The short-time behavior of an unstable particle is examined in a realistic field theoretical model. The onset of linear decreasing of the nondecay probability is shown to be extremely rapid, so rapid as to exclude any relevance of regimes quadratic in time (``Zeno'' quantum paradox). The result applies to super-renormalizable, renormalizable, and nonrenormalizable cases, particularly to the proton decay problem. We discuss also the deviations from the exponential decay law in small-Q-value decays.The short-time behavior of an unstable particle is examined in a realistic field theoretical model. The onset of linear decreasing of the nondecay probability is shown to be extremely rapid, so rapid as to exclude any relevance of regimes quadratic in time (``Zeno'' quantum paradox). The result applies to super-renormalizable, renormalizable, and nonrenormalizable cases, particularly to the proton decay problem. We discuss also the deviations from the exponential decay law in small-Q-value decays.

Physical Review LettersOct 25, 199361 citations

Decay of a quantum-mechanical state described by a truncated Lorentzian energy distribution

Kathleen M. Sluis, Eric A. Gislason

The nondecay probability Q(t) for a Lorentzian energy distribution peaked at ${\mathit{E}}_{0}$ with full width at half maximum equal to \ensuremath{\Gamma} is given in atomic units by ${\mathit{Q}}_{\mathit{L}}$ (t)=exp(-\ensuremath{\Gamma}t). However, the Lorentzian distribution includes energies in the range -\ensuremath{\infty}\ensuremath{\le}E\ensuremath{\le}\ensuremath{\infty}. On physical grounds, this distribution must be truncated at some minimum energy E=0. In this paper, Q(t) is derived for a properly normalized truncated Lorentzian energy distribution. To aid the analysis, the average lifetime \ensuremath{\tau}( ${\mathit{t}}_{\mathrm{min}}$ ) for that fraction of the system that has not decayed by ${\mathit{t}}_{\mathrm{min}}$ is also calculated for this system. For a given value of the ratio ${\mathit{E}}_{0}$ /\ensuremath{\Gamma}, the function Q(t) shows different behavior in different time regimes; this behavior is presented and the time periods are determined. It is shown that Q(t) falls off linearly at very small t, but at a rate only half as fast as for ${\mathit{Q}}_{\mathit{L}}$ (t). Thus this system cannot exhibit the behavior known as Zeno's paradox, but rapid measurements on this system that collapse the wave function back to the original state could, in principle, reduce the decay rate by a factor of 2.The nondecay probability Q(t) for a Lorentzian energy distribution peaked at ${\mathit{E}}_{0}$ with full width at half maximum equal to \ensuremath{\Gamma} is given in atomic units by ${\mathit{Q}}_{\mathit{L}}$(t)=exp(-\ensuremath{\Gamma}t). However, the Lorentzian distribution includes energies in the range -\ensuremath{\infty}\ensuremath{\le}E\ensuremath{\le}\ensuremath{\infty}. On physical grounds, this distribution must be truncated at some minimum energy E=0. In this paper, Q(t) is derived for a properly normalized truncated Lorentzian energy distribution. To aid the analysis, the average lifetime \ensuremath{\tau}(${\mathit{t}}_{\mathrm{min}}$) for that fraction of the system that has not decayed by ${\mathit{t}}_{\mathrm{min}}$ is also calculated for this system. For a given value of the ratio ${\mathit{E}}_{0}$/\ensuremath{\Gamma}, the function Q(t) shows different behavior in different time regimes; this behavior is presented and the time periods are determined. It is shown that Q(t) falls off linearly at very small t, but at a rate only half as fast as for ${\mathit{Q}}_{\mathit{L}}$(t). Thus this system cannot exhibit the behavior known as Zeno's paradox, but rapid measurements on this system that collapse the wave function back to the original state could, in principle, reduce the decay rate by a factor of 2.

Physical Review AMay 1, 199132 citations

On the Significance of Quasi-Probability in Quantum Mechanics

Kazuki Fukuda

Mar 24, 20160 citations

On the Absence of the Zeno Effect in Relativistic Quantum Field Theory

R. F. Álvarez-Estrada, José L. Sánchez-gómez

We study the time evolution of decaying particles in renormalizable models of Relativistic Quantum Field Theory. Significant differences between the latter and Non Relativistic Quantum Mechanics are found —in particular, the Zeno effect seems to be absent in such RQFT models. Conventional renormalization yields finite time behaviour in some cases but fails to produce finite survival probabilities in others. The analysis of the finite time evolution in relativistic quantum field theory (RQFT) and, in particular, the behaviour at short times of the survival probability of unstable particles in RQFT is a relevant matter as far as the foundations of quantum mechanics are concerned and has some peculiarities which make it to be worth studying. But before going to RQFT, we will briefly recapitulate the main features of the survival amplitude (SA) at short times in non relativistic quantum mechanics (QM henceforth); see, for instance, [1] for a comprehensive review.We study the time evolution of decaying particles in renormalizable models of Relativistic Quantum Field Theory. Significant differences between the latter and Non Relativistic Quantum Mechanics are found —in particular, the Zeno effect seems to be absent in such RQFT models. Conventional renormalization yields finite time behaviour in some cases but fails to produce finite survival probabilities in others. The analysis of the finite time evolution in relativistic quantum field theory (RQFT) and, in particular, the behaviour at short times of the survival probability of unstable particles in RQFT is a relevant matter as far as the foundations of quantum mechanics are concerned and has some peculiarities which make it to be worth studying. But before going to RQFT, we will briefly recapitulate the main features of the survival amplitude (SA) at short times in non relativistic quantum mechanics (QM henceforth); see, for instance, [1] for a comprehensive review.

Jan 1, 20087 citations

Problems of Probability Theory in the Light of Quantum Mechanics

Martin Strauss

Springer eBooksJan 1, 19722 citations

The nondecay probability Q(t) for a Lorentzian energy distribution peaked at

{\mathit{E}}_{0}

with full width at half maximum equal to \ensuremath{\Gamma} is given in atomic units by

{\mathit{Q}}_{\mathit{L}}

(t)=exp(-\ensuremath{\Gamma}t). However, the Lorentzian distribution includes energies in the range -\ensuremath{\infty}\ensuremath{\le}E\ensuremath{\le}\ensuremath{\infty}. On physical grounds, this distribution must be truncated at some minimum energy E=0. In this paper, Q(t) is derived for a properly normalized truncated Lorentzian energy distribution. To aid the analysis, the average lifetime \ensuremath{\tau}(

{\mathit{t}}_{\mathrm{min}}

) for that fraction of the system that has not decayed by

{\mathit{t}}_{\mathrm{min}}

is also calculated for this system. For a given value of the ratio

{\mathit{E}}_{0}

/\ensuremath{\Gamma}, the function Q(t) shows different behavior in different time regimes; this behavior is presented and the time periods are determined. It is shown that Q(t) falls off linearly at very small t, but at a rate only half as fast as for

{\mathit{Q}}_{\mathit{L}}

(t). Thus this system cannot exhibit the behavior known as Zeno's paradox, but rapid measurements on this system that collapse the wave function back to the original state could, in principle, reduce the decay rate by a factor of 2.