PATH-INTEGRAL FOR THE RELATIVISTIC PARTICLE AND HARMONIC-OSCILLATORS

The action for a massive particle in special relativity can be expressed as the invariant proper length between the endpoints. In principle, one should be able to construct the quantum theory for such a system by the path integral approach using this action. On the other hand it is well known that the dynamics of a free, relativistic, spinless massive particle is best described by a scalar field which is equivalent to an infinite number of harmonic oscillators. We clarify the connection between these two-apparently dissimilar-approaches by obtaining the Green function for the system of oscillators from that of the relativistic particle. This is achieved through defining the path integral for a relativistic particle rigorously by two separate approaches. This analysis also shows a connection between square real Lagrangians and the system of harmonic oscillators which is likely to be of value in more general context.