24/04 - Ricardo Grande

On the mathematical discovery of new physical phenomena

The objective of this lecture is to try to develop an answer to the following question: how is it possible to physicists to preview new physical phenomena only by looking to some specific terms that lie in the mathematical formalism of some physical theory? Our starting point will be the historical analysis of the prediction of a quantum phenomenon by David Bohm and Yakir Aharonov (Bohm e Aharonov, 1959), the so called Aharonov-Bohm effect. Then we’ll discuss some aspects of the mathematical and physical foundations of quantum theory in opposition to classical theory of electromagnetism.  Finally, we are going to show our philosophical point of view about the subject. 
As it’s well known, that effect was previewed when Bohm and Aharonov suggested a special kind of interpretation to a vector function called  vector potential  that entered Schödinger equation. To understand what they had in mind, we need to look very briefly to the way potentials enter in the formulation of classical 
electrodynamics. Potentials (e.g., the vector potential) were introduced in the theory of classical electromagnetism only as a mathematical tool to compute the fields whilst the fields (solely) were the responsible for the physical effects. For instance, the electromagnetic field     was intended to be obtained by calculating the rotational of the potential, i.e.,- but in quantum mechanics, based on Aharonov and Bohm views, the potential    could have a brand new physical interpretation leading to a brand new and non-classical phenomena.  That one could be tested empirically. Its existence was empirically tested and rigorously demonstrated by Tonomura (Tonomura, 1989). Our approach will be guided by some of da Silva’s ideas (da Silva, 2010) on the foundations of mathematics we’ve been studying since our PhD work (Grande 2011) and our current studies on the philosophical aspects of the Aharonov-Bohm effect. 

BOHM, D. E AHARONOV, Y.  Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485-491 (1959).
DA SILVA, J. J.“Structuralism and the applicability of mathematics”, Axiomathes 20 – 229-253 (2010).
GRANDE, R. M. A aplicabilidade da matemática à física. Tese de doutorado apresentada ao instituto de 
filosofia e ciências humanas da Unicamp, Campinas (2011).

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