An effect generated by the nonexponential behavior of the survival amplitude of an unstable state at the long time region is considered. It is known that this amplitude tends to zero as t goes to the infinity more slowly than any exponential function of t. Using methods of asymptotic analysis we find the asymptotic form of this amplitude in the long time region in a general model independent case. We find that the long time behavior of this amplitude affects the form of the instantaneous energy of unstable states: This energy should be much smaller for suitably long times t than the energy of this state for t of the order of the lifetime of the considered unstable state.
Approximate formulas are given for the effective Hamiltonian Hll(t} governing the time evolution in a subspace JVil of the state space %. It is proved that this approximation is correct for any Hamiltonian H
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 has 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 seems 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 p = 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-exponential form of the survival probability, 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 and that the survival probabilities should 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 computed decay curves of the moving particles have fluctuating form at relatively short times including times of order of the lifetime.
Abstract:An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as → ∞ more slowly than any exponential function of . This can be described in terms of the time-dependent decay rate γ( ) which, when considered with the Khalfin result, means that this γ( ) is not a constant for large but that it tends to zero as → ∞. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the "instantaneous energy" of the unstable state. This energy should be much smaller for suitably larger values of than when is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long ( → ∞) and relatively short (lifetime of the state) time regions, are different. This is a purely quantum mechanical effect. It is hypothesized that there is a possibility to detect this effect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.
By analyzing the survival probability amplitude of an unstable state we show that the energy corrections to this state in the long (t → ∞) and relatively short (lifetime of the state) time regions, are different. It is shown that in the considered model the above corrections decrease to zero as t → ∞. It is hypothesized that this property could be detected by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of possible deviations of the fine structure constant α as well as other astrophysical and cosmological parameters.
The Bell-Steinberger relation is analyzed. The questionable points of the standard derivation of this relation are discussed. It is shown that the use of a more accurate approximation than the one usually used in the derivation of this relation can lead to corrections to the right hand side of the standard Bell-Steinberger relation.
We find that charged unstable particles as well as neutral unstable particles with non-zero magnetic moment which live sufficiently long may emit electromagnetic radiation. This new mechanism is connected with the properties of unstable particles at the post exponential time region. Analyzing the transition times region between exponential and non-exponential form of the survival amplitude it is found that the instantaneous energy of the unstable particle can take very large values, much larger than the energy of this state for times from the exponential time region. Based on the results obtained for the model considered, it is shown that this purely quantum mechanical effect may be responsible for causing unstable particles to emit electromagnetic-, X-or γ-rays at some time intervals from the transition time regions.
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