Subspace quantum dynamics and the quantum action principle
Authors: Bader Richard F. W. , Srebrenik Shalom , Nguyen-Dang T. Tung 
Published: February 2003

Schwinger’s quantum action principle is used to obtain a quantum mechanical description of a subspace and its properties. The subspaces considered are those regions (Ω) of real space as defined by a property of the system’s charge distribution ρ (r), namely, that they be bounded by a surface S (r) through which the flux in ∇ρ (r) is zero at every point on S (r). Through the variation of the action integral, expressed in terms of the appropriately defined Lagrangian integral operator for a subspace, one obtains the quantum equations of motion and an expression for the change in the subspace action integral operator, Δ𝒲̂ (Ω). This expression obeys a principle of stationary action, and thus, as for a system with boundaries at infinity, it defines the generators of infinitesimal unitary transformations. The change in a subspace property 𝒪̂ (Ω) as induced by such an infinitesimal unitary transformation is investigated and related to the corresponding change as described by the calculus of variations. The latter consists of two contributions, one from the variation over the domain of the subspace, the other from a variation of its surface. It is found that the change in 𝒪̂ (Ω) caused by the domain variation is expressible in terms of the commutator of 𝒪̂ (Ω) and ℱ̂ (t), the generator of the infinitesimal unitary transformation. Through the definition of a subspace projector in the coordinate representation this commutator is shown to contain the subspace projection of the all space transformation and a term which corrects for the nonhermiticity of the projected generator. The contribution to the change in 𝒪̂ (Ω) from the surface variation is not expressible in terms of a commutator. However, this change is still quantitatively determined by the action of the generator as a result of the subspace boundary being defined in terms of the observable charge distribution. The Heisenberg equation of motion for a subspace property is a particular result obtained from the general analysis of the change induced in a subspace when the total system is subjected to a canonical transformation. The effect of the temporal generator−ℋ̂δt on a subspace property is expressible as a projection of the usual all space result and a term describing the flux in the vector current of the property density across the boundary of the subspace (the domain variation), plus a contribution arising from the change in the boundary with time (the surface variation). Schwinger’s quantum action principle is re-expressed as a sum of the changes in the action integral operator for each subspace in the system, changes which assume their simplest physical form for the particular class of subspaces studied here. Since the application of the zero-flux boundary condition to a molecular system partitions it into a collection of chemically identifiable atomiclike fragments, the total change in the transformation function as given by the action principle, may be expressed in terms of a sum over the change in action for each atom in a molecule.