Abstract :
In the present paper, we derive some relations in the framework of Nottaleʹs theory of scale-relativity. To establish them, we use the formal tool of this theory, namely a complex derivative operator associated to a complex velocity. First, Section 1, we connect it to the quantum derivative and to the canonical commutation relations. We point out the possibility to recover some differential relations given by the correspondence principle (Sections 2–4). Then, we compare the complex equations that we obtained with their well-known classical counterparts. In particular, we determine the expression of energy in the theory of scale-relativity (Sections 2 and 5). The original result, with respect to the ‘scale-covariance’ statements (Def. 1 3), is the following: energy is not a pure quadratic expression in terms of velocities, as in classical mechanics; it contains also a term of velocity divergence, which is present intrinsically in its expression. Then, we infer the Dirac and Pauli equations from our energy formulas, using the matrix complex velocities introduced in Section 8. Previously, we generalised our formulas for a massless particle (Section 6), and for the Riemannian case (Section 7). Equations with electromagnetic field are considered (Sections 2.9, 3.1 and 5.5). Principally, we find that the ‘form’ of the equations of motion also changes, in spite of (Def. 1 3): they contain an additional term of current, which is absent in the classical equations. Then, we build the complex derivative operator in the momentum space. From it, we derive the Schrödinger equation in p-representation for the harmonic oscillator (Section 4). We also propose to extend the statement (Def. 3) [3] by introducing (Section 1) a notation suggested by the ‘symmetric product’ of Ikeda and Watanabe and Itô. We show that it allows us to restore scale-covariance for formulas that contain derivatives, and especially to recover the usual form of the Leibniz rule written with the derivative operator.
