Scale relativity and Quantization of the Planetary System Around the Pulsar PSR B 1257 + 12

The theory of scale relativity suggests that, at very large time scales, classical laws should be replaced by a non-deterministic, quantum-like description. The theory gives up individual, localized trajectories, but it yields the probability density of families of virtual trajectories, as deduced from a generalized Schr6dinger equation. In this paper, we compare the system of three planets recently discovered around the pulsar PSR B1257+ 12 to the predictions of the theory. We expect the planet periods to be quantized according to the formula 1",, = 2rcGMes R (4n 2 + n/2#7>o) 3 , where ~0 is a multiple of 144 km/s (as determined from the Solar System and from galactic and extragalactic data). The observed periods confirm the theory with a remarkably high precision (relative differences of some 10 4) and in a highly significant way (probability 3 × 10 5), allowing us to successfully check second and third order terms. We finally predict the periods of possible additional planets in this system, in particular T2 = 1.958 days.