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QUANTUM OSCILLATION PHENOMENA

Stefano De Leo, C. C. Nishi, PIETRO P. ROTELLI

Published Jan 1, 2002• 3 citations
OpenAlex

Abstract

We give a detailed analysis of the oscillation formula within the context of the wave packet formalism. Particular attention is made to insure flavor eigenstate creation in the physical cases (Delta p not equal 0). This requirement imposes non instantaneous particle creation in all frames. It is shown that the standard formula is not only exact when the mass wave packets have the same velocity, but it is a good approximation when minimal slippage occurs. For more general situations the oscillation formula contains additional arbitrary parameters, which allows for the unknown form of the wave packet envelope.

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We give a detailed analysis of the oscillation formula within the context of the wave packet formalism. Particular attention is made to insure flavor eigenstate creation in the physical cases (Delta p not equal 0). This requirement imposes non instantaneous particle creation in all frames. It is shown that the standard formula is not only exact when the mass wave packets have the same velocity, but it is a good approximation when minimal slippage occurs. For more general situations the oscillation formula contains additional arbitrary parameters, which allows for the unknown form of the wave packet envelope.
We give a detailed analysis of the oscillation formula within the context of the wave packet formalism. Particular attention is made to insure flavor eigenstate creation in the physical cases (Δp≠0). This requirement imposes noninstantaneous particle creation in all frames. It is shown that the standard formula is not only exact when the mass wave packets have the same velocity, but it is a good approximation when minimal slippage occurs. For more general situations the oscillation formula contains additional arbitrary parameters, which allows for the unknown form of the wave packet envelope.
We introduce an approximation scheme to perform an analytic study of the oscillation phenomena in a pedagogical and comprehensive way. By using Gaussian wave packets, we show that the oscillation is bounded by a time-dependent vanishing function which characterizes the slippage between the mass-eigenstate wave packets. We also demonstrate that the wave packet spreading represents a secondary effect which plays a significant role only in the nonrelativistic limit. In our analysis, we note the presence of a new time-dependent phase and calculate how this additional term modifies the oscillating character of the flavor conversion formula. Finally, by considering box and sine wave packets we study how the choice of different functions to describe the particle localization changes the oscillation probability.
We seek a quantum-theoretic expression for the probability that a "fermionic" particle which is initially in a well-defined flavor, linear combination of mass-eigenstates, will be found, at later times, in another flavor state. We approach this problem by using the Dirac equation as evolution equation for the mass-eigenstates. The Dirac formalism is useful and essential in keeping clear many of the conceptual aspects of quantum oscillation phenomena that naturally arise in a relativistic spin one-half particle theory. Our study leads to the conclusion that the fermionic nature of the particles and the interference between positive and negative frequency components of mass-eigenstate wave packets modify the standard oscillation probability, obtained by implicitly assuming a "scalar" nature of the mass-eigenstates. Nevertheless, under particular assumptions, i.e. ultra-relativistic particles, strictly peaked momentum distributions and minimal slippage, these modifications introduce correction factors proportional to [Formula: see text] which are practically un-detectable by any experimental analysis.
In this paper we discuss some aspects of the theory of wave packets. We consider a popular noncovariant Gaussian model used in various applications and show that it predicts too slow a longitudinal dispersion rate for relativistic particles. We revise this approach by considering a covariant model of Gaussian wave packets, and examine our results by inspecting a wave packet of arbitrary form. A general formula for the time dependence of the dispersion of a wave packet of arbitrary form is found. Finally, we give a transparent interpretation of the disappearance of the wave function over time due to the dispersion—a feature often considered undesirable, but which is unavoidable for wave packets. We find, starting from simple examples, proceeding with their generalizations and finally by considering the continuity equation, that the integral over time of both the flux and probability densities are asymptotically proportional to the factor 1/| x | 2 in the rest frame of the wave packet, just as in the case of an ensemble of classical particles.
The controversies regarding the oscillation formula are reviewed. The different existing quantum field treatments of oscillations are reconciled by a thorough study of the external wave packet model. Stationary models are shown to be a particular case of this model. All decoherence terms are computed explicitly, so that it becomes clear that the coherence length tends to infinity in the stationary limit. Decoherence is shown to originate not only in the separation of wave packets, but also in their dispersion. Relativistic and nonrelativistic oscillating particles are treated on the same footing. The oscillation formula is derived for an unstable particle within the same formalism. Finally, correlated oscillations are discussed in detail.
It has been firmly established that neutrinos change their flavour during propagation. This feature is attributed to the feature that each flavour eigenstate is a superposition of three mass eigenstates, which propagate with different frequencies. This picture, although widely accepted, is wrong in the simplest approach and requires quite sophisticated treatment based on the wave-packet description within quantum field theory. In this communication we present a novel, much simpler explanation and show that oscillations among massive particles can be obtained in a natural way. We use the framework of quantum mechanics with time being a physical observable, not just a parameter.