Path Integrals

Abstract

The path integral formalism of quantum physics, as an alternative to the operator approach, was introduced by Feynman1 in 1948. In recent years, it has been applied in almost all areas of physics and has become a powerful and essential tool to do quantum physics. There is an intimate connection between this formalism and its classical counterpart, and the action, as the time integral of the Lagrangian, appears naturally in it. In its simplest form, it involves in developing an expression for the amplitude ‹xt | x′t′› for a particle initially at x′ at time t′ to be found at x at a later time t > t ′ as a sum over all paths beginning at x′ and ending up at x. Quantum physics being probabilistic, this expression involves, in general, in addition to the classical path joining x′ to x, an (uncountable) infinite number of possible paths joining these two points. The importance of a so-called Lagrangian formulation of quantum physics was emphasized by Dirac and fully exploited by Feynman in his classic work.