ar X iv : h ep - t h / 93 03 13 7 v 1 2 4 M ar 1 99 3 RU - 93 - 20 - B March 1993 On the Quantizations of the Damped Systems

Based on a simple observation that a classical second order differential equation may be decomposed into a set of two first order equations, we introduce a Hamiltonian framework to quantize the damped systems. In particular, we analyze the system of a linear damped harmonic oscillator and demonstrate that the time evolution of the Schrödinger equation is unambiguously determined. * This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DOE91ER40651.B The search for field theoretic formulations of fundamental anyons has refocused attention on the long outstanding problem of the quantization of (non-Hamiltonian) systems in which only the equation of motion is explicitly known. The common feature of this class of systems is its history dependence. The most notable such system is certainly that of the damped harmonic oscillator. Apart from requiring the knowledge of quantum aspects of this class of systems for practical reasons (for example, the laser system), this problem has its own right to be investigated both for the mathematical interests and exploring the new (perhaps deeper) fundamental features of the quantum theory. There have been quite a number of attempts to solve this problem, which may be classified into two classes. One is to start with a modified quantization scheme or a bigger system with introduction of additional degrees of freedom such that the conventional Schrödinger or Heisenberg dynamical description is valid. The other is to formulate the quantum theory in a modified Schrödinger dynamics (a nonlinear one). Clearly, attempts in the second class do not respect the superposition principle as the nonlinearity enters as a consequence of wavefunction dependent potentials. Similarly, the first class of the attempts also has some disadvantages. The introduction of the non-hermitian or time-dependent Hamiltonian in this class often leads to the conclusion of non-existence of the Schrödinger wave description. As is usually done in quantum mechanics, the time evolution of the Schrödinger dynamics may be carried out only if one insures the completeness of the Hamiltonian eigenstates. The hermitian Hamiltonian in our standard theory guarantees unique unitary time evolution. For dissipative systems, the time evolution is certainly no longer unitary, yet one 2 may hope that the Schrödinger quantum mechanics may be still used for unambiguously determining the time evolution. The purpose of this paper is to present an alternative way of modifying the quantization framework in which the Lagrangians are introduced naturally and to show that the Schrödinger dynamics is indeed achieved. Thus the above mentioned puzzle will be resolved. We shall begin with a general damped system on the line in an external potential, whose classical motion is govern by the non-linear second order differential equation: ẍ+ k(x)ẋ+ g(x) = 0, (1) where the dots denote time derivatives and k(x) and g(x) are real functions. We observe that a reduction of Eq. (1) to ẋ+ f(x) = 0 (2) may be defined if f satisfies f(f ′ − k) + g = 0, (3) where the prime represents an x-derivative. Note the complex conjugate f of f will be also a solution of Eq. (3) if f is a solution. One may verify this reduction by directly differentiating Eq. (2) with respect to time and using of the condition (3). Roughly speaking, this reduction reflects the fact that a second order differential equations may be decomposed into a set of two first order equations. For the obvious reasons, this reduction # During the preparation of this paper, we receive a paper by Feshbach and Tikochinsky, in which this puzzle was also resolved in a different way. We thank Roman Jackiw for bringing this reference to our attention. 3 has at least two advantages. Classically, a first order equation (linear or not) is of course simpler than the second order one,* while quantum mechanically, the first order equations may be regarded as the Hamiltonian equations by properly defining the canonical variables. For later convenience, we denote xi(i = 1, 2) as the two (independent) solutions of Eq. (1) or the solutions of the follow set of the first order equations: ẋi + fi(xi) = 0, (4) where fi are two (independent) solutions of Eq. (3). One may easily show that x = x1+x2 will be a solution of Eq. (1) only for the linear damped case, namely, k=constant and g is a linear function of x. The Lagrangians L L = 2