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-This paper describes the present state of an attempt at understanding the quantum behaviour of microphysics in terms of a nondifferentiable space-time continuum having fractal (i.e., scale-dependent) properties. The fundamental principle upon which we rely is that of scale relativity, which generalizes to scale transformations Einstein's principle of relativity. After having related the fractal and renormalization group approaches, we develop a new version of stochastic quantum mechanics, in which the correspondence principle and the Schrödinger equation are demonstrated by replacing the classical time derivative by a "quantumcovariant" derivative. Then we recall that the principle of scale relativity leads one to generalize the standard "Galilean" laws of scale transformation into a Lorentzian form, in which the Planck length-scale becomes invariant under dilations, and so plays for scale laws the same role as played by the velocity of light for motion laws. We conclude by an application of our new framework to the problem of the mass spectrum of elementary particles.

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The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrödinger and Klein-Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrödinger and Klein-Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds ↔ −ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx µ ↔ −dx µ ) and by requiring invariance under parity and time inversion. The Pauli equation is recovered as a nonrelativistic-motion approximation of the Dirac equation.