Research Goals

In this work we will develop techniques for multi-resolution analysis of classical and quantum chaos and turbulence, contexts for which our understanding of the system's characteristics is less rooted on analytical results and more dependent of numerical simulations. With this we intend not only to achieve a better understanding of the behaviour of physical systems described by non-linear equations, but also to generalize the idea that it is possible to develop a local analysis that overcomes the limitations of the usual paradigms of Fourier and linear superposition. That is, in this project we target systems that are beyond the Inverse Scattering methods for evolution equations and Non-Linear Fourier Transforms.
This research will require a comparative study involving the different wavelet representations of differential operators describing the evolution of the system and a similar multi-resolution analysis of actual pictures or data from experimental settings where the system is evolving, which could be obtained through the use of digital photographic camera, digitizing equipment and software,  data acquisition cards. This way, we will find out what resolution in the representation will cause a significant loss of information, either because of the boundary conditions specifications or due to a loss of information for relevant cooperative smaller-scale phenomena.


The meaning of quantization for non-linear, non-integrable systems with few symmetries becomes generally obscure due to the inexistence of Lie subalgebras of observables to yield a linear representation theory. Furthermore, the standard paradigms of Quantum Mechanics, i.e the Complementarity Principle and Fourier Analysis, cannot be clearly understood when we deal with other systems where the Weyl-Heisenberg Algebra is not crucial.
There is also the issue of the superposition principle: non-linearity means that superposition is apriori excluded. However, that that need not necessarily be the case has been known by Lie itself in the context of systems of ordinary diferential equations, and more recently by the realization that soliton solutions that superpose exist and can be found, either via the Inverse Scattering Methods, Bäcklund transformations or Non-Linear Fourier transform methods.
Understanding in this new 'spectral' setting the usual dispersion relations of a non-linear quantum evolution expressed by nonlinear terms in Schrödinger's equation is also necessary.


Most of the rigorous mathematical results concerning the multifractal formalism have been obtained in the context of dynamical system theory. However in physics as well as in other applied sciences, fractals appear not only as singular measures, but also as singular functions. The wavelet transform is a method to achieve a unified thermodynamical description of singular distributions including measures and functions, and scaling the inverse fractal problem. The self-similarity properties of fractal objects can be  expressed in terms of a dynamical system which leaves the object invariant. The inverse fractal problem consists in recovering this dynamical system from the data. In this context, the wavelet transform can be used to extract a one-dimensional map which accounts for its construction process.[5,6]
In renormalization group QFT and variational description of lattice statistical field theories, wavelet coefficients express in few numbers the rescalable properties of the theory, and renormalization group transformations have been implemented with these coefficients as basic dynamic variables for summation at each scale, which is here naturally defined.


Recent time-correlation experiments by Franson [7], Mandel [8] and others lead us to consider the possibility of wave-packet interference without their spatial superposition.


On the other hand, interferometric experiments with neutrons and post-superposition spatial filtering, carried by the Vienna group led by Prof. Rauch [9], have shown that we can in fact observe interferences in cases where the Fourier wave-packets describing the neutron are not spatially superposed in the interferometer. We are therefore interested in explaining this phenomenon, and our own work in progress points to a possible explanation of these experiences using locality arguments and wavelet analysis. On the other hand, we have in mind the proposal of new experiments which will allow us to weight the advantages of wavelet local analysis versus non-local Fourier analysis in applications to specific physical problems.

Interference is not an exclusively wave phenomena. Diffusion processes and random discrete maps could be made to display interference-like patterns which resemble those observed experimentally in quantum mechanics.
As is well known, iterated function systems are time-discrete maps whose image can give rise to fractal-like point sets. It is this property of being able to generate chaotic points as time passes which suggested to us that these maps in which randomness plays a key role, may be analogous to the two-slit interference experiment with photons or others particles obeying the rules of conventional quantum mechanics.[3]

Specifically, by simulation and analysis of non-linear Schrödinger equations, Burger's type equations, chaotic  and strange attractor equations and Hamilton-Jacobi equations, we want to determine if, in the evolution of dynamical systems controlled by sets of parameters, a coherent-state analysis could establish new complementarity relations between the resolution for neighbouring initial conditions, both in parameter and in physical-space, and the 'separation' between state-space portraits determined by the corresponding wavelet coefficients. We will have to define appropriate wavelet transforms of non-linear partial differential equations or at least coherent-state decompositions of evolution operators, and find out wether there is a better, 'adapted', wavelet frame connected to each particular system.
We are attempting the conversion to wavelet analysis of spectral resolution methods for the Schrödinger equations with potentials that classically exhibit chaotic behaviour (Feit,Fleck & Steiger, J.Comp.Phys. 47, 412-`82)  (Carlos Ramos & Amaro Rica da Silva)
In quantum chaos, coherent-sate entropy (S _ CS) (SBomczy Overscript[n, '] ski & Overscript[Z, ·]yczkowski '97) has been recently proposed as the appropriate notion for quantum analogs to classical chaotic maps in compact spaces: Baker's, Arnold's cat (T^2), the periodically kicked top (S^2). These are classical systems with a positive Kolmogorov-Sinai entropy (S _ KS) which go to zero in the quantum extension.

With the concept of quantum 'fuzzy' measure, each measurement naturally leads the system to a mixture of coherent states, and the coherent-sate entropy S _ CS = S _ Dyn + S _ Msr. In the semi-classical limit one can obtain S _ Msr -> 0, S _ Dyn -> S _ KS.

Bibliographic References

  • Feit,Fleck & Steiger,J.Comp.Phys.47,412-`82
  • Geometrical Dequantization and the Correspondence Problem, G.G.Emch, Int'l J.Theor.Phys,22,5 (1983)
  • G.N. Ord, in Present status of the quantum theory of light, eds. S. Jeffers et al., Kluwer Academic (1997).
  • Arneodo et al., Fractals  1 (1993) 629
  • Arneodo et al., Europhys. Lett. 25 (1993) 479
  • J.D. Franson, Phys. Rev. Lett. 62(1989)2205
  • X. Y. Zou, T. P. Grayson and L. Mandel, Observation of quantum interference effects in the frequency domain, Phys. Rev. Lett. 69 (1992) 3041.
  • H. Rauch, Superposition experiments in neutron interferometry, Il Nuovo Cimento B110, (1995) 557.

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