The Klein-gordon’s Field. a Counter-example of the Classical Limit

In this work we will study the classical limit of Klein-Gordon’s field, with an homogeneous potential which does not depend on Planck’s constant. First we will see that, in this case, the Klein-Gordon’s equation is equivalent to a hamiltonian system, composed by an infinite number of harmonic oscillators with frequencies which depend on time. Once we have seen this equivalence, we will quantize these oscillators and we will obtain the energy and the electric charge operators. With the energy operator, we will obtain the quantum equation of KleinGordon’s field. We will also see that we can find all the eigenfunctions of the energy and the electric charge operators. Consequently, with all those eigenfunctions we can construct the Fock’s space. After that, we will study the quantum dynamic of vacuum state. We will see that, if the space dimension is 1, when ~ → 0, the probability that does not exist any particle-antiparticle pair, converges to 1. However, in dimension 2 or 3, we will prove that, when ~ → 0, this probability does not converge to 1. Consequently, in dimension 2 or 3, the classical limit is not true. The notation that we are going to use, is the following <,> euclidean scalar product. <,>2 scalar product of L2. ||.||2 norm L2. ||.||2 norm ∞.