## 2. Quantum probability and complex wavefunctionsNottale's major statement is that the CPSS trajectories become
nondifferentiable (and hence fractal) after their very short inverse
Lyapounov exponent of about 5 Myr, thus forcing one to jump to a
probabilistic description Note that, recently, this link between the Schroedinger equation and fractal trajectories has been strengthened, although the two were not equated (Hermann, 1997). ## 2.1. Continuity equationThe question we wish to address in the present paper is the following: once we have the CPSS Schroedinger equation, what does it mean? How does the key-concept of quantum probability enter macroscopic quantization? Nottale's Scale-Relativity theory proposes that matter concentrates where the infinite set of geodesics of the fractal space-time get denser. Quantitatively, he relates this statement to the so-called Madelung-Bohm-DeBroglie equation of continuity (Madelung, 1926; Bohm, 1952a,b, 1953; DeBroglie, 1956; Nelson, 1966; Nottale, 1993, 1996a,b, 1997) which is but the imaginary part of the Schroedinger equation: once its wavefunction solution is assumed complex-valued: The quantity is Nottale's geodesics density,
while ## 2.2. Divergence of for stationary statesThere is another serious difficulty related to the identification
of with a probability density, and it is the
following. In the case of a stationary and separable quantum state,
the quantity where is
defined by Eq. (2) and is Tackling this problem actually amounts to recovering Einstein's
definition of a ## 2.3. Schroedinger equation's irregular eigenmodeThe goal of the present paper is to tentatively resolve the
conflict between the Schroedinger equation and CM on the scale of the
Solar System. The argument is that the Schroedinger spectrum is also
the spectrum of the nonlinear Madelung system that is obtained from
the Schroedinger equation by choosing an ansatz solution of the type
Eq. (2) and then separating the real and imaginary parts (Madelung,
1926). The corresponding system of two coupled Ordinary Differential
Equations (ODE's) yields a single nonlinear second-order ODE, known as
the Ermakov-Milne-Pinney (EMP) equation (Ermakov, 1880; Milne, 1930;
Pinney, 1950), in the stationary and separable case (Reinisch, 1994;
1997). This EMP equation is related to the linear second-order
Schroedinger eigenproblem by a nonlinear superposition formula making
use of the fundamental set of solutions of Schroedinger's equation
(Milne, 1930; Common & Musette, 1997). Indeed, to each normalized
(regular) eigenfunction of the discrete
Schroedinger spectrum that is labeled by the (main) quantum number
## 2.4. Schroedinger equation's Hamilton-Jacobi dynamicsTherefore the present theory makes an extensive use of this irregular Schroedinger solution that is completely discarded in the `standard' (i.e. Schroedinger-Born) interpretation of QM. However there is a strong tendency in the very recent literature to re-introduce these irregular Schroedinger solutions in the investigation of the properties of the discrete (Richter & Wuensche, 1996; Leonhardt & Raymer, 1996) as well as of the continuum (Leonhardt & Schneider, 1997) energy spectrum. The technical interest of taking into account the fundamental set of solutions to the Schroedinger equation is that one can build the general wavefunction solution of the type Eq. (2) as . This yields the phase shift that is related to the actual momentum field by use of the classical Hamilton-Jacobi equation. Therefore, to each value in the configuration space that parametrizes the `trajectory' of the system in the plane, there exists the polar angle that unambiguously defines the actual momentum field . And the classical (i.e. causal) dynamics that is derived from this momentum field allows us to give its local classical statistical meaning which, thus, extrapolates Born's postulate to the case where is not integrable. ## 2.5. Schroedinger equation's new parameterThe constant Wronskian One might wonder whether the present theory is not a mere remake of
the several Madelung-Bohm-DeBroglie hydrodynamical pictures of QM (see
Holland, 1993, for an extensive account of these theories). The answer
is clearly negative, for there is a crucial difference between
Let us stress as a final remark that the Madelung-DeBroglie-Bohm
"causal" description which is used in the present paper is rigorously
equivalent to the complex Hamilton-Jacobi equation that is provided by
Nottale's Scale Relativity theory, although their conceptual
foundations are quite different. Indeed, while the former makes use of
real quantities (amplitude and action © European Southern Observatory (ESO) 1998 Online publication: August 6, 1998 |