\documentclass{Rinton-P9x6}


 

1 

 

Equivalence of Electromagnetic Fluctuation and Nuclear (Yukawa) Forces:            

the 𝝅𝟎 Meson, its Mass and Lifetime 

 Barry W. Ninham1,*, Iver Brevik2,* and Mathias Boström3,*  

1  Department of  Materials Physics , Research School of Physics , Australian National University, 

Canberra, Australia, 0200. 

2 Department of Energy and Process Engineering, Norwegian University of Science and 

Technology, NO-7491 Trondheim, Norway. 

3Centre for Materials Science and Nanotechnology, Department of Physics, University of Oslo, P. 

O. Box 1048 Blindern, NO-0316 Oslo, Norway. 

* E-mail addresses: Barry.Ninham@anu.edu.au; iver.h.brevik@ntnu.no; 

mathias.bostrom@smn.uio.no  

 

Received: Sep 09, 2022 Revised: Dec 03, 2022 Just Accepted Online: Dec 05, 2022 Published: Xxx 

 

This article has been accepted for publication and undergone full peer review but has not been 

through the copyediting, typesetting, pagination and proofreading process, which may lead to 

differences between this version and the Version of Record. 

 

Please cite this article as: 

Ninham B.W., Brevik I., Boström M. (2022) Equivalence of Electromagnetic Fluctuation and 

Nuclear (Yukawa) Forces: the π0 Meson, its Mass and Lifetime. Substantia. Just Accepted. DOI: 

10.36253/Substantia-1807 

mailto:Barry.Ninham@anu.edu.au
mailto:iver.h.brevik@ntnu.no
mailto:mathias.bostrom@smn.uio.no


 

2 

 

 

Abstract  

It is shown how Maxwell’s equations for the electromagnetic field with Planck quantisation of 

allowed modes appears to provide a semiclassical account of nuclear interactions. The mesons 

emerge as plasmons, collective excitations in an electron positron pair sea. The lifetime and mass 

of 𝜋0 mesons are predicted.  

 

Keywords: Casimir-effect, Meson-theory, positron-electron-plasma, lifetime 

 

 

 

1. Electrodynamic fluctuation forces. 

Feynman is reported to have believed that  there had to be a connection between electromagnetic 

theory and nuclear forces.  [1].  He never found such a link. It is shown how such a connection might 

come about. 

 

Historical background: Where do mesons come from??  

A hundred years ago Rutherford’s team at Cambridge had shown that the atomic nucleus was 

comprised of protons and neutrons. The particles had a mass, 2000 times that of an electron ; protons 

were positively charged. A neutron could transform into a proton and a negatively charged electron. 

Electrostatic forces played a role in the interactions between nucleons. But whatever other forces 

held them together remained a mystery [2]. Quantum mechanics in its various manifestations, from 

Planck, Sommerfeld and Bohr, Schroedinger, Heisenberg, Dirac; and later quantum electrodynamics 



 

3 

 

promised insights. In 1935 Yukawa had came up with a  characterisation of this so called “weak 

nuclear interaction” that worked. The force was mediated by “particles” called mesons, mass 273 

times that of an electron and variously charged. They were detected from cosmic ray decay by Powell 

in 1937. There were 𝜇 and 𝜋 and later bigger 𝐾 mesons.  Particle physics developed subsequently 

culminating in the prediction of Higgs unifying boson.  But the fundamental physics embodied in 

Maxwell’s equations for the electromagnetic field and quantum mechanics seemed to have nothing 

much to do with it. Somehow something was apparently missing. The electromagnetic forces seemed 

too small. But protons and electrons were charged. The mystery and disjunction remained.  

 

 

Theory in words without equations 

The classic paper of Casimir in 1948 [3] on relativistic effects on the attractive forces between 

colloidal particles was motivated by, and applied to the newly developed Deryaguin-Overbeek theory 

of Colloid stability. Overbeek had posed the problem of these ”retardation” effects to his friend 

Casimir in Utrecht (BWN private communication with Overbeek). Casimir derived the forces due to 

quantisation of zero temperature electromagnetic fluctuations in the vacuum between two ideal metal 

plates. It stimulated a huge literature that still flourishes. It seemed to bear on our problem. But it 

could not due to its limitation to zero temperature. This and the term retardation are incorrect and 

unphysical [4-6].  

Further insights into the nature of the electromagnetic vacuum had to wait on the 

development of Lifshitz theory for interactions between and across dielectric media [7] and 

included temperature. This theory at time appeared to be the culmination and triumph of quantum 

electrodynamics. It had been foreshadowed by P. N. Lebedev who discovered light radiation 



 

4 

 

pressure. He was a friend of J. Clerk Maxwell and the stepfather of Deryaguin. Deryaguin had asked 

Lifshitz to work on the problem.  In 1894 Lebedev wrote: ”If the solution of this problem ever 

becomes possible we shall be able, from the results of spectral analysis, to calculate in advance the 

values of the intermolecular forces due to molecular inter-radiation, deduce the laws of their 

temperature dependence, and, by comparing the values obtained with experimental results, solve 

the fundamental problem of molecular physics whether all the so-called ’molecular forces’ are 

confined to the already known mechanical action flight radiation mentioned above, to 

electromagnetic forces, or whether some forces of hitherto unknown origin are involved” as quoted 

by Deryaguin [8]. However the triumph was illusory. The generalisation of the Casimir effect 

involved some sleight of hand that approximated a non linear problem by a linear one [9].  

This theory applied to Casimir’s two plate problem gives out automatically: the binding 

energy of two nucleons in nucleus in equilibrium and automatically replaces the problem by one with 

a virtual intervening electron positron pair sea with known density and therefore plasma frequency 

of excitations. The renormalisation is identical to the Klein Gordon equation for scalar mesons with 

mass identified from the plasma frequency. The implication is that positive and negatively charged 

𝜋 mesons are identifiable with bound electron-plasma and positron-plasma excitations. And 

what used to be called K mesons are higher order double plasma excitations known from solid 

state physics. What is quite new is that the identification of the scalar 𝜋0 meson with a collective 

excitation in the electron positron sea allows us to calculate its lifetime , correctly. Taken 

together, binding energy, scalar meson mass , and lifetime all  seem to add plausibility to our 

case. The simplified version of Lifshitz theory we have used is the same Lifshitz theory at the 

foundations of physical chemistry, molecular and colloidal particle interactions in the DLVO 



 

5 

 

theory. There the limitations due to the linearisation approximation are very clear. If the 

equivalence we have drawn is correct so too must present theories of particle physics. 

 

We first outline what we mean by electromagnetic forces. 

A 1961 paper of Dzyaloshinski, Lifshitz and Pitaevski [7] applied quantum electrodynamics to the 

problem of molecular forces. It extended earlier work on electromagnetic fluctuation forces 

between molecules and colloidal particles of Casimir and Lifshitz to include effects of an intervening 

medium between the interacting particles. This impressive advance turned out later to be flawed. 

An approximation made in the derivation meant that the formidable mathematical formalism 

collapsed to a semi-classical theory. By this we mean Maxwell’s equations with boundary conditions 

and quantisation of allowed modes [6,9,10]. 

Technically the reason for this is that in the development of the theoretical formalism 

there occurs an integral equation for the polarisation operator that involves a non-linear coupling 

constant integration. An approximate solution can be found by linearising. The true polarization 

operator is then replaced by the macroscopic dielectric susceptibility.  A detailed exposition can be 

found in Eq. 2.9 and Eq. 3.1 in Ref. [7]. 

 

2. Theory 

2.1 Model Assumptions and the Casmir Energy  

We assume that the nucleons have a structure which involves electromagnetic forces somehow as 

protons have a positive charge and a magnetic moment. Then, how much of a role could 

electromagnetic forces play in nuclear interactions?  Consider two nucleons. If the nucleons were 



 

6 

 

perfectly reflecting spheres, calculation of the electromagnetic fluctuation forces would require an 

analytic solution of the Helmholtz equation. This is complicated [11,12]. So we simplify the model 

and approximate the nucleons by perfectly reflecting planes with the same cross sectional area as 

the (spherical) nucleons.  

Then the attractive electromagnetic fluctuation energy of interaction (all energies in 

this work are given per unit area) across a vacuum at zero temperature is [3]  

 

𝐸 = −
𝜋2

720

ℏ𝑐

𝑑3
.    (1) 

 

Here d is the distance between the plates,  ℏ is Planck’s constant and c the velocity of light. We take 

d to be the distance between surfaces of the protons. The effective surface area is  𝐴 = 𝜋r2, 

r=proton radius~0.8 fermi. A typical nucleon-nucleon surface to surface distance is of the order of 

one fermi.  Then the available two nucleon-nucleon energy for binding in a nucleus from vacuum 

fluctuations is about 5 MeV. The implication is that there is enough electromagnetic energy 

available in the zero-point Casimir energy to account for nuclear interactions.  The binding energy 

per nucleon varies in different atomic nuclei but is typically in the range from 1.1 MeV to 8.8 MeV.  

 

2.2 Temperature Dependence of Electromagnetic Forces  

The observation that the zero temperature Casimir vacuum fluctuation energy is enough to provide 

the binding energy of nucleons in a nucleus is suggestive. To take matters further we need to 

consider the effects of temperature. The Gibbs free energy extension of Casimir’s result that does 

so is due to Lifshitz, it is [7,9,10,13], 



 

7 

 

 

𝐺(𝑑, 𝑇) =
𝑘𝑇

𝜋
∑ ′ ∫ 𝑑𝑞𝑞𝑙𝑛 [1 − 𝑒 

−2𝑑√𝑞2+𝜉𝑛
2 /𝑐2

]
∞

0
∞
𝑛=0 , (2) 

 

where k is Boltzmann’s constant, T is temperature, q is the wavevector, and 𝜉𝑛 = 2𝜋𝑛𝑘𝑇/ℏ . The 

prime indicates that the zero frequency term carries a factor of one half. Explicitly, at small 

distances, or high temperatures, this has the expansion [14], 

 

G(d, T) ≈
 − π2 ℏ𝑐

720d3 
−  

 ζ(3) k3 T3 

2 πℏ2 c2 
+  

π2 d k4 T4 

45ℏ3 c3 
 +..,  (3) 

 

ζ(3) ≈ 1.202 is a zeta function. Here the first term is the attractive (zero temperature) Casimir 

result. The third term is the equilibrium black body radiation energy in the vacuum between the 

plates. It opposes the attractive Casimir term. Additional exponentially decaying terms are 

negligible in the regime of interest and have been omitted. Leaving aside the second term for the 

moment, we suppose that the first and third terms are equal at equilbrium. This then provides us 

with a temperature determined by the distance d between the two plates, T=
ℏc

2kd
, at which the 

attractive and repulsive forces balance.  

 

The Electron—Positron  Sea   

The second term, is a chemical potential term in the Gibbs free energy. We can recognise it explicitly 

as due to an electron positron pair sea formed from the photons in the gap by the reaction 

𝑒 ++𝑒− ↔ γ [11]. From the temperature at distance d we can calculate the density for this  electron 



 

8 

 

positron pair sea. As discussed by Landau and Lifshitz [15] the number of electrons and positrons 

are very nearly identical and both very large, even at temperatures of the order of  𝑚𝑐 2. (An 

electron-positron plasma becomes more nearly perfect with increasing density so we can use 

perfect gas formulae and ignore correlations .) 

The second term can then be re-written as 

 

 ζ(3) k3 T3 

2 πℏ2 c2 
=

π(ρ− +ρ+)ℏ𝑐

6
,    (4) 

 

where we use the expression for the density (𝜌 = ρ− + ρ+ =
3ζ(3)(kT)3

π2ℏ3 c3
) of the electron-positron 

plasma [15]. The interpretation of the chemical potential term (the second term in Eq. (3)) is the 

key to the equivalence we seek.  

 

2.3 Reformulation: the Klein Gordon Equation and Meson Mass 

The imposition of a balance between the vacuum fluctuation and black body radiation forces has 

reformulated the problem to be that of an electromagnetic fluctuation force in which there are two 

metal plates separated by a medium. This medium, an   electron-positron plasma, has the 

permittivity 

 

𝜀(ω) = 1 −
𝜔𝑝

2

𝜔2
;    𝜔𝑝

2 =  
4𝜋(ρ−+ρ+)𝑒

2

𝑚𝑒
,  (5) 

 



 

9 

 

where e is the unit electric charge and 𝑚𝑒  is the mass of the electron. The electromagnetic 

fluctuation interaction energy between two perfectly conducting plates across a plasma can be 

derived from the equation for the scalar potential, in Maxwell’s equations [9], which after a Fourier 

transform reduces to,  

 

∇2∅ +
𝜔2

𝑐2
(1 −

𝜔𝑝
2

𝜔2
) ∅ = 0,   (6) 

  

 Yukawa [16] proposed that the nuclear interaction could be derived from the Klein-Gordon 

equation, 

 

(
1

𝑐2
𝜕2

𝜕𝑡2
− ∇2 + 𝜇2 )∅𝜋 = 0.   (7) 

 

This equation has the solution ∅𝜋~ ±
𝑔2𝑒 −𝜇𝑑

𝑑
 [16]. The range of the Yukawa potential is inversely 

proportional to the meson mass (𝑚𝜋 ): 𝑑𝜋 =
1

𝜇
= ℏ/(𝑚𝜋𝑐).  One can proceed from this known 

relationship between 𝑑𝜋 and  𝑚𝜋 [2]. But, for later work on the lifetime it is useful to recall first the 

very basic physical assumptions used to relate the meson mass to the Yukawa decay length.  As 

discussed by Wick [2], mesons act via emission and absorption processes of virtual excitations, and 

the time required for the excitation to travel between a pair of nucleons is of the order ∆𝑡~
𝑑𝜋

𝑐
. The 

relativistic energy, ∆𝐸 ≥ 𝑚𝜋 𝑐
2, obeys the Heisenberg uncertainty principle for energy [17]:  

∆𝐸∆𝑡 ≥ ℏ. These expressions for energy and time lead us to the required relationship: 𝑑𝜋 ≅

ℏ/(𝑚𝜋𝑐).   



 

10 

 

The Klein-Gordon equation for Yukawa potential (∅) after a Fourier transform may be 

cast into the form [16], 

 

∇2∅𝜋 +
𝜔2

𝑐2
(1 −

1

𝜔2
(

𝑚𝜋𝑐
2

ℏ
)

2

) ∅𝜋 = 0.  (8) 

 

We identify this equation with equation (6). Thus, we obtain after identification of Eq. (6) with Eq. 

(8)  𝜔𝑝
2 =

4𝜋𝜌𝑒2

𝑚𝑒
=

𝑐2

𝑑𝜋
2 = (

𝑚𝜋𝑐
2

ℏ
)

2

. The meson mass follows as 

 

 𝑚𝜋 =
2𝑒ℏ

𝑐
√

𝜋𝜌

𝑚𝑒𝑐
2
.     (9) 

 

This gives 𝑚𝜋 = 267𝑚𝑒 in surprising agreement with the experimental result (264𝑚𝑒). We discuss 

this result further in section 3.  In this scenario the charged 𝝅− and 𝝅+mesons would emerge as 

electron-plasmon and positron-plasmon bound states.  

 

2.4 Binding Energy of Nucleons Casimir-Lifshitz theory  

Returning to the model of Sec. 2.1, we have for the interaction of two perfectly conducting plates 

across an intervening plasma [14,18], the Gibbs free energy 

 

𝐺(𝑑, 𝑇) =
𝑘𝑇

𝜋
∑ ′ ∫ 𝑑𝑞𝑞𝑙𝑛 [1 − 𝑒 

−2𝑑√𝑞2+𝜅2+𝜉𝑛
2 /𝑐2

]
∞

0
∞
𝑛=0 , (10) 

 



 

11 

 

where 𝜅 = 𝜔𝑝/𝑐. For high temperatures at fixed separation, or large separation at fixed 

temperature, it follows [19,20] it has an expansion of the form: 

 

𝐺(𝑑, 𝑇) = −
𝑘𝑇𝜅

4𝜋

𝑒 −2𝜅𝑑

𝑑
[1 +

1

2𝑑𝜅
] −

(𝑘𝑇)2𝑒 −2η𝑑

ℏ𝑐

𝑒−𝜌
∗η𝑑

𝑑
+ 𝑂(𝑒 −4η𝑑 ), (11) 

 

where 𝜌∗ = 𝜌𝑒 2ℏ2/(𝜋𝑚𝑒𝑘
2 𝑇 2), η =

2𝑘𝑇

ℏ𝑐
 and 𝜅 is defined above. 

Both the n=0 and n>0 terms behave similarly to the Yukawa potential [16]. Both provide a 

contribution to our model nuclear binding energy that agrees very well with the experimentally 

observed binding energy per nucleon. We will compare our theoretical results with the typical 

experimental results in Sec. 3. 

 

2.5 Lifetime of  Plasmons and  Mesons   

Our assumption is that at equilibrium the zero point fluctuation energies of the vacuum and the 

black body radiation energy cancel out. What is left are collective excitations, plasmons in the 

remaining electron -positron sea. These can be identified as pions. This allows us to estimate the 

lifetime of the  𝜋0 meson. The lifetime is that for the decay of a plasmon into two electron-positron 

pairs [21]. These can decay to produce two photons. The theory of collective electron excitations 

plasmons is known. The broadening (Δ𝐸) of the plasmon peak and its lifetime (𝜏 ≥ 1/∆𝐸) is known 

analytically and measured [22], 

 

Δ𝐸~
6𝜋𝜀𝐹

5ℏ
(

𝑞𝜋

𝑞𝐹
)

2
(

ℏ𝜔𝑝

2𝜀𝐹
)

3

[10 ln(2) + 2 − 4.5
ℏ𝜔𝑝

2𝜀𝐹
+ 𝑂 (

ℏ𝜔𝑝

2𝜀𝐹
)

2

. . ], (12) 



 

12 

 

 

The entities involved are the Fermi energy (𝜀𝐹 ∝ 𝜌
2/3), plasma frequency (𝜔𝑝 ∝ 𝜌

1/2), and Fermi 

wavevector (𝑞𝐹 ∝ 𝜌
1/3). These depend on density and (in our case) on the distance between the 

nucleons. The lifetime dependence upon the electron-positron plasma density can be deduced once 

we have a model for the neutral pi meson (plasmon) wave vector. In order to calculate the lifetime 

of the plasmon we need an estimate for the 𝑞𝜋-vector. We use the relationship between q-vector 

and energy. The relativistic energy of the plasmon excitation (meson with mass 𝑚𝜋), 𝐸 ∼ 𝑚𝜋𝑐
2, 

[2], is assumed spread into kinetic energy (
ℏ2𝑞𝜋

2

2𝑚𝑒
) for each particle of two electron-positron pairs (in 

general not all energy turns into the kinetic energy of these particles). This leads to an order of 

magnitude estimate for the wave vector of the plasmon: 𝑞𝜋 ≤ 𝑐√
𝑚𝜋𝑚𝑒

2
ℏ⁄ . As we have shown in Eq. 

(9) that 𝑚𝜋 ∝ 𝜌
1/2, the broadening and lifetime is apparently independent of electron-positron 

density (and independent of separation between nucleon pairs). A possibly better estimate 

subtracts off the relativistic energy for each of the particles created in the two electron-positron 

pairs from the relativistic energy of the plasmon. This leads to: 𝑞𝜋 ≤ 𝑐√
(𝑚𝜋−4𝑚𝑒)𝑚𝑒

2
ℏ⁄ ,  with only a 

slight density dependence for the lifetime. The estimate will be seen to lead to the same numerical 

value (to the first decimal place) as the “naive” (Weinberg’s word [23]),   QFT (Quantum Field 

Theory) approximation for the uncharged pion lifetime.  Both our result for lifetime and the “naive” 

one have the same order of magnitude (~0.2 × 10−16s). This can be compared with  the state-of-

the-art QFT result (0.80-0.85 × 10−16s) which agrees with the experimental value (0.834 ×

10−16s), cf. Sec. 3.  

 



 

13 

 

 

3. Results 

3.1 Numerical, Experimental and selected QFT Results for 𝝅𝟎 Mesons 

Meson Mass 

The equivalent black box at a nucleon pair separation of 1 fermi or closer contains very nearly the 

maximum number of electron positron pairs. If we take d~1.5 fermi, the equivalent temperature is 

𝑘𝑇~128𝑚𝑒𝑐
2. This leads via Eq. (9) to a meson mass of 267𝑚𝑒 which compares remarkably 

favorably with the experimental results [24,25], 𝑚𝑒 ≈ 0.511 𝑀𝑒𝑉 and 𝑚𝜋 ≈ 134.97 𝑀𝑒𝑉 ≈

264𝑚𝑒. The dependence of the estimated meson mass on nucleon separation will be shown in 

Table 1. 

 

Meson Lifetime 

Using this distance for the lifetime in the equation given by Ninham [22], we obtain the 𝜋0 lifetime 

≥ 0.16 × 10−16s. Noteworthy, as we mentioned earlier the predicted lifetime is stable for different 

nucleon-nucleon separations unlike binding energy (which increases with decreasing separations). 

This is a curious consequence of the density dependence of the plasmon wavevector. This is 

applicable only at the very high temperatures we predict (corresponding to a plasmon with energy 

high enough to create particles). The experimental textbook result [24] is around 0.83 × 10−16s. 

Our result is of the right order of magnitude. A “simple” QFT approximation [23] leads to an 

estimate for the lifetime around 0.22 × 10−16s. (A theoretically plausible improvement of the 

“simple” QFT result discussed by Weinberg [23] leads to 0.52 × 10−13s which is different by a factor 

1000 from the experimental result). A better theoretical approximation, assuming among other 



 

14 

 

things the number of colors for the quarks, leads to an estimated QFT lifetime for the neutral pion 

of ~0.9 × 10−16s [23].  

The decay of the neutral pion into two photons has its basis in the explicit breaking of 

the axial symmetry by quantum fluctuations of quark and gluon fields. The first four decay pathways 

[21] are: (1) 𝜋0 → 𝛾𝛾, (2) 𝜋0 → 𝛾 + 𝑒
++𝑒−, (3) 𝜋0 → 𝛾 +positronium; (4) 𝜋0 → 𝑒

++𝑒− + 𝑒 ++𝑒−. 

Our theory, taken with the reactions 𝑒 ++𝑒− ↔ γ and 𝑒 ++𝑒 − → positronium → γ, could account 

for the 𝜋0 particle being able to produce these four decay pathways. Precise measurements of the 

decay width of the 𝜋0 → 𝛾𝛾 process give an average of 7.80 eV. This gives a lifetime of  

0.834 × 10−16s [26,27]. This is in good agreement with previous theoretical results and with its 

estimated 1.5% accuracy offers a benchmark test for the most sophisticated theoretical estimates 

including the prediction 0.804 × 10−16s by Kampf and Moussallam [28]. High accuracy calculations 

of the lifetime also include those discussed by Larin et al. [26] and by Bernstein and Holstein [29]. 

These authors [26,29] discuss how the axial, chiral, anomaly originating from quantum fluctuations 

of quark and gluon field, and exploiting the number of QCD quark colors, drives the 𝜋0 meson decay 

with a lifetime around 0.849 × 10−16s. 

 

Nuclear Binding Energy 

 Furthermore, the Lifshitz-Yukawa binding energy at this separation receives -0.9 MeV from the n=0 

term and -3.6 MeV from the n>0 term leading to a total binding energy from electromagnetic 

fluctuation interaction of 4.5 MeV. The binding energy increases with decreasing nucleon-nucleon 

separation in line with the fact that binding energies of nucleons are different in different nuclei 

[30-32], and also in line with the fact that local surroundings influence the local structure of the 



 

15 

 

nucleons [30-32]. The binding energy per nucleon varies in different atomic nuclei from 1.1 MeV for 

deuterium to 8.8 MeV for Nickel-62.  Also, the structure of neutrons and protons within different 

nuclei depends on the local environment (for references see the work by Feldman [31]). (Note also 

in passing the experimental data on nucleon binding energies in Ref. [33]. In that (controversial) 

paper the authors infer that neutron-neutron, just as proton-proton interactions are repulsive, 

whereas the neutron-proton interaction is attractive.) 

 

 

Summary of Numerical Results  

We summarize our numerical results in Table 1. The equivalent temperatures (note that: 𝑚𝑒 𝑐
2/𝑘 ≈

5.9 × 109𝐾) are high enough to generate the electron-positron plasma. The effective surface area 

is taken to be 𝐴 = 𝜋r2 with r=proton radius~0.8 fermi. Improved estimates would, for example, 

require an expansion of our planar estimate to consider a pair of perfectly conducting spheres in a 

high-density electron-positron plasma. 

 

Table 1: The lifetime, meson mass and binding energy versus separation between a pair of neutrons 

(or protons).  Recall the approximations in our model (Sec. 2.1), implying the nucleons to be 

replaced by conducting plates. 

Separation Lifetime Meson Mass Binding Energy kT 

1.0 fermi 1.61 × 10−17s 491𝑚𝑒  13.6 MeV 193 𝑚𝑒𝑐
2 

1.5 fermi 1.62× 10−17s 267𝑚𝑒  4.5 MeV 128 𝑚𝑒 𝑐
2 



 

16 

 

2.0 fermi 1.64× 10−17s 173𝑚𝑒  2.0 MeV 97 𝑚𝑒𝑐
2 

 

 

4. Summary and Conclusion 

We began this enquiry with the idea that if Feynman believed there ought to be a link between 

electromagnetic theory and nuclear forces, there might be something in it. It seems there is. From 

our semi-classical theory we have been able to predict better than order of magnitude estimates 

for the basic properties of the neutral pion, namely its decay length, mass, and lifetime. In the 

picture a high-density electron-positron plasma emerges quantitatively and naturally as a key player 

in nuclear interactions. A defect is the modelling of nucleon interactions by planar perfectly 

reflecting surfaces. There are two free length parameters, area and distance between the model 

”nucleons”. But they are close to actual distance scales. 

 

It would be more convincing if the theory also predicted the various decay modes for 𝜋0, in terms 

of 𝑒 +𝑒− pairs and photons. Further, in such a theory the charged mesons, 𝝅−/𝝅+, would emerge 

as an electron/positron bound to a plasmon.  

 

One thing is clear. There is certainly enough energy available to account for nucleon 

interactions. And if the claim that our theory is not equivalent to the canonical theory, where has 

that energy gone? It is possible to push matters further by including magnetic suceptibilities in the 

formalism for interactions using a fully relativistic electron-positron plasma.  There is a useful 

analytic framework available in the work of Daicic, Kowalenko, Frankel and co-workers [34-36]. In 



 

17 

 

connection with this we observe that Larin et al. [26] performed some of the most precise 

measurements of the lifetime of the 𝜋0 meson. Their weighted average final result for the 𝜋0 → 𝛾𝛾 

decay width defines the new lifetime to be 8.337 × 10−17s.  Such surprisingly short lifetimes can in 

the QCD framework be obtained once axial anomaly is accounted for. The axial anomaly, which 

historically provided strong evidence in favor of the color-charge concept in QCD, seems to present 

us with state of the art knowledge about some of most fundamental aspects of nature—for example, 

by constraining the fundamental physics beyond the Standard Model and presenting opportunity to, 

e.g., measure the light quark mass ratio [26,27]. However, using a much simpler semi-classical theory 

we have found results that turn out to have exactly the right order of magnitude. This suggests an as 

yet unexplored link between our theory (expanded to magnetic anisotropic media) and one of the 

most profound theories in physical science. 

There are wider implications: If the equivalence we seek can be firmed up, the 

consequences would be significant. The full QFT of interactions of DLP involves a nonlinear coupling 

constant integral equation for the polarisation operator. That awkward difficulty was resolved by 

replacing that by a linear integral, and the whole formalism collapsed to a semi classical theory. 

The consequences of these mathematical simplifications have been a serious  obstacle 

to progress in the biological and engineering sciences that depend on molecular forces in the  

disciplines of physical, colloid and surface chemistry [37-40]. The theories inconsistently treat 

electrostatic forces in a nonlinear theory and quantum fluctuation (dispersion) forces in a linear 

theory [37,41]. So central specific ion (Hofmeister) effects, and hydration effects are lost. The 

problem is being partially rectified [40]. But a proper fundamental theory requires the complete 

non linear theory to go further.  



 

18 

 

The same would be true for the theory of nuclear interactions. It should also  be a non 

linear theory and not linear as it is now.   

A partial version of this mss (BWN and Colin Pask, unpublished) was written in 1969. 

A brief version was published by two of us (BWN and MB) in 2003, but it omitted the meson lifetime. 

This version is more detailed and includes this important result. In the following 55 years what is 

new has been the application of Lifshitz theory to the foundations of physical chemistry. The 

literature is extensive. Classical theories ignore all important Hofmeister (specific ion effects). 

The problem can be traced to the same linearization approximation and rectified. The 

equivalence established between Lifshitz theory and pi zero mesons implies that particle 

physics suffers the same difficulties. 

 

References 

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equations”, Am. J. Phys. 58, 209 (1990). A related observation was made by R. P. Feynman, Phys. 

Rev. 80, 440 (1950): ‘‘high energy potentials could excite states corresponding to other eigenvalues, 

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[2] G. C. Wick, “Range of nuclear forces in Yukawa's theory”, Nature 142, 293-294 (1937). 

[3] H. B. G. Casimir, “On the attraction between two perfectly conducting plates”, Proc. K. Ned. 

Akad. Wet. 51, 793 (1948). 

[4] B. W. Ninham and V. A. Parsegian, “van der Waals forces: special characteristics in lipid-water 

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[5] V. A. Parsegian and B. W. Ninham , “Temperature-dependent van der Waals forces.”, Biophys. 

J. 10, 664 (1970). 



 

19 

 

[6] D. J. Mitchell, B. W. Ninham and P. Richmond, “On black body radiation and the attractive force 

between two metal plates”, Am. J. Phys. 40, 674 (1972).  

[7] I. E. Dzyaloshinskii, E. M. Lifshitz, and P. P. Pitaevskii, ”The general theory of van der Waals 

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(2009). 

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anomaly”, Rev. Mod. Phys. 85, 49 (2013). 



 

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[41] M. Boström, D. R. M. Williams, and B. W. Ninham, “Specific ion effects: why DLVO theory fails 

for biology and colloid systems”, Phys. Rev. Lett. 87, 168103 1-4 (2001). 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  


	Equivalence of Electromagnetic Fluctuation and Nuclear (Yukawa) Forces:            the ,𝝅-𝟎. Meson, its Mass and Lifetime
	2.1 Model Assumptions and the Casmir Energy
	2.2 Temperature Dependence of Electromagnetic Forces