[Paper Review] Gauge Invariant Quantization of Dissipative Systems of Charged Particles in Extended Phase Space

Siamak Khademi, S. Nasiri|ArXiv.org|Nov 14, 2005

Spectroscopy and Quantum Chemical Studies1 references3 citations

TL;DR

This paper proposes an extended phase space (EPS) formalism to resolve gauge dependence in quantum dissipative systems of charged particles under time-dependent electric fields. By generalizing gauge transformations to canonical transformations in EPS, the authors show that solutions in different gauges (Coulomb and Lorenz) yield identical physical results, including gauge-invariant conductivity, resolving inconsistencies found in prior work using standard Schrödinger quantization.

ABSTRACT

Recently, it is shown that the extended phase space formulation of quantum mechanics is a suitable technique for studying the quantum dissipative systems. Here, as a further application of this formalism, we consider a dissipative system of charged particles interacting with an external time dependent electric field. Such a system has been investigated by Buch and Denman, and two distinct solutions with completely different structure have been obtained for Schrödinger's equation in two different gauges. However, by generalizing the gauge transformations to the phase space and using the extended phase space technique to study the same system, we demonstrate how both gauges lead to the same conductivity, suggesting the recovery of gauge invariance for this physical quantity within the extended phase space approach.

Motivation & Objective

• To resolve the long-standing issue of gauge dependence in quantum dissipative systems, particularly for charged particles in time-dependent electric fields.
• To address the inconsistency in physical results—especially for conductivity—obtained in different gauges using conventional Schrödinger quantization.
• To demonstrate that the extended phase space (EPS) formalism restores gauge invariance by making gauge transformations canonical transformations in the extended space.
• To show that solutions in different gauges are unitarily equivalent, ensuring physical observables like conductivity are gauge invariant.
• To generalize conventional gauge transformations to the EPS framework, enabling consistent quantization of dissipative systems with energy loss.

Proposed method

• Employ the extended phase space (EPS) formulation, introducing conjugate momenta πₚ and π_q for momentum and coordinate variables, forming a 4D phase space (p, q, πₚ, π_q).
• Define an extended Lagrangian and Hamiltonian to describe the dynamics of a damped harmonic oscillator system, incorporating the Kanai Hamiltonian for dissipation.
• Generalize gauge transformations to the EPS by defining extended gauge transformations as canonical transformations in the 4D phase space, preserving the symplectic structure.
• Apply the EPS formalism to a system of charged particles in a time-dependent electric field, using both the A-gauge (vector potential) and φ-gauge (scalar potential) formulations.
• Derive time-evolved wavefunctions in both gauges and show they are related by a unitary transformation, ensuring physical equivalence.
• Calculate the conductivity using the time-averaged current response and demonstrate its invariance under extended gauge transformations.

Experimental results

Research questions

• RQ1 Can the extended phase space formalism resolve the gauge dependence of physical observables in quantum dissipative systems?
• RQ2 Do different gauges (Coulomb and Lorenz) yield physically equivalent results when using the EPS approach?
• RQ3 Is conductivity a gauge-invariant quantity in the EPS formulation of dissipative quantum systems?
• RQ4 How do extended gauge transformations in EPS differ from conventional gauge transformations in standard quantum mechanics?
• RQ5 Can the EPS formalism consistently describe both the dissipative system and its mirror image, ensuring energy conservation?

Key findings

• The extended phase space formalism enables a consistent quantization of dissipative systems by introducing a mirror image system that absorbs energy at the same rate as the original system.
• Extended gauge transformations in EPS are canonical transformations, ensuring the mathematical consistency of the formalism.
• Solutions to the Schrödinger equation in both the A-gauge and φ-gauge are unitarily equivalent, differing only by a phase factor, thus guaranteeing physical equivalence.
• The conductivity calculated in both gauges is identical, demonstrating that it is a gauge-invariant physical observable in the EPS framework.
• The formalism resolves the contradiction found by Buch and Denman, who obtained physically distinct solutions in different gauges using standard quantization.
• The order of extension and gauge transformation does not affect the final result for this system, indicating a robust and consistent formalism for this class of problems.