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Introduction to path integrals - McGill University

1 Introduction to path integralsv. January 22, 2006 Phys 719 - M. HilkeCONTENT Classical stochastic dynamics Brownian motion (random walk) Quantum dynamics Free particle Particle in a potential Driven harmonic oscillator Semiclassical approximation Statistical description (imaginary time) Quantum dissipative systemsINTRODUCTIONPath integrals are used in a variety of fields, includingstochastic dynamics, polymer physics, protein folding,field theories, quantum mechanics, quantum field theo-ries, quantum gravity and string theory. The basic ideais to sum up all contributing paths. Here we will overviewthe technique be starting on classical dynamics, in par-ticular, the random walk problem before we discuss thequantum case by looking at a particle in a potential, andfinish with an example of a quantum dissipative system,where this technique bares all its STOCHASTIC DYNAMICSThe classical equation of motion for a particle in afluctuating environment is simply given bymdvdt=F(t) v/B(1)wheremis the mass,vthe velocity andBdescribes thefriction of the particle.

In this spirit, Sect. 2 gives an introduction to path integrals. Some aspects discussed in this section are not necessarily closely related to the problem of dissipative systems. They rather serve to illustrate the path integral approach and to convey to the reader the beauty and power of this approach. In Sect. 3

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