[Paper Review] Dissipation in Quantum Mechanics, Scalar and Vector Field Theory

Fardin Kheirandish, M. Amooshahi|ArXiv.org|Oct 17, 2006

Advanced Thermodynamics and Statistical Mechanics3 references3 citations

TL;DR

This paper introduces a consistent, field-theoretic framework for quantum dissipative systems by modeling the environment as massless Klein-Gordon fields coupled via position- and momentum-dependent functions. It derives generalized Langevin-Schrödinger equations with explicit noise and memory terms, calculates energy flow via transition probabilities, and establishes energy conservation, extending the formalism to scalar and vector field theories with susceptibility functions linked to coupling functions.

ABSTRACT

A new minimal coupling method is introduced. A general dissipative quantum system is investigated consistently and systematically. Some coupling functions describing the interaction between the system and the environment are introduced. Based on coupling functions, some susceptibility functions are attributed to the environment explecitly. Transition probabilities relating the way energy flows from the system to the environment are calculated and the energy conservation is explecitly examined. This new formalism is generalized to the dissipative scalar and vector field theories along the ideas developed for the quantum dissipative systems

Motivation & Objective

• To develop a consistent, systematic quantum mechanical description of dissipative systems beyond phenomenological Hamiltonians.
• To address the limitations of the Caldirola-Kanai model, particularly the violation of uncertainty relations over time.
• To model the environment as a quantum field (Klein-Gordon) with minimal coupling, enabling explicit derivation of noise and memory effects.
• To generalize the formalism to scalar and vector field theories while preserving energy conservation.
• To derive transition probabilities that explicitly track energy flow from system to environment.

Proposed method

• Model the environment as a massless Klein-Gordon field interacting with the system via position- and momentum-dependent coupling functions.
• Derive the total Hamiltonian including system, environment, and interaction terms, with canonical quantization applied to both system and field degrees of freedom.
• Use Heisenberg equations of motion to derive the generalized Langevin-Schrödinger equation with memory and noise terms.
• Introduce susceptibility functions χ and χ̃ via Fourier transforms of coupling functions, linking them to the environment's response.
• Explicitly construct noise operators RN and R̃N in terms of field ladder operators and coupling functions.
• Apply Laplace transforms to solve the generalized Langevin equation for both positive and negative times.

Experimental results

Research questions

• RQ1 How can a consistent quantum mechanical description of dissipative systems be formulated without violating fundamental principles like the uncertainty relation?
• RQ2 What is the explicit form of the noise and memory terms in the equation of motion when the environment is modeled as a quantum field?
• RQ3 How are the susceptibility functions of the environment related to the underlying coupling functions in the interaction Hamiltonian?
• RQ4 What is the role of transition probabilities in tracking energy flow from the system to the environment?
• RQ5 How can the formalism be generalized from quantum mechanics to scalar and vector field theories?

Key findings

• The formalism successfully avoids the violation of uncertainty relations observed in the Caldirola-Kanai model by using a field-theoretic environment.
• The generalized Langevin-Schrödinger equation is derived directly from Heisenberg equations, incorporating both memory and noise terms with explicit operator forms.
• The noise fields RN and R̃N are explicitly expressed in terms of field ladder operators and coupling functions, ensuring consistency with quantum fluctuations.
• Susceptibility functions χ and χ̃ are derived as integrals over frequency-dependent coupling strengths, linking them to the environment's linear response.
• Transition probabilities are calculated, showing that all initial energy of a damped harmonic oscillator is fully absorbed by the environment, confirming energy conservation.
• The framework is generalized to vector field theories, yielding a generalized Langevin-Schrödinger equation with two distinct memory and noise contributions for momentum and position coupling.