We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system functions from classical Lagrangians, Cadzow's variational principle applied to the action sum, Maeda - Noether and Logan invariants of motion, elliptic and hyperbolic harmonic oscillator behaviour, gauge invariant electrodynamics and charge conservation, and the Grassmannian oscillator. First quantized discrete time mechanics is discussed via the concept of system amplitude, which permits the construction of all quantities of interest such as commutators and scattering amplitudes. We discuss stroboscopic quantum mechanics, or the construction of discrete time quantum theory from continuous time quantum theory and show how this works in detail for the free Newtonian particle. We conclude by applying the Schwinger action principle to the important case of the quantized discrete time inhomogeneous oscillator.
We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system functions from classical Lagrangians, Cadzow's variational principle applied to the action sum, Maeda - Noether and Logan invariants of motion, elliptic and hyperbolic harmonic oscillator behaviour, gauge invariant electrodynamics and charge conservation, and the Grassmannian oscillator. First quantized discrete time mechanics is discussed via the concept of system amplitude, which permits the construction of all quantities of interest such as commutators and scattering amplitudes. We discuss stroboscopic quantum mechanics, or the construction of discrete time quantum theory from continuous time quantum theory and show how this works in detail for the free Newtonian particle. We conclude by applying the Schwinger action principle to the important case of the quantized discrete time inhomogeneous oscillator.
We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system functions from classical Lagrangians, Cadzow’s variational principle applied to the action sum, Maeda-Noether and Logan invariants of the motion, elliptic and hyperbolic harmonic oscillator behaviour, gauge invariant electrodynamics and charge conservation, and the Grassmannian oscillator. First quantised discrete time mechanics is discussed via the concept of system amplitude, which permits the construction of all quantities of interest such as commutators and scattering amplitudes. We discuss stroboscopic quantum mechanics, or the construction of discrete time quantum theory from continuous time quantum theory and show how this works in detail for the free Newtonian particle. We conclude with an application of the Schwinger action principle to the important case of the quantised discrete time inhomogeneous oscillator. 1
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.
The theory of discrete extension, introduced here, demonstrates that various re-sults from elementary, non-relativistic quantum mechanics can be obtained without recourse to eigenvalue problems, operators, or state vectors in Hilbert space. Us-ing a mathematical structure similar to that of Newtonian mechanics, the theory generates correct, quasi-discrete values of action increments and energy levels for bound states of quantum-level systems. Quantization of the harmonic oscillator provides a simple illustration. Like Newtonian mechanics the theory provides intuitive conceptualizations for basic entities and kinematics. However, in place of the classical concepts of particle and field, the non-classical, unitary concept of a discretely extended ob-ject emerges from the theory as an alternative characterization of the nature of quantum-level entities. Because such an object is well-defined and unitary, the theory avoids the dilemmas of wave-particle dualism and complementarity. The theory offers insight into the nature of a number of quantum effects such as the zero-point energy of a harmonic oscillator. It also makes a number of testable predictions that distinguish it from standard quantum theory and from Bohmian mechanics.
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.
A generalization of quantum mechanics is given in which the central mathematical concept is the analogue of the action in classical mechanics. It is therefore applicable to mechanical systems whose equations of motion cannot be put into Hamiltonian form. It is only required that some form of least action principle be available. It is shown that if the action is the time integral of a function of velocity and position (that is, if a Lagrangian exists), the generalization reduces to the usual form of quantum mechanics. In the classical limit, the quantum equations go over into the corresponding classical ones, with the same action function. As a special problem, because of its application to electrodynamics, and because the results serve as a confirmation of the proposed generalization, the interaction of two systems through the agency of an intermediate harmonic oscillator is discussed in detail. It is shown that in quantum mechanics, just as in classical mechanics, under certain circumstances the oscillator can be completely eliminated, its place being taken by a direct, but, in general, not instantaneous, interaction between the two systems. The work is non-relativistic throughout.
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