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System-free quantum mechanical description of particle decay processes

George Jaroszkiewicz

Published Aug 31, 2006• 0 citations
OpenAlex

Abstract

Using the formalism of system-free quantum mechanics, we show how quantum mechanical particle decay probabilities can be discussed rigorously within a framework that conserves total probability, requiring neither non-Hermitian Hamiltonians nor the ad-hoc introduction of complex energies. We apply our formalism to single channel decays, the ammonium molecule, and neutral Kaon decay processes.

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Using the formalism of system-free quantum mechanics, we show how quantum mechanical particle decay probabilities can be discussed rigorously within a framework that conserves total probability, requiring neither non-Hermitian Hamiltonians nor the ad-hoc introduction of complex energies. We apply our formalism to single channel decays, the ammonium molecule, and neutral Kaon decay processes.
Utilizing the GKLS master equation we show that the decay property of a particle can be straightforwardly incorporated. In standard particle physics the decay is often described by an efficient non-hermitian Hamiltonian, in accord with the seminal Wigner-Weisskopf approximation. We show that by enlarging the Hilbert space and defining specific GKLS operators we have attained a formalism with a hermitian Hamiltonian and probability conserving states. This proves that the Wigner-Weisskopf approximation is Markovian and completely positive. In addition, this formalism allows a straightforward generalization to many-particle decays. Last, but not least, some impacts of the GKLS master equation onto systems at high energies are reported, such as for neutral meson, neutrino and hyperon systems.
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 KL,SK_{L,S}KL,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.
The purpose of this dissertation is to review the salient features of non-Hermitian quantum mechanics. An introduction to Hermitian quantum mechanics is included to make this review as accessible as possible. Attempts at formulating a consistent physical theory are introduced, before examining non-Hermitian theories’ uses as convenient computational frameworks. Particular emphasis is placed on recent developments in open quantum systems that utilise non-Hermitian Hamiltonians. Chapter four introduces a logic that maps a non-Hermitian Hamiltonian onto a non-Hamiltonian algebra that has a Hermitian Hamiltonian. This was put forward by Sergi, who then goes on to show its application to a two level system. The time evolution is then derived in terms of the density matrix model. This system can then be used to analyse different types of decay such as coherence and population difference. This serves to illustrate the usefulness of the approach.