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arXiv:1208.1242 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Higher time derivatives in effective equations of canonical quantum systems

Martin Bojowald A, Suddhasattwa Brahma, Elliot Nelson

Published Nov 2, 2012• 27 citations
arXivHTMLPDF

Abstract

Quantum-corrected equations of motion generically contain higher time derivatives, computed here in the setting of canonically quantized systems. The main example in which detailed derivations are presented is a general anharmonic oscillator, but conclusions can be drawn also for systems in quantum gravity and cosmology.
DOI
10.1103/physrevd.86.105004

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The problem of how to obtain a quantum equation of motion with higher derivatives is discussed. First the basic ideas of generalized classical mechanics with higher derivatives are reviewed. The generalized canonical conjugate momenta and a function N, corresponding to the Hamiltonian function, are constructed. Then certain reduced momenta are introduced, which in absence of higher derivatives are identical to the canonical conjugate momenta. The function N is expressed in terms of the reduced momenta. In order to obtain a quantum equation of motion, the reduced momenta in N are replaced by operators fulfilling some given conditions; thus the corresponding operator N̂ can be identified with the Hamiltonian operator. With its help, and introducing a wave function, the quantum equation of motion with higher derivatives is obtained.
One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.
A major difficulty in comparing quantum and classical behavior resides in the struc- tural differences between the corresponding mathematical languages. The Heisenberg equations of motion are operator equations only formally identical to the classical equations of motion. By taking the expectation of these equations, the well-known Ehrenfest theorem provides identities which, however, are not a closed system of equations which allows to evaluate the time evolution of the system. The formalism of the effective action seems to offer a possibility of comparing quantum and classical evolutions in a system- atic and logically consistent way by naturally providing approximation schemes for the expectations of the coordinates which at the zeroth order coincide with the classical evolution [1]. The effective action formalism leads to equations of motion which differ from the classical equations by the addition of terms nonlocal in the time variable. This means that for these equations an initial value problem is not meaningful and they have to be interpreted in an appropriate way. Here we analyze situations in which the nonlocal terms can be reasonably approximated by local ones, so that the quantum corrections do not modify the locality of classical equations. In the simplest approximation, the effective Lagrangian differs from the corresponding classical one by a renormalization of both the potential- and the kinetic-energy terms. We shall not discuss the causal formalism used, for example, in refs. [2-4], as in the approximation considered this would lead to the same local equations. The present contribution describes the beginning of a systematic study of semiclassical evolutions using the effective action formalism. In the first part, after introducing the formalism of the effective action and its expansion in powers of hbar (loop expansion) in the context of quantum mechanics, we concentrate on the structure of the first-order corrections in li. These corrections are evaluated to the second order in the derivative expansion [5], by two different methods. The first is based on a Euclidean approach [6], the second one on an adiabatic approximation in evaluating functional determinants. In the second part of the article we put the formalism at work, choosing as our case study a two-dimensional (2D) anharmonic oscillator of the kind considered in molecular physics. The results of the simulations show that by increasing lithe effective dynamics tends to regularize the classical motion and becomes qualitatively very similar to the quantum evolution provided the energy is sufficiently small. The evaluation of the effective dynamics in more general cases will be presented in a forthcoming paper.
The solution of the quantum dynamical equation $\frac{i\ensuremath{\hbar}\mathrm{dT}}{\mathrm{dt}}=HT$, for the time-displacement operator TTT, is given, when the Hamiltonian HHH is a polynomial of the second degree in canonically conjugate variables, with arbitrary time-dependent coefficients. Heisenberg's equations of motion are then solved, and the general integral of Schr\"odinger's equation in coordinate space is expressed by the Green's function corresponding to TTT. An example is given.
In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and geometrical picture are developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state. The same methods are used to describe dynamical coherent states, which in turn provide means to compute quantum corrections to the symplectic structure of an effective system.