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Astron. Astrophys. 337, 299-310 (1998)

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2. Quantum probability and complex wavefunctions

Nottale'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 à-la-Born/Schroedinger (Nottale, 1993). This link between nondifferentiable particle trajectories (explicitely involving, or not, fractal space-time) and the Schroedinger equation, first pioneered by Feynman (Feynman & Hibbs, 1965), and shortly after followed by (amongst others) Nelson (1966), Abbot & Wise (1981), Ord (1983), Nottale (1989), Sornette (1990), is indeed a crucial progress in the conceptual frame of the Schroedinger equation, for it opens the possibility of applying the basic quantum rules, and amongst them the orbit and/or energy quantization formulas, to dynamical systems, the dimensions, energy and action of which lie far from those of the microscopical world. Schroedinger's equation is no longer restricted to atoms. It may concern much larger dynamical systems, such as astronomical systems in the present case.

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 equation

The 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 [FORMULA] is Nottale's geodesics density, while µ is the (reduced, in the case of a central-field problem) particle mass. Making use of the continuity equation Eq. (1) in order to confirm the identification of [FORMULA] with a probability density, as Nottale does, is indeed very attractive and has been proposed by numerous authors (see Holland, 1993 for an extensive review). The pioneering works originate from Bohm et al.'s classical (causal) statistical interpretation of QM that has the dual sense of the probability distribution of the initial conditions characterizing an ensemble of particle trajectories in the configuration space, as well as of a real `quantum-potential' field present in a single experiment (Bohm, 1952a,b, 1953; Bohm & Vigier, 1954; DeBroglie, 1956, Holland, 1993). However this statistical interpretation was criticized by Keller (1953) and Pauli (1953) amongst others, who showed that Eq. (1) is indeed a necessary condition for [FORMULA] to be a quantum probability density of the classical statistical type, but not a sufficient one.

2.2. Divergence of [FORMULA] for stationary states

There is another serious difficulty related to the identification of [FORMULA] with a probability density, and it is the following. In the case of a stationary and separable quantum state, the quantity [FORMULA] where [FORMULA] is defined by Eq. (2) and [FORMULA] is not integrable. It diverges to infinity at the boundaries of the system (Reinisch, 1994; 1997). This divergence emphasizes the ambiguity of the probabilistic Born postulate of QM which has given rise to many of the conceptual difficulties associated with the subject (Rae, 1992). Indeed, while on the one hand this postulate yields as a general statement that [FORMULA] is the density of probability to find the particle at a given position (Born, 1926), on the other hand, the Born probability density is rather [FORMULA] than [FORMULA] in the case of stationary and separable quantum systems that ultimately reduce to 1-dimensional Schroedinger eigenproblems, precisely for the sake of normalization of the probability density (Reinisch, 1994; 1997).

Tackling this problem actually amounts to recovering Einstein's definition of a real understanding of quantum physics, namely that the probabilistic description of the whole quantum process should naturally (i.e. self-consistently ) emerge from the `complete' dynamical description of the system itself (Pais, 1982). In our opinion, there is no doubt that, whereas this point of view opened sophisticated debates in quantum microphysics by the endless controversy between the Copenhagen interpretation of QM and its causal alternative (Einstein & Born, 1969), it is the clue to any convincing macro-Schroedinger model of chaos. Hence the credo of our present approach: if it is macro-Schroedinger, then it should be possible to tell it in classical terms.

2.3. Schroedinger equation's irregular eigenmode

The 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 [FORMULA] of the discrete Schroedinger spectrum that is labeled by the (main) quantum number N, there exists a second linearly independent solution [FORMULA] to the same eigenvalue [FORMULA], which is, however, non-normalizable, or `irregular' (Richter & Wuensche, 1996). We show that this irregular solution [FORMULA] (which is divergent to infinity at the boundaries of the system) allows us to give [FORMULA] a clear local classical statistical sense in terms of the time interval that is spent by the system between two spheres of radius r and [FORMULA], although the quantity [FORMULA] obviously diverges at the boundaries of the system since [FORMULA]. Note that such a simple classical statistical meaning of [FORMULA] has been suggested by White (1931; 1934) in the early thirties.

2.4. Schroedinger equation's Hamilton-Jacobi dynamics

Therefore the present theory makes an extensive use of this irregular Schroedinger solution [FORMULA] 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 [FORMULA]. This yields the phase shift [FORMULA] that is related to the actual momentum field [FORMULA] by use of the classical Hamilton-Jacobi equation. Therefore, to each value [FORMULA] in the configuration space that parametrizes the `trajectory' of the system in the [FORMULA] plane, there exists the polar angle [FORMULA] that unambiguously defines the actual momentum field [FORMULA]. And the classical (i.e. causal) dynamics that is derived from this momentum field allows us to give [FORMULA] its local classical statistical meaning which, thus, extrapolates Born's postulate to the case where [FORMULA] is not integrable.

2.5. Schroedinger equation's new parameter

The constant Wronskian A of the fundamental set of solutions [FORMULA] appears to be the fundamental new parameter of the present theory. It is independent of the energy eigenvalue [FORMULA] and its choice, as a mere constant of integration, is free. Its actual physical dimension is a matter flux. It has been shown (Reinisch, 1994; 1997) that standard microphysics amounts to taking [FORMULA]. On the other hand, the present paper shows that the value [FORMULA] (in appropriate reduced units) corresponds to the macro-Schroedinger CM.

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 all these stationary hydrodynamical theories and the present one. In the case of the Kepler problem, for instance, Madelung et al. assume [FORMULA] (Holland, 1993). Then [FORMULA] and there is no irregular mode in the corresponding Madelung description. Therefore this latter trivially degenerates into the standard Schroedinger eigenproblem related to the single remaining regular eigenfunction [FORMULA]. As a consequence, the classical dynamical information that is provided by the gradient of the phase [FORMULA] is lost and, in order to replace it, one is then forced to artificially introduce Born's postulate like a sort of deus ex machina (Reinisch, 1994). On the other hand, when the Wronskian A is non-zero, it can be shown that the Born postulate can indeed be recovered by the purely classical interpretation of the steady-state matter-flow dynamics that is related to the momentum field [FORMULA] (Reinisch, 1997). Therefore the Born postulate is actually contained in the non-degenerated ([FORMULA]) Madelung system.

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 [FORMULA] and action S) in the complex description (Eq. (2)) of the wave function [FORMULA], the latter simply assumes [FORMULA] and S complex. The interest of such a formal complex action is its gradient. It yields a complex momentum field which, in turn, defines the complex covariant derivative operator associated with the scale covariance postulate (Nottale, 1997; Pissondes, 1997).

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Online publication: August 6, 1998