Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 19Green's Functions and Path Integrals

In 1827, Brown investigates the random motions of pollen suspended in water under a microscope. The irregular movements of the pollen particles are due to their random collisions with the water molecules. Later, it becomes clear that many small objects interacting randomly with their environment behave the same way. Today, this motion is known as the Brownian motion and forms the prototype of many different phenomena in diffusion, colloid chemistry, polymer physics, quantum mechanics, and finance. During the years 19201930, Wiener approaches Brownian motion in terms of path integrals. This opens up a whole new avenue in the study of many classical systems. In 1948, Feynman gives a new formulation of quantum mechanics in terms of path integrals. In addition to the existing Schrödinger and Heisenberg formulations, this new approach not only makes the connection between quantum and classical physics clearer but also leads to many interesting applications in field theory. In this chapter, we introduce the basic features of this intriguing technique, which not only has many interesting existing applications but also has tremendous potential for future uses. In conjunction with the anomalous diffusion phenomena and the path integrals over Lévy paths, we also introduce the Fox's -functions; a versatile and an elegant tool of applied mathematics. ...