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The stochastic quantization method for systems with dissipation

Rodanthy Tzani

Published Jan 1, 1994• 1 citation

Abstract

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

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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

Phenomenological Equations of Motion for Dissipative Systems Lagrangian Hamiltonian and Hamilton-Jacobi Formulation of the Classical Dissipative Systems Noether's Theorem and Non-Noether Conservation Laws Dissipative Forces Derived from Many-Body Problems A Particle Coupled to a Field and the Damped Motion of a Central Particle Coupled to a Heat Bath Quantization of Dissipative Systems in General and of Explicitly Time-Dependent Hamiltonians in Particular Density Matrix and the Wigner Distribution Function for Damped Systems Path Integral Formulation of a Damped Harmonic Oscillator Quantization of the Motion of an Infinite Chain Heisenberg's Equations of Motion for a Particle Coupled to a Heat Bath Quantum Mechanical Models of Dissipative Systems and the Concept of Optical Potential.

This paper investigated the basic formalism for treating the dissipative Hamiltonian systems within the framework of the canonical method using the path integral quantization. The Hamiltonian treatment of the dissipative systems leads to obtain the equations of motion as a total differential equation in many variables which require the investigation of integrability conditions on the action and the equations of motion. In this formalism, the action integral for dissipative systems is obtained from equations of motion and written in phase space. Besides, the quantization of these systems is investigated using the action to construct the wave function in terms of the canonical phase space coordinates. Two examples are examined, the free particle and the simple harmonic oscillator.

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 connections between standard theoretical tools used to study open quantum systems can sometimes seem opaque. Whether it is a Lindblad master equation, the equation of motion for the Wigner function, or a dissipative Keldysh action, features evident in one formalism are often masked in another. Here, we reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. This symmetry can then be used to easily diagonalize these models, and it provides an intuitive way to demonstrate the separation of dissipation and fluctuations in linear systems. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ``wave functions'' of the density matrix in the eigenbasis of the third-quantized superoperators we introduce. The field-theory representation of the time-evolution operator in this basis is then the Keldysh path integral. To highlight the utility of our approach, we apply our version of third quantization to a dissipative nonlinear oscillator, and we use it to obtain new exact results.

The Newton‐Lagrange equations of motion represent the fundamental law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses classical and quantal dynamics of such systems and presents some irresponsible speculations by introducing a certain canonical two‐form Ω. By its construction Ω embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle is introduced, where variation is performed over a set of “umbilical surfaces” instead of system histories. It provides correct Newton‐Lagrange equations of motion and something more. The quantization is inspired by the Feynman functional integral approach. The quintessence is to rearrange path integral into an “umbilical world‐sheet” integral in accordance with the proposed variational principle. In the case of potential‐generated forces, the new approach reduces to the standard quantum mechanics.