Consistent Probabilistic Description of the Neutral Kaon System

TL;DR

This paper presents a consistent probabilistic quantum mechanical description of the neutral Kaon system using a Lindblad approach, resolving issues with non-orthogonal states and providing results aligned with traditional methods.

Contribution

It introduces a Lindblad-type open quantum system framework for the Kaon system, improving the understanding of decay and CP violation phenomena.

Findings

Provides a consistent probabilistic description avoiding non-physical states

Aligns decay rate results with Weisskopf-Wigner approach

Models the system as an open quantum system with density matrices

Abstract

The neutral Kaon system has both CP violation in the mass matrix and a non-vanishing lifetime difference in the width matrix. This leads to an effective Hamiltonian which is not a normal operator, with incompatible (non-commuting) masses and widths. In the Weisskopf-Wigner Approach (WWA), by diagonalizing the entire Hamiltonian, the unphysical non-orthogonal "stationary" states $K_{L,S}$ are obtained. These states have complex eigenvalues whose real (imaginary) part does not coincide with the eigenvalues of the mass (width) matrix. In this work we describe the system as an open Lindblad-type quantum mechanical system due to Kaon decays. This approach, in terms of density matrices for initial and final states, provides a consistent probabilistic description, avoiding the standard problems because the width matrix becomes a composite operator not included in the Hamiltonian. We consider the dominant-decay channel to two pions, so that one of the Kaon states with definite lifetime becomes stable. This new approach provides results for the time dependent decay rates in agreement with those of the WWA.