The Lagrangian description of a quantum system with a finite number of noncanonical operators is presented. Using this description the dynamics of the operators of the system are derived. The class of variations, commutation rules, and the form of Lagrangian are shown to be self-consistent.
The classical and quantum theories of dynamical systems whose Lagrangians are linear in the co-ordinate derivatives are studied, with a view to clarifying a n-uiber of points in relation to Schwinger's quantum-mechanical variational principle. In particular, linear variations of the dynamical variables are considered, and it is shown that they may be consistently included in Schwinger's formalism. It is also found to be possible to obtain Green's generalized commutation rules by excluding c number variations from the theory, for a suitably restricted class of Lagrangian.
Abstract The classical and quantum theories of dynamical systems whose Lagrangians are linear in the co-ordinate derivatives are studied, with a view to clarifying a number of points in relation to Schwinger’s quantum-mechanical variational principle. In particular, linear variations of the dynamical variables are considered, and it is shown that they may be consistently included in Schwinger’s formalism. It is also found to be possible to obtain Green’s generalized commutation rules by excluding c number variations from the theory, for a suitably restricted class of Lagrangian.
Quantum theory is developed from a q-number (operator) action principle with a representation-invariant technique for limiting the number of independent system variables. It is shown that in a q-number theory such a limitation on the number of variations is necessary, since a completely arbitrary q-number variation implies an infinite number of conditions to be satisfied. This implication arises because, in a q-number theory, the system variables are matrices which, in most cases, have an infinite number of elements, each of which is independently varied in a completely arbitrary q-number variation. A review of Schwinger's quantum field theory formalism shows that the limitation to c- number variations, while accomplishing the desired limitation on the number of independent variations, is not representation-invariant, i.e., under a change of representation, c-number variations applied to the original system variables transform to q-number variations of the new system variables. In the Hamiltonian formulation of classical mechanics, invariant infinitesimal canonical transformations are generated by constants of the motion. It is also possible to generate infinitesimal invariant transformations from constants of the motion in the classical Lagrangian formulation. This latter possibility is exploited to obtain a q-number theory. It is demanded that the variations of the system variablesmore » in the action principle be the invariant infinitesimal transformations generated by constants of the motion. This requirement accomplishes the desired limitation of the number of independent variations in a representation-invariant manner. The resulting theory is developed in detail for a certain class of Lagrangian functions. A set of commutation relations is derived and a dynamical principle for determining the temporal development of the system is postulated. A discussion of the results, including consideration of the self-consistency of the formalism developed, is presented. Finally, a specific problem is worked through and it is shown that the results obtained correspond with those of the canonical method of quantization.« less
A previous q-number variational method is extended to be applicable to any quantum system for which the classical Lagrangian is given by Lc=½gij\dotqi\dotqj+ui\dotqi-v. The q-number variation is necessary for the formulation to be form-invariant under a general space-time transformation. From the action principle, not only the Euler-Lagrange equation but also the commutation relations up to a constant factor are obtained. Futhermore the form of the quantum Lagrangian is, to some extent, decided in order for solutions to exist for the action principle. It is shown that the first Noether theorem holds and the quantization is consistent with the canonical formalism.
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.
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.