In the context of the sensational results concerning superluminal velocities, announced recently by the OPERA Collaboration, we have proposed a classical model yielding a statistically calculated measured velocity of a beam, higher than the velocity of the particles constituting the beam. The two key elements of our model, necessary and sufficient to obtain this curious result, are a time-dependent "transmission" function and statistical method of the maximum-likelihood estimation.
Inspired by the amazing results of the OPERA Collaboration [1], claiming that the velocity of light c has been beaten by a beam of neutrinos, we propose a simple classical model, which yields the curious effect of a seeming increase of "effective" velocity (see [2], for an earlier independent approach, and also [3], for a wave version). There are dozens of papers, which have recently appeared, adopting various attitudes towards the results presented by the OPERA Collaboration, for example [4,5,6,7,8] (see also an example of an older work on superluminal velocities [9]). The attitude of our paper is definitely skeptical. In view of the fact that we have found a ("mathematical") model providing an "artificial" increase of real velocity, we are forced to put the results announced by the OPERA Collaboration in doubt. Moreover, we have also proposed a simple working example, such that appropriately fitting its parameters one can simulate the controversial results of the OPERA Collaboration. Strictly speaking, our paper does not indisputably invalidates the conclusions drawn by the OPERA Collaboration, but it seriously weakens their argumentation, indicating a logical gap in their reasoning. Our model is purely classical and dynamics free. No new physics, nor quantum mechanics, nor even (classical) wave mechanics is involved. Only standard classical kinematics notions as well as the statistical method of the maximum-likelihood estimation (MLE) are used in our approach. The assumptions of our model are the following. A spatially homogeneous, lasting the period T , beam of classical particles ("extraction") moving with a constant speed u (≦ c) travels from a source to a detector. The distance between the source and the detector is d. The probability density function (PDF) of the time of emission of the particles within the duration T of production of the beam is given by the function w(t). In an ideal situation (none of the particles is lost) we would obtain, according to classical kinematics, the measured data waveform y(t) = w(tt 0 ), where t 0 = d/u. Now, let as suppose that the fraction of the particles emitted, measured by the detector, due to some physical mechanism, is given by the (non-negative) "transmission" function
where N e (t) is the number of the particles emitted at the time instant t, and N d (t) is the number of the particles detected, which have been sent at the same time instant t. In other words, in general, not all particles emitted are detected (f (t) < 1) -obvious, and moreover f (t) = const -conceivable.
For simplicity, we will assume that the numbers of the particles, N d (t) and N e (t), are so large that we are allowed to use a continuous approximation. Then, the transmission function
To be able to draw conclusions from experimental data, we should implement some statistical methodology. To be so precise as possible, in our approach, we adopt the method of the MLE, as the OPERA Collaboration has done [1]. In the framework of the MLE, we introduce the likelihood function L.
The logarithm l of L is given by the formula
where t j are the time instants corresponding to the measurement events at the detector, and the time deviation δt we are interested in (see [1], for details) provides the maximum of l (and also of L). As the numbers of the measured events t j are large, in our continuous approach, instead of summation in (2), we should use integration with an appropriate integration measure z(t)dt, where z(t) represents the time distribution of the experimental events detected by the OPERA. In fact, z(t) is determined by product of two factors. The first factor, y, is proportional to the number of the particles sent, i.e. y(t + t 0 ) = w(t), and the second one is proportional to the transmission function f (t). Then,
Finally,
To demonstrate that our idea actually works, we propose a specific example. The parameters of the example are so fitted that it yields the time deviation δt ≈ +75.5 ns (for comparison, the OPERA result is 60.7 ns). For calculational simplicity, we assume the Gaussian form of the PDF (see the solid line in Fig. 1),
as well as the Gaussian form of of the transmission function,
where the one tenth in front of the exponent is reminiscent of “10% variation” in [2]. The time distribution of the “detected experimental data” z(t), corresponding to w(t) of the form (5) and to the transmission function f (t) of the form ( 6) is presented in Fig. 1 by the dashed curve. In this (doubly) Gaussian case it is even possible to solve the problem analytically (see Appendix), but
The presented specific example is intended to “mathematically” qualitatively simulate the OPERA experiment, yielding the time deviation δt ≈ +75.5 ns. The solid curve represents the probability density function (PDF) w(t), whereas the dashed one corresponds to the time distribution of the “detected experimental data” z(t).
for our purposes a numerical value will do. It appears, that the maximum of (4)
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