Covariance in General Relativity and Scale-Covariance in Scale-Relativity Theory, Quadratic Invariants and Leibniz Rule

We recall the principle of general covariance which allows us, in general relativity, to deduce laws of physics which hold in the presence of gravitation from laws which hold without gravitation, and which is implemented by the minimal prescription (ημν,∂α)→ (gμν, α). Then, we examine the formal prescription of scale-relativity theory from which we should be able to deduce laws which hold at quantum scales from laws which hold at classical scales. This prescription requires us to replace, in classical equations, the total time derivatives d/dt by a scale-covariant derivative d/dt. We show that such a prescription has actually to be extended. Indeed, it does not allow us to obtain a whole set of complex scale-covariant equations, which yield, in all cases, the right corresponding quantum equations. We exhibit many basic cases for which the substitution (ημν, ∂α)→ (gμν, α) is efficient in general relativity, while the prescription d/dt→d/dt does not lead to the right equations in scale-relativity. These cases concern the Hamilton–Jacobi equation, the form of energy — more generally the quadratic invariants — and the electromagnetic case. Indeed, we find that the usual quadratic form of nonrelativistic and relativistic invariants does not hold in this framework and that a divergence term appears in addition to the quadratic term. Moreover, we find that a current term is present with the Lorentz force in the equations of motion with an electromagnetic field. Finally, we point out that the operator d/dt does not fulfil the Leibniz rule. We show that this fact may be related to the canonical commutation relations in quantum mechanics. Moreover, we show how it would be possible to connect, in a systematic way, the use of this operator with many relations of quantum mechanics, in particular, those where differential relations appear to be given by the correspondence principle. Finally, we show that we can recover the usual form of many equations — especially the fundamental Leibniz rule — by extending the above-mentioned prescription by introducing a symmetric product .