Hierarchy of symplectic forms for the Schr�dinger and the Dirac equations on a line

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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The Dirac Electron Consistent with Proper Gravitational and Electromagnetic Field of the Kerr–Newman Solution

Galaxies ◽

10.3390/galaxies9010018 ◽

2021 ◽

Vol 9 (1) ◽

pp. 18

Author(s):

Alexander Burinskii

Keyword(s):

Electromagnetic Field ◽

Quantum Theory ◽

Gravitational Field ◽

Dirac Equation ◽

Higgs Field ◽

Vacuum State ◽

Relativistic String ◽

Dirac Equations ◽

Dirac Electron ◽

Relativistic Scattering

The Dirac electron is considered as a particle-like solution consistent with its own Kerr–Newman (KN) gravitational field. In our previous works we considered the regularized by López KN solution as a bag-like soliton model formed from the Higgs field in a supersymmetric vacuum state. This bag takes the shape of a thin superconducting disk coupled with circular string placed along its perimeter. Using the unique features of the Kerr–Schild coordinate system, which linearizes Dirac equation in KN space, we obtain the solution of the Dirac equations consistent with the KN gravitational and electromagnetic field, and show that the corresponding solution takes the form of a massless relativistic string. Obvious parallelism with Heisenberg and Schrödinger pictures of quantum theory explains remarkable features of the electron in its interaction with gravity and in the relativistic scattering processes.

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Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽

10.3390/sym13081373 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1373

Author(s):

Louis H. Kauffman

Keyword(s):

Dirac Equation ◽

Schrödinger Equation ◽

Schrodinger Equation ◽

Clifford Algebras ◽

Discrete Systems ◽

Complex Numbers ◽

Majorana Fermions ◽

Matrix Algebras ◽

Dirac Equations ◽

Points Of View

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

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QUANTUM SINGULARITIES IN STATIC SPACETIMES

International Journal of Modern Physics D ◽

10.1142/s0218271811019062 ◽

2011 ◽

Vol 20 (05) ◽

pp. 729-743 ◽

Cited By ~ 17

Author(s):

JOÃO PAULO M. PITELLI ◽

PATRICIO S. LETELIER

Keyword(s):

Quantum Mechanics ◽

Wave Operator ◽

Point Of View ◽

Quantum Mechanical ◽

Mathematical Framework ◽

Physical Content ◽

Dirac Equations ◽

Quantum Particles ◽

Klein Gordon ◽

Quantum Singularities

We review the mathematical framework necessary to understand the physical content of quantum singularities in static spacetimes. We present many examples of classical singular spacetimes and study their singularities by using wave packets satisfying Klein–Gordon and Dirac equations. We show that in many cases the classical singularities are excluded when tested by quantum particles but unfortunately there are other cases where the singularities remain from the quantum mechanical point of view. When it is possible we also find, for spacetimes where quantum mechanics does not exclude the singularities, the boundary conditions necessary to turn the spatial portion of the wave operator to be self-adjoint and emphasize their importance to the interpretation of quantum singularities.

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Solutions of super linear Dirac equations with general potentials

Differential Equations and Dynamical Systems ◽

10.1007/s12591-009-0018-6 ◽

2009 ◽

Vol 17 (3) ◽

pp. 235-256 ◽

Cited By ~ 2

Author(s):

Jian Ding ◽

Junxiang Xu ◽

Fubao Zhang

Keyword(s):

Dirac Equations

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TOPOLOGICAL INSULATOR AND THE DIRAC EQUATION

SPIN ◽

10.1142/s2010324711000057 ◽

2011 ◽

Vol 01 (01) ◽

pp. 33-44 ◽

Cited By ~ 71

Author(s):

SHUN-QING SHEN ◽

WEN-YU SHAN ◽

HAI-ZHOU LU

Keyword(s):

Dirac Equation ◽

Topological Insulator ◽

Bound States ◽

Topological Insulators ◽

Mass Term ◽

Point Of View ◽

Quadratic Term ◽

General Description ◽

Dirac Equations ◽

Topological Quantum

We present a general description of topological insulators from the point of view of Dirac equations. The Z2 index for the Dirac equation is always zero, and thus the Dirac equation is topologically trivial. After the quadratic term in momentum is introduced to correct the mass term m or the band gap of the Dirac equation, i.e., m → m − Bp2, the Z2 index is modified as 1 for mB > 0 and 0 for mB < 0. For a fixed B there exists a topological quantum phase transition from a topologically trivial system to a nontrivial system when the sign of mass m changes. A series of solutions near the boundary in the modified Dirac equation is obtained, which is characteristic of topological insulator. From the solutions of the bound states and the Z2 index we establish a relation between the Dirac equation and topological insulators.

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