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.
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.
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.
Abstract Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
The conventional correspondence basis for quantum dynamics is here replaced by a self-contained quantum dynamical principle from which the equations of motion and the commutation relations can be deduced. The theory is developed in terms of the model supplied by localizable fields. A short review is first presented of the general quantum-mechanical scheme of operators and eigenvectors, in which emphasis is placed on the differential characterization of representatives and transformation functions by means of infinitesimal unitary transformations. The fundamental dynamical principle is stated as a variational equation for the transformation function connecting eigenvectors associated with different spacelike surfaces, which describes the temporal development of the system. The generator of the infinitesimal transformation is the variation of the action integral operator, the spacetime volume integral of the invariant lagrange function operator. The invariance of the lagrange function preserves the form of the dynamical principle under coordinate transformations, with the exception of those transformations which include a reversal in the positive sense of time, where a separate discussion is necessary. It will be shown in Sec. III that the requirement of invariance under time reflection imposes a restriction upon the operator properties of fields, which is simply the connection between the spin and statistics of particles. For a given dynamical system, changes in the transformation function arise only from alterations of the eigenvectors associated with the two surfaces, as generated by operators constructed from field variables attached to those surfaces. This yields the operator principle of stationary action, from which the equations of motion are obtained. Commutation relations are derived from the generating operator associated with a given surface. In particular, canonical commutation relations are obtained for those field components that are not restricted by equations of constraint. The surface generating operator also leads to generalized Schr\"odinger equations for the representative of an arbitrary state. Action integral variations which correspond to changing the dynamical system are discussed briefly. A method for constructing the transformation function is described, in a form appropriate to an integral spin field, which involves solving Hamilton-Jacobi equations for ordered operators. In Sec. III, the exceptional nature of time reflection is indicated by the remark that the charge and the energy-momentum vector behave as a pseudoscalar and pseudovector, respectively, for time reflection transformations. This shows, incidentally, that positive and negative charge must occur symmetrically in a completely covariant theory. The contrast between the pseudo energy-momentum vector and the proper displacement vector then indicates that time reflection cannot be described within the unitary transformation framework. This appears most fundamentally in the basic dynamical principle. It is important to recognize here that the contributions to the lagrange function of half-integral spin fields behave like pseudoscalars with respect to time reflection. The non-unitary transformation required to represent time reflection is found to be the replacement of a state vector by its dual, or complex conjugate vector, together with the transposition of all operators. The fundamental dynamical principle is then invariant under time reflection if inverting the order of all operators in the lagrange function leaves an integral spin contribution unaltered, and reverses the sign of a half-integral spin contribution. This implies the essential commutativity, or anti-commutativity, of integral and half-integral field components, respectively, which is the connection between spin and statistics.
Abstract This chapter introduces Schrödinger’s equation and the action integral from which the differential equation is derived by the principle of stationary action. This principle is central to almost all of physics. The discretization of the integral with finite elements and interpolation with polynomials scaled by interpolation coefficients at the ends of the elements (nodes) allows us to integrate out the spatial dependence leaving the action as a binomial expression in the coefficients. Now the action is varied with respect to these coefficients to obtain a discretized matrix version of the Schrödinger’s equation. This typically leads to matrix equation or a generalized eigenvalue problem to solve. In simple terms, this is the essence of the finite element method. The boundary conditions for the wavefunction and the derivative of the wavefunction can be very easily imposed at the nodes prior to the solution of the matrix equations. The probability current has to be continuous at interfaces in the interior of the physical domain. The form of the current is obtained by a gauge-variational method in an unambiguous manner. Going from the differential equation to the action integral, and vice versa, are treated in great detail so that the reader can gain confidence in working with variational principles.
Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian Lc(\dotx, x) = (M/2) f-1(x)\dotx2 - v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lagrangian operator is essential in our formulation.
For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one-dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.