Theory of Dirac Equation without Negative Energies

摘要

It is shown that the well-known Hermitean operator ’sign of frequency’ for the freeDirac equation has the physical meaning of ’sign of charge’. Since the kinetic energy of a freeparticle should not depend on its charge state, this identification requires a modification of thetraditional quantum mechanical 4-momentum operators when used with Dirac spinors. Due tothe new 4-momentum operators the Dirac equation has no negative energy solutions and thecomplex of problems associated with the latter disappears from the theory. The quantumnumber ’sign of charge’ rigorously defines electronic and positronic plane waves. Secondquantization of the free Dirac equation does not need the traditional amendments requiredby the negative energy values. As an example for the application of the theory the relativistichydrogen ground state wave function is analyzed with respect to the quantum number ’sign ofcharge’. Since the operator ’sign of charge’ does not commute with the Coulomb potential thewave function is only an approximate eigenfunction of the operator ’sign of charge’. It is shownhow one can construct ’effective potentials’ that commute with the operator ’sign of charge’and thus are able to produce eigenfunctions of charge when used in the Dirac equation.