A follow-up to 'e and pi'.

One more curious fact, before I leave the subject of those marvellous numbers e and π, at least for now.

            Quantum electrodynamics employs a theoretical coupling constant, instead of the actual, empirical one, which as we know is α = 0.00729735308.  The QED version is called j² and is exactly equal to 0.01; j = 0.1 is the QED equivalent of electric charge.  (The reason being that α^½ = e/√(4πε₀ħc) = 0.085424546 is a way of expressing electric charge in dimensionless terms, with ε₀ = 1/4π, and ħ and c both equal to 1 in ‘natural units’.)

            It so happens that:

 

m_p/m_e  =  8παφμ_e/j⁴  =  1836.152755656 ,

 

where φ is a dimensionless constant, equal to 1.0000005765078.

            We have already seen that

 

m_p/m_e  =  eπ²ζ/2αμ_e ,

 

and

 

α  =  eξ/12π³μ_e ,

 

where ζ = 1.0000305692511 and ξ = 1.0000116993595, respectively.

            It is easy to see from this that:

 

4αξφμ_e/3ζπ⁴j⁴  =  1 ,

 

which relates all of the (apparently arbitrary) dimensionless constants to the values of α, μ_e, j and π.  (Interestingly, e disappears.)

            The one constant we have left is the largest of them, κ = 1.000249044275, which appeared, it will be recalled, in the equation:

 

m_n/m_e  =  eπ²κ/2α .

 

Simple re-arranging gives us

 

2α  =  eπ²κm_e/m_n ,

 

and substituting in the above, we have

 

2eκξφμ_em_e/3ζπ²j⁴m_n  =  1 ,

 

and so e makes its re-appearance here.

            The expression κξφ/ζ = 1.0002307469, which constant I shall label η.  Ergo, the above may be re-written:

 

2eημ_em_e/3π²j⁴m_n  =  1 ,

 

or:

 

m_n/m_e  =  2eημ_e/3π²j⁴  =  1838.68366066 .

These results regarding the proton-electron and neutron-electron mass ratios are quite remarkable.  I have known about them for some time, but I have been unable to interest the scientific community in either of them, because they do not fit into any of the existing paradigms.  That is unfortunate for me, perhaps, but doubly unfortunate for the scientific community, which is yet again showing how it is apt to be blinded by its own prejudices.

            I may well have more to say about this on another occasion.

 

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