Path Integral Quantization of Regular Lagrangian

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.


Introduction
The path integral is an expression for the propagator in terms of an integral over an infinite dimensional space of paths in configuration space.Many important applications of path integral have been found in statistical physics, in the theory of phase transitions, super fluidity, super conductivity, quantum optics, and plasma physics.
The path integral concept was introduced for the first time by (Wiener, 1921) as a method to solve problems in the theory of diffusion and Brownian motion.This integral which is now also called the Wiener integral has played a central role in the further development of the subject of path integration.
It was reinvented in a different form by (Feynman, 1948), for the reformulation of quantum mechanics.The Feynman approach was inspired by Dirac's paper on the role of the Lagrangian and the least-action principle in quantum mechanics (Dirac, 1933).
The purpose of the present work is to construct the Hamilton Jacobi function for regular Lagrangian using separation of variables technique in order to quantize the regular systems using path integral method.This paper is organized as follows.In section 2, Hamilton Jacobi formulation and path integral quantization were discussed.In section 3, illustrative example is examined.In section 4, the work closes with some concluding remarks.

Hamilton Jacobi Formulation and Path Integral Quantization
The Lagrangian formulation of classical system requires that, the system is formulated by the generalized coordinates i q and velocities i q , according to that, we can write the Lagrangian as a function of these , and the corresponding Euler Lagrange equation is given by (Thornton, 2004).
( 1 ) According to that, the canonical Hamiltonian can be written as (2) where, the momentum can be obtained from As a result, the Hamiltonian is constructed in the system, and one can write the Hamilton Jacobi equation like where The momentum now is It is known that in standard formulation the Hamilton Jacobi problem focuses on finding ) , ( t q S , which satisfies equation( 4) and one can achieve that by using the method of separation of variables, by assuming , (Goldstein,1980), where α is a constant, and ) , ( α q w is time independent and called Hamilton's characteristic function, it follows that In the canonical method the action function and the equations of motion are written as total differential equations as follows is the Hamilton Jacobi function, which is obtained in terms of the canonical coordinates.
The path integral representation may be written as

Illustrative Example
Our formalism can be illustrated by discussing the following example: Consider a particle of mass m falls vertically under the influence of gravity, without frictional forces (Jarab'ah, 2013).The Lagrangian is given by The canonical Hamiltonian is The solution of the Hamiltonian Jacobi equation can be constructed as Then, Using equation ( 6), the function Taking the first time derivative of equation ( 18) this yields Inserting equation (20) and equation ( 21) into equation ( 17) we obtain Matching power of y, we get Making use of equation ( 25) and ( 29), the Hamilton Jacobi function takes the following form Now we come to the quantization of our system using path integral representation.
Using equation ( 11) the path integral for this example is The path integral representation can be written as

Conclusion
The path integral formulation of regular systems was studied within the framework of Hamilton Jacobi equation.The Hamiltonian treatment of regular systems gives the Hamilton Jacobi equation, which leads to obtain the action function S using the technique of separation of variables.Then the path integral is obtained directly as an integration over the canonical phase space coordinates i q and i q .An illustrative example was examined.