This article purports to be an html version of the PDF file — OPERA anomaly analysis, of the intended research paper, originally publicized in this place, on 15th Jan 2012.
The author is in the process of converting to a good web based version of the above PDF.
Author review II, 07 Feb 2012;
At near c = 1 we get a very big uncertainty on speed, our beta and gamma correspond to gamma*beta ~ 0.9673, if you check by hand or by Wolfram, Wolfram does not lead to the required level of accuracy of less than 25 ppm, one can take a very precise calculator and see that, but the idea is clear with present methods we have beta given by gamma*beta = 0.9673, gamma = 3.94, c_b= 15.
So with beta close to 1 we have gamma goes to 700 => 1000 => 10000 and so on. We stop at a place where beta is only 1 ppm less than 1, that is beta = .999999, we are done, gamma one gets by simple calculation from any high precise calculator with a required precision at this level.
This gives a huge number and c_b becomes much larger. Then we have delta_beta which is the uncertainty on speed a very large value even for say 100 KeV, which means at that speed of beta =0.999999 one just does not have any way to see superluminal speed. Our method is a good way to iterate computationally.
Author review I, 03 Feb 2012;
In our OPERA anomaly analysis our idea is to check the value of Lorentz factor gamma for beta as close to c=1. Then any excess will either correspond to an uncertainty of measurements given to the minimum imposed by nature, Quantum Mechanics or the excess is real and we have a problem with the validity of Theory of Relativity.
I cross-checked in Wolfram and our analysis constraint — given in our paper, see PDF version appended above, seems to correspond to beta ~0.9673. This value is not as close as we want. One way is to add more terms into our binomial summation instead of k=10, eg say k=25 or larger.
One of the things I am worried about is the fact that Michael Lugo’s summation on their mathoverflow.net site is perhaps restricted to 0 ≤ k ≤ N. So we need to find the correct way of adding the coefficients in the expansion of Lorentz factor gamma. In-fact from the cross-check with Wolfram it seems this is yet another confirmation that our result is also correct, given that we obtain gamma*beta ~ 0.9673.
We had obtained this in beta -> 1 limit, so the limit must be this accurate given that Wolfram possibly has an accurate formula in their software. For doing the calculations consistently one needs to adjust for the beta with its Wolfram value in the methods given in our paper.
We see that with this adjustment we get instead of 71 KeV as a null superluminal excess about 500 KeV constraint. In any case OPERA does not report any uncertainties on energy, so its not clear they have a 100 KeV error on energy of neutrino or even 100 MeV or a GeV error on the neutrino energy.
Once we have the value of either beta or gamma — gamma is convenient perhaps but gamma and beta are correlated, we have all other constraints and conclusions firmly in place. Then we can confidently say about the anomaly.
As you know OPERA anomaly is 0.0025% = 0.000025 ppm of speed of light. So we need to develop gamma, beta to be more accurate than this. Presently my Wolfram cross-check results suggest that instead of 71 KeV one has ~500 KeV for 7.5 km/s to be a null excess as its merely an uncertainty.
But then beta is 0.9673 which is much below c = 1. This is not a problem as any excess with energy uncertainty predicted by our method does not take us above speed of light until we know the actual uncertainty from OPERA.
I check from Mathoverflow.net another method and bigger k in old method. This gives a much higher value for gamma at beta~1 so 1.23 km/s error on speed as reported by OPERA itself is incurred by 11.65 GeV. Note that this is a huge error and if — wolfram+our method, is correct, then, OPERA is doing something crazy. In any case I need to review everything again.
In the absence of our method one has to fall back on Weinberg’s or Landau’s equations which are basic quantum mechanics, but perhaps they are not as accurate as our summation. One needs to cross check all the methods and predict the energy uncertainties.