Action

action, in law: see procedure.

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Action

Any one of a number of related integral quantities which serve as the basis for general formulations of the dynamics of both classical and quantum-mechanical systems. The term has been associated with four quantities: the fundamental action S, for general paths of a dynamical system; the classical action S_C, for the actual path; the modified action S^′, for paths restricted to a particular energy; and action variables, for periodic motions.

A dynamical system can be described in terms of some number N of coordinate degrees of freedom that specify its configuration. As the vector q whose components are the degrees of freedom q₁, q₂, …, q_N varies with time t, it traces a path q(t) in an N-dimensional space. The fundamental action S is the integral of the lagrangian of the system taken along any path q(t), actual or virtual, starting from a specified configuration q₁ at a specified time t₁, and ending similarly at configuration q₂ and time t₂. The value of this action S[q(t)] depends on the particular path q(t). The actual path q_C(t) which is traversed when the system moves according to newtonian classical mechanics gives an extremum value of S, usually a minimum, relative to the other paths. This is Hamilton's least-action principle. The extremum value depends only on the end points and is called the classical action S_C(q₁, q₂; t₁, t₂).

An important variant of Hamilton's principle applies when the virtual paths q(t) are restricted to motions all of the same energy E, but no longer to a specific time interval, t₁ - t₂. The modified action S^′ = S - E(t₁ - t₂) obeys a modified least-action principle, usually called Maupertuis' principle, namely, that the classical path gives again an extremal value of S^′ relative to all paths of that energy. Maupertuis' principle is closely related to Fermat's principle of least time in classical optics for the path of light rays of a definite frequency through a region of inhomogeneous refractive index. See Hamilton's principle, Minimal principles

In quantum mechanics, as originally formulated by E. Schrödinger, the state of particles is described by wave functions which obey the Schrödinger wave equation. States of definite energy in, say, atoms are described by stationary wave functions, which do not move in space. Nonstationary wave functions describe transitory processes such as the scattering of particles, in which the state changes. Both stationary and nonstationary state wave functions are determined, in principle, once the Schrödinger wave propagator (also called the Green function) between any two points q₁ and q₂ is known. In a fundamental restatement of quantum mechanics, R. Feynman showed that all paths from q₁ to q₂, including the virtual paths, contribute to the wave propagator. Each path contributes a complex phase-term exp i (&phgr;[q(t)]), where the phase &phgr; is proportional to the action for that path. The resulting sum over paths, appropriately defined, is the path integral (or functional integral) representation of the Schrödinger wave propagator. The path integral has become the general starting point for most formulations of quantum theories of particles and fields. The classical path q_C(t) of least action now plays the role in the wave function as being the path of stationary phase. See Propagator (field theory)