Dirac Equation in Scale Relativity

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 ($dx^{\mu} \leftrightarrow - dx^{\mu}$) and the parity and time reversal symmetries.Comment: 33 pages, 4 figures, latex. Submitted to Phys. Rev.

Patches in a timeline with ossia score

Handling of time and scores in patchers such as PureData, Max/MSP has been an ongoing concern for composers and users of such software. We introduce an integration of PureData inside the ossia score interactive and intermedia sequencer, based on libpd. This integration allows to score precisely event that are being sent to a PureData patch, and process the result of the patch’s computations afterwards in score. This paper describes the way this integration has been achieved, and how it enables composers to easily add a temporal dimension to a set of patches, by leveraging both the computational power of PureData and the temporal semantics of the ossia system, in order to create complex compositions

Patches in a timeline with ossia score

Handling of time and scores in patchers such as PureData, Max/MSP has been an ongoing concern for composers and users of such software. We introduce an integration of PureData inside the ossia score interactive and intermedia sequencer, based on libpd. This integration allows to score precisely event that are being sent to a PureData patch, and process the result of the patch's computations afterwards in score. This paper describes the way this integration has been achieved, and how it enables composers to easily add a temporal dimension to a set of patches, by leveraging both the computational power of PureData and the temporal semantics of the ossia system, in order to create complex compositions

A scale-relativistic derivation of the Dirac Equation

The application of the theory of scale relativity to microphysics aims at recovering quantum mechanics as a new non-classical mechanics on a non-derivable space-time. This program was already achieved as regards the Schr\"odinger and Klein Gordon equations, which have been derived in terms of geodesic equations in this framework: namely, they have been written according to a generalized equivalence/strong covariance principle in the form of free motion equations $D^2x/ds^2=0$, where $D/ds$ are covariant derivatives built from the description of the fractal/non-derivable geometry. Following the same line of thought and using the mathematical tool of Hamilton's bi-quaternions, we propose here a derivation of the Dirac equation also from a geodesic equation (while it is still merely postulated in standard quantum physics). The complex nature of the wave function in the Schr\"odinger and Klein-Gordon equations was deduced from the necessity to introduce, because of the non-derivability, a discrete symmetry breaking on the proper time differential element. By extension, the bi-quaternionic nature of the Dirac bi-spinors arises here from further discrete symmetry breakings on the space-time variables, which also proceed from non-derivability.Comment: 13 pages, accepted for publication in Electromagnetic Phenomena, Special issue dedicated to the 75th anniversary of the discovery of the Dirac equatio