Principle of stationary action and the definition of a proper open system

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Abstract
The generalization of the variation of the action-integral operator introduced by Schwinger in the derivation of the principle of stationary action enables one to use this principle to obtain a description of the quantum mechanics of an open system. It is shown that augmenting the Lagrange-function operator by the divergence of the gradient of the density operator, a process which leaves the equations of motion unaltered, leads to a class of generators whose associated infinitesimal transformations yield variations of the action-integral operator for an open system, similar in form and content to those obtained for the total, isolated system. The augmented Lagrange-function operator and the associated action-integral operator are termed proper operators, since only their variation yields equations of motion for the observables of an open system, in agreement with the expressions obtained from the field equations. Modifying the generator in this manner is shown to be equivalent to requiring that the open system be bounded by a surface through which there is a local zero flux in the gradient vector field of the electron density. Only the observables of such properly defined open systems are described by the correct equations of motion. The physical significance of such proper open systems is discussed.

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