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Cambridge University Press eBooks

Lee's approach to discrete time mechanics

George Jaroszkiewicz

Published Apr 17, 2014• 0 citations
Publisher

Abstract

Lee's discretization Time is a continuous parameter in standard Schrödinger wave mechanics, and quantum wavefunctions are differentiable functions of that parameter. We shall refer to this as temporal differentiability . Temporal differentiability is assumed in Dirac's more abstract formulation of quantum mechanics, where state vectors in some abstract Hilbert space and Hermitian operators over that space depend on continuous time (CT) (Dirac, 1958). The same is assumed in relativistic quantum field theories, where quantum field operators in the Heisenberg picture are differentiable over time and space. Temporal differentiability is a necessary condition for the existence of the quantum wave equations and operator field equations found in such theories. Unfortunately, these differentiable equations are usually impossible to solve exactly, so various techniques such as perturbation theory and computer simulation are employed to make suitable approximations. Numerical techniques on their own, however, are insufficient to answer questions of principle, such as the meaning of quantized spacetime. A more principled approach to the solution of quantum differential equations, namely the path integral (PI), was developed, exploited and popularized principally by Feynman (Feynman and Hibbs, 1965). In some situations, such as quantum gravity, the PI was found to be virtually the only technology available to discuss the quantum physics. We discuss this approach in greater detail in Chapter 18.

DOI
10.1017/cbo9781139525381.011

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Lee's discretization Time is a continuous parameter in standard Schrödinger wave mechanics, and quantum wavefunctions are differentiable functions of that parameter. We shall refer to this as temporal differentiability . Temporal differentiability is assumed in Dirac's more abstract formulation of quantum mechanics, where state vectors in some abstract Hilbert space and Hermitian operators over that space depend on continuous time (CT) (Dirac, 1958). The same is assumed in relativistic quantum field theories, where quantum field operators in the Heisenberg picture are differentiable over time and space. Temporal differentiability is a necessary condition for the existence of the quantum wave equations and operator field equations found in such theories. Unfortunately, these differentiable equations are usually impossible to solve exactly, so various techniques such as perturbation theory and computer simulation are employed to make suitable approximations. Numerical techniques on their own, however, are insufficient to answer questions of principle, such as the meaning of quantized spacetime. A more principled approach to the solution of quantum differential equations, namely the path integral (PI), was developed, exploited and popularized principally by Feynman (Feynman and Hibbs, 1965). In some situations, such as quantum gravity, the PI was found to be virtually the only technology available to discuss the quantum physics. We discuss this approach in greater detail in Chapter 18.
Quantization We now discuss the quantization of discrete time (DT) classical mechanics (CM). We restrict our attention in this chapter to a system under observation (SUO) consisting of a single point particle of mass m moving in one spatial dimension with Cartesian coordinate x . The generalization to two or more degrees of freedom is as straightforward here as in CT quantum mechanics (QM). The Dirac bra-ket notation will be used for convenience. The standard principles of QM have proven remarkably successful and consistent over the years (Peres, 1993) and we have no reason to alter them apart from changing from CT to DT. This is a significant change. Quantum mechanics became dominant in physics and chemistry because of the success of the Schrödinger equation, which is a first-order-in-time differential equation. Therefore, we should take care to ensure that discretizing time does not undermine the successes of CT QM. There are two reasons why discretizing time in QM might be considered. First, the Schrödinger equation is hard, or even impossible, to solve exactly in many situations and temporal discretization might be seen as a step towards numerical simulation by computer. This motivated the work of Bender (Bender et al. , 1985a, 1985b, 1993) and others on the DT Schrödinger equation. The second reason is one of principle: we may want to explore the properties of DT QM as a self-consistent theory in its own right rather than as an approximation to CT QM.
18.1 Introduction The Feynman path-integral (PI) approach to quantum mechanics (QM) has proven to be an invaluable technical tool with applicability far beyond the non-relativistic QM from which it sprang (Feynman and Hibbs, 1965). The PI approach is based on a fundamental quantum principle, which is that nothing can or should be said with certainty about a system under observation (SUO) when it is not actually being observed. This is in complete contrast to classical mechanics (CM), which asserts that an SUO exists in a well-defined state at all times, regardless of whether it is being observed or not. It is a remarkable fact that what appears to be a metaphysical point actually underpins the principles of quantum physics. The quantum principle that we cannot say precisely in classical terms what an SUO is doing when we are not observing it has a powerful corollary: if we cannot say what the SUO is doing when we are not observing it, then we have to allow for the possibility that it could be doing anything, or at least anything that would be consistent with required conservation laws, behind our backs. Following a lead from Dirac (1933), Feynman developed and popularized a mathematical way of expressing this idea: the path integral. We shall discuss his approach in the case of an SUO consisting of a Newtonian point particle moving in one spatial dimension. Discrete time (DT) comes into the picture right at the start of the formulation.
To introduce the formalism of the functional integral in quantum mechanics, this chapter starts from the definition of the temporal evolution operator. It is important to emphasize that in the classical limit is not only the classic trajectory that have the grater contribution, but all the paths close to this also contribute. The chapter shows how integral formulation is perfectly consistent with Schrödinger's time-dependent differential equation. It retrieves this equation from definition of transition amplitude as a path integral. The chapter discusses functional derivative generalized to the case of the ordinary functional derivative. A very useful tool to solve problems related to functional integrals is the Wick rotation or Euclidean rotation. In quantum mechanics there are few problems we can solve exactly, and very often, when possible, we use perturbation theory to find an approximate solution. The chapter follows this approach with formalism of functional integration and construct a perturbation theory from which can be obtained a physical interpretation.
An interpretation and re-formulation of modern physics which removes the presumption of the space-time continuum, and bases physical theory on a small number of rational and empirical principles. After briefly describing the philosophical principles underlying the theory, we rigorously construct a discrete model of quantum mechanics. Kets are names, or labels which categorise states of matter but do not directly describe them. The principle of superposition is seen as a definitional truism in the categorisation of states. This resolves the measurement problem of quantum mechanics by attributing the collapse of the wave function to information. The probability interpretation has a natural meaning in which the configuration of interacting particles plays the role of a hidden variable. The model supports a form of relativistic quantum field theory which does not depend on quantisation or second quantisation from classical mechanics. Continuous laws of wave mechanics are found in a discrete metaphysic which does not involve waves. Classical law is the expected behaviour of many elementary particles. The constraints on the theory are sufficient to establish that particles are point-like entities with specific properties found in nature. Newton's first law and conservation of momentum are established from the principle of homogeneity. Maxwell's equations are derived from the simple interaction in which a Dirac particle emits or absorbs a photon. Feynman rules are calculated for the discrete theory and differ from the standard rules by the removal of the ultraviolet divergence and the use of proper loop integrals. They give finite and unambiguous results.
Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.
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.