A new perspective on renormalization: invariant actions, a dynamical DNA

Details A new perspective on renormalization: invariant actions, a dynamical DNA A new perspective on renormalization: invariant actions, a dynamical DNA

2011-06-22

arxiv.org/abs/1106.4369v1

Physics

General Physics

15 pages, 6 figures

Scientific paper

This is a note to a quantum field theorist, interpreting the "Mallat" transformation (i.e., group invariant scattering). First the characteristics of the transformation are examined, and related to some well known issues with renormalization. The Lagrangian density and action for the texture ID problem that is the focus of Mallat is derived. This is the key translation in order to make the correspondence to the mathematics of quantum field theory and path integrals. Next Mallat's transformation is developed from the perspective of a "renormalization'" of the actions. That is, a change of coordinates to "smoothed" paths. The effective action is then calculated utilising a stationary phase approximation to the usual Gaussian integrals that are encountered. It should be noted that this evaluation is only the leading term in an asymptotic series. The fundamental excitations of the system are now well identified through their relationships to the generating function and the corresponding currents. The resulting Feynman diagrams are presented. The difference between two states of the system are then quantified by excitations needed to scatter one state into the other. The renormalization is based on an iterative wavelet transformation where the wavelet family respects the group symmetries of both the base manifold and the field. Finally, a connection to the analysis of complex nonlinear systems will be made where the difference in these states of complex systems will form a natural metric.

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