Mathematical Relation between the Proton Mass , the Proton Radius , the Fine Structure Constant , the Compton Wavelength and the Hagedorn Temperature

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The Proton Radius, the Proton Mass, and the Fine Structure Constant
The proton radius is given by NIST CODATA as 0.8751×10 −15 meters, with a standard uncertainty of 0.0061 × 10 −15 .There has been considerable discussion about the proton radius in the recent years; see [1][2][3][4].
The proton mass is given by CODATA as 1.672621898×10 −27 .Arnold Sommerfeld [5] introduced the fine structure constant in relation to spectral lines.The fine structure constant, α ≈ 0.0072973525664 (2014 CODATA recommended values), plays an important role in modern physics.There is no consensus on exactly why the fine structure constant has the "magical" value it has, but there are a number of interesting, speculative ideas about what it could represent; see [6][7][8][9], for example.
Here we speculate that the proton radius could be connected to the proton mass, m P , through the fine structure constant by the following formula This is well inside the uncertainty in the proton radius given by CODATA.However, it does not necessarily mean there is anything to this relationship.Many readers will recognize that m P c is the reduced Compton wavelength formula, so we could alternatively have written the formula simply as where λP is the reduced Compton wavelength of the proton.Naturally, there is uncertainty here, just as there is some uncertainty in the proton mass and even, to some degree, in the fine structure constant.
Another interesting point is that if we had a hypothetical particle with a reduced Compton wavelength equal to half the proton radius given by the formula above, then if this particle travelled at a velocity equal to v = c−αc = c(1−α), it would have a relativistic mass exactly equal to the proton mass.
It is well-known that any elementary mass can be described as Assume now we have λ = 2r p , then we get a relativistic mass of where m P is the proton mass.That would be rather "fun"-a particle traveling at speed c(1 − α) and relativistic mass equal to the proton surrounded by a particle (electron) traveling at speed cα.
Looking at this mathematical relationship, if what it appears to hold is more than a coincidence, then we can also find the fine structure constant from the proton radius and the reduced Compton wavelength of the proton and only the negative solution seems to make sense, so this would mean Again, this solution is based on the idea that there may be a fundamental relationship here; granted this is a speculative suggestion on a possible relationship with the fine structure constant.We leave it up to future inquiry to see whether this will lead to anything useful or not.

The Fine-Structure Constant and the Hagedorn Temperature
The Hagedorn temperature [10] is a theoretical temperature where quarks can be extracted (evaporated) from a proton.It is essentially the temperature where hadronic matter is unstable.The Hagedorn temperature is extremely high; according to traditional grand-unified string models, it is said to be about 10 30 kelvin, see [11].This is about two orders of magnitude smaller than the Planck temperature, which is ≈ 1.03391 × 10 30 kelvin and we divide this temperature by the Planck temperature [12,13], we would get the fine structure constant.Actually, there is considerable uncertainty, even regarding the size of the Planck mass.However, we can indirectly measure the Planck mass from a Cavendish experiment without any knowledge of the Newton gravitational constant; see [14].Further, the uncertainty in the Planck mass is half the uncertainty of the Newton gravitational constant.
It is important to note that the Hagedorn temperature has never been measured and if its true temperature is different than the speculation here, then there is no such connection.

Conclusions
In this paper we have suggested a possible interesting relationship between the proton mass and the proton radius "through" the fine structure constant.We have also suggested a possible connection between the Hagedorn temperature, the Planck temperature, and the fine structure constant.We do not know if there are truly such connections or not, and more observational and theoretical work is needed.
We will end this note with a quote from Feynman about the fine structure constant: It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.

mpc 2 k
b ≈ 1.42 × 10 32 kelvin.If the Hagedorn temperature should be mpc 2 α k b