Functional Integration; Path Integrals

David Bailin, A. Love

There are two widely used approaches to quantum field theory. The first is based on field operators and the canonical quantisation of these field operators. The second approach involves path integrals over classical fields. This chapter develops the idea of path integrals (or functional integrals) in a very intuitive way without any attempt at mathematical precision or rigour. It exploits the analogy between vectors and functions, and between matrices and differential operators on functions. Since very few path integrals can be performed exactly, the chapter concentrates on Gaussian path integrals. These are important in their own right, but much more so because they can be used in approximation schemes when the exact path integral is intractable.

James P. Edwards, Urs Gerber, Christian Schubert +2 more

We introduce two integral transforms of the quantum mechanical transition kernel that represent physical information about the path integral. These transforms can be interpreted as probability distributions on particle trajectories measuring respectively the relative contribution to the path integral from paths crossing a given spatial point (the hit function) and the likelihood of values of the line integral of the potential along a path in the ensemble (the path-averaged potential).

Ivo Sachs, Siddhartha Sen, James Sexton

Michael Stone

Ola A. Jarab’ah

Path integral formulation based on the canonical method is discussed. The Hamilton Jacobi function for regular Lagrangian is obtained using separation of variables method. This function is used to quantize regular systems using path integral method. The path integral is obtained as integration over the canonical phase space coordinates. One illustrative example is considered to demonstrate the application of our formalism.