Path Integral and the Induction Law

We show how the induction law is correctly used in the path integral computation of the free particle propagator. The way this primary path integral example is treated in most textbooks is a little bit missleading. ⋆ e-mail: [email protected] † e-mail: [email protected] 1 The path integral quantization method was developed in detail by Feynman [1] in 1948. Feynman developed some earlier ideas introduced by Dirac [2]. Since then, path integral methods have provided a good alternative procedure to quantization and many books have been written on this subject, not only in quantum mechanics [3, 4, 5, 6, 7], but also in quantum field theory [8, 9, 10] (many modern textbooks in quantum field theory devote a few chapters to functional methods). We can surely say that in the last decades, Feynman’s method has been recognized as a very convenient and economic mathematical tool for treating problems in a great variety of areas in physics, from ordinary quantum mechanics and statistical quantum mechanics to quantum field theory and condensed matter field. It is a common feature of almost all texts which introduce the Feynman quantization prescription to use the unidimensional free particle propagator as a first example. In many cases, this simple example is the only one that is explicitly evaluated. The reason for that is simple: after the free particle propagator has been presented, it is usual to introduce the semiclassical method, which is exact for quadratic lagrangians, so that examples like the oscillator propagator or the propagator for a charged particle in a uniform magnetic field can be obtained without the explicit calculation of the Feynman path integral (for the oscillator propagator, the reader may find both calculations, that is, the explicit one and the semiclassical one in Ref.[11]; see also references therein). Curious as it may seem, the free particle propagator is not treated as it should, regarding the correct use of the mathematical induction law. It is the purpose of this note to show how the induction law shold be applied to the free particle propagator in the context of path integrals. In what follows, we first make some comments about the usual way of obtaining this propagator and then we show how one should proceed if the use of induction law is taken seriously. The Feynman prescription for the quantum mechanical transition amplitude K(xN , x0; τ) of a particle which was localized at x0 at time t = 0, to be at the position xN at time t = τ (called Feynman propagator) is given by the path integral [3]: K(xN , x0; τ) = lim N→∞ Nε=τ √ m 2πih̄ε ∫ N−1