rapidly decreasing with distance. Other forces that would be expressed as similar
products of α and δ p appear too large or too small to be observable.
Another link between the electric and gravific forces [27] results from the integer
parts of a and d p being the sums of Catalan numbers in the Mersenne series: 2
2
−
1 = 3, 2
3
− 1 = 7, 2
7
− 1 = 127, then M’ 7 = 3 + 7 + 127 = 137 ≈ a, and the
sum M’ 127 up to 2
127
− 1 ≈ d p where d p ≡ δ
− 1
p and δ p is defined as δ e but with the
electron mass replaced by that of the proton (or the neutron). The magnitude of the
strong nuclear force, although it is a short distance one, seems consistent with the
sum M’ 3 = 10.
In this paper, we have discussed the deviations, from their integer approximates,
of the measured values of the free electron gyromagnetic factor g e , the fine-structure
constant inverse a, and the proton gravitational invariant inverse d p . After recalling
that the relative deviation of the measured g e from 2 (∼ 0.1%) was explained by
quantum field and size effects, we have tried to express that of a from 137 and that
of d p from M’ 127 as similar expansions.
Surprisingly, the relative deviation of measured a from 137 (< 0.03%) could
even better be expressed (with 0.4 ppb accuracy) as a simple 2-term series
involving even powers of π/137. Similarly, the relative deviation of measured d p
from M’ 127 (< 0.50%) could be expressed (with 0.6 ppm accuracy) as 2-term series
involving powers of either 2/137 ð ≈ g e ̸ aÞ or g e ̸ aπ ð ≈ 1 ̸ 6
3
Þ. For reasons yet
unexplained, the numbers g e , a, d p and π seem to be deeply related.
Acknowledgements I wish to thank the colleagues who helped me clarify these ideas at QSCP
meetings and elsewhere. Erkki Brändas, Uzi Kaldor, John Macken, Francis Sanchez, and Ivan
Todorov made especially useful comments. Thanks are due to my wife, Marja Rantanen, for
stimulating my speculations with inspiring piano playing.
References
1. Maruani J (2012) The Dirac electron: spin, Zitterbewegung, the Compton wavelength, and the
kinetic foundation of rest mass. Prog Theor Chem Phys B 26:23–46
2. Maruani J (2013) The Dirac electron as a massless charge spinning at light speed: implications
on some basic physical concepts. Prog Theor Chem Phys B 27:53–74
3. Maruani J (2015) The Dirac electron as a privileged road to the understanding of quantum
matter. Quantum Matter 4:3–11
4. Maruani J (2016) The Dirac electron: from quantum chemistry to holistic cosmology. J Chin
Chem Soc 63:33–48 and references therein
5. Thaller B (1992) The Dirac equation. Springer, Berlin
6. Sakurai J (1967) Advanced quantum mechanics. Addison-Wesley, Reading, MA, ch. 3
7. Feynman RP (1998) Quantum electrodynamics. Addison-Wesley, Reading, MA
8. Weinberg S (1995) The quantum theory of fields. Cambridge U P
9. Dirac PAM (1928) Quantum theory of the electron. Proc Roy Soc (London) A 117:610–624;
Quantised singularities in the electromagnetic field: ibid (1931) A 133:60–72; Theory of
electrons and positrons. Nobel lectures (1933) pp 320–325
10. Dirac PAM The principles of quantum mechanics. Clarendon Press, Oxford, 1st edn 1930,
4th edn 1958, chs 11–12
378
J. Maruani
products of α and δ p appear too large or too small to be observable.
Another link between the electric and gravific forces [27] results from the integer
parts of a and d p being the sums of Catalan numbers in the Mersenne series: 2
2
−
1 = 3, 2
3
− 1 = 7, 2
7
− 1 = 127, then M’ 7 = 3 + 7 + 127 = 137 ≈ a, and the
sum M’ 127 up to 2
127
− 1 ≈ d p where d p ≡ δ
− 1
p and δ p is defined as δ e but with the
electron mass replaced by that of the proton (or the neutron). The magnitude of the
strong nuclear force, although it is a short distance one, seems consistent with the
sum M’ 3 = 10.
In this paper, we have discussed the deviations, from their integer approximates,
of the measured values of the free electron gyromagnetic factor g e , the fine-structure
constant inverse a, and the proton gravitational invariant inverse d p . After recalling
that the relative deviation of the measured g e from 2 (∼ 0.1%) was explained by
quantum field and size effects, we have tried to express that of a from 137 and that
of d p from M’ 127 as similar expansions.
Surprisingly, the relative deviation of measured a from 137 (< 0.03%) could
even better be expressed (with 0.4 ppb accuracy) as a simple 2-term series
involving even powers of π/137. Similarly, the relative deviation of measured d p
from M’ 127 (< 0.50%) could be expressed (with 0.6 ppm accuracy) as 2-term series
involving powers of either 2/137 ð ≈ g e ̸ aÞ or g e ̸ aπ ð ≈ 1 ̸ 6
3
Þ. For reasons yet
unexplained, the numbers g e , a, d p and π seem to be deeply related.
Acknowledgements I wish to thank the colleagues who helped me clarify these ideas at QSCP
meetings and elsewhere. Erkki Brändas, Uzi Kaldor, John Macken, Francis Sanchez, and Ivan
Todorov made especially useful comments. Thanks are due to my wife, Marja Rantanen, for
stimulating my speculations with inspiring piano playing.
References
1. Maruani J (2012) The Dirac electron: spin, Zitterbewegung, the Compton wavelength, and the
kinetic foundation of rest mass. Prog Theor Chem Phys B 26:23–46
2. Maruani J (2013) The Dirac electron as a massless charge spinning at light speed: implications
on some basic physical concepts. Prog Theor Chem Phys B 27:53–74
3. Maruani J (2015) The Dirac electron as a privileged road to the understanding of quantum
matter. Quantum Matter 4:3–11
4. Maruani J (2016) The Dirac electron: from quantum chemistry to holistic cosmology. J Chin
Chem Soc 63:33–48 and references therein
5. Thaller B (1992) The Dirac equation. Springer, Berlin
6. Sakurai J (1967) Advanced quantum mechanics. Addison-Wesley, Reading, MA, ch. 3
7. Feynman RP (1998) Quantum electrodynamics. Addison-Wesley, Reading, MA
8. Weinberg S (1995) The quantum theory of fields. Cambridge U P
9. Dirac PAM (1928) Quantum theory of the electron. Proc Roy Soc (London) A 117:610–624;
Quantised singularities in the electromagnetic field: ibid (1931) A 133:60–72; Theory of
electrons and positrons. Nobel lectures (1933) pp 320–325
10. Dirac PAM The principles of quantum mechanics. Clarendon Press, Oxford, 1st edn 1930,
4th edn 1958, chs 11–12
378
J. Maruani