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arXiv:1803.07709 · Advances in High Energy Physics

Time Dilation in Relativistic Quantum Decay Laws of Moving Unstable Particles

Filippo Giraldi

Published May 29, 2018• 4 citations
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Abstract

The relativistic quantum decay laws of moving unstable particles are analyzed for a general class of mass distribution densities which behave as power laws near the (nonvanishing) lower bound μ0 of the mass spectrum. The survival probability Pp(t) , the instantaneous mass Mp(t) , and the instantaneous decay rate Γp(t) of the moving unstable particle are evaluated over short and long times for an arbitrary value p of the (constant) linear momentum. The ultrarelativistic and nonrelativistic limits are studied. Over long times, the survival probability Pp(t) is approximately related to the survival probability at rest P0(t) by a scaling law. The scaling law can be interpreted as the effect of the relativistic time dilation if the asymptotic value Mp∞ of the instantaneous mass is considered as the effective mass of the unstable particle over long times. The effective mass has magnitude μ0 at rest and moves with linear momentum p or, equivalently, with constant velocity 1/1+μ02/p2 . The instantaneous decay rate Γp(t) is approximately independent of the linear momentum p , over long times, and, consequently, is approximately invariant by changing reference frame.

DOI
10.1155/2018/7308935

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The relativistic quantum decay laws of moving unstable particles are analyzed for a general class of mass distribution densities which behave as power laws near the (nonvanishing) lower bound μ0 of the mass spectrum. The survival probability Pp(t) , the instantaneous mass Mp(t) , and the instantaneous decay rate Γp(t) of the moving unstable particle are evaluated over short and long times for an arbitrary value p of the (constant) linear momentum. The ultrarelativistic and nonrelativistic limits are studied. Over long times, the survival probability Pp(t) is approximately related to the survival probability at rest P0(t) by a scaling law. The scaling law can be interpreted as the effect of the relativistic time dilation if the asymptotic value Mp∞ of the instantaneous mass is considered as the effective mass of the unstable particle over long times. The effective mass has magnitude μ0 at rest and moves with linear momentum p or, equivalently, with constant velocity 1/1+μ02/p2 . The instantaneous decay rate Γp(t) is approximately independent of the linear momentum p , over long times, and, consequently, is approximately invariant by changing reference frame.
Abstract The decay of a moving system is studied in case the system is initially prepared in a superposition of two, approximately orthogonal, unstable quantum states. The survival probability is evaluated over short and long times in the reference frame where the unstable system moves with constant linear momentum p . The mass distribution densities, which describe the two unstable states of the superposition, exhibit thresholds near the non-vanishing lower bounds and of the mass spectra. If these lower bounds coincide, the long-time survival probability shows a dominant inverse-power-law decay and is approximately related to the survival probability at rest via a time dilation. The scaling factor reads . If the lower bounds and differ, damped oscillations of the survival probability appear over long times and the scaling relation is lost. These oscillations are enveloped in an inverse-power-low profile which is determined by the thresholds. By changing reference frame, the period T 0 of the oscillations at rest transforms in the longer period T p according to a factor which is the weighted mean of the factors and , with non-normalized weights and . The oscillations tend to vanish in the ultrarelativistic limit.
Abstract The transformation of canonical decay laws of moving unstable quantum systems is derived from the basic principles of quantum theory and special relativity by approximating, over intermediate times, the decay laws at rest with superpositions of exponential modes via the Prony analysis. The survival probability , which is detected in the laboratory reference frame where the unstable system moves with constant linear momentum p , is represented by the transformed form of the survival probability at rest . The transformation of the intermediate times, which is induced by the change of reference frame, is obtained by evaluating the function . Under determined conditions, this function grows linearly and the survival probability transforms, approximately, according to a scaling law over an estimated time window. The relativistic dilation of times holds, approximately, over the time window if the mass of resonance of the mass distribution density is considered to be the effective mass at rest of the moving unstable quantum system.
Late time properties of moving relativistic particles are studied. Within the proper relativistic treatment of the problem we find decay curves of such particles and we 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 amplitude, occur much earlier than it follows from the classical standard approach boiled down to replace time t by t/γL (where γL is the relativistic Lorentz factor) in the formula for the survival probability. The consequence is that fluctuations of the corresponding decay curves can appear much earlier and much more unstable particles have a chance to survive up to these times or later. It is also shown that fluctuations of the instantaneous energy of the moving unstable particles have a similar form as the fluctuations in the particle rest frame but they are seen by the observer in his rest system much earlier than one could expect replacing t by t/γL in the corresponding expressions for this energy and that the amplitude of these fluctuations can be even larger than it follows from the standard approach. All these effects seem to be important when interpreting some accelerator experiments with high energy unstable particles and the like (possible connections of these effects with GSI anomaly are analyzed) and some results of astrophysical observations.
We study the survival probability of moving relativistic unstable particles with definite momentum p⃗≠0\vec{p} \neq 0p​=0. The amplitude of the survival probability of these particles is calculated using its integral representation. 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-expo\-nen\-tial form of the survival probability, should occur much earlier than it follows from the classical standard approach resolving itself into replacing time ttt by t/γt/γt/γ (where γγγ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities should tend to zero as t→∞t\rightarrow \inftyt→∞ 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 computed decay curves of the moving particles have fluctuating form at relatively short times including times of order of the lifetime.
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.
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.