DUALITY OF TIME:

Complex-Time Geometry and Perpetual Creation of Space

by Mohamed Haj Yousef

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3.5.2  Dirac Wave Equation

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In 1928, Paul Dirac extended the Pauli equation, which described spinning electrons, and he made it relativistic in order to dealt properly with the high speed at which an electron orbits the nucleus. By using the simplest electromagnetic interaction, Dirac was able to predict the value of the spin magnetic moment of the electron, in good agreement with the experimentally observed value, which was too large if calculated by classical physics. Subsequently, Dirac was able to solve for the spectral lines of the hydrogen atom, and apply physical first principles to produce Sommerfeld’s successful formula for the fine structure of the hydrogen spectrum.

Dirac equation is a relativistic wave equation describing electromagnetic interactions of all spin-half massive particles, such as electrons and quarks, for which parity is a symmetry. This equation was the first theory to account fully for Special Relativity in the context of Quantum Mechanics.

Furthermore, based on his new equation which sometimes yielded a negative value for energy, Dirac posited the existence of an anti-electron, or the positron. This subsequently led to the many-particle Quantum Field Theory. The existence antimatter was experimentally confirmed after several years. In 1933, Dirac shared the Nobel Prize with Schroedinger, “for the discovery of new productive forms of atomic theory”.

In contrast to the Schroedinger equation, which described wave-functions of only one complex value, the Dirac wave-function are vectors of four complex numbers, known as bispinors, two of which resemble the Pauli wave-function in the non-relativistic limit. In the limit of zero mass, Dirac equation also reduces to the Weyl equation, which describes massless spin-half particles called Weyl fermions.

The Dirac equation in the form originally proposed by Dirac is:

[Sorry. Ignored egin{multline} ... end{multline}]

Here:is the wave-function for the electron of rest masswith space-time coordinates, and the,,are the components of the momentum, understood to be the momentum operator in the Schroedinger equation. Also,is the speed of light, andis the reduced Planck constant.

The new elements in this equation are thematricesand, and the four-component wave-function, because the evaluation of it at any given point in configuration space is a bispinor; it is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron.

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