The stochastic quantization method for systems with dissipation

Fernando Haas

Fernando Haas

Rodanthy Tzani

The stochastic quantization of dissipative systems is discussed. It is shown that in order to stochastically quantize a system with dissipation, one has to restrict the Fourier transform of the space-time variable to the positive half domain in the complex plane. This breaks the time-reversal invariance, which manifests in the formulation through the resulting noninvariant forms for the propagators. The relation of the stochastic approach with the Caldeira and Leggett path-integral method is also analyzed. † tzani@vana.physto.se; address after September 1994: Imperial College, Physics Dept., Prince Consort Road, London SW72BZThe quantization of a classical system may, in general, be hindered by several obstacles, such as nonrenormalizability, topological obstructions, anomalies etc. There exists, moreover, a whole class of systems for which there is no available method of quantization, namely non-hamiltonian systems. The quantization of systems with no lagrangian or hamiltonian formulation, apart from the wellknown examples of systems with dissipation, includes several interesting physical problems, such as the Bargmann-Wigner higher spin field equations [1] and field theories of fundamental anyons [2]. For the Bargmann-Wigner system of equations there is no action which directly reproduces this set of equations, although the quantization for higher spin systems has been done using more complicated and indirect ways [3]. The case of anyonic systems is known to be unsolved. The most common example of a non-hamiltonian system is, nevertheless, the case of a particle on the line with friction. Its equation of motion is m¨x + γ ˙x + dV dx = 0 (1) where overdot denotes time derivative and V (x) is the potential of the particle. If we assume an action of the form S = f(x, ˙x, t) dt (2) for the above equation, the variation of the action gives δS δx = dt ( ∂f ∂ ∂f ∂ ∂f

Kunio Yasue

The stochastic quantization procedure, proposed by Nelson, is further developed to include the problem of field quantization. To maintain the mathematical rigor in treating infinitely many degrees of freedom, basic notions of the nonstandard analysis are adopted. As an application of the present method, a quantum mechanical description of dissipatively interacting fields, such as the laser electric field in the lossy cavity, is investigated.