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arXiv:0803.0935

Unitarity Constraint upon Kinematical Analyses of the GSI Time-Modulated Radioactive Decay Experiment

Murray Peshkin

Published Mar 6, 2008• 1 citation
arXivHTMLPDF

Abstract

It is tempting to try to explain the reported time-modulated decay rate in the GSI experiment by arguing that the matrix element for decay at any time results in an interference between two slightly different momentum values in the parent ion beam. I show here that a unitarity theorem of quantum mechanics rules out a wide class of such explanations.

DOI
10.48550/arxiv.0803.0935

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It is tempting to try to explain the reported time-modulated decay rate in the GSI experiment by arguing that the matrix element for decay at any time results in an interference between two slightly different momentum values in the parent ion beam. I show here that a unitarity theorem of quantum mechanics rules out a wide class of such explanations.
We study the deviations from the usual exponential decay law for quantum mechanical systems. We show that simple and physically motivated deviations from the Breit-Wigner energy distribution of the unstable state are sufficient to generate peculiar deviations from the exponential decay law. Denoting with p(t)p(t)p(t) the survival probability, its derivative h(t)h(t)h(t) shows typically an oscillating behavior on top of the usual exponential function. We argue that this can be a viable explanation of the observed experimental results at GSI Darmstadt, where the function h(t)h(t)h(t) has been experimentally measured for electron capture decays of Hydrogen-like ions. Moreover, if our interpretation is correct, we predict that by measuring h(t)h(t)h(t) at times close to the initial one, the number of decays per second rapidly drops to zero.
The basic requirement that, in quantum theory, the time evolution of any state is determined by the action of a unitary operator, is shown to be the underlying cause for certain ``exact'' results that have recently been reported about the time dependence of transition rates in quantum theory. Departures from exponential decay, including the ``quantum Zeno effect,'' as well as a theorem by Khalfin about the ratio of reciprocal transition rates, are shown to follow directly from such considerations. At sufficiently short times, unitarity requires that reciprocity must hold, independent of whether T invariance is valid. If T invariance does not hold, unitarity restricts the form of possible time dependence of reciprocity ratios. \textcopyright{} 1996 The American Physical Society.
The quantum-mechanical theory of the decay of unstable states is revisited. We show that the decay is non-exponential both in the short-time and long-time limits using a more physical definition of the decay rate than the one usually used. We report results of numerical studies based on Winter's model that may elucidate qualitative features of exponential and non-exponential decay more generally. The main exponential stage is related to the formation of a radiating state that maintains the shape of its wave function with exponentially diminishing normalization. We discuss situations where the radioactive decay displays several exponents. The transient stages between different regimes are typically accompanied by interference of various contributions and resulting oscillations in the decay curve. The decay curve can be fully oscillatory in a two-flavor generalization of Winter's model with some values of the parameters. We consider the implications of that result for models of the oscillations reported by GSI.
In this article, I provide a pedagogical discussion of the GSI anomaly. Using two different formulations, namely the intuitive quantum-field-theory language of the second quantized picture as well as the language of amplitudes, I clear up the analogies and differences between the GSI anomaly and other processes (the double-slit experiment using photons, ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ scattering, and charged-pion decay). In both formulations, the conclusion is reached that the decay rate measured at GSI cannot oscillate if only standard model physics is involved and the initial hydrogen-like ion is no coherent superposition of more than one state (in case there is no new, yet unknown, mechanism at work). Furthermore, a discussion of the quantum beat phenomenon will be given, which is often assumed to be able to cause the observed oscillations. This is, however, not possible for a splitting in the final state only.