Derivation of the Dirac Equation by Conformal Differential Geometry

Foundations of Physics 43 (5):631-641 (2013)

Francesco Martini
Università degli Studi di Bologna
A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation
Keywords Relativistic top  Quantum spin  Conformal geometry
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DOI 10.1007/s10701-013-9703-y
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The Nature of Space and Time.Stephen Hawking & Roger Penrose - 1996 - Princeton University Press.
Quantum Gravity.Carlo Rovelli - 2007 - Cambridge University Press.
Space-Time-Matter.Hermann Weyl - 1922 - E.P. Dutton and Company.

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Citations of this work BETA

Proof of the Spin–Statistics Theorem.Enrico Santamato & Francesco De Martini - 2015 - Foundations of Physics 45 (7):858-873.
Formulation of Spinors in Terms of Gauge Fields.S. R. Vatsya - 2015 - Foundations of Physics 45 (2):142-157.

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