Dirac equation in scale relativity

Dec, 2001
33 pages
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Abstract:
The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows to recover quantum mechanics as mechanics on a non-differentiable (fractal) space-time. The Schr\odinger and Klein-Gordon equations have already been demonstrated as geodesic equations in this framework. We propose here a new development of the intrinsic properties of this theory to obtain, using the mathematical tool of Hamilton's bi-quaternions, a derivation of the Dirac equation, which, in standard physics, is merely postulated. The bi-quaternionic nature of the Dirac spinor is obtained by adding to the differential (proper) time symmetry breaking, which yields the complex form of the wave-function in the Schr\odinger and Klein-Gordon equations, the breaking of further symmetries, namely, the differential coordinate symmetry (dxμdxμdx^{\mu} \leftrightarrow - dx^{\mu}) and the parity and time reversal symmetries.
Note:
  • Submitted to Phys.Rev.D
  • Dirac equation
  • relativity theory
  • quaternion
  • symmetry breaking
  • Schroedinger equation
  • Klein-Gordon equation
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