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.
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.
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.
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.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the 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
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 Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.