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Here is a latest status from the author-review of the paper

*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.

—————————————————–

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.

## Manmohan Dash

## mdash@vt.edu, manmohan.dash@willgood.org

## Mikael Franzén

## mikael.franzen@willgood.org, mikael.franzen@i3tex.com

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

*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.

*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 {

*S*

_{r}= 2.

*GM*

_{e}} 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.

**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) × 10

^{6}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;

*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.**

*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.

# 2 Relativistic kinematics and Quantum mechanics

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

*E*

^{2}=

*m*

^{2}+

*p*

^{2}, so

**E = (m**

^{2}+ p^{2})^{1 ⁄ 2},*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.

*E*= {

^{1}⁄

_{2}} × (

*m*

^{2}+

*p*

^{2})

^{ − 1 ⁄ 2}× 2 × {

*m*Δ

*m*+

*p*Δ

*p*}.

*δ*= central difference.

*energy*−

*time*uncertainty relationship, Δ

*E*.Δ

*t*≥ ℏ;

*E*.Δ

*t*= (

*m*

^{2}+

*p*

^{2})

^{ − 1 ⁄ 2}(

*m*Δ

*m*+

*p*Δ

*p*).Δ

*t*≥ ℏ,

*m*

^{2}+

*p*

^{2})

^{ − 1 ⁄ 2}

*m*.Δ

*m*.Δ

*t*+ (

*m*

^{2}+

*p*

^{2})

^{ − 1 ⁄ 2}

*p*.Δ

*p*.Δ

*t*≥ ℏ.

*γ*

^{2}

*β*

^{2})

^{ − 1 ⁄ 2}Δ

*m*.Δ

*t*+ (1 +

*γ*

^{2}

*β*

^{2})

^{ − 1 ⁄ 2}

*γ*

*β*.Δ

*p*.Δ

*t*≥ ℏ,

*p*=

*m*

*γ*

*β*, so naturally

*p*= (Δ

*m*)

*γ*

*β*+

*m*.Δ(

*γ*

*β*).

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 Δ*m*.Δ*t* ~ ℏ limit which says that any uncertainty on *m*is a minimum in that limit, so we have

*γ*

^{2}

*β*

^{2})ℏ +

*m*.

*c*

_{b}.

*γ*

*β*.Δ

*β*.Δ

*t*≥ ℏ√(1 +

*γ*

^{2}

*β*

^{2}).

*m*.Δ

*t*~ ℏ does not make the minimum Δ

*E*.Δ

*t*~ ℏ, in other words eqn(4↑) is not an equality yet, and this is consistent.

So;

**m.c**

_{b}.γβ.Δβ.Δt**≥**

**ℏ(√(1 + γ**

^{2}β^{2}) − 1 − γ^{2}β^{2})

**(m.c**

_{b}.γβ)/(√(1 + γ^{2}β^{2}) − (1 + γ^{2}β^{2})).Δβ.Δt ≥ ℏ**(m.c**

_{b}.γβ)/(√(1 + d^{2}_{b}) − (1 + d^{2}_{b})).Δβ.Δt ≥ ℏ**Δβ.Δt ≥ (ℏ)/(m.c**

_{b}.d_{b}).(√(1 + γ^{2}β^{2}) − 1 − γ^{2}β^{2})**Δβ.Δt ≥ (λ**

_{c})/(c_{b}.d_{b}).(√(1 + γ^{2}β^{2}) − 1 − γ^{2}β^{2})

*m*.Δ

*t*~ ℏ . Also, it is worthwhile to mention here that Δ

*β*in the above equations is Δ

*β*

_{c}= causality violation uncertainty which is necessarily −

*ve*. We can intuit this if we say ±

*abs*(Δ

*β*

_{c}) = Δ

*β*where Δ

*β*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

*eq*: − 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 Δ

*m*.Δ

*t*~ 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

*c*

_{b},

*d*

_{b}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

*c*

_{b}= 15.006 and

*d*

_{b}= 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}. *λ*_{c} = reduced Compton wavelength. We evaluate the above equations in terms of known values and we have

**Δβ**

_{c}.Δt ≥ − 0.211.(λ_{ν})This is not only valid for neutrinos but also for any particle that is moving at or near the *speed* = *β* = 1 ≡ 3.0 × 10^{8}*m* ⁄ *s*. We derived *c*_{b}, *d*_{b} to the order *β*^{10} 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 Δ*β* 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 Δ*β*is restricted by the Compton wavelength.With that in mind

**Δβ.Δt ≥ 0.211.(λ**

_{ν})or

*β*≥ (0.211 × 6.6 × 10

^{ − 7}×

*eV*.

*ns*)/(2 × 10 ×

*eV*.

*ns*)

or Δ*β* ≥ 0.696 × 10^{ − 8}, for *c* = 3.0 × 10^{8} *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^{ − 4}eV**, 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 × 10

^{8}and for

*energy*−

*order*and then one multiplies the

*momentum*−

*order*by

*c*= 3 × 10

^{8}. So for a 2.09

*km*⁄

*s*uncertainty we have;

**ΔE ~ 19.8 KeV**, Δ

*p*~ 660

*ppm*

*of*1

*eV*, Δ

*m*~ 0.22 × 10

^{ − 12}

*eV*.

**ΔE ~ 19.8 KeV**, Δ

*p*~ 660

*ppm*

*of*1

*eV*, Δ

*m*~ 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**, Δ

*p*~ 2.37

*meV*or Δ

*m*~ 0.79 × 10

^{ − 12}

*eV*. We refer to this in fig. (1↓)

## 2.3 Errors on energy from recent experiments

*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].

*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.

*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

- 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*. . - 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 ℏ, 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. . - The uncertainty of d and t go opposite to the uncertainty of the other variables
*E*,*p*, v,*m*to the order of either ℏ or in consistency of units ℏ ⁄*m*_{0}. ℏ ⁄*m*_{0}, in speed-of-light units is called a reduced Compton wavelength of the particle represented by mass*m*_{0}. . 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 vin the set of equivalent variables. - How do we sum our binomial coefficients? We use a bound on summation of binomial coefficients given by Michael Lugo [6]:

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 Δ*E*, as binomial coefficients are fractions that we summed to a very high degree already. - 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)*.*

We want

*f*(1).(14)According to a bound given in

*MathOverFlow*.*Net*as refered in Note-(3↑), [4](15)

**.**Let us take

*k*= 10, i.e. ~*β*^{20}. then;.

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 × 10^{8}*m*⁄*s*everywhere; there is a*m*−*term*, which is the case for*f*(1) = ∑_{β → 1}(*γ**β*). So we have(16)Now we evaluate the constant

,

,(17)We define*c*_{b}=*g*(*β*) +*h*(*β*) with (18)*h*(*β*) = 2*β*^{2}(1 −*β*^{2})^{3 ⁄ 2}and

*g*(*β*) =*f*(*β*) if*β*= 1,(19)(20)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*. - 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) ,note the sign of inequality here and the fact that

So,

(22)where Δ

*β*_{c}is the speed-excess or causality violation in terms of speed if*β*→ 1. Since 0 ≤*β*≤ 1the last (in)equation means(23)as (

*λ*_{c})/(Δ*t*) is +*ve*. If*β*→ 1 this means

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 Δ*β*_{c} is a causality violation excess at this limit of *β* → 1, so we can change its sign and we have

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 *λ*_{c}. 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 0.211but 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

(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 Δ*β* which then makes the

valid/sufficient for inferring Δ*β*_{c} < (*λ*_{c})/(Δ*t*). This “additive” correction factor is − (1 − *β*), eg if *β* = 0.5 then Δ*β*_{c} − 0.5 < (*λ*_{c})/(Δ*t*) or Δ*β*_{c} < (*λ*_{c})/(Δ*t*) + 0.5 and Δ*β* ≥ (*λ*_{c})/(Δ*t*) + 0.5. Notice that Δ*β* and Δ*β*_{c} are different: Δ*β*_{c} refers to causality violation uncertainty, hence Δ*β*_{c} is in the *β* → 1 limit but Δ*β* refers to actual uncertainty on the speed. So in general if 0 ≤ *β* < 1;

**Δβ ≥ (λ**

_{c})/(Δt) + (1 − β), 0 ≤ β < 1.

- 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:***δ**τ*= 0,*dt*^{2}−*dx*^{2}= 0{for the distance and time to be equal only light has this property}.*δ**τ*= 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*ν*’*s*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

^{5}= 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

*B*

^{0}

_{s}→

*hh*Decays at the Υ(5

*S*) Resonance”, arXiv:1006.5115v2 [hep-ex] (2011).

*B*

^{±}→

*K*

^{ + }

*π*

^{ + }

*π*

^{ − }

*J*⁄

*ψ*Decays”, arXiv:0308029v1[hep-ex], (2003).

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

## Manmohan Dash

## mdash@vt.edu, manmohan.dash@willgood.org

author’s mail: Mahisapat, Dhenkanal, Odisha, India, 759001

## Mikael Franzén

## mikael.franzen@willgood.org, mikael.franzen@i3tex.com

*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

*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.

*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 {

*S*

_{r}= 2.

*GM*

_{e}} 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.

**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) × 10

^{6}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;

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.

**“energy, mass, momentum”**equation usually called the

**energy-momentum**relation [2] is expressed in

**speed-of-light**= 1 units as:

*E*

^{2}=

*m*

^{2}+

*p*

^{2}, so

**E = (m**

^{2}+ p^{2})^{1 ⁄ 2}*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:

*E*= {

^{1}⁄

_{2}} × (

*m*

^{2}+

*p*

^{2})

^{ − 1 ⁄ 2}× 2 × {

*m*Δ

*m*+

*p*Δ

*p*}.

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, Δ*E*.Δ*t* ≥ ℏ;

*E*.Δ

*t*= (

*m*

^{2}+

*p*

^{2})

^{ − 1 ⁄ 2}(

*m*Δ

*m*+

*p*Δ

*p*).Δ

*t*≥ ℏ, so,

*m*

^{2}+

*p*

^{2})

^{ − 1 ⁄ 2}

*m*.Δ

*m*.Δ

*t*+ (

*m*

^{2}+

*p*

^{2})

^{ − 1 ⁄ 2}

*p*.Δ

*p*.Δ

*t*≥ ℏ.

So we have now,

*γ*

^{2}

*β*

^{2})

^{ − 1 ⁄ 2}Δ

*m*.Δ

*t*+ (1 +

*γ*

^{2}

*β*

^{2})

^{ − 1 ⁄ 2}

*γ*

*β*.Δ

*p*.Δ

*t*≥ ℏ

in the left we used *p* = *m**γ**β*, so naturally

*p*= (Δ

*m*)

*γ*

*β*+

*m*.Δ(

*γ*

*β*).

where

We also define *d*_{b} = (*γ**β*)_{β → 1}.

*β*→ 1 limit and re-evaluate the constants. Let us take the Δ

*m*.Δ

*t*~ ℏ limit which says any uncertainty on

*m*is a minimum in that limit, so we have

**(1 +**

*γ*^{2}*β*^{2})ℏ +*m*.*c*_{b}.*γ**β*.Δ*β*.Δ*t*≥ ℏ√(1 +*γ*^{2}*β*^{2})*m*.Δ

*t*~ ℏ still does not make the minimum Δ

*E*.Δ

*t*~ ℏ, in other words eqn(4↑) is not an equality yet, which is only consistent.

**m.c**

_{b}.γβ.Δβ.Δt**≥**

**ℏ(√(1 + γ**

^{2}β^{2}) − 1 − γ^{2}β^{2})**(**

*m*.*c*_{b}.*γ**β*)/(√(1 +*γ*^{2}*β*^{2}) − (1 +*γ*^{2}*β*^{2})).Δ*β*.Δ*t*≥ ℏ**(**

*m*.*c*_{b}.*γ**β*)/(√(1 +*d*^{2}_{b}) − (1 +*d*^{2}_{b})).Δ*β*.Δ*t*≥ ℏ**Δ**

*β*.Δ*t*≥ (ℏ)/(*m*.*c*_{b}.*d*_{b}).(√(1 +*γ*^{2}*β*^{2}) − 1 −*γ*^{2}*β*^{2})**Δ**

*β*.Δ*t*≥ (*λ*_{c})/(*c*_{b}.*d*_{b}).(√(1 +*γ*^{2}*β*^{2}) − 1 −*γ*^{2}*β*^{2})We give a general description of this in the note where we do not set Δ*m*.Δ*t* ~ ℏ . Also it is worthwhile to mention here that Δ*β* in the above equations is Δ*β*_{c} = causality violation uncertainty which is necessarily − *ve*. We can intuit this if we say ±*abs*(Δ*β*_{c}) = Δ*β* where Δ*β*is actual uncertainty on speed which can be blown up by errors from a variety of sources.

## 2.2 The OPERA neutrino speed excess

**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 Δ

*m*.Δ

*t*~ 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

*c*

_{b},

*d*

_{b}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

*c*

_{b}= 15.006 and

*d*

_{b}= 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}. *λ*_{c} is reduced **Compton Wavelength**. We evaluate the above equations in terms of known values and we have

**Δβ**

_{c}.Δt ≥ − 0.211.(λ_{ν})*speed*=

*β*= 1 ≡ 3.0 × 10

^{8}

*m*⁄

*s*. We derived

*c*

_{b},

*d*

_{b}to the order

*β*

^{10}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 Δ

*β*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 Δ

*β*is restricted by the Compton wavelength.

With that in mind

**Δβ.Δt ≥ 0.211.(λ**

_{ν})or,

*β*≥ (0.211 × 6.6 × 10

^{ − 7}×

*eV*.

*ns*)/(2 × 10 ×

*eV*.

*ns*)

or Δ*β* ≥ 0.696 × 10^{ − 8}, for *c* = 3.0 × 10^{8} *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 ^{ − 4}eV**, 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 × 10

^{8}and for

**energy-order**one multiplies the

**momentum-order**by

*c*= 3 × 10

^{8}. So we have for 2.09

*km*⁄

*s*uncertainty,

**ΔE ~ 19.8 KeV**, Δ

*p*~ 660

*ppm*

*of*1

*eV*, Δ

*m*~ 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**, Δ*p* ~ 660 *ppm* *of* 1*eV*, Δ*m* ~ 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**, Δ*p* ~ 2.37 *meV* or Δ*m* ~ 0.79 × 10^{ − 12} *eV*.

## 2.3 Errors on energy from recent experiments

*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

- 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*. - 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 ℏ, 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. - The uncertainty of
*d*and*t*go opposite to the uncertainty of the other variables {*E*,*p*, v,*m*} to the order of either ℏ or in consistency of units ℏ ⁄*m*_{0}. ℏ ⁄*m*_{0}in speed-of-light units is called a**Reduced Compton Wavelength**of the particle represented by mass*m*_{0}. 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 vin the set of equivalent variables. - How do we sum our binomial coefficients? We use a bound on summation of binomial coefficients given by Michael Lugo: [4];
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 Δ*E*as binomial coefficients are fractions, we summed to a very high degree already. - 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].We want*f*(1);(8)According to a bound given in

**MathOverFlow.Net**as refered in Note-(3↑), [4](9)Let us take

*k*= 10, i.e. ~*β*^{20}.

As we had noted earlier a**mass-order**value is in the**𝕆(10**and we need to multiply for consistency of^{ − 8})**speed-of-light,***c*= 3.0 × 10^{8}*m*⁄*s*everywhere there is a**m-term**, which is the case for*f*(1) = ∑_{β → 1}(*γ**β*). So we have(10)Now we evaluate the constant

We differentiate

We define*c*_{b}=*g*(*β*) +*h*(*β*) with(12)*h*(*β*) = 2*β*^{2}(1 −*β*^{2})^{3 ⁄ 2}and

*g*(*β*) =*f*(*β*) if*β*= 1.(13)(14)We multiply here

*c*= 3.0 × 10^{8}*m*⁄*s*like earlier, this brings**mass-terms**and**momentum-terms**to the same order. We obtain(15)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**. - 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), note the sign of inequality here and the fact that

So,

(17)where Δ

*β*_{c}is the speed-excess or causality violation in terms of speed if*β*→ 1. Since 0 ≤*β*≤ 1the last (in)equation means(18)as (

*λ*_{c})/(Δ*t*) is +*ve*. If*β*→ 1this means(19), or

(20)**Δβ**_{c}< (λ_{c})/(Δt)One can see that the last expression is “exactly” what we had obtained in our analysis at the top. In-fact Δ

*β*_{c}is a causality violation excess at this limit of*β*→ 1, so we can change its sign and we have(21)**Δβ > (λ**_{c})/(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*λ*_{c}. 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)(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 Δ*β*which then makes thevalid/sufficient for inferring

Δ*β*_{c}< (*λ*_{c})/(Δ*t*), this “additive” correction factor is − (1 −

*β*), e.g. if*β*= 0.5thenΔ*β*_{c}− 0.5 < (*λ*_{c})/(Δ*t*)or

Δ*β*_{c}< (*λ*_{c})/(Δ*t*) + 0.5and

Δ

*β*≥ (*λ*_{c})/(Δ*t*) + 0.5Notice that Δ

*β*and Δ*β*_{c}are different, Δ*β*_{c}refers to causality violation uncertainty hence Δ*β*_{c}is in the*β*→ 1 limit but Δ*β*refers to actual uncertainty on the speed. So in general if 0 ≤*β*< 1;**Δβ ≥ (λ**_{c})/(Δt) + (1 − β), 0 ≤ β < 1

Figure 1 Basic Quantum Mechanics: The super-luminal neutrinos need to have

an energy precision better than energy uncertainty corresponding to

reduced Compton wavelength of the neutrinosFigure 2 **The situation in a super-luminal paradigm, theory of Relativity**that corresponds to other particles. {nothing takes lesser time than light}

**On the shell:***δ**τ*= 0,*dt*^{2}−*dx*^{2}= 0

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

*δ**τ*= 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*ν*’*s*are super-luminal 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

^{5}= 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

**3-18 b**, (2010)

*B*

^{0}

_{s}→

*hh*Decays at the Υ(5

*S*) Resonance”, arXiv:1006.5115v2 [hep-ex] (2011).

*B*

^{±}→

*K*

^{ + }

*π*

^{ + }

*π*

^{ − }

*J*⁄

*ψ*Decays”, arXiv:0308029v1[hep-ex] (2003).

(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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