4. Scale-covariant equations of physics
4.1. Generalized Newton-Schrödinger equation: particle in scalar field
4.1.1. Lagrangian approach
Let us now specialize our study, and consider Newtonian mechanics, i.e., the general case when the structuring field is a scalar field. The Lagrange function of a closed system, , is generalized as , where denotes a scalar potential. The Euler-Lagrange equations keep the form of Newton's fundamental equation of dynamics
which is now written in terms of complex variables and complex operators. In the case when there is no external field, the scale-covariance is explicit, since Eq. (42) takes the form of the equation of inertial motion, . The complex momentum now reads:
so that from Eq. (41) we arrive at the conclusion that, in this case, the complex velocity is a gradient, namely the gradient of the complex action:
We may now introduce a complex wave function which is nothing but another expression for the complex action ,
It is related to the complex velocity as follows:
which is the correspondence principle of quantum mechanics for momentum and energy, but here demonstrated and written in terms of exact equations. We have now at our disposal all the mathematical tools needed to write the fundamental equation of dynamics (Eq. (42)) in terms of the new quantity . It takes the form
Standard calculations with differential operators (Nottale 1993a) transform this expression to:
up to an arbitrary phase factor which may be set to zero by a suitable choice of the phase of . In the very particular case when is inversely proportional to mass, , we recover the standard form of Schrödinger's equation:
and this theory (assuming complete nondifferentiability) yields quantum mechanics (Nottale 1993a).
It is remarkable that, in this approach, we have obtained the Schrödinger equation without introducing a probability density (since expectations are taken on the beam of virtual geodesics) and without writing any Kolmogorov nor Fokker-Planck equation. In this regard our theory differs profoundly from Nelson's (1966, 1984) stochastic mechanics, in which one works with a real Newton equation and with real backward and forward Fokker-Planck equations; these equations are combined to yield two real equations, which are finally identified with the real part and the imaginary part of the complex Schrödinger equation. In our theory, we use only one complex equation of dynamics from the beginning of our calculation; as a consequence, the real and imaginary parts of our Schrödinger equation is not a pasting of two real equations, but instead involve combinations of terms through the complex product, so that obtaining in this way a Schrödinger equation was not a priori evident.
The statistical meaning of the wave function (Born postulate) can now be deduced from the very construction of the theory. Even in the case of only one particle the virtual family of geodesics is infinite (this remains true even in the zero particle case, i.e. for the vacuum field). The particle is one random geodesic of the family, and its probability to be found at a given position must be proportional to the density of the fluid of geodesics. This density can now be easily calculated from our variables, since the imaginary part of Eq. (50) writes:
where V is the real part of the complex velocity, and has already been identified with the classical velocity (at the classical limit). This equation is recognized as the equation of continuity, implying that represents the fluid density which is proportional to the density of probability, and then ensuring the validity of Born's postulate. The remarkable new feature here that allows us to obtain such a result is that the equation of continuity is not written as an additional a priori equation, but is now a part of our generalized equation of dynamics.
4.1.2. Fractal potential and Energy equation
Let us reexpress the effect of the fractal fluctuation in terms of an effective "force". We shall separate the two effects of nondifferentiability, namely, doubling of time derivative expressed in terms of complex numbers, and fractalization, expressed by the occurence of nonclassical second order terms in the total time derivative, then treat them in a different way.
We are led in the following calculation by the well-known way allowing to recover a Newtonian, force-like interpretation of the equation of geodesics in Einstein's general relativity theory. Start with the covariant form of the geodesics equations, , develop the covariant derivative and obtain , which generalizes Newton's equation, in terms of a "force" .
Once complex numbers are introduced , we write the time derivative as a partially covariant derivative:
The equation of a free particle still takes the form of Newton's equation of dynamics, but including now a right-hand member:
In our scale-relativistic, fractal-space-time approach, this "force" is assumed to come from the very structure of space-time. When applied to the microphysical domain, we can require it to be universal, independent of the mass of the particle. Then must be a universal constant:
This result provides us with a new definition of , and implies that must be the Compton length of the particle:
Once the Compton length obtained, it is easy to get the de Broglie length, that arises from it through a Lorentz transform (see Nottale 1994 for more detail).
The introduction of this potential allows us to derive the Schrödinger equation in a very fast way, by the Hamilton-Jacobi approach (see Pissondes 1996 for a more detailed development of this approach in the scale-relativistic framework). Such a derivation explains the standard quantum mechanical "derivation" via the correspondence principle. We simply write the expression for the total energy, including the fractal potential plus a possible external potential ,
then we replace and by their expressions (47) and (58). This yields (with )
which is nothing but the standard Schrödinger equation, now obtained in a direct way rather than integrated from the Lagrange equation, i.e.
4.1.3. Quantization of Newtonian gravitation
A preferential domain of application of our new framework is gravitation. Indeed, gravitation is already understood, in Einstein's theory, as the various manifestations of the geometry of space-time at classical scales. Now our proposal may be summarized by the statement that space-time is not only Riemannian but becomes also fractal at very large scales. The various manifestation of the fractal geometry of space-time could therefore be attributed to new effects of gravitation (this becomes a matter of definition).
We shall give herebelow our system of equation for the motion of particles in a Newtonian gravitational field. Paper II of the present series will be devoted to the study of some of its solutions.
As a first step toward writing a general equation of structure formation by a gravitational potential, we shall consider the special case of an "external" gravitational field that can be considered as unaffected by the evolution of the structure considered. Such a situation corresponds to a structuring field that can be considered as global with respect to the structures that it will contribute to form. Typical examples of such a case are the two-body problem, i.e., test particles in the potential of a central more massive body (e.g., planetary systems, binary systems in terms of reduced mass), and cosmology (particles embedded into a background with uniform density). For this type of problem, the equations of evolution are the classical Poisson equation and the Schrödinger-Newton equation:
Here the mass density is assumed to remain undisturbed whatever the evolution of the test-particles described in Eq. (63), so that the potential can be found from Eq. (62) and inserted in Eq. (63). Solving for these equations will yield a probability density for the test particles subjected to the potential .
Since this probability density is that of all the possible
positions of the test-particle, as described by the density of its
virtual trajectories (of which the actual trajectory is one particular
random achievement), it will be interpreted as a tendency for the
system to make structures (Nottale 1996b; Nottale, Schumacher &
Gay 1997). To get an understanding of its meaning, one should keep in
mind that the above theory holds only at very large time scales, and
that at ordinary time scales the classical theory and its predictions
must still be used. Such structures may therefore be achieved (and
observed) in several different ways.
A detailed treatment of the gravitational case, including an analysis of the main solutions to Eqs. (62)-(63), will be the subject of paper II of this series.
A more general situation can be considered, when the gravitational potential is precisely due to the particles whose evolution is looked for. In this case, the particles can no longer be considered as test-particles. When the particles have equal mass, the mass density in the Poisson equation is proportional to the probability density given by our generalized Schrödinger equation. The equations of evolution of such a system write:
(See below a still more general fluid-like approach and our quantization of the Euler and Navier-Stokes equations). This is now a "looped", highly non-linear system with feedback. While in microphysical standard quantum mechanics is constrained to be (but see Nottale 1996a for a possible special scale-relativistic generalization to high energy particles), the situation is far more complicated in the macrophysical case. To be fully general, we may also consider the case when the parameter becomes a "field", itself dependent on position, time, resolution-scale (as implied by a fractal dimension different from 2, and possibly on the local value of the probability density [see Nottale 1994, 1995, for a first treatment of the case of a variable coefficient ]. In this regard, the theory remains incomplete, since the problem of constructing the equation for this new field remains essentially open.
The field equation and the particle trajectory equation are no longer independent from each other. The gravitational potential and the probability density are now present in both equations. They can therefore be combined in terms of a unique fourth-order equation for the probability distribution of matter in the Universe, in which the potential has now disappeared:
These various systems of equations are too much complicated to be solved in general, so that only simplified situations will be considered when looking for analytical solutions, in Paper II of this series. However, the universal properties of gravitation allows one to reach a general statement about the behavior of these equations and their solutions. The always attractive character of the gravitational potential (except when considering the contribution of a cosmological constant, see Paper II) implies that it acts as a potential well, so that the energy of systems described by Eqs. (64)-(65)-(66) will always be quantized. This equation is then expected to yield definite structures in position and velocity, which are given by the probability densities constructed from its solutions. We therefore suggest that it may stand out as a general equation for the formation and evolution of gravitational structures.
4.2. Particle in vectorial field
Our theory can be tentatively generalized to the case when the structuring field is vectorial, as, e.g., in the case of an electromagnetic field (Nottale 1994b, 1996a). Once again, it is easy to make classical mechanics scale-covariant. The generalized momentum and energy of a particle in a vectorial potential A write:
The resulting equations have the form of the Schrödinger equation in presence of an electromagnetic field (of vector potential A and scalar potential ):
Such an equation may be relevant for a large set of still unsolved astrophysical problems where magnetic fields play an important role (see e.g. Zeldovich et al. 1983). We shall consider its application in subsequent papers of this series.
4.3. Particle in tensorial field: Einstein-Schrödinger geodesics equation
The application of our theory to a particle in a gravitational field plays a particular role in its development. Indeed, while Newton's description of gravitation remains in terms of field and potential, gravitation is identified in the more profound vision of Einstein with the various manifestations of the Riemannian nature of space-time. In this case, our problem corresponds no longer to studying the effect of the fractal geometry of space-time on a particle embedded in an outer field. As recalled above, it now amounts to study the motion of a free particle in a space-time whose geometry would be both fractal (at large scale) and Riemannian (in the mean).
Let us use the general-relativistic and scale-relativistic covariances in order to write the geodesics equations in such a space-time. Einstein's covariant derivative writes:
Using this covariant derivative, Einstein's geodesics equations are written in terms of the free particle equation of motion :
This equation can now be made scale-covariant, by replacing by at all levels of the construction. We define a scale+Einstein-covariant derivative:
However, such an equation remains incomplete, as shown by Pissondes (1997), in agreement with Dohrn and Guerra (1977). A geodesics correction must be added to the usual parallel displacement, that leads to add to Eq. (75) a term , now involving the Ricci tensor in the new geodesics equation. Moreover, one must be cautious with the interpretation of this equation. It is obtained by assuming that the two (Einstein and scale) covariances do not interact one on each other. This can be only a rough approximation. Indeed, in order to solve the problem of the motion in a general, non flat fractal space-time (which is nothing but the problem of finding a theory of quantum gravity in our framework), one should strictly examine the geometrical effects of curvature and fractality at the level of the construction of the covariant derivatives, not only once they are constructed. This problem reveals to be extraordinarily complicated (Nottale 1997), and will not be considered further in the present paper.
A second problem with Eq. (75) concerns the interpretation of the scale-covariant derivative in the motion-relativistic case. It is obtained by assuming that not only space but space-time is fractal, which implies that the trajectories of particles can go backward in time. This is not a problem in microphysics: on the contrary, it is even needed by the existence of virtual pairs of particle-antiparticles, through Feynman's interpretation of antiparticles as particles going backward in time. (See Ord 1983; Nottale 1989, 1993a for a development of the fractal approach to this question). It is more difficult to make a similar interpretation in the macroscopic case, so that we shall only consider the non(motion)-relativistic limit of Eq. (75) for comparison with actual data (Paper III of this series). This is nothing but the above generalized Newton's equation of dynamics,
that can be integrated in terms of the generalized Schrödinger equation (Eq. 50).
4.4. Euler-Schrödinger equation
Our approach can be generalized to fluid mechanics in a straightforward way. Actually we have already partly adopted a fluid description when introducing a velocity field . Applying scale-covariance, the Euler equation for a fluid in a gravitational potential ,
In the general case is not a gradient, and we cannot transform this equation into a Schrödinger-like equation. However, in the case of an incompressible fluid ( = cst), and more generally in the case of an isentropic fluid (including perfect fluids), is the gradient of the enthalpy by unit of mass w (see, e.g., Landau & Lifchitz 1971)
In this approximation Eq. (78) becomes the Euler-Lagrange equation constructed from the Lagrange function . Therefore it derives from a stationary action principle working with the complex action . Our whole previous formalism is now recovered. We introduce the probability amplitude (now defined for a unit mass):
In terms of is a gradient:
This equation can now be integrated, leading to a generalized Schrödinger-like equation:
4.5. Navier-Schrödinger equation
A similar work can be performed with the Navier-Stokes equations, at least formally. Our scale-covariant generalized Navier-Stokes equations write:
It is quite remarkable that the viscosity term in the Navier-Stokes equation plays a role similar to the coefficient . This suggest to us to combine them into a new complex parameter
In terms of , the complex Navier-Stokes equation recovers the form of the complex Euler equation:
Once again, in the incompressible or isentropic cases, this equation can be integrated to yield a Schrödinger-like equation:
This equation is also valid in the presence of a gravitational field or in the presence of any field that is the gradient of a potential . It becomes:
However, its interpretation is more difficult than in previous calculations. Indeed the complex nature of prevents the imaginary part of this equation to be an equation of continuity. We shall no longer consider the viscous case in the present paper. We intend to study this situation in more detail in forthcoming works.
4.6. Motion of solids
The equation of the motion of a solid body can be given the form of Euler-Lagrange equations, and therefore comes in a very easy way under our theory. The role of the variables is now played by the rotational coordinates, where denotes three rotational Euler angles and is the corresponding rotational velocity. The Euler-Lagrange equation writes (Landau & Lifchitz, 1969):
in terms of the Lagrange function L of the solid, that writes:
where is the tensor of inertia of the body and U a potential term. We use throughout this section the tensorial notation where a sum is meant on two repeated indices. The right-hand member of Eq. (88) writes:
which identifies with the total torque, i.e., the sum of the moments of all forces acting on the body. In the left-hand member one recognizes the angular momentum about the center of mass,
Let us consider the rotational motion of the solid at very large time scales. We are in similar conditions as in the case of translational motion, but now the position angles have replaced the coordinates. In our nondeterministic approach, we definitively give up the hope to make strict predictions on the values of these angles, and we now work in terms of probability amplitude for these values. Once again, by this way we become able to predict (angular) structures, since all values of the angles will no longer be equivalent, but instead some of them will be favored, corresponding to peaks of probability density.
The angular velocity can be decomposed in terms of a backward and forward mean, leading to define a mean complex angular velocity and a fluctuation such that , where is now a tensor. We then build a scale-covariant derivative:
We now introduce the wave function as another expression for the action , , where is a constant having the dimension of an angular momentum. Provided this constant is given by (this is a generalization of the previous scalar relation ), Eq. (94) can be integrated in terms of a generalized Schrödinger equation acting on the rotational Euler angles:
An example of application of this equation to the Solar System (quantization of the obliquities and inclinations of planets and satellites) has been given in (Nottale 1996c).
4.7. Dissipative systems: first hints
One can generalize the Euler-Lagrange equations to dissipative systems thanks to the introduction of a dissipation function f (see e.g. Landau & Lifchitz 1969):
where f is linked to the energy dissipation by the equation . This becomes in the Newtonian case:
We shall only consider here briefly the simplified isotropic case:
and its complex generalization:
We obtain a new generalized equation (Nottale 1996a):
which is still Schrödinger-like (and remains scale-invariant under the transformation , up to an arbitrary energy term), since it corresponds to a perturbed Hamiltonian: , with the operator V such that . The same problem has also been recently considered by Ahmed and Mousa (1996), with equivalent results. The standard methods of perturbation theory in quantum mechanics can then be used to look for the solutions of this equation. This will be presented in a forthcoming work.
4.8. Field equations
As is well-known, the profound unity of physics manifests itself by the fact that field equations can also be given the form of Lagrange equations. The potentials play the role of the generalized coordinates, the fields play the role of the time-derivatives of coordinates and the coordinates play the role of time:
Namely, field equations take the same form as the equations of motion of particles, once this substitution is made. For simplicity of the argument we work with only one x variable in what follows (the generalization to any dimension will be given in a forthcoming work). One defines a Lagrange function then an action from this Lagrange function. The action principle leads to Euler-Lagrange equations that write:
For example, the Lagrange equation constructed from is the Poisson field equation, . This well-known structure of present physical theories allows us to apply our method to fields themselves. This leads to a quantization of classical fields, but in a new way and with an interpretation quite different from that of the second quantization in standard quantum mechanics.
Here we consider a field potential whose evolution with time is known to be chaotic. On a very long time-scale, far larger than its chaos time, it can be described in terms of a long-term, differentiable mean and a non-differentiable fluctuation . We are once again led to the same quantum-like method: we give up the possibility to strictly know the value of the potential at any point or instant, but we introduce a probability amplitude for it, , such that the probability of a given value of is given by . The combined effect of fractal fluctuations and passage to complex numbers due to the breaking of the reflexion invariance leads to defining a complex field , then a scale-covariant derivative that writes:
Using this covariant derivative, Eq. (104) can be quantized. We obtain:
In the particular case of a scalar field considered above, this equation can be integrated under the form of a generalized Schrödinger equation for the probability amplitude of the potential :
A study of the solutions of this equation and its writing for any number of dimensions will be presented in a forthcoming work.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998