A Toy Model of Renormalization and Reformulation

Physics – General Physics

Scientific paper

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18 pages, 2 figures, slightly improved text

Scientific paper

I consider a simple mechanical problem where the "particle acceleration" due to an external force creates sound waves. Theoretical description of this phenomenon should provide the total energy conservation. In order to better couple seemingly insufficiently coupled "mechanical" and "wave" equations at disposal, I advance an "interaction Lagrangian" similar to that of the Classical Electrodynamics (a self-action ansatz). New coupled "mechanical" and "wave" equations manifest unexpectedly wrong dynamics. I show that renormalization of the fundamental constants in the wrong equations works: the original "inertial" properties of solutions are restored. The corresponding exactly renormalized equations contain only physical fundamental constants, describe the experimental data, and reveal a deeper physics -- that of permanently coupled constituents of the "particle". The perturbation theory is then just a routine calculation. Thus, I show that with advancing a self-action Lagrangian we make a mathematical and physical error, that renormalization is just illegitimately discarding harmful corrections compensating this error, that the exactly renormalized equations may sometimes accidentally coincide with the correct equations, and that the right theoretical formulation of permanently coupled constituents can be fulfilled directly, if realized.

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