remark scalar product Schrodinger and Klein-Gordon Sf(X solutions of problem subgroup subset Supp symplectic tangent theorem 7.1 torus transforms transverse unique universal covering space values vanishing phase zero