Harmonic potential as an effective limit of a discrete classical interaction

Abstract

Motivated by improving the understanding of the quantum-to-classical transition we use a simple model of classical discrete interactions for studying the discrete-to-continuous transition in the classical harmonic oscilator. A parallel is traced with gravity for stressing the relevance of such discrete interaction models. Fundamental interactions, according to quantum field theory, are realized through the exchange of interaction quanta-packets of matter-energy with defined quantum numbers, viz. momentum-energy, spin, electric charge, etc. Thei are discrete interactions, in contradistinction to the classical continuous picture. The Bohr's correspondence principle, a useful guideline in the early days of quantum mechanics, states that in the limit of very large quantum numbers the classical idea of continuity must result from the quantum discreteness as an effective description. It would be very interesting to see in a clear way how this transition discrete-to-continuous occurs. This is the objective of the present letter with the use of a simple model of discrete classical interaction for studying this transition in the classical simple harmonic oscillator. We should not forget, however, that the harmonic potential, although being an extremely useful tool in all branches of modern physics, is not itself a fundamental interaction, which, as well known, are just the gravitational, the electromagnetic, the weak and the strong interactions; actually it is just an effective description. This may just valorize the importance of studying how it can be understood as an effective * PIVIC-UFES

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