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Journal of Mathematical Physics

Time displaced interactions: Classical dynamics and path integral quantization

Lawrence S. Schulman

Published May 1, 1995• 2 citations
Publisher

Abstract

The path integral was created to quantize systems whose dynamics are nonlocal in time and for which a Hamiltonian formulation could not be found. In this article we quantize a linear oscillator with an interaction that depends on the oscillator’s position at past and future times. The central issue is the classical boundary value problem for such an oscillator. Because two-time functional boundary conditions are used to implement the quantization, we take up related questions, such as causality in this context, and the general issue of stability for forward and inward data specifications.

DOI
10.1063/1.531050

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The path integral was created to quantize systems whose dynamics are nonlocal in time and for which a Hamiltonian formulation could not be found. In this article we quantize a linear oscillator with an interaction that depends on the oscillator’s position at past and future times. The central issue is the classical boundary value problem for such an oscillator. Because two-time functional boundary conditions are used to implement the quantization, we take up related questions, such as causality in this context, and the general issue of stability for forward and inward data specifications.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.
Although the variational principles of classical mechanics lead to two-time boundary value problems, when dealing with the real world everyone knows you should use initial conditions. Not surprisingly, the eighteenth-century statements of classical variational principles were invested with religious notions of teleology: guidance from future paths not taken could only be divine. The Feynman path integral formulation of quantum mechanics makes it less preposterous that a particle can explore non-extremal paths; moreover, it is most naturally formulated using information at two times. In the previous chapter, use of the two-time boundary value problem was proposed as a logical prerequisite for considering arrow-of-time questions. Perhaps this is no less teleological than Maupertuis's principle, except that we remain neutral on how and why those conditions are imposed. In this chapter we deal with the technical aspects of solving two-time boundary value problems. In classical mechanics you get into rich existence questions—sometimes there is no solution, sometimes many. For stochastic dynamics the formulation is perhaps easiest, which is odd considering that this is the language most suited to irreversible behavior. Our ultimate interest is the quantum two-time problem and this is nearly intractable. Later in the book I will propose that the universe is most simply described as the solution of a two-time boundary value problem. A natural reaction is to wonder whether this is too constraining. Given that our (lower entropy) past already cuts down the number of possible microstates, are there sufficient microstates to meet a future condition as well?
The path integral approach to quantum mechanics requires a substantial generalisation to describe the dynamics of systems confined to bounded domains. Nonlocal boundary conditions can be introduced in Feynman's approach by means of boundary amplitude distributions and complex phases to describe the quantum dynamics in terms of the classical trajectories. The different prescriptions involve only trajectories reaching the boundary and correspond to different choices of boundary conditions of selfadjoint extensions of the Hamiltonian. One dimensional particle dynamics is analysed in detail.
The Lagrangian formulation of quantum dynamics in terms of path-integrals due to Feynman describes systems for which the Hamiltonian is classical in form and quantization is carried out in terms of commutators rather than anticommutators. The difficulty with the Feynman method is the actual evaluation of the pathintegral itself. An explicit evaluation is given for classical wave motion in one dimension. This requires an extension of the Feynman method which was introduced by Tobocman and studied in detail by Davies. The work of Corson on the question of a unified formulation of dynamics is discussed. (Author)
The path integral formulation can reproduce the right energy spectrum of the harmonic oscillator potential, but it cannot resolve the Coulomb potential problem. This is because the path integral cannot properly take into account the boundary condition, which is due to the presence of the scattering states in the Coulomb potential system. On the other hand, the Sommerfeld quantization can reproduce the right energy spectrum of both harmonic oscillator and Coulomb potential cases since the boundary condition is effectively taken into account in this semiclassical treatment. The basic difference between the two schemes should be that no constraint is imposed on the wave function in the path integral while the Sommerfeld quantization rule is derived by requiring that the state vector should be a single-valued function. The limitation of the semiclassical method is also clarified in terms of the square well and δ(x)δ(x)δ(x) function potential models.