## Sunday, January 15, 2012

### OPERA anomaly analysis

SCROLL_DOWN to read the paper.
Here is a latest status from the author-review of the paper

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 let 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 comupatationally.

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) 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. Infact 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 unceratainty. Butt 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 untill 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.
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If you want to cite our version

Citation: "OPERA neutrino anomaly is a result of not interpreting energy uncertainty.", Manmohan Dash and Mikael Franzen, Communicating Science, April 11, 2011, publisher: Invariance Publishing House, MDash Foundation

Link to permaweblink in citation or copy/archive material as it is with Creative Commons and other copyrights in this web-site in mind: permaweblink
Our citations might slightly change depending on exact source where we placed our ideas first. eg we may want to cite our online journals "Various Musings" if we formalize it and make it permanent.

# OPERA neutrino anomaly is a result of not interpreting energy uncertainty.

## First publicized/Available Online: January 15, 2011 or earlier (Linked to other articles/copies/files Article-1, Article-2)

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This is Version-1.7, Ver-1.6 at the bottom
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# OPERA neutrino anomaly is a result of not interpreting energy uncertainty.

Informal association with Willgood Institute, registered in Sweden,
author’s mail: Mahisapat, Dhenkanal, Odisha, India, 759001
previously affiliated with VT, USA and KEK, Japan.
Willgood Institute, Luckvägen 5, 517 37 Bollebygd, Sweden
i3tex AB, Klippan 1A , 414 51 Gothenburg, Sweden
Abstract
In this paper we bring out a remarkable consistency of theory of Relativity in explaining the anomalous excess of speed of neutrinos observed in the recent baseline experiment of OPERA. The OPERA experiment is performed by shooting neutrinos produced from protons at SPS, CERN to the laboratory at Gran-Sasso where OPERA has placed its neutrino brick detectors. We believe that we have found the reason why this result was misinterpreted to claim superluminal neutrinos.. The energy uncertainties inherently present in the OPERA neutrino measurement have not been reported on the claims of speed excess. The basics of Quantum Mechanics on the kinematic aspects of these neutrinos is pointed out in this paper. We make a minimal review of this negligence of uncertainties which is sufficient to see where OPERA has lacked a cautious sight in claiming superluminal neutrinos. We perform a rigorous check of Quantum Mechanics uncertainty principle in terms of Energy-Time to make our claim of lack of any evidence of superluminal neutrino.

Key-words: OPERA experiment, neutrino speed anomaly, Special Relativity, energy-time uncertainty, speed-time uncertainty, Compton wavelength of neutrino, superluminal neutrino

# 1 Introduction

In this paper we provide a stringent condition on the minimum uncertainty on energy one deals with, on any particle with energy E momentum p and rest mass m. We find a relation between the uncertainty on speed and time following directly from the uncertainty of energy and time. Our relation is general and expressed in terms of the Compton wavelength of any particle, in particular the neutrino from OPERA experiment [6]. This is a very accurate form of speed-time uncertainty relationship derived from energy-time uncertainty relationship. In our calculations we have made careful attempts to be consistent with the units of speed-of-light. Our result is valid for ultra-relativistic conditions of OPERA as much as it is valid for any particle speed., down to the lowest β one can theorize. All we do for OPERA situation is let our β → 1. We do not use ultra-relativistic conditions except when evaluating constants in the case of OPERA neutrinos. Our expressions are valid for a relativistic treatment of general nature.
We take note of the fact that experiment and theory of nature must, be consistent with what we know, and thus where applicable, results must confirm the famous uncertainty principle of Heisenberg.
From the logical edifice of Relativity theory and it’s most popular concepts follows that there are 3 uncertainty principles, but in terms of equivalence, only 1 uncertainty principle is chosen as per the necessity of the physical problem at hand. Here we chose the explicit form of Energy-Time uncertainty relationship, because a baseline speed measurement rests on such a situation.
The OPERA experiment measures it’s neutrino speed by claiming a very precise timing aided by the GPS satellite system for such measurements. This entails them a millimeter level accuracy in distance and a ns level accuracy in time at-least as per the specification of their GPS receivers. Since we have done much prior analysis that shows that GPS satellites in their circular orbits are very very well understood as per special and general theory of relativity, we do not ascertain any source of inaccuracy here. We mention that gravity of earth size objects is {Sr = 2.GMe} in itself a millimeter level accuracy. The exact value depends on the specific parameters of the problem and the separation from the gravity-source. The 25 ppm speed-of-light excess of the OPERA experiment in terms of absolute speed is a  ~ 7.5 km ⁄ s excess. Such a large fallout in the speed-of-light is an unexpectedly large fallout with respect to the theory of Relativity.
The conclusion we draw is one that has a millimeter level accuracy on GPS distance and nanosecond level accuracy on GPS time. Hence one must see a millimeter level speed excess as in other cases of theory of Relativity paradigm if interpreted correctly. This indicates that the further complicacies in OPERA situation comes for two reasons. i. We are dealing with elementary particles whose masses are the smallest we know in the physical world ii. These particles have speeds that are immensely relativistic. For these two reasons one does not see a minimum in the millimeter range. In-fact reason-i is dominant as masses can vary over a wider scale. The relativistic factor; reason-ii, does not vary as much. eg the OPERA neutrinos and any electrons moving at about the same speed have the same factor. But, for these two cases the minimum neutrino uncertainty is at 2.09 − meters ⁄ seconds where as for electrons this will be (0.511 ⁄ 2) × 106 times less. This is for a 10 − ns GPS aided time precision {and any type of time precision in general}. The electrons moving at about speed-of-light will be uncertain of their speed at-least by;
$\dpi{150} \bg_green \left(\frac{4.18}{0.511}\right)\times10^{-6}\, m/s=\mathbf{8.18\,\mu m/s;\; electron's-minimum.}$
We mention in advance that in this paper we determine for a 2 eV neutrino a minimum of  ~ 2.09 m ⁄ s uncertainty in speed either below or above speed of light.
2.09 m ⁄ s; neutrino’s − minimum.
A reinterpretation of OPERA paper would suggest that the millimeter level distances {of the GPS} were blown up in the actual data-analysis. This is because distances and energies are correlated in theory of Relativity. by the well established energy-momentum relationship. This relationship assumes further degrees of complicacy in particle physics experiments when kinematic relations of various energy channels and detector responses are added. So we need to factor in all sources of energy uncertainties in order to see why OPERA is seeing an anomaly of sorts is quite explainable by basic Physics. OPERA sets their neutrino masses to a nominal value of 2 eV which means a 0 uncertainty on the mass. On the other hand, the total Energy ⁄ momentum uncertainties do not vanish that way and increases the mass error again.
We also refer to a more general case of kinematic errors on neutrino mass. This resembles more to the method of the MINOS experiment on neutrino speed [9]. MINOS assigns their neutrinos a mass from the procedure of reconstruction in the detector itself. Our treatment is a general form for any sophisticated analysis applicable to any kind of particle physics experiment, or even to a particle reaction, out side of accelerators or detectors.

# 2 Relativistic kinematics and Quantum mechanics

## 2.1 Uncertainty Relation of speed − time from energy − time

The “energy, mass, momentum” equation usually called the energy-momentum relation [2] is expressed in speed-of-light  = 1 units as:
E2 = m2 + p2, so
(1) E = (m2 + p2)1 ⁄ 2,
where m is the rest mass of the neutrino or any relativistic particle. We note that m can itself be a nominal value as used by OPERA, or a further kinematic sequence as used by MINOS, e.g. from various combinatorial sources. Given this difference, we suspect that this is why MINOS does not see a significant anomalous effect as the uncertainties, if present, automatically take care of the validity of the uncertainty minimums. For a stronger claim, one needs to factor in all the kinematic contribution of energy uncertainty on m and it follows the same path as carried out in this analysis.
We have given a general form of this here. One needs the exact kinematic channels so as to iterate correctly in the relativistic equations inherently present in eqn (1↑). The errors associated with energy from other sources can be placed by hand in our derived result later, if one knows such with precision. In general any result on speed is dominated by errors of distance/speed/energy as these are equivalents, given a fixed precision on time. MINOS has its kinematic neutrino mass errors included in its analysis, so some of the errors might be canceling each other out although they do not have a statistically significant result. We do not know if MINOS also suffers the same errors as neglected by OPERA. MINOS needs to check our analysis predictions in their experimental method to see if this actually explains their findings. It is interesting to follow the exact channels from MINOS experiment and apply our methods.
Reverting to our analysis, we differentiate the above, eqn (1↑) to see the relation between any shift or error in the above equation. That is the errors will be related in the differentials, given by:
ΔE = {12} × (m2 + p2) − 1 ⁄ 2 × 2 × {mΔm + pΔp}.
This analysis does not differentiate between the forward, backward or central differentials, so you can use any; Delta  = Δ =  forward, anadelta  = ∇ =  backward and delta  = δ =  central difference.
Now let us apply the Heisenberg’s energy − time uncertainty relationship, ΔEt ≥ ℏ;
ΔEt = (m2 + p2) − 1 ⁄ 2(mΔm + pΔp).Δt ≥ ℏ,
so,
(m2 + p2) − 1 ⁄ 2mmt + (m2 + p2) − 1 ⁄ 2ppt ≥ ℏ.
We have therefore,
(1 + γ2β2) − 1 ⁄ 2Δmt + (1 + γ2β2) − 1 ⁄ 2γβpt ≥ ℏ,
in the preceding equation we used p = mγβ, so naturally
(2) Δp = (Δm)γβ + m.Δ(γβ).
Applying equation(2↑) we obtain:

(3) $\dpi{150} \bg_green \mathbf{\frac{[(1+\gamma^{2}\beta^{2}).\Delta m.\Delta t+m.c_{b}.\gamma\beta.\Delta\beta.\Delta t]}{(1+\gamma^{2}\beta^{2}){}^{1/2}}\geq\hbar}$,
where $\dpi{150} \bg_green c_{b}=\left(\frac{d(\gamma\beta)}{d\beta}\right)_{\beta\to1}=\left(\frac{\Delta(\gamma\beta)}{\Delta\beta}\right)_{\beta\to1}$.
We also define db = (γβ)β → 1.
These definitions do not take away the generality as long as they have not been evaluated. So we can change our β → 1 limit and re-evaluate the constants. Let us take the Δmt ~ ℏ limit which says that any uncertainty on m is a minimum in that limit, so we have

(4) (1 + γ2β2)ℏ + m.cb.γββt ≥ (1 + γ2β2).
Note that setting Δmt ~ ℏ does not make the minimum ΔEt ~ ℏ, in other words eqn(4↑) is not an equality yet, and this is consistent.
So;
(5) m.cb.γβ.Δβ.Δt ≥ ℏ((1 + γ2β2) − 1 − γ2β2)

(6) (m.cb.γβ)/((1 + γ2β2) − (1 + γ2β2)).Δβ.Δt ≥ ℏ
(7) (m.cb.γβ)/((1 + d2b) − (1 + d2b)).Δβ.Δt ≥ ℏ
(8) Δβ.Δt ≥ ()/(m.cb.db).((1 + γ2β2) − 1 − γ2β2)
(9) Δβ.Δt ≥ (λc)/(cb.db).((1 + γ2β2) − 1 − γ2β2)

(10)$\dpi{150} \bg_green \Delta\beta.\Delta t\geq\lambda_{c}.\left(\frac{\sqrt{1+d_{b}^{2}}}{c_{b}.d_{b}}-\frac{1+d_{b}^{2}}{c_{b}.d_{b}}\right)$
We give a general description of this in the Notes we append in the end, see {see NOTE(3↓)} , where we do not set Δmt ~ ℏ . Also, it is worthwhile to mention here that Δβ in the above equations is Δβc =  causality violation uncertainty which is necessarily  − ve. We can intuit this if we say ±absβc) = Δβ where Δβ is the actual uncertainty on speed which can be blown up by errors from a variety of sources.

## 2.2 The OPERA neutrino speed excess

All the above 5 equations, followed by equation \eqrefeq: − 3, that is, eqn (6↑) to eqn (10↑) are general forms of speed − time uncertainty relation. Also, we have lost the generality of uncertainty on mass m at this point. The generality can be reverted by not employing the uncertainty relation Δmt ~ ℏ. These 5 equations are chosen to a given accuracy and in a given relativistic limit. We have employed the summation of binomial [1] coefficients to determine cb,  db hence the subscript b. Later {see NOTE-(3↓)} we will give details of how we determined these constants for OPERA neutrino situation. They are for OPERA neutrinos, given by cb = 15.006 and db = 3.942, which reminds us that β and γ are ultra-relativistic.
A note of caution; these constants have been adjusted for a momentum − order calculation. These may therefore change for mass − order and energy − order calculations. For mass − order they are found to be  ~ 10 − 8. λc =  reduced Compton wavelength. We evaluate the above equations in terms of known values and we have
Δβc.Δt ≥  − 0.211.(λν)
This is not only valid for neutrinos but also for any particle that is moving at or near the speed = β = 1 ≡ 3.0 × 108m ⁄ s. We derived cb,  db to the order β10 at the limit β → 1. We will attempt a more rigorous review of the evaluation of these constants in a later communication. But, for now, after several iterations and the fact that summing of the binomial coefficients must in the end give only a value that does not change widely, it is enough to make a claim that our result is correct. The  − ve sign comes because Δβ is a causality violation limit. In this limit the particle is going below β = 1. It’s an uncertainty. One can also say the minimum uncertainty Δβ is restricted by the Compton wavelength.

With that in mind
(12) Δβ.Δt ≥ 0.211.(λν)
or

Δβ ≥ (0.211 × 6.6 × 10 − 7 × eV.ns)/(2 × 10 × eV.ns)
or Δβ ≥ 0.696 × 10 − 8, for c = 3.0 × 108 m ⁄ s this is Δv ≥ 2.09 m ⁄ s.

One then concludes that OPERA must see a minimum of 2.09 m ⁄ s at a precision of 10 ns. {for 1 ns we must multiply by 10, for speed, energy and momentum}. We see that {Δβ,  Δp} ≈ 6.6 × 10 − 4eV, for a 10 ns precision to see a 2.09 km ⁄ s uncertainty in the speed of neutrinos. This uncertainty is for momentum − order, for mass − order one divides by c = 3 × 108and for energy − order and then one multiplies the momentum − order by c = 3 × 108. So for a 2.09 km ⁄ s uncertainty we have; ΔE ~ 19.8 KeV , Δp ~ 660 ppm of 1 eV , Δm ~ 0.22 × 10 − 12 eV.
One sees therefore that if OPERA incurs an uncertainty on its energy, mass or momentum measurement of the neutrinos, a very small value given by ΔE ~ 19.8 KeV, Δp ~ 660 ppm of 1eV, Δm ~ 0.22 × 10 − 12 eV the super-luminal claim of 7.5 km ⁄ s is valid. One needs to tighten the uncertainties on E,  p,  m a little more, to be consistent with 7.5 km ⁄ s, not the lesser 2.09 km ⁄ s. So, one needs to blow up the constraint by multiplying a factor (7.5)/(2.09) = 3.589. This means OPERA’s superluminal claims vanish at ΔE ~ 71.06 KeV, Δp ~ 2.37 meV or Δm ~ 0.79 × 10 − 12 eV. We refer to this in fig. (1↓)

## 2.3 Errors on energy from recent experiments

Recent particle physics experiments such as belle at KEK,  Japan have an uncertainty of  ~ 500 KeV on their center of mass energy of  ~ 5.4 GeV. This is a result from 2010 − 2011 [7]. The belle uncertainty at same center of mass energy was  ~ 800 KeV, in a highly cited paper from 2003 [8].
This suggests that while the techniques of reconstruction have improved it has not come down to a value of 100 KeV. OPERA could survive 10 KeV but not 100 KeV. This raises a very pertinent question on what OPERA could achieve in terms of their energy uncertainty since that is also a particle physics experiment. Without the actual statement of uncertainty on their energy it is not at all safe to make a superluminal claim.
In other words Quantum Mechanics does not exclude superluminal neutrinos, it imposes an extremely harsh condition on the precision of energy. One needs to be able to see in the entirety of one’s analysis, if there is any superluminal excess or not. Following our discussion from subsection − B above, in-fact, OPERA claims a 1.23 km ⁄ s uncertainty, this is possible only when they have slightly worse uncertainty than 10 KeV on their energy of 17 GeV as you can see above. {actually11.65 KeV\thickapprox1.23 km ⁄ s and corresponds to a 0.685 ppm uncertainty on their energy.}. At 100 KeV OPERA superluminal claim “vanishes”. In-fact as we just showed it is not valid at 71 KeV uncertainty on their 17 GeV energy which is a 4.18 ppm uncertainty on energy.

# 3 Notes

1. We need a subtle point of relativity in our calculations to be consistent with the overall results of our analysis. Here is the actual equivalence: we realize that they are present in various important equations established by theory of relativity and used frequently in relativistic applications/investigations. SO the total set of equivalence is E, p, v, m, d, t. The 3 Quantum Mechanics uncertainty principles are all constituted from among these variables, hence they are all equivalently only 1 equation but appear in different forms if we start from one and derive another. We notice that d, t are also equivalenced like the famous E, m. The other important property to notice is d and t are chosen for the parametrization of kinematics and all the other variables here are paired for canonical commutation with respect to d and t. .

2. When the set of equivalent parameters {E, p, v, m, d, t} is commuted with either d or t, one at a time and excluding them from the main set of parameters, the commutation produces the uncertainty in the order of Planck’s constant , which is the reduced Planck’s constant. The uncertainty of commutation bears a very simple inequation for 3 specific cases and these are called the uncertainty relationship or the (in)equations of Quantum Mechanics. But, in conjugation with relativity, as we mentioned already, one can start with only one uncertainty relation and observe the other two by employing the “classical” or relativistic definitions of the other parameters as per suitability of the problem. Then, from a simple relationship of commutation-uncertainty, one arrives at more complicated relationships which in specific cases and in the limits of minimum uncertainty returns to the simpler form again. We also note that the simpler form of uncertainty can be rendered more complicated in the realm of particle physics as more than one kinematic contributions appear and as detector responses are factored into the uncertainty behavior. .

3. The uncertainty of d and t go opposite to the uncertainty of the other variables E, p, v, m to the order of either or in consistency of units ℏ ⁄ m0 . ℏ ⁄ m0, in speed-of-light units is called a reduced Compton wavelength of the particle represented by mass m0 . . Additional speed-of-light unit consistency check is needed at several levels of the calculation. From an equivalence we do not always use an uncertainty equation in the same variable. They need to be equated in a correct dimensional analysis. This can easily be done by employing a speed-of-light unit. We were careful in this paper with the units and dimensions so as not to incur incorrect values. The parameters d and t define the speed v, hence we have also included v in the set of equivalent variables.

4. How do we sum our binomial coefficients? We use a bound on summation of binomial coefficients given by Michael Lugo [6]:
$\dpi{150} \bg_green f\,(n,\, k)\,\le \,\dbinom{n}{k}\,\frac{n(k-1)}{n(2k-1)}.$
On MathOverFlow.Net {Sum of the first k binomial coefficients} Michael Lugo gives two bounds on summing the binomial coefficients, one for a fixed k which we use in this paper and one for k = N ⁄ 2 + α(N). Because of the method of summing the binomial coefficient and the exact order of β we will incur a very slight error on the constraint we provide on the uncertainty of E. This does not change the order of the energy uncertainty ΔE, as binomial coefficients are fractions that we summed to a very high degree already.

5. Here we give the details of summing the binomial coefficients in the expansion of the Lorentz Factors in the limit of β → 1. We refer to an analysis we have done in determining the binomial expansion of the Lorentz Factors and their power functions [5] .
(13) $\dpi{150} \bg_green f\,(\beta)=\beta(1-\beta^{2})^{-1/2} & =\beta\left\{ \sum_{k=0}^{\infty}\,\dbinom{-\frac{1}{2}}{k}\,(-\beta^{2})^{k}\right\}$.
We want f (1).
(14) $\dpi{150} \bg_green f\,(1)=\sum_{k=0}^{\infty}\,\dbinom{-\frac{1}{2}}{k}\,(-1)^{k}.$
According to a bound given in MathOverFlow.Net as refered in Note-(3↑), [4]
(15)   $\dpi{150} \bg_green \mathbf{f\,(1)=\sum_{k=0}^{\infty}\,\dbinom{-\frac{1}{2}}{k}\,(-1)^{k}} & \leq\dbinom{-\frac{1}{2}}{k}\,\frac{-\frac{1}{2}-(k-1)}{-\frac{1}{2}-(2k-1)}\{(-1)^{k}\}$.

Let us take k = 10, i.e.  ~ β20. then;
$\dpi{150} \bg_green f\,(1)=\dbinom{-\frac{1}{2}}{10}\frac{-\frac{1}{2}-(10-1)}{-\frac{1}{2}-(20-1)}\{(-1)^{10}\}=\frac{9.5}{19.5}\dbinom{-\frac{1}{2}}{10}$
$\dpi{150} \bg_green f\,(1)=\frac{9.5}{19.5}\frac{(-\frac{1}{2})!}{(-\frac{1}{2}-10)!(10)!}=\frac{9.5}{19.5}\frac{(-0.5)}{(-10.5)\times3628800}$$\dpi{150} \bg_green f\,(1)=-\frac{95\times5\times10^{-2}\times2.76\times10^{-5}}{195\times105}=1.314\times10^{-8}$.
As we had noted earlier, a mass − order value is in the O(10 − 8) and we need to multiply for consistency of speed − of − light c = 3.0 × 108 m ⁄ s everywhere; there is a m − term, which is the case for f (1) = β → 1(γβ). So we have
(16)  $\dpi{150} \bg_green \mathbf{f\,(1)=\sum_{\beta\rightarrow1}(\gamma\beta)=d_{b}=3.942.}$
Now we evaluate the constant
$\dpi{150} \bg_green c_{b}=\left(\frac{d(\gamma\beta)}{d\beta}\right)_{\beta\rightarrow1}=\left(\frac{\Delta(\gamma\beta)}{\Delta\beta}\right)_{\beta\rightarrow1}.$
$\dpi{150} \bg_green \gamma\beta=\frac{\beta}{\sqrt{1-\beta^{2}}}$,
$\dpi{150} \bg_green \frac{d}{dx}\left(\frac{F_{1}}{F_{2}}\right)=\frac{F_{2}.F_{1}^{'}-F_{1}.F_{2}^{'}}{F_{2}^{2}}$,
(17)
We define cb = g (β) + h (β) with (18) h (β) = 2β2(1 − β2)3 ⁄ 2
and g (β) = f (β) if β = 1,

(19)  $\dpi{150} \bg_green h\,(\beta)=2\beta^{2}\left\{ \sum_{k=0}^{\infty}\,\dbinom{-\frac{3}{2}}{k}\,(-\beta^{2})^{k}\right\}$

$\dpi{150} \bg_green h\,(1)=2\left\{ \sum_{k=0}^{\infty}\,\dbinom{-\frac{3}{2}}{k}\,(-1)^{k}\right\} ^{k=10} & \leq2\,\dbinom{-\frac{3}{2}}{10}\,\dfrac{-\frac{3}{2}-9}{\frac{3}{2}-19}$

$\dpi{150} \bg_green h\,(1)=2\times\frac{10.5}{20.5}\frac{(-\frac{3}{2})!}{(-\frac{3}{2}-10)!(10)!}=\frac{9.5}{19.5}\frac{(-0.5)}{(-10.5)\times3628800}$

$\dpi{150} \bg_green h\,(1)=\frac{105\times2\times15\times10^{-2}\times2.76\times10^{-5}}{115\times205}=3.688\times10^{-8}$

$\dpi{150} \bg_green g\,(1)+h\,(1)=\sum_{\beta\rightarrow1}\frac{d}{d\beta}(\gamma\beta)=c_{b}=(3.688+1.314)\times10^{-8}$
(20) $\dpi{150} \bg_green \mathbf{g\,(1)+h\,(1)=\sum_{\beta\rightarrow1}\frac{d}{d\beta}(\gamma\beta)=}\mathbf{c_{b}=15.006}$
This concludes our method of evaluating followed by summing the binomial coefficients in the expansion of the Lorentz Factors and their power functions. As noted, we have evaluated these in the momentum − order.

6. Despite our rigorous calculations to look for a possible explanation for OPERA experiment anomaly we find that one of our previous analysis would have led to the same conclusion if we were to correctly interpret that equation, [3]. This can be considered an uncertainty on propertime. In the text this is not mentioned in terms of Compton Wavelength of the participating particle or an uncertainty on proper-time. We change the notations slightly and use it for an uncertainty on space and time;
(21) $\dpi{150} \bg_green \mathbf{(\Delta x)^{2}-(\Delta t)^{2}\leq\left(\dfrac{\hslash}{m_{0}}\right)^{2}}$,
note the sign of inequality here and the fact that
$\dpi{150} \bg_green \left(\frac{\hslash}{m_{0}}\right)^{2}=\lambda_{c}^{2}.$

So,
(22) $\dpi{150} \bg_green (\Delta t)^{2}.\left\{ \left(\frac{\mathrm{v}}{c}\right)^{2}-1\right\} \leq\left(\frac{\hslash}{m_{0}}\right)^{2}, & \,(\beta+\Delta\beta_{c})^{2}\leq1+\frac{\lambda_{c}^{2}}{(\Delta t)^{2}}$
where Δβc is the speed-excess or causality violation in terms of speed if β → 1. Since 0 ≤ β ≤ 1 the last (in)equation means
(23)$\dpi{150} \bg_green \beta^{2}+(\Delta\beta_{c}){}^{2}+2.\beta.\Delta\beta_{c} & <1+\left(\frac{\lambda_{c}}{\Delta t}\right)^{2}+2.\frac{\lambda_{c}}{\Delta t}$
as (λc)/(Δt) is  + ve. If β → 1 this means
(24) $\dpi{150} \bg_green (1+\Delta\beta_{c})^{2}<\left(1+\frac{\lambda_{c}}{\Delta t}\right)^{2}$,
or
(25) Δβc < (λc)/(Δt).

One can see that the last expression is “exactly” what we had obtained in our analysis at the top in section (2.2↑). In-fact Δβc is a causality violation excess at this limit of β → 1, so we can change its sign and we have
(26) Δβ > (λc)/(Δt).

By summing binomially and seeing the fact that the mass m can be really small, such as that of neutrino, we see that we have a factor 0.211 in front of λc. This is to be expected, since for the equality above that we had removed will now be valid. So we approach the equality; binomially and asymptotically. The Compton − wavelength − limit corresponds to an inequality. The sources of errors, such as detector response, will add − on to the Compton-wavelength. In-fact, there never is an equality since it is asymptotic. That is, the factor 0.211 is only approximate although very very accurate. So, these lighter particles are capable of offsetting the causality violation excess by say a factor 0.211 but not completely make it zero. One can continue to sum the binomial coefficients to a larger and larger value and have a lighter and lighter mass but never completely breakaway from the photon speed.
For the electron which is much more massive than the neutrino, therefore, one would expect a factor which is  < 0.211 and one must see speed − uncertainties larger than any speed-excess above speed − of − light. Although
(27) $\dpi{150} \bg_green (\beta+\Delta\beta_{c})^{2}<\left(1+\frac{\lambda_{c}}{\Delta t}\right)^{2} & \Rightarrow\Delta\beta_{c}<\frac{\lambda_{c}}{\Delta t},\;\beta\to1$,
(in the β → 1 limit), this is also valid in the contrary since 0 ≤ β ≤ 1. This is evident because if β≪1 one needs an “additive” correction factor to Δβ which then makes the $\dpi{150} \bg_green (\beta+\Delta\beta_{c})^{2}<\left(1+\frac{\lambda_{c}}{\Delta t}\right)^{2}$ valid/sufficient for inferring Δβc < (λc)/(Δt). This “additive” correction factor is  − (1 − β), eg if β = 0.5 then Δβc − 0.5 < (λc)/(Δt) or Δβc < (λc)/(Δt) + 0.5 and Δβ ≥ (λc)/(Δt) + 0.5. Notice that Δβ and Δβc are different: Δβc refers to causality violation uncertainty, hence Δβc is in the β → 1 limit but Δβ refers to actual uncertainty on the speed. So in general if 0 ≤ β < 1;
Δβ ≥ (λc)/(Δt) + (1 − β),  0 ≤ β < 1.

7. We show here 2 figures, fig. (2↓) and fig. (3↓) which address a hypothetical superluminal situation of the neutrino Vs the photon. We briefly describe the implications of this.
Figure 2 The situation in a superluminal paradigm, theory of Relativity
 Figure 2

For the same amount of distance light produced a circle of time lesser in diameter to the length that corresponds to other particles.

{nothing takes lesser time than light}

On the shell:

δτ = 0,  dt2dx2 = 0

{for the distance and time to be equal only light has this property}.

δτ = 0,  dt = dx,  dx ⁄ dt = 1 = c

Inside circle/shell:

nothing, not even light moves here, it takes less time than light for same distance

NEUTRINOS?

If νs are superluminal by default they will constitute a circle like this and photons will be arrows outside the circle hovering in time−like regions defined by maximal ν speed, but are there maximal ν speeds that can go above speed of light??

# 4 Summary/Conclusion

The OPERA experiment is a particle physics experiment which has for 3 years accumulated a galore of neutrinos produced from the decay of protons. These protons were shoot from the laboratory at CERN, Geneva and the neutrinos produced from these passed through the earth-crust to another laboratory at Gran-Sasso, Italy which is located about 733 kms, same as the coastline of HongKong or the distance of Oslo, Norway to Mo I Rana, Norway. The experiment recently claimed that as per its analysis aided by time precision of GPS satellites to about 10 nano-seconds they observe that their 15000 neutrinos would travel faster than light particles known as photons. Such neutrinos are called as super-luminal neutrinos as they can travel past lumina, that is light. This is not a very welcome observation since it puts into question Albert Einstein’s firmly established Theory of Relativity and almost all the other branches of Modern Physics. In the last 3 months the world has seen about a 100 applications of the various knowledge segments of the physical world to explain or refute the findings. None of the findings were sufficient to disprove OPERA experiment’s findings which had performed an analysis with an impressive statistical significance that is hardly ever achieved in such measurements.
We have deduced from basic energy-time uncertainty relation of Quantum Mechanics a very accurate form of speed-time uncertainty relation. This speed-time uncertainty relation imposes stringent conditions at what energy uncertainty OPERA can claim super-luminal neutrino and at what uncertainties it cannot. We follow basic conditions of special Theory of Relativity and Quantum Mechanics to claim that OPERA must have total energy uncertainties below  ~ 71 kilo-electron-volts to consistently conclude that special theory of Relativity is inconsistent with their finding. We cite latest measurement uncertainties in some recent experiments of similar nature and similar energy scale to claim that such precision has yet not been achieved. The way forward for OPERA is to study and disclose their energy uncertainties. This will rest the matter of whether neutrino speed can go above the speed limit of nature imposed by special theory of Relativity of Albert Einstein. According to this theory only light particles known as photons travel at the speed limit of nature since they have no mass. Neutrinos are known to have some mass, although very very small compared e.g. to electrons, because they can change their form from one type to another. According to Theory of Relativity anything that has mass can never go above the photon’s speed of  ~ (3 × 105 = 3, 00, 000) km ⁄ s.

We are thankful to the free world for the resources which enabled us to discuss our ideas and communicate the research. We are also grateful to collegues, friends and family who have provided valuable feedback and support. This research was in part supported by i3tex and the Willgood Institute.

# References

[1] G. Arfken & H.Weber, "Binomial Theorem", 6th Indian edition, chapter 5.6, (1988).
[2] R. Resnick, "Introduction to Special Relativity", Chapter 3.5, equation 3 − 18b, (2010).
[3] S. Weinberg, “Gravitation and Cosmology: Special Relativity”, Wiley Indian Edition, chapter 2.13, (2009).
[4] Michael Lugo (mathoverflow.net/users/143), Sum of ’the first k’ binomial coefficients for fixed n , http://mathoverflow.net/questions/17236 (version: 2010-03-05).
[5] Dash, M. & Franzén, M. (2012). Time dilation have opposite signs in hemispheres of recession and approach. PHILICA.COM Article number 312.
[6] The OPERA Collaboration, “Measurement of the neutrino velocity with the OPERA detector in the CNGS beam”, arXiv:1109.4897v1 [hep-ex], (2011).
[7] The Belle Collaboration, “Search for B0s → hh Decays at the Υ(5S) Resonance”, arXiv:1006.5115v2 [hep-ex] (2011).
[8] The Belle Collaboration, “Observation of a New Narrow Charmonium State in Exclusive B± → K + π + π − J ⁄ ψ Decays”, arXiv:0308029v1[hep-ex], (2003).
[9] The MINOS Collaboration, “Measurement of neutrino velocity with the MINOS detectors and NuMI neutrino beam”, arXiv:0706.0437v3 [hep-ex], (2007).

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Version 1.6
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# OPERA neutrino anomaly is a result of not interpreting energy uncertainty.

Informal association with Willgood institute, registered in Sweden,
author’s mail: Mahisapat, Dhenkanal, Odisha, India, 759001
previously affiliated with VT, USA and KEK, Japan.
Willgood Institute, Luckvägen 5, 517 37 Bollebygd, Sweden
i3tex AB, Klippan 1A , 414 51 Gothenburg, Sweden

Abstract

In this paper we bring out a remarkable consistency of theory of Relativity in explaining the anomalous excess of speed of neutrinos observed in the recent baseline experiment of OPERA. The OPERA experiment is performed by shooting neutrinos produced from protons at SPS, CERN to the laboratory at Gran-Sasso where OPERA has placed its neutrino brick detectors. We believe this result was misinterpreted to claim a super-luminal neutrino. The energy uncertainties inherently present in the OPERA neutrino measurement have not been reported on the claims of speed excess. The basics of Quantum Mechanics on the kinematic aspects of these neutrinos have been pointed out in this paper. We make a minimal review of this negligence of uncertainties which is sufficient to see where OPERA has lacked a cautious sight in claiming super-luminal neutrinos. We perform a rigorous check of Quantum Mechanics uncertainty principle of Energy-Time to make our claim of lack of any evidence of super-luminal neutrino.

OPERA experiment, neutrino speed anomaly, Special Relativity, energy-time uncertainty, speed-time uncertainty, Compton Wavelength of neutrino, super-luminal neutrino

# 1 Introduction

In this paper we provide a stringent condition on the minimum uncertainty on energy one deals with, on any particle with energy E momentum p and rest mass m. We find a relation between the uncertainty on speed and time following directly from the uncertainty of energy and time. Our relation is general and expressed in terms of the Compton wavelength of any particle, in particular the neutrino from OPERA experiment [6]. This is a very accurate form of speed-time uncertainty relationship from energy-time uncertainty relationship. In our calculations we have made careful attempts to be consistent with the units of speed-of-light. Our result is valid for ultra-relativistic conditions of OPERA as much as it is valid for any speed of the particle, down to the lowest β one can theorize. All we do for OPERA situation is let our β → 1. We do not use ultra-relativistic conditions except when evaluating constants in the case of OPERA neutrinos. Our expressions are valid for a relativistic treatment of general nature.

We mention that any experiment and theory of nature in Physics to be consistent with what we know must confirm to the famous uncertainty principle of Heisenberg, where applicable. From the logical edifice of Relativity theory and it’s most popular concepts follows that there are 3 uncertainty principles, but in terms of equivalence, only 1 uncertainty principle is chosen as per the necessity of the physical problem at hand. Here we chose the explicit form of Energy-Time uncertainty relationship, because a baseline speed measurement rests on such a situation.

The OPERA experiment measures it’s neutrino speed by claiming a very precise timing aided by the GPS satellite system for such measurements. This entails them a millimeter level accuracy in distance and a ns level accuracy in time at-least as per the specification of their GPS receivers. Since we have done much prior analysis that shows that GPS satellites in their circular orbits are very very well understood as per special and general theory of relativity, we do not ascertain any source of inaccuracy here. We mention that gravity of earth size objects is {Sr = 2.GMe} in itself a millimeter level accuracy. The exact value depends on the exact parameters of the problem and the separation from the gravity-source. The 25 ppm speed-of-light excess of OPERA experiment in terms of absolute speed is a  ~ 7.5 km ⁄ s excess. Such a large fallout in the speed-of-light is an unexpectedly large fallout with respect to the theory of Relativity.

The conclusion we draw is one has a millimeter level accuracy on GPS distance and nanosecond level accuracy on GPS time. Hence one must see a millimeter level speed excess as in other cases of theory of Relativity Paradigm if interpreted correctly with the recognition that the further complicacies in OPERA situation comes for two reasons. i. We are dealing with elementary particles whose masses are the smallest we know in the physical world ii. These particles have speeds that are immensely relativistic. For these two reasons one does not see a minimum in the millimeter range. In-fact the reason-i is dominant as masses can vary over a wider scale. The relativistic factor; reason-ii, does not vary as much. e.g. the OPERA neutrino and any electron moving at about the same speed, reason-ii is the same factor. But for these two cases for neutrino the minimum uncertainty we find is at 2.09 − meters ⁄ seconds where as for electrons this will be (0.511 ⁄ 2) × 106 times lesser than what it is for neutrinos. This is for a 10 − ns GPS aided time precision {and any type of time precision in general}. The electrons moving at about speed-of-light will be uncertain of their speed at-least by;
$\dpi{150} \bg_green \left(\frac{4.18}{0.511}\right)\times10^{-6}\, m/s=\mathbf{8.18\,\mu m/s;\; electron's-minimum.}$

We mention in advance that in this paper we determine for a 2 eV neutrino a minimum of  ~ 2.09 m ⁄ s uncertainty in speed either below or above speed of light.
2.09 m ⁄ s; neutrino’s − minimum.
A reinterpretation or rather a correct analysis of OPERA paper would suggest that the millimeter level distances {of the GPS} were blown up in the actual data-analysis of the OPERA measurement since distances and energies are correlated in theory of Relativity by the well established energy-momentum relationship. This relationship assumes further degrees of complicacy in particle physics experiments when kinematic relations of various energy channels and detector responses are added. So we need to factor in all sources of energy uncertainties to see why OPERA seeing an anomaly of sorts is quite explainable by basic Physics. OPERA sets their neutrino masses to a nominal value of 2 eV which means a 0 uncertainty on the mass but the total Energy/Momentum uncertainties do not vanish that way which increases the mass error again. We refer to a more general case of kinematic errors on neutrino mass. This resembles more to the method of MINOS experiment on neutrino speed[9]. MINOS assigns their neutrinos a mass from the procedure of reconstruction in the detector itself. Our treatment is a general form for any sophisticated analysis in any kind of particle physics experiment or even a particle reaction out side of accelerators or detector.

# 2 Relativistic kinematics and Quantum mechanics

## 2.1 Uncertainty Relation of speed-time from energy-time.

The “energy, mass, momentum” equation usually called the energy-momentum relation [2] is expressed in speed-of-light  = 1 units as: E2 = m2 + p2, so
(1) E = (m2 + p2)1 ⁄ 2

where m is the rest mass of the neutrino or any relativistic particle. We note that m can itself be a nominal value as used by OPERA or a further kinematic sequence as used by MINOS. This is actually the reason we suspect why MINOS does not see the anomalous effect with a higher significance as the uncertainties if present automatically take care of the validity of the uncertainty minimums. For a stronger claim one needs to factor in all the kinematic contribution of energy uncertainty on m and it follows the same analysis path as presented in this paper. We have given a general form of this in this report. One needs the exact kinematic channels so as to iterate correctly in the relativistic equations inherently present in eqn (1↑). The errors associated with energy from other sources can be placed by hand in our derived result later, if one knows such with precision. In general any result on speed is dominated by errors of distance/speed/energy as these are equivalents, given a fixed precision on time. MINOS has it’s kinematic neutrino mass errors included in it’s analysis, so some of the errors might be canceling each other although they do not have a statistically significant result. We do not know if MINOS also suffers the same errors as neglected by OPERA or not.

We differentiate the above, eqn (1↑) to see the relation between any shift or error in the above equation. That is the errors will be related in the differentials, given by:
ΔE = {12} × (m2 + p2) − 1 ⁄ 2 × 2 × {mΔm + pΔp}.

This analysis does not differentiate between the forward, backward or central differentials, so you can use any; delta  = Δ =  forward, anadelta  = ∇ =  backward and delta = δ =  central difference.

Now let us apply the Heisenberg’s energy-time uncertainty relationship, ΔEt ≥ ℏ;
ΔEt = (m2 + p2) − 1 ⁄ 2(mΔm + pΔp).Δt ≥ ℏ, so,
(m2 + p2) − 1 ⁄ 2mmt + (m2 + p2) − 1 ⁄ 2ppt ≥ ℏ.

So we have now,
(1 + γ2β2) − 1 ⁄ 2Δmt + (1 + γ2β2) − 1 ⁄ 2γβpt ≥ ℏ

in the left we used p = mγβ, so naturally
(2) Δp = (Δm)γβ + m.Δ(γβ).

Using this, eqn(2↑);

(3) $\dpi{150} \bg_green \mathbf{\frac{[(1+\gamma^{2}\beta^{2}).\Delta m.\Delta t+m.c_{b}.\gamma\beta.\Delta\beta.\Delta t]}{(1+\gamma^{2}\beta^{2}){}^{1/2}}\geq\hbar}$

where
$\dpi{150} \bg_green c_{b}=\left(\frac{d(\gamma\beta)}{d\beta}\right)_{\beta\to1}=\left(\frac{\Delta(\gamma\beta)}{\Delta\beta}\right)_{\beta\to1}$
We also define db = (γβ)β → 1.

These definitions do not take away the generality as long as they have not been evaluated. So we can change our β → 1 limit and re-evaluate the constants. Let us take the Δmt ~ ℏ limit which says any uncertainty on m is a minimum in that limit, so we have

(4) (1 + γ2β2)ℏ + m.cb.γββt ≥ (1 + γ2β2)

Note that setting Δmt ~ ℏ still does not make the minimum ΔEt ~ ℏ, in other words eqn(4↑) is not an equality yet, which is only consistent.

(5) m.cb.γβ.Δβ.Δt ≥ ((1 + γ2β2) − 1 − γ2β2)

(m.cb.γβ)/((1 + γ2β2) − (1 + γ2β2))βt ≥ ℏ
(m.cb.γβ)/((1 + d2b) − (1 + d2b))βt ≥ ℏ
Δβt ≥ ()/(m.cb.db).((1 + γ2β2) − 1 − γ2β2)
Δβt ≥ (λc)/(cb.db).((1 + γ2β2) − 1 − γ2β2)

$\dpi{150} \bg_green \Delta\beta.\Delta t\geq\lambda_{c}.\left(\frac{\sqrt{1+d_{b}^{2}}}{c_{b}.d_{b}}-\frac{1+d_{b}^{2}}{c_{b}.d_{b}}\right)$
We give a general description of this in the note where we do not set Δmt ~ ℏ . Also it is worthwhile to mention here that Δβ in the above equations is Δβc =  causality violation uncertainty which is necessarily  − ve. We can intuit this if we say ±absβc) = Δβ where Δβ is actual uncertainty on speed which can be blown up by errors from a variety of sources.

## 2.2  The OPERA neutrino speed excess

All the above 5 equations followed by eqn (5↑) are the general form of speed-time uncertainty relation. Also we have lost the generality of uncertainty on mass m at this point. The generality can be reverted by not employing the uncertainty relation Δmt ~ ℏ. These 5 equations are chosen to a given accuracy and in a given relativistic limit. We have employed the summation of binomial [1] coefficients to determine cb,  db hence the subscript b. Later we will give the details how we determined these constants for OPERA neutrino situation. They are for OPERA neutrino, given by cb = 15.006 and db = 3.942 which reminds us that β and γ are ultra-relativistic.

A note of caution; these constants have been adjusted for a momentum-order calculation. These may change therefore for mass-order and energy-order calculations. For mass-order they are found to be  ~ 10 − 8. λc is reduced Compton Wavelength. We evaluate the above equations in terms of known values and we have
Δβc.Δt ≥  − 0.211.(λν)

This is not only valid for neutrinos but any particle that is moving at or near speed = β = 1 ≡ 3.0 × 108m ⁄ s. We derived cb,  db to the order β10 at the limit β → 1. We will attempt a more rigorous review of the evaluation of these constants in a later communication. But for now after several iterations and the fact that summing of binomial coefficients must in the end give only a value that does not change widely is enough to make a claim that our result is correct. The  − ve sign comes because Δβ is a causality violation limit. In this limit the particle is going below β = 1. It’s an uncertainty. One can also say the minimum uncertainty Δβ is restricted by the Compton wavelength.

With that in mind
(6) Δβ.Δt ≥ 0.211.(λν)
or,

Δβ ≥ (0.211 × 6.6 × 10 − 7 × eV.ns)/(2 × 10 × eV.ns)
or Δβ ≥ 0.696 × 10 − 8, for c = 3.0 × 108 m ⁄ s this is Δv ≥ 2.09 m ⁄ s.

One concludes OPERA must see a minimum of 2.09 m ⁄ s at a precision of 10 ns. {for 1 ns one just multiplies by 10, for speed, energy, momentum} We see that {Δβ,  Δp} ≈ 6.6 × 10 − 4eV, for a 10 ns precision to see a 2.09 km ⁄ s uncertainty in the speed of neutrino. This uncertainty is for momentum-order, for mass-order one divides by c = 3 × 108 and for energy-order one multiplies the momentum-order by c = 3 × 108. So we have for 2.09 km ⁄ s uncertainty, ΔE ~ 19.8 KeV, Δp ~ 660 ppm of 1 eV, Δm ~ 0.22 × 10 − 12 eV.

One sees therefore that if OPERA incurs an uncertainty on it’s energy, mass or momentum measurement of the neutrino a very small value given by ΔE ~ 19.8 KeV, Δp ~ 660 ppm of 1eV, Δm ~ 0.22 × 10 − 12 eV the super-luminal claim of 7.5 km ⁄ s can still be valid. One needs to tighten the uncertainties on E,  p,  m a little more, as we need 7.5 km ⁄ s not the lesser 2.09 km ⁄ s as we did. So, one needs to blow up the constraint by multiplying a factor (7.5)/(2.09) = 3.589, in other words OPERA’s super-luminal claims vanish at ΔE ~ 71.06 KeV, Δp ~ 2.37 meV or Δm ~ 0.79 × 10 − 12 eV.

## 2.3 Errors on energy from recent experiments

The recent particle physics experiments such as belle at KEK, Japan have an uncertainty of  ~ 500 KeV on their center of mass energy of  ~ 5.4 GeV. This is a result from 2010 − 2011 [7]. The belle uncertainty at same center of mass energy was  ~ 800 KeV, in a highly cited paper from 2003 [8]. This suggests that while the techniques of reconstruction have improved it has not come down to a value of 100 KeV. OPERA could survive 10 KeV but not 100 KeV. This raises a very pertinent question on what OPERA could achieve in terms of their energy uncertainty since that is also a particle physics experiment. Without the actual statement of uncertainty on their energy it is not at all safe to make a super-luminal claim.

In other words Quantum Mechanics does not exclude super-luminal neutrinos, it imposes an extremely harsh condition on the precision of energy. One needs in the entirety of one’s analysis to be able to see if there is any super-luminal excess or not. Following our discussion from subsection-B (2.2↑) above, in-fact, OPERA claims a 1.23 km ⁄ s uncertainty, this is possible only when they have slightly worse uncertainty on their energy of 17 GeV than 10 KeV as you can see above {actually 11.65 KeV ≈ 1.23 km ⁄ s}. At 100 KeV their super-luminal claim vanishes, actually as we just showed it vanishes at 71 KeV on their 17 GeV energy which is a 4.18 ppm uncertainty on energy. 11.65 KeV ≈ 1.23 km ⁄ s is a 0.685 ppm uncertainty on their energy.

# 3 Notes

1. We need a subtle point of relativity in our calculations, to be consistent with the overall results of our analysis. Here is the actual equivalence: we realize that they are present in various important equations established by theory of relativity and used frequently in relativistic applications/investigations. SO the total set of equivalence is E, p, v, m, d, t. The 3 Quantum Mechanics uncertainty principles are all constituted from among these variables, hence they are all equivalently only 1 equation but appear in different forms if we start from one and derive another. We notice that d, t are also equivalenced like the famous E, m. The other important property to notice is d and t are chosen for the parametrization of kinematics and all the other variables here are paired for canonical commutation with respect to d and t.

2. When The set of equivalent parameters {E, p, v, m, d, t} is commuted with either d or t, one at a time and excluding them from the main set of parameters the commutation produces the uncertainty in the order of Planck’s constant , which is the reduced Planck’s constant. The uncertainty of commutation bears a very simple inequation for 3 specific cases when these are called the uncertainty relationship or (in)equations in Quantum Mechanics. But in conjugation with relativity as we mentioned already one can start with only one uncertainty relation and observe the other two by employing the “classical” or relativistic definitions of the other parameters as per suitability of the problem. Then from a simple relationship of commutation-uncertainty one arrives at more complicated relationships which in specific cases and in the limits of minimum uncertainty returns to the simpler form again. We also note that the simpler form of uncertainty can be rendered more complicated in the realm of particle physics as more than one kinematic contributions appear and as detector responses are factored into the uncertainty behavior.

3. The uncertainty of d and t go opposite to the uncertainty of the other variables {E, p, v, m} to the order of either or in consistency of units ℏ ⁄ m0. ℏ ⁄ m0 in speed-of-light units is called a Reduced Compton Wavelength of the particle represented by mass m0. Additional speed-of-light unit consistency check is needed at several levels of the calculation. From an equivalence we do not always use an uncertainty equation in the same variable. They need to be equated in a correct dimensional analysis. This can easily be done by employing a speed-of-light unit. We were careful in this paper with the units and dimensions so as not to incur incorrect values. d, t define the speed v, hence we have also included the v in the set of equivalent variables.

4. How do we sum our binomial coefficients? We use a bound on summation of binomial coefficients given by Michael Lugo: [4];
$\dpi{150} \bg_green f\,(n,\, k)\,\le \,\dbinom{n}{k}\,\frac{n(k-1)}{n(2k-1)}.$
On MathOverFlow.Net {Sum of the first k binomial coefficients} Michael Lugo gives two bounds on summing the binomial coefficients, one for a fixed k which we use in this paper and one for k = N ⁄ 2 + α(N). Because of the method of summing the binomial coefficient and the exact order of β we will incur a very slight error on the constraint we provide on the uncertainty of E. This does not change the order of the energy uncertainty ΔE as binomial coefficients are fractions, we summed to a very high degree already.

5. Here we give the details of summing the binomial coefficients in the expansion of the Lorentz Factors in the limit of β → 1. Here we refer to an analysis we have done in determining the binomial expansion of the Lorentz Factors and their power functions [5] .
$\dpi{150} \bg_green f\,(\beta)=\beta(1-\beta^{2})^{-1/2} & =\beta\left\{ \sum_{k=0}^{\infty}\,\dbinom{-\frac{1}{2}}{k}\,(-\beta^{2})^{k}\right\}$

We want f (1);
(8) $\dpi{150} \bg_green f\,(1)=\sum_{k=0}^{\infty}\,\dbinom{-\frac{1}{2}}{k}\,(-1)^{k}.$
According to a bound given in MathOverFlow.Net as refered in Note-(3↑), [4]

(9)  $\dpi{150} \bg_green \mathbf{f\,(1)=\sum_{k=0}^{\infty}\,\dbinom{-\frac{1}{2}}{k}\,(-1)^{k}} & \leq\dbinom{-\frac{1}{2}}{k}\,\frac{-\frac{1}{2}-(k-1)}{-\frac{1}{2}-(2k-1)}\{(-1)^{k}\}$

Let us take k = 10, i.e.  ~ β20.
$\dpi{150} \bg_green f\,(1)=\dbinom{-\frac{1}{2}}{10}\frac{-\frac{1}{2}-(10-1)}{-\frac{1}{2}-(20-1)}\{(-1)^{10}\}=\frac{9.5}{19.5}\dbinom{-\frac{1}{2}}{10}$
$\dpi{150} \bg_green f\,(1)=\frac{9.5}{19.5}\frac{(-\frac{1}{2})!}{(-\frac{1}{2}-10)!(10)!}=\frac{9.5}{19.5}\frac{(-0.5)}{(-10.5)\times3628800}$
$\dpi{150} \bg_green f\,(1)=-\frac{95\times5\times10^{-2}\times2.76\times10^{-5}}{195\times105}=1.314\times10^{-8}$
As we had noted earlier a mass-order value is in the 𝕆(10 − 8) and we need to multiply for consistency of speed-of-light, c = 3.0 × 108 m ⁄ s everywhere there is a m-term, which is the case for f (1) = β → 1(γβ). So we have
(10)  $\dpi{150} \bg_green \mathbf{f\,(1)=\sum_{\beta\rightarrow1}(\gamma\beta)=d_{b}=3.942.}$
Now we evaluate the constant
$\dpi{150} \bg_green c_{b}=\left(\frac{d(\gamma\beta)}{d\beta}\right)_{\beta\rightarrow1}=\left(\frac{\Delta(\gamma\beta)}{\Delta\beta}\right)_{\beta\rightarrow1}.$
We differentiate
$\dpi{150} \bg_green \gamma\beta=\frac{\beta}{\sqrt{1-\beta^{2}}}$
$\dpi{150} \bg_green \frac{d}{dx}\left(\frac{F_{1}}{F_{2}}\right)=\frac{F_{2}.F_{1}^{'}-F_{1}.F_{2}^{'}}{F_{2}^{2}}$,

We define cb = g (β) + h (β) with
(12) h (β) = 2β2(1 − β2)3 ⁄ 2
and g (β) = f (β) if β = 1.
(13)  $\dpi{150} \bg_green h\,(\beta)=2\beta^{2}\left\{ \sum_{k=0}^{\infty}\,\dbinom{-\frac{3}{2}}{k}\,(-\beta^{2})^{k}\right\}$
(14) $\dpi{150} \bg_green h\,(1)=2\left\{ \sum_{k=0}^{\infty}\,\dbinom{-\frac{3}{2}}{k}\,(-1)^{k}\right\} ^{k=10} & \leq2\,\dbinom{-\frac{3}{2}}{10}\,\dfrac{-\frac{3}{2}-9}{\frac{3}{2}-19}$
$\dpi{150} \bg_green h\,(1)=2\times\frac{10.5}{20.5}\frac{(-\frac{3}{2})!}{(-\frac{3}{2}-10)!(10)!}=\frac{9.5}{19.5}\frac{(-0.5)}{(-10.5)\times3628800}$
$\dpi{150} \bg_green h\,(1)=\frac{105\times2\times15\times10^{-2}\times2.76\times10^{-5}}{115\times205}=3.688\times10^{-8}$
$\dpi{150} \bg_green g\,(1)+h\,(1)=\sum_{\beta\rightarrow1}\frac{d}{d\beta}(\gamma\beta)=c_{b}=(3.688+1.314)\times10^{-8}$
We multiply here c = 3.0 × 108 m ⁄ s like earlier, this brings mass-terms and momentum-terms to the same order. We obtain
(15) $\dpi{150} \bg_green \mathbf{g\,(1)+h\,(1)=\sum_{\beta\rightarrow1}\frac{d}{d\beta}(\gamma\beta)=}\mathbf{c_{b}=15.006}$
This concludes our method of evaluating followed by summing the binomial coefficients in the expansion of the Lorentz Factors and their power functions. As noted we have evaluated these in the momentum-order.
6. After we performed our calculations and see that OPERA experiment not citing the uncertainty on their energy as a reason of why they see this anomaly we also realized that one of our previous analysis would have led to the same conclusion if we were to correctly interpret that equation. This equation we are referring is used in the text of Weinberg [3]. This can be considered an uncertainty on proper-time. This is not mentioned in terms of Compton wavelength of the participating particle or an uncertainty on proper-time. We change the ideas slightly and use it for an uncertainty on space and time;
(16) $\dpi{150} \bg_green \mathbf{(\Delta x)^{2}-(\Delta t)^{2}\leq\left(\dfrac{\hslash}{m_{0}}\right)^{2}}$
, note the sign of inequality here and the fact that
$\dpi{150} \bg_green \left(\frac{\hslash}{m_{0}}\right)^{2}=\lambda_{c}^{2}.$
So,
(17)  $\dpi{150} \bg_green (\Delta t)^{2}.\left\{ \left(\frac{\mathrm{v}}{c}\right)^{2}-1\right\} \leq\left(\frac{\hslash}{m_{0}}\right)^{2}, & \,(\beta+\Delta\beta_{c})^{2}\leq1+\frac{\lambda_{c}^{2}}{(\Delta t)^{2}}$
where Δβc is the speed-excess or causality violation in terms of speed if β → 1. Since 0 ≤ β ≤ 1 the last (in)equation means
(18) $\dpi{150} \bg_green \beta^{2}+(\Delta\beta_{c}){}^{2}+2.\beta.\Delta\beta_{c} & <1+\left(\frac{\lambda_{c}}{\Delta t}\right)^{2}+2.\frac{\lambda_{c}}{\Delta t}$
as (λc)/(Δt) is  + ve. If β → 1 this means
(19) $\dpi{150} \bg_green (1+\Delta\beta_{c})^{2}<\left(1+\frac{\lambda_{c}}{\Delta t}\right)^{2}$
, or
(20) Δβc < (λc)/(Δt)
One can see that the last expression is “exactly” what we had obtained in our analysis at the top. In-fact Δβc is a causality violation excess at this limit of β → 1, so we can change its sign and we have
(21) Δβ > (λc)/(Δt)

By summing binomially and seeing the fact that the mass m can be really small such as that of neutrino we see that we have a factor 0.211 in front of λc. This is to be expected since for the equality above that we had removed will now be valid. So we approach the equality; binomially and asymptotically. The Compton-wavelength-limit corresponds to an inequality. The sources of errors such as detector response will add-on to the Compton-Wavelength. In-fact there never is an equality since it is asymptotic. That is, the factor 0.211 is only approximate although very very accurate. So, these lighter particles are capable of offsetting the causality violation excess by say a factor 0.211 but not completely make it zero. One can continue to sum the binomial coefficients to a larger and larger value and have a lighter and lighter mass but never completely breakaway from the photon speed.
For electron which is much more massive than the neutrino, therefore, one would expect a factor which is  < 0.211 and one must see speed-uncertainties larger than any speed-excess above speed-of-light. Although

(22) $\dpi{150} \bg_green (\beta+\Delta\beta_{c})^{2}<\left(1+\frac{\lambda_{c}}{\Delta t}\right)^{2} & \Rightarrow\Delta\beta_{c}<\frac{\lambda_{c}}{\Delta t},\;\beta\to1$
(in the β → 1 limit) this is also valid in contrary since 0 ≤ β ≤ 1. This is evident because if β≪1 one needs a “additive” correction factor to Δβ which then makes the
$\dpi{150} \bg_green (\beta+\Delta\beta_{c})^{2}<\left(1+\frac{\lambda_{c}}{\Delta t}\right)^{2}$
valid/sufficient for inferring
Δβc < (λc)/(Δt)
, this “additive” correction factor is  − (1 − β), e.g. if β = 0.5 then
Δβc − 0.5 < (λc)/(Δt)
or
Δβc < (λc)/(Δt) + 0.5
and
Δβ ≥ (λc)/(Δt) + 0.5
Notice that Δβ and Δβc are different, Δβc refers to causality violation uncertainty hence Δβc is in the β → 1 limit but Δβ refers to actual uncertainty on the speed. So in general if 0 ≤ β < 1;
Δβ ≥ (λc)/(Δt) + (1 − β),  0 ≤ β < 1

# 4 Summary/Conclusion

The OPERA experiment is a particle physics experiment which has for 3 years accumulated a galore of neutrinos produced from the decay of protons. These protons were shoot from the laboratory at CERN, Geneva and the neutrinos produced from these passed through the earth-crust to another laboratory at Gran-Sasso, Italy which is located about 733 k.m.s, same as the coastline of Hong Kong or the distance of Oslo, Norway to Mo I Rana, Norway. The experiment recently claimed that as per its analysis aided by time precision of GPS satellites to about 10 nano-seconds they observe that their 15000 neutrinos would travel faster than light particles known as photons. Such neutrinos are called as super-luminal neutrinos as they can travel past lumina, that is light. This is not a very welcome observation since it puts into question Albert Einstein’s firmly established Theory of Relativity and almost all the other branches of Modern Physics. In the last 3 months the world has seen about a 100 applications of the various knowledge segments of the physical world to explain or refute the findings. None of the findings were sufficient to disprove OPERA experiment’s findings which had performed an analysis with an impressive statistical significance that is hardly ever achieved in such measurements.

We have deduced from basic energy-time uncertainty relation of Quantum Mechanics a very accurate form of speed-time uncertainty relation. This speed-time uncertainty relation imposes stringent conditions at what energy uncertainty OPERA can claim super-luminal neutrino and at what uncertainties it cannot. We follow basic conditions of special Theory of Relativity and Quantum Mechanics to claim that OPERA must have total energy uncertainties below  ~ 71 kilo-electron-volts to consistently conclude that special theory of Relativity is inconsistent with their finding. We cite latest measurement uncertainties in some recent experiments of similar nature and similar energy scale to claim that such precision has yet not been achieved. The way forward for OPERA is to study and disclose their energy uncertainties. This will rest the matter of whether neutrino speed can go above the speed limit of nature imposed by special theory of Relativity of Albert Einstein. According to this theory only light particles known as photons travel at the speed limit of nature since they have no mass. Neutrinos are known to have some mass, although very very small compared e.g. to electrons, because they can change their form from one type to another. According to Theory of Relativity anything that has mass can never go above the photon’s speed of  ~ (3 × 105 = 3, 00, 000) km ⁄ s.

# Acknowledgement

We are thankful to the free world for the resources which enabled us to discuss our ideas and communicate the research. In special we would like to mention free software from various sources such as source-forge and LyX, X-fig, googledocs repository and share, and a social site where we frequently find it very easy to make our communication: Face Book. One of the author is also sincerely thankful to the blog-site word-press. They have been a constant and supportive source for our communication without which the communication and brainstorming prior to professional sharing was not possible for us. This research was in part supported by i3tex and the Willgood institutes. Both authors also thank their families for a remarkable support they have availed from their respective families.

# References

[1] G.Arfken; H.Weber, "Binomial Theorem", 6th Indian edition, chapter 5.6, (1988)
[2] R.Resnick, "Introduction to Special Relativity", Chapter 3.5, equation 3-18 b, (2010)
[3] S. Weinberg, “Gravitation and Cosmology: Special Relativity”, Wiley Indian Edition, chapter 2.13, (2009)
[4] Michael Lugo (mathoverflow.net/users/143), Sum of ’the first k’ binomial coefficients for fixed n , http://mathoverflow.net/questions/17236 (version: 2010-03-05)
[5] Dash, M. & Franzén, M. (2012). “Time dilation have opposite signs in hemispheres of recession and approach”, PHILICA.COM, Article number 312
[6] The OPERA Collaboration, “Measurement of the neutrino velocity with the OPERA detector in the CNGS beam”, arXiv:1109.4897v1 [hep-ex], (2011).
[7] The Belle Collaboration, “Search for B0s → hh Decays at the Υ(5S) Resonance”, arXiv:1006.5115v2 [hep-ex] (2011).
[8] The Belle Collaboration, “Observation of a New Narrow Charmonium State in Exclusive B± → K + π + π − J ⁄ ψ Decays”, arXiv:0308029v1[hep-ex] (2003).
[9] The MINOS Collaboration, “Measurement of neutrino velocity with the MINOS detectors and NuMI neutrino beam”, arXiv:0706.0437v3 [hep-ex](2007).
(TYPESET note: one can also use personal installation of Latex Equation Editor from Source-forge.net, although the codecogs.com plugin works pretty well, oen may remove the split environments in latex to see this, did not work for me)

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