Dirac and Klein-Gordon Equations in Curved Space

Source: arXiv


We introduce matrix operator algebra involving a universal curvature
constant. Using elements of the algebra, we write the Dirac equation without
the need for spin connections or vierbeins. Iterating the equation and using
the algebra leads to the Klein-Gordon equation in curved space in its canonical
from (without first order derivatives). We obtain exact solutions of the Dirac
and Klein-Gordon equations for a static diagonal metric.

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    • "That said, there are several curved spacetime versions of the Dirac equation (cf. [9] [10] [12]-[17]). In our modest view; save for the introduction of a seemingly mysterious four vector potential A µ , what makes the curved spacetime version of the Dirac equations presented in the reading [18] stands-out over other attempts in that the method used in arriving at these curved spacetime Dirac equations [18] is exactly the same as that used by Professor [19] [20]. "
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    ABSTRACT: The present reading is part of our on-going attempt at the foremost endeavour of physics since man began to comprehend the heavens and the earth. We present a much more improved unified field theory of all the forces of Nature i.e. the gravitational, the electromagnetic, the weak and the strong nuclear forces. The proposed theory is a radical improvement of Professor Hermann Weyl [1– 3]'s supposed failed attempt at a unified theory of gravitation and electromagnetism. As is the case with Professor Weyl's theory, unit vectors in the resulting/proposed theory vary from one point to the next, albeit, in a manner such that they are compelled to yield tensorial affinities. In a separate reading [4], the Dirac equation is shown to emerge as part of the description of the these variable unit vectors. The nuclear force fields – i.e., electromagnetic, weak and the strong – together with the gravitational force field are seen to be described by a four vector field Aµ, which forms part of the body of the variable unit vectors and hence the metric of spacetime. The resulting theory very strongly appears to be a logically consistent and coherent unification of classical and quantum physics and at the same time a grand unity of all the forces of Nature. Unlike most unification theories, the present proposal is unique in that it achieves unification on a four dimensional continuum of spacetime without the need for extra-dimensions.Imagination will often carry us to worlds that never were. But without it, we go nowhere[5]." – Carl Edward Sagan (1934 − 1996) INTRODUCTION
    Full-text · Article · Jul 2014 · Journal of Modern Physics
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    ABSTRACT: Brane model of universe is considered for a free particle. Conservation laws on the brane are obtained using the symmetry properties of the brane. Equation of motion is derived for a particle using variation principle from these conservation laws. This equation has a form of Klein-Gordon equation. Comparison of squared Dirac-Fok-Ivanenko equation for a spin particle with Klein-Gordon equation in curved space has given an expression for chiral spin current variation through the derivative of spin connectivity. This chiral spin current is anomalous spin current corresponding to spontaneous chiral symmetry breaking of massive particle in the space of KG equation solutions.
    Full-text · Article · Jul 2013