Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1104.1815

On the Quantum Deviations from Einstein Dilation of Unstable Quanton Decay Evolution and Lifetimes

Gordon N. Fleming

Published Apr 10, 2011• 0 citations
arXivHTMLPDF

Abstract

For over a decade several workers have argued for the existence of quantum deviations from the classical, Einstein dilation of the decay evolution of moving or Lorentz boosted unstable particles. While the general claim is correct, the discussions have been incomplete and, sometimes, misleading. The discussions have been of three kinds. Type 1 examines the time dependence of the survival probability for 3-momentum eigenstates of the unstable quanton (Khalfin). Type 2 does the same for velocity eigenstates, obtaining an outrageous result which then discredits velocity eigenstates (Shirokov / Hegerfeldt). Type 3 examines arbitrary boosts of 3-momentum eigenstates (Stefanovich). Type 1 is incomplete since the momentum eigenstates are not the boosts of one another. Type 2 is misleading since the outrageous result is due to misinterpreting the initial conditions of the velocity eigenstates (as I have previously argued). Type 3 is the most satisfactory, but has failed to recognize and implement the unification of all three types of discussion that can be achieved. In this paper I will provide that unified treatment, beginning with a recapitulation of Type 1 and offering further clarification of Type 2 in the process. The unified treatment fully reinstates velocity eigenstates as essential contributors to unstable quanton states. Besides discussing the time evolution of survival probabilities I also focus on the concept of lifetime defined as the average time of decay. This quantity is helpful in order to display the inequivalent dependence of dilation on momentum and boosts most sharply and the deviation from Einstein dilation most cleanly.

DOI
10.48550/arxiv.1104.1815

Related papers

1 On the Quantum Deviations from Einstein Dilation of Unstable Quanton * Decay Evolution and Lifetimes

Gordon N. Fleming

Abstract: For over a decade several workers have argued for the existence of quantum deviations from the classical, Einstein dilation of the decay evolution of moving or Lorentz boosted unstable particles. While the general claim is correct, the discussions have been incomplete and, sometimes, misleading. The discussions have been of three kinds. Type 1 examines the time dependence of the survival probability for 3-momentum eigenstates of the unstable quanton (Khalfin). Type 2 does the same for velocity eigenstates, obtaining an outrageous result which then discredits velocity eigenstates (Shirokov / Hegerfeldt). Type 3 examines arbitrary boosts of 3-momentum eigenstates (Stefanovich). Type 1 is incomplete since the momentum eigenstates are not the boosts of one another. Type 2 is misleading since the outrageous result is due to misinterpreting the initial conditions of the velocity eigenstates (as I have previously argued). Type 3 is the most satisfactory, but has failed to recognize and implement the unification of all three types of discussion that can be achieved. In this paper I will provide that unified treatment, beginning with a recapitulation of Type 1 and offering further clarification of Type 2 in the process. The unified treatment fully reinstates velocity eigenstates as essential contributors to unstable quanton states. Besides discussing the time evolution of survival probabilities I also focus on the concept of lifetime defined as the average time of decay. This quantity is helpful in order to display the inequivalent dependence of dilation on momentum and boosts most sharply and the deviation from Einstein dilation most cleanly.

•Oct 31, 2015•0 citations

Decay Law and Time Dilatation

Francesco Giacosa

We study the decay law for a moving unstable particle. The usual time-dilatation formula states that the decay width for an unstable state moving with a momentum $p$ and mass $M$ is $\tildeΓ_{p}=ΓM/\sqrt{p^{2}+M^{2}}$ with $Γ$ being the decay width in the rest frame. In agreement with previous studies, we show that in the context of QM as well as QFT this equation is \textit{not} correct provided that the quantum measurement is performed in a reference frame in which the unstable particle has momentum $p$ (note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM). We then give, to our knowledge for the first time, an analytic expression of an improved formula and we show that the deviation from $\tildeΓ_{p}$ has a maximum for $p/M=\sqrt{2/3},$ but is typically \textit{very} small. Then, the result can be easily generalized to a momentum wave packet and also to an arbitrary initial state. Here, we give a very general expression of the non-decay probability. As a next step, we show that care is needed when one makes a boost of an unstable state with zero momentum/velocity: namely, the resulting state has zero overlap with the elements of the basis of unstable states (it is already decayed!). However, when considering a spread in velocity, one finds again that $\tilde{Γ}_{p}$ is typically a very good approximation. The study of a velocity wave-packet represents an interesting subject which constitutes one of the main outcomes of the present manuscript.\textbf{ }In the end, it should be stressed that there is no whatsoever breaking of special relativity, but as usual in QM, one should specify which kind of measurement on which kind of state is performed.

arXiv:1512.00232•Acta Physica Polonica B•Jan 1, 2016•17 citations

Shirokov's contracting lifetimes and the interpretation of velocity eigenstates for unstable quantons

Gordon N. Fleming

This paper is concerned with the interpretation of velocity eigenstates for unstable quantons, their relationship to space like momentum eigenstates for such quantons and the explanation of Shirokovs contracting lifetimes for such velocity eigenstates. It is an elaboration of a portion of the authors earlier study.

arXiv:0910.5448•Oct 28, 2009•1 citation

Properties of decaying relativistic particles: "GSI anomaly" like effects

K. Urbanowski

We study the survival probability of moving relativistic unstable particles and we found the integral representation of the survival amplitude for such particles moving with constant momentum ~ 6 0. Using this representation of the amplitude we found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival probability, occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t/γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities tend to zero as t → ∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the properly computed decay curves of the moving particles have analogous fluctuating form as the decay curves obtained in the GSI experiment.

•Jan 1, 2015•1 citation

Evolution in Time of Moving Unstable Systems

M. I. Shirokov

Relativistic quantum theory shows that the known Einstein time dilation (ED) approximately holds for the decay law of the unstable particle having definite momentum p (DP). I use a different definition of the moving particle as the state with definite velocity v (DV). It is shown that in this case the decay law is not dilated. On the contrary, it is contracted as compared with the decay law of the particle at rest. It is demonstrated that ED fails in both DP and DV cases for time evolution of the simple unstable system of the kind of oscillating neutrino. Experiments are known which show that ED holds for mesons. The used theory may explain the fact by supposing that the measured mesons are in DP state.

arXiv:quant-ph/0508087•Aug 11, 2005•11 citations

Nonclassical face of quantum decay processes

K. Urbanowski

A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).

Journal of Physics Conference Series•Aug 1, 2017•0 citations
For over a decade several workers have argued for the existence of quantum deviations from the classical, Einstein dilation of the decay evolution of moving or Lorentz boosted unstable particles. While the general claim is correct, the discussions have been incomplete and, sometimes, misleading. The discussions have been of three kinds. Type 1 examines the time dependence of the survival probability for 3-momentum eigenstates of the unstable quanton (Khalfin). Type 2 does the same for velocity eigenstates, obtaining an outrageous result which then discredits velocity eigenstates (Shirokov / Hegerfeldt). Type 3 examines arbitrary boosts of 3-momentum eigenstates (Stefanovich). Type 1 is incomplete since the momentum eigenstates are not the boosts of one another. Type 2 is misleading since the outrageous result is due to misinterpreting the initial conditions of the velocity eigenstates (as I have previously argued). Type 3 is the most satisfactory, but has failed to recognize and implement the unification of all three types of discussion that can be achieved. In this paper I will provide that unified treatment, beginning with a recapitulation of Type 1 and offering further clarification of Type 2 in the process. The unified treatment fully reinstates velocity eigenstates as essential contributors to unstable quanton states. Besides discussing the time evolution of survival probabilities I also focus on the concept of lifetime defined as the average time of decay. This quantity is helpful in order to display the inequivalent dependence of dilation on momentum and boosts most sharply and the deviation from Einstein dilation most cleanly.
Abstract: For over a decade several workers have argued for the existence of quantum deviations from the classical, Einstein dilation of the decay evolution of moving or Lorentz boosted unstable particles. While the general claim is correct, the discussions have been incomplete and, sometimes, misleading. The discussions have been of three kinds. Type 1 examines the time dependence of the survival probability for 3-momentum eigenstates of the unstable quanton (Khalfin). Type 2 does the same for velocity eigenstates, obtaining an outrageous result which then discredits velocity eigenstates (Shirokov / Hegerfeldt). Type 3 examines arbitrary boosts of 3-momentum eigenstates (Stefanovich). Type 1 is incomplete since the momentum eigenstates are not the boosts of one another. Type 2 is misleading since the outrageous result is due to misinterpreting the initial conditions of the velocity eigenstates (as I have previously argued). Type 3 is the most satisfactory, but has failed to recognize and implement the unification of all three types of discussion that can be achieved. In this paper I will provide that unified treatment, beginning with a recapitulation of Type 1 and offering further clarification of Type 2 in the process. The unified treatment fully reinstates velocity eigenstates as essential contributors to unstable quanton states. Besides discussing the time evolution of survival probabilities I also focus on the concept of lifetime defined as the average time of decay. This quantity is helpful in order to display the inequivalent dependence of dilation on momentum and boosts most sharply and the deviation from Einstein dilation most cleanly.
We study the decay law for a moving unstable particle. The usual time-dilatation formula states that the decay width for an unstable state moving with a momentum ppp and mass MMM is Γ~p=ΓM/p2+M2\tildeΓ_{p}=ΓM/\sqrt{p^{2}+M^{2}}Γ~p​=ΓM/p2+M2​ with ΓΓΓ being the decay width in the rest frame. In agreement with previous studies, we show that in the context of QM as well as QFT this equation is \textit{not} correct provided that the quantum measurement is performed in a reference frame in which the unstable particle has momentum ppp (note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM). We then give, to our knowledge for the first time, an analytic expression of an improved formula and we show that the deviation from Γ~p\tildeΓ_{p}Γ~p​ has a maximum for p/M=2/3,p/M=\sqrt{2/3},p/M=2/3​, but is typically \textit{very} small. Then, the result can be easily generalized to a momentum wave packet and also to an arbitrary initial state. Here, we give a very general expression of the non-decay probability. As a next step, we show that care is needed when one makes a boost of an unstable state with zero momentum/velocity: namely, the resulting state has zero overlap with the elements of the basis of unstable states (it is already decayed!). However, when considering a spread in velocity, one finds again that Γ~p\tilde{Γ}_{p}Γ~p​ is typically a very good approximation. The study of a velocity wave-packet represents an interesting subject which constitutes one of the main outcomes of the present manuscript.\textbf{ }In the end, it should be stressed that there is no whatsoever breaking of special relativity, but as usual in QM, one should specify which kind of measurement on which kind of state is performed.
We study the survival probability of moving relativistic unstable particles and we found the integral representation of the survival amplitude for such particles moving with constant momentum ~ 6 0. Using this representation of the amplitude we found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival probability, occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t/γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities tend to zero as t → ∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the properly computed decay curves of the moving particles have analogous fluctuating form as the decay curves obtained in the GSI experiment.
Relativistic quantum theory shows that the known Einstein time dilation (ED) approximately holds for the decay law of the unstable particle having definite momentum p (DP). I use a different definition of the moving particle as the state with definite velocity v (DV). It is shown that in this case the decay law is not dilated. On the contrary, it is contracted as compared with the decay law of the particle at rest. It is demonstrated that ED fails in both DP and DV cases for time evolution of the simple unstable system of the kind of oscillating neutrino. Experiments are known which show that ED holds for mesons. The used theory may explain the fact by supposing that the measured mesons are in DP state.
A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).