Applications of quaternions to field equations
Author(s)
Gursey, Feza
Type
Thesis
Abstract
The object of the thesis is to show that the quaternion algebra can be applied successfully to the discussion and solution of Dirac’s equation and that it can be extended to general relativity to give an invariant formulation of field equations.
In the first three chapters which deal with the quaternion formulation of the wave equation, a connection is established with the matrix formalism of Dirac on one hand and with different quaternion formulations on the other, Attention is drawn to two space-like vectors, which form an orthogonal frame of reference together with the current and spin density four—vectors, and new divergence equations and algebraic identities satisfied by tensorial densities involving the charge conjugate function are derived. A generalization of the wave equation by means of a space-like four-vector is given. In the two chapters that follow a new method for solving the wave equation is developed and exact solutions are obtained in several problems in a rapid and concise manner. The formalism is also shown to be useful in the non-relativistic approximation. In the remaining chapters wave equations for particle* of spin 1/2 and 1 are put in a quaternion form invariant for general coordinate transformations and valid in the affine space-time of distant parallelism. By means of the new concept of covariant quaternions these invariant equations way be also expressed in covariant form. Application to Dirac's equation leads to new tensor form for this equation and throws some light on its connection with general relativity. A new form for Dirac's matrix equation. In a general coordinate system is found which only involves the torsion tensor associated with the basis vectors that have been chosen.
In the first three chapters which deal with the quaternion formulation of the wave equation, a connection is established with the matrix formalism of Dirac on one hand and with different quaternion formulations on the other, Attention is drawn to two space-like vectors, which form an orthogonal frame of reference together with the current and spin density four—vectors, and new divergence equations and algebraic identities satisfied by tensorial densities involving the charge conjugate function are derived. A generalization of the wave equation by means of a space-like four-vector is given. In the two chapters that follow a new method for solving the wave equation is developed and exact solutions are obtained in several problems in a rapid and concise manner. The formalism is also shown to be useful in the non-relativistic approximation. In the remaining chapters wave equations for particle* of spin 1/2 and 1 are put in a quaternion form invariant for general coordinate transformations and valid in the affine space-time of distant parallelism. By means of the new concept of covariant quaternions these invariant equations way be also expressed in covariant form. Application to Dirac's equation leads to new tensor form for this equation and throws some light on its connection with general relativity. A new form for Dirac's matrix equation. In a general coordinate system is found which only involves the torsion tensor associated with the basis vectors that have been chosen.
Version
Open Access
Date Awarded
1950-07
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Jones, H.
Sponsor
Turkish Ministry of Education
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)