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Generalized Relativistic Hamiltonian in an Alpha Field

An alpha field is a potential field that can be presented by two dimensionless field parameters α and α ′ . These parameters should satisfy the field equations of the related potential field in each concrete case. The problem is to derive the generalized relativistic Hamiltonian form that can be applied to an alpha field. In the derivation of the generalized relativistic Hamiltonian the new General Lorentz Transformation model (GLTα – model, presented in CASYS'99 ) is employed. Starting with variation principle, a linear Dirac's like relativistic Hamiltonian has been derived as the function of the field parameters α and α ′ . It is also shown that the same Hamiltonian can be derived starting with the generalized line element ds, since ds is a fundamental invariant of the four dimensional space-time continuum (IJCAS-2001, vol.10). The obtained form of the Hamiltonian is equal to the general covariant energy equation that can bee derived by employing the null component of the covariant four-momentum vector in an alpha field. The obtained result gives the possibility to compare the coefficients of the well known Dirac’ s Hamiltonian with the related coefficients of the linear Dirac's like relativistic Hamiltonian as the function of the field parameters α and α ′ . This comparison illuminates the new properties of the coefficients of the Dirac’ s Hamiltonian and helps to understand their physical sense. The nonlinear relativistic Hamiltonian for an alpha field has been derived starting with the linear one and using quadratic operation. This form of the Hamiltonian is a function of the relativistic invariant term α α ′ and belongs to the usual structure of the relativistic Hamiltonian. The main shortage of this form of the Hamiltonian is the fact that this form is a nonlinear function of the extended momentum. Usually, one can introduce the nonrelativistic approximation of the nonlinear relativistic Hamiltonian. It also has been done in this paper. The explanation of the occurrence of the four possible components of the relativistic Hamiltonian in an alpha field, like in the Dirac’ s theory, has also been presented and discussed. Including the solutions of the field parameters α and α ′ in the electromagnetic and gravitational fields, we obtain the related Hamiltonians for an electron in the electromagnetic field and for a particle in the gravitational field. The obtained Hamiltonians can also be applied for the case where two or more potential fields are present at the same time. This is about the Hamiltonian of the multipotential field, of course.

Dimensionless Field Parameters; Relativistic Hamiltonian; Linear Dirac's like Hamiltonian; Nonlinear Hamiltonian; Hamiltonians Comparison.

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