Classical Action
To connect classical physics to quantum physics, we start with a review of the concept of action and its variation [1,2]. The Lagrangian L of the system is defined as the difference between the kinetic energy and the potential energy U of the system integrated over time resulting in a functional S, called the action of the system, for that particular path [1,2]:
The action 3.25 is a functional that takes as argument a function q(t) and returns a number, the action S[q]. Here, q(ta) = qtl andq(fi,) = qb are the starting and ending coordinates of the path; mi is the mass associated with the i"' generalized coordinate. The path q(t) actually chosen by the system is called the classical path or true path and nearby path not chosen by the system denoted by q(t) is called the false path. The classical path has the property of extremizing the action S in comparison with all neighbouring paths,
having the same end points q(ta) = q„ andq(tb) = qb. Here, Sq(t) = y(t) may be interpreted as quantum fluctuations around the classical path (see Figure 3.1). The representation in 3.27 will be shown to separate the action into a classical and a fluctuating part. For force-driven systems, each of the terms contains a force-free and the other a force term.
We see that the kinetic and potential energies are functions of time t and so for each possible path we obtain a different action. Hence, there are many possible paths in configuration space. However, in classical mechanics only one path is actually realized: the classical path or true path that is determined from the principle of stationary action, so-called calculus of variations [1,2]:
This variation of the action is done over all possible paths of the system in configuration space during the time evolution when the fluctuating part y(t) vanishes at the boundaries, that in mathematics are referred to as Dirichlet boundary conditions:
This yields the Euler-Lagrange equations of motion for q(t):
It is a set of s equations, one for each (q,q)-pair with solution being the classical path q(f). So, it is instructive to note that the particle’s motion is governed by the principle that the action is stationary, the so-called the Least Action Principle from which the Euler-Lagrange equations of classical mechanics are derived.
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