**Version X6-b**

Jacques
Chauveheid

**Abstract**

**A variant of the Einstein-Cartan theory [1] details the relation between short-range forces and nuclear reactions, according to masses of fermions being proportional to the Brout-Englert-Higgs scalar field [2]. The asymmetry between electron and nucleon sustains the primary concept of electrical neutrality through a weak nuclear force mechanism in the Widom-Larsen theory.**

__Key-words:__

*Cartan's program, strong-nuclear force, weak interaction, two-body problem, Newton's third principle, neutron stars.*

__I. INTRODUCTION__
Eddington mentioned the concept of asymmetric affine connection in
1921 and pointed out applications in microphysics but did not pursue his idea

**[3]**. In 1922, Elie Cartan introduced geometric torsion, as the antisymmetric part of an asymmetric affine connection. In May 1929, Cartan wrote a letter to Einstein**[3]**in which he recommended the use of the differential formalism he developed, however Einstein did not follow Cartan's advice. Einstein-Cartan theories constitute an important group of theoretical approaches following Cartan's geometric program and linking torsion to spin**[1, 3]**.
Between 1944 and 1950, J. Mariani published four papers dealing with
astrophysical magnetism

**[4]**and introduced an "ansatz" structurally similar to that used in the present theory. The German word "ansatz" refers to a supposed relationship between fields of distinct origin, for example geometric contrasting with physical. Einstein also used an ansatz when he identified gravitation with the 4-space metric, but he did not put it in the form of an equation, presumably because looking too trivial.
The organization of the paper is as follows

**:**Section II details the Lagrangian formulation and the calculus of variations. Section III is about field equations and quantitative expressions of forces. Section IV introduces short-range forces between charged particles, first referred to as strong-nuclear between nucleons. Section V is on Yukawa and complexity. Section VI describes short-range forces in both systems electron-proton and electron-neutron, evidencing a weak nuclear mechanism in the Widom-Larsen theory.
When not stated otherwise, mathematical conventions are those of
reference

**[1]**.

__II. THE THEORY AND ITS METHOD__
The Lagrangian of this theory is mainly gravitational and
electromagnetic. Besides a new constant, five fields include geometric torsion.
This Lagrangian also produces a theory of motion in the form of quantitative
expressions of short-range forces such as nuclear and weak.

The procedure retrieves Einstein's equation for gravitation and Maxwell's
linear electromagnetism in vacuum. The torsion structure is part of an extended
4-dimensional Einstein-Hilbert Lagrangian, whose full affine connection
includes the torsion T schematically introduced through the ansatz T = FJ
(without indices), where F is the electromagnetic field and J is the electric
current density. Torsion produces three Lagrangian terms evidencing quadratic
electromagnetic couplings, subsequently describing short-range forces and
magnetic moments coupled with electromagnetism

**[1]**.
A simpler Lagrangian, in reduced form for motion, produces the
classical forces acting on massive matter, through coordinate variation. These
forces include short-range forces interpreted as strong-nuclear and weak,
besides spin forces

**[5]**not calculated so far. A third quantum Lagrangian is also part of the theory but does not include yet the specific quadratic field couplings relative to the short-range forces, which might lead to strong-nuclear and weak forces expressed in terms of probability packets**[6]**.
Due to the local character of physical forces, implied by the
presence of the current J and a non-vanishing torsion in massive matter, the
first and second Lagrangians only define field equations and forces inside
massive matter, here synonymous to electrically charged particles such as
electrons and quarks. This is so because forces act on massive matter at its
precise location, not close to it in vacuum if one accepts that massive
particles are not singularities of fields in vacuum. Nevertheless, this
formalism is easily extended to the usual description of forces in vacuum,
together with cosmology

**[1]**.
Somewhat summarizing the nuclear problem, a coupling of torsion to
massive matter, via electrodynamics, evidences the quadratic structure of torsion
in an enlarged Einstein-Maxwell theory implying squared electromagnetism
covering short-range forces such as nuclear and weak.

About equations,
attempting to extract all interactions from a system of field equations is not
applicable in this theory, because one derives field equations by varying
separately all fields representing variable entities in a field Lagrangian.
From a second Lagrangian for motion, one obtains a unique equation of motion,
containing all interactions, by varying the 4-coordinates of one massive
(charged) particle. There is therefore no reason to confuse both processes,
apart from their common field Lagrangian. Moreover, the same Lagrangian field
couplings were introduced in a third quantum Lagrangian that includes
Schrödinger's field, which produces a wave equation retrieving Schrödinger's
equation in a non-relativistic weak field approximation. This operation leads
to energy definition through Hamiltonian wave-solutions of Schrödinger's
time-dependent equation (of evolution)

**[1]**.
In other words, these
three Lagrangians describe fields inside matter and the forces acting in vacuum
on this massive matter, which here is electrically charged. Physics thus
appears quite far from a unique theoretical model, besides the Standard Model
and a rather long list covering classical mechanics, thermodynamics, numbers of
quantum approaches and more.

__III. EQUATIONS OF MOTION__
In general relativity,
the trajectories of matter are geodesics. This is the consequence of the
variational postulate δS = 0, whose action S is defined by the line-integral
implying the gravitational force

**[7]**in a curved 4-space in general relativity**:**
S =

**∫**- mcds =**∫**p_{k}dx^{k}, (1a)
where p

_{k}is the 4-momentum defined by
p

_{k}≡ mcdx_{k}/ds ; (ds^{2}≡ - dx_{k}dx^{k}). (1b)
For other interactions,
gravity is switched off

**[3]**and the formalism of special relativity is used. Such interactions are derivable from the action**[7]**
S =

**∫**(p_{k}+ eN_{k})dx^{k}, (2)
where e is the electric
charge and N

_{k}is the enlarged vector potential constructed from the Lagrangian densities containing the electric current density. An ensuing equation of motion, encompassing all interactions besides gravity, is derived from (2) by the sole variation of coordinates. One verifies
δ(ds

^{2}) = 2ds(δds) = - 2dx_{k}δdx^{k}. (3)
Eq. (3) implies

(δp

_{k})dx^{k}≡ 0, (4)
obtained by replacing p

_{k}with its expression (1b). For the variation of (2), one uses δdx^{k}= dδx^{k}and δN_{k}= ∂_{i}N_{k}δx^{i}, integrating by parts according to the known procedure, which gives
δS =

**│**(p_{k}+ eN_{k})δx^{k}**│**-**∫**[dp_{k}- e(∂_{k}N_{i}- ∂_{i}N_{k})dx^{i}]δx^{k}(5)
(from points A fo B).

In accordance with usual
limit conditions, the null-variation of S (δS = 0) implies the equation of
motion

F

_{k}≡ dp_{k}/dt = e(∂_{k}N_{i}- ∂_{i}N_{k})dx^{i}/dt, (6)
where F

_{k}is the generalization of the Newtonian force in special relativity. Since dt is not an invariant, F_{k}is not a vector.
Since (2) avoids the
summation symbol for various particles, the equation of motion (6) is
restricted to a system formed by different fields and only one particle, at the
approximation that one moving body does not affect the fields. The aim is
delimiting the general problem of interactions in the simplest case, being
aware that this technique is apparently limited to a 2-body problem. This
method is now applied to the field Lagrangian £

**[1]**defined by
£/√g ≡ (1/2K)[R + T

^{a,bc}(T_{a,bc}+ ΦJ_{(a}F_{bc)})] + A_{i}J^{i}
- (1/4μ

_{o})F_{ik}F^{ik}+^{ }αΦJ_{i}J^{i}, (7)
where g

_{ab}is the symmetric metric tensor with g ≡ - det(g_{ab}). T_{a,bc}is the torsion tensor and parentheses around three indices mean their cyclic permutation, J_{k}is the electric current density. A_{k}is the 4-vector potential, F_{ik}is the electromagnetic tensor and R is Riemann's scalar. K is Einstein's constant of gravitation and μ_{o}is the magnetic permeability of vacuum. Φ is a scalar field and α is a (new) constant. Furthermore, agreeing with Poincaré's definition of science as a system of relations**[8]**, only the relation between torsion and physical fields is meaningful regarding the relation between geometry and physics.
In line with next
equations (16) to (18), the line-integral (2) for motion will arise as a
4-volume integration of the Lagrangian terms, which in (7) include the electric
current density J

^{i }containing the 4-velocity c(dx^{i}/ds), used for coordinate variation. For motion, one thus discards the terms not containing J^{i}**.**
Consequently, in special
relativity (gravitation switched off

**[3]**) this procedure leads to the second reduced Lagrangian scalar L for motion, given by
L ≡ (1/2K)T

^{a,bc}ΦJ_{(a}F_{bc)}+ A_{i}J^{i}+ αΦJ_{i}J^{i}, (8)
where

T

_{a,bc}= - (Φ/2)J_{(a}F_{bc) }(9)
is the
equation-definition for torsion

**[1].**One details
J

_{k}≡ ρ_{o}u_{k}= ρ_{o}c(dx_{k}/ds) = ρ(dx_{k}/dt) (10)
(u

_{k}is the 4-velocity), where ρ_{o }is the rest electric charge density and ρ is this charge density in the referential of the observer, so that (10) implies
ρ = ρ

_{o }/ (1 - v^{2}/c^{2})^{1/2}. (11)
Using (9), one puts (8)
in the form

L = N

_{i}J^{i}+ αΦJ_{i}J^{i}, (12)
with

N

_{a}≡ A_{a}- (3Φ^{2 }/ 4K)J_{(a}F_{ik)}F^{ik}, (13)
due to the substitution
of torsion by the right member of (9) and the identity

J

_{(a}F_{bc).}J^{(a}F^{bc)}_{ }= 3J_{a}F_{bc }J^{(a}F^{bc)}. (14)
One first shows that the
second "nuclear" term including J

_{i}J^{i }in the right member of (12) produces the line-integral (1a) after 4-volume integration, according to the mass condition**[1]**
m = αΦρ

_{o}e, (15)
where the mass m
characterizes an infinitesimal quantity of massive matter.

Electrically charged
matter and massive matter being synonyms here, the extension by integration of
Eq. (15) at the particle level implies that the mass of any massive particle is
roughly proportional to the average value of the scalar field Φ inside matter.
The Φ-field is therefore the Brout-Englert-Higgs scalar in classical form,
without which no massive matter would exist in a nonsensical universe. This
scalar field plays a key-role in next equations (16) to (18) clarifying the
calculation of acceptable, although approximate, quantitative expressions of
non-gravitational forces. Since the existence of Einstein's solutions for
gravitation in a static, spherically symmetric space-time implies the existence
of Φ

**[1]**, the mass m in Eq, (15) is at the same time inertial and gravitational, respecting the equivalence principle. The Einstein solutions express that gravity unites all forms of mass-energy and imply that αΦ is not constant.
The Φ-field was
introduced as a variable physical quantity uniting in matter the mass m with
its corresponding electric charge e, both quantities varying according to a
common radius, which is seen in (15) put at the particle level in the
approximate form

m = αΦe

^{2 }/ V, (15b)
where the scalar Φ provides the necessary quantization of
particle radius, without which the particle size would be indeterminate in a
continuum theory with a continuous
metric across the border matter-vacuum. In addition, gauge invariance of the field
equations results from infinitesimal variations of Φ, variations subsequently identified to the quantum field Ψ (Schrödinger or Dirac field)

**[9]**.
Using (15) and the
notation u

_{k}≡ J_{k}/ρ_{o}for relativist 4-velocity, one writes**∫**dt(αΦJ

_{k}J

^{k})d

^{3}x =

**∫**dt(αΦρ

_{o}u

_{k}ρ)(dx

^{k}/dt)d

^{3}x =

**∫**(m/e)u

_{k}ρdx

^{k}d

^{3}x

≈ ∫p

_{k}dx^{k}, (16)
at the particle level
where m and e are now the respective mass and electric charge of a particle. N

_{i}J^{i}in (12) then gives**∫**dt(N

_{k}J

^{k})d

^{3}x =

**∫**dt(N

_{k}ρdx

^{k}/dt)d

^{3}x ≈

**∫**eN

_{k}dx

^{k}, (17)

and

S ≡

**∫**(L.d^{3}x)dt ≈**∫**(p_{k}+ eN_{k})dx^{k}. (18)
where dx

^{k}refers now to the particle trajectory.
Field equations and equations of motion are distinct objects. From
the first field Lagrangian £, one gets field equations by varying the fields.
From the second matter Lagrangian L, simplified for motion, one derives an
equation of motion by varying the coordinates representing the location of
matter.

In this theory of
motion, massive charged particles are not point-like. However, the framework of
motion relative to such particles has no interest in a variable charge density
inside matter. From now, one will therefore interpret ρ

_{o}as the average value of the rest charge density in matter.

__IV. SHORT-RANGE FORCES AT THE ELECTRO-NUCLEAR APPROXIMATION__
If quantum mechanics can
provide quantitative expressions of forces according to Erhenfest's work and
the principle of correspondence

**[6],**recognized quantitative expressions for nuclear and weak forces are not available**[10].**
To evidence short-range
forces, one discards the Lorentz force generated by A

_{a}in the right member of (13), whose second term will produce an attractive force in 1/r^{5}, of intensity proportional to the square of the non-relativistic momentum multiplied by the particle volume at rest (see further). This short-range force is distance-dependent**[11, 12]**, in opposition to spin forces of greater local character.
One now calculates the
components f

_{a}of this short-range force in the approximation of electromagnetism reduced to its electric components, due to the non-relativistic neglect of the magnetic field**[6]**. Accordingly, this short-range force of electro-nuclear character reads
f

_{a}≡ dp_{a}/dt = e(∂_{a}Q_{b}- ∂_{b}Q_{a})dx^{b}/dt, (19)
with

Q

_{a}= - (3Φ^{2 }/ 4K)J_{(a}F_{ik)}F^{ik}, (20)
from (13), thus without
A

_{a}implying the Lorentz force. Using (15), Eq. (20) then reads
Q

_{a}= - (3m^{2 }/ 4Kα^{2}e^{2}ρ_{o}^{2})J_{(a}F_{ik)}F^{ik}. (21)
Due to the existence of
quarks, the interaction between nucleons is a 6-body problem. However, one will
treat the system nucleon-nucleon as a 2-body problem in a first approximation.

Rectangular coordinates
characterize the referential Oxy where a first static proton is located at the
origin O. A second proton moves above the x-axis, its velocity

**v**being parallel to the y-axis. Using x^{o}= ct in the approximation of point-like protons, the components E_{x}and E_{y}of Maxwell's electric field in vacuum, produced by the proton at rest are
E

_{x}= cF_{xo}= ex/r^{3}; E_{y}= cF_{yo}= ey/r^{3}(22)
(r

^{2}≡ x^{2}+ y^{2}; v_{y}≡ v).
The electric current
density reads

J

_{y}= ρ_{o}v_{y }/ Γ**;**J_{x}= 0, (23)
where Γ ≡ (1 - v

^{2}/c^{2})^{1/2}. (24)
One calculates

J

_{(x}F_{ik)}F^{ik}= 2J_{y}F_{xo}F_{yo}**;**J_{(y}F_{ik)}F^{ik }= - 2J_{y}(F_{xo})^{2}, (25)
and finds

Q

_{x}= (-3m^{2}v / 2Kα^{2}ρ_{o}Γc^{2})(xy / r^{6}), (26a)
Q

_{y}= (3m^{2}v / 2Kα^{2}ρ_{o}Γc^{2})(x^{2 }/ r^{6}), (26b)
Q

_{o}= 0. (27).
The components of f

_{a}then read
f

_{x}= e(∂_{x}Q_{y}- ∂_{y}Q_{x})v, (28a)
f

_{y}= 0. (28b)
Writing x/r = sinδ,
where δ is here the angle between the straight line defined by the two proton
centers and the velocity of the moving proton, Eq, (28a) gives

f

_{x}= (-9m^{2}v^{2}Vsinδ**/**2Kα^{2}c^{2}Γ)/r^{5}(29)
(f

_{y}= 0) where V ≡ e/ρ_{o}is the proton volume at rest, previously defined in the approximation of a constant rest electric charge density ρ_{o}.
In the more general case of two
particles with respective charges e

_{1}and e_{2}, the particle 1 being at rest, (29) goes over into
f

_{x}= -[9(m_{2})^{2}(v_{2})^{2}V_{2}sinδ / 2Kα^{2}c^{2}Γ ](e_{1}**/**e_{2})^{2}/r^{5}, (30)
where m

_{2}, v_{2}, V_{2}, e_{2}are the respective mass, velocity, volume at rest (V_{2 }≡ e_{2}/ρ_{o}), and electric charge of the moving particle, e_{1}being the charge of the particle at rest, r being the distance between the two particles. Eqs. (28) and (30) determine the short-range force exerted by particle 1 on particle 2, which is perpendicular to the velocity of particle 2. The quark structure also implies that (e_{1})^{2 }and (e_{2})^{2}are sums of squared quark charges in the case of nucleons. Accordingly the expression "summed charge squared", worth 1 for a proton and 2/3 for a neutron, figures in ref**[13]**. Moreover, Eq. (30) applies to all massive particles, here built on electricity (Mie's idea, see below).
These sums of squared
quark charges appear concerned by the equation e = Cr, unnumbered formula
between Eqs. (45) and (46) in ref.

**[1]**,**where e is the electric charge, C is a coefficient characterizing a determined particle, or a group of them, r is the radius of a fundamental charged particle such as a quark. This equation is related to the words "neutron mean squared intrinsic charge radius" in ref.****[14]**. Moreover, the 1/ r^{5}dependence of the strong interaction came out in 1926 after unsuccessful attempts with 1/ r^{2}and 1/ r^{4}**[5, 15]**, references in which A. Pais recalls the discovery of a non-central component of the nuclear force, discovered by Schwinger and Bethe in 1939. This non-central character was confirmed in the forties**[15]**. Attractive forces are not necessarily central and the range of the strong-nuclear force is infinite, however its intensity rapidly decreases with distance, reason why this force is referred to as short-range.
There is more on the
subject of fundamentals, briefly recalled now. As dynamics is an essential
feature of physical theories

**[6]**, Mariani's ansatz toward field unification**[4]**centered on the motion of massive matter, but the problem of motion has little to do with a field theory in vacuum where particles are singularities of the fields (see above). One thus sees that the problem of motion essentially resides inside massive matter, which implies interior solutions of field equations in matter**[1]**. In 1912, Gustav Mie introduced this idea of matter constituted by fields. Hermann Weyl detailed Mie's theory in his book*Space-Time-Matter*(Dover, NY 1952, p. 206), in which Weyl reproduced Mie's words when writing**:***matter is "purely" electrical in nature*. Einstein and Leopold Infeld retook this idea of matter constituted by fields in*The Evolution of Physics*(Simon & Schuster, NY 1938, p. 242).
The present theory in matter reproduces field theories in
torsion-free situations. Equations of motion for gravitation and
electromagnetism (geodesics and Lorentz's force) are also retrieved.
Furthermore, one retrieves quantum mechanics for the hydrogen atom from
identical field couplings, including Dirac's magnetic dipole and spin-orbit
energies, by introducing additional constants such as the electron charge and
mass, besides Planck's constant

**[1]**. Accordingly, the quantum treatment of strong-weak couplings has been done regarding magnetic moments**[1]**, but this procedure needs to be extended to short-range forces, strong-nuclear and weak, since quantum theory is over-imposed on classical structures and models.

__V. YUKAWA AND COMPLEXITY__
Since perpendicular to
velocity in Eqs. (28) and (30), the short-range force does not produce energy
(see below). Moreover, this force is firmly non-central, which contrasts with
central forces oriented along straight lines connecting two particle centers

**[11, 12]**. A complex and chaotic kinematics, together with a rather unexpected dynamics characterize therefore strong-nuclear forces manifesting a tendency toward unpredictability and instability**[15]**, besides radioactivity opposing nuclear stability.
Apart from this
stability issue, nuclear forces are velocity-dependent and complex

**[15]**, so that Lev Landau's suggested to limit the study of the strong-nuclear force to binary nuclear interactions ("two by two"**[5]**). This was probably due to the Yukawa distance and the short-range character of strong-nuclear forces, to what A. Pais added "*the nuclear 2-body problem is just too complicated*"**[15]**. However, Pais' words mean that the complexity resides in the nuclear problem, not necessarily in theories describing nuclear phenomena.
About this issue of
complexity, the present theory may also look complicated, example of its
non-linear version of Maxwell's theory, whose approximation for the electric
field in Eqs. (22) may be responsible for the vanishing work (energy) produced
by the strong-nuclear force, according to Eqs. (29) and (30). From another
standpoint, perpendicularity does not materialize exactly in the real world
because 3-dimensional orthogonality is not an invariant in 4-dimensional
relativity theory. In contrast, the attractive character of forces is invariant
according to a definite arrow of time. To keep things simple, we will from now
neglect spin-spin and spin-orbit effects.

Within a
non-relativistic approximation (Γ= 1), one makes (e

_{1})^{2}= (e_{2})^{2}and takes the absolute value of the strong-nuclear force between two protons or two neutrons from Eq. (30), equaling its right member to mv^{2}/(R/2) for motion around the center of mass and assuming a circular motion. The factor v^{2}then simplifies in both members, which gives
R

^{4}= 9mV / 4Kα^{2}c^{2}, (31)
(sinδ = 1).

The binding of two
protons, or two neutrons, consequently implies their fixed separation defining
the Yukawa distance R figuring in Eq. (31), for a common particle volume (see
further). R is currently valued at 1.4 fermi, which allows the calculation of
the constant α according to a nucleon radius of 0.7 fermi (approximation).

__VI. WEAK AND STRONG-NUCLEAR FORCES IN THE SYSTEM ELECTRON-NUCLEON__**A. Electrons and nucleons**

Landau wrote

**:**"...quantum mechanics occupies a very original position in the range of physical theories; it contains classical mechanics as a limiting case and at the same time needs this limit to be founded" (translation**[5]**). In agreement with Bohr's correspondence, one then sees the importance of a classical theory of motion and forces, before a quantum treatment evidencing probability packets**[6]**bringing new elements, besides reassuring the calculation of the important tunnel effect within an appropriate framework.
In a first
approximation, the strong-nuclear force presented here does not produce energy
to realize nuclear fusion. This situation looks a bit deceiving but leads to a
somewhat realistic deduction. Nuclear fusion, based on the unique role of the
strong-nuclear force, still does not present a high degree of probability after
many years of hot fusion experiments. Other phenomena could therefore
intervene.

Before neutron formation
with emission of a neutrino, the electron capture by a proton is a bound system
electron-proton constituting a two-body system respecting Newton's principle of
equality between action and reaction, in an energy conservative bound system
not interfering with energy-momentum conservation for the fields around it. The
phenomenon of orbital electron capture from atomic
levels K, L, M exists
in neutron stars

**[16]**. Its classical description implies the presence of two short-range forces, detailed now without writing the electrostatic force between proton and electron, because this force cancels out in both members of next Eq. (33) expressing equality of action and reaction for motion around the center of mass.
Quark charges in a proton are 2/3, 2/3, -1/3, whose sum of squared
charges = 1 is the same as the squared charge of the electron, which implies (e

_{1})^{2}= (e_{2})^{2 }in Eq.(30). Within a non-relativistic approximation (Γ =1), one writes an equation whose left member is the intensity of the short-range force exerted by a proton on an electron, the right member expressing the converse. Simplifying both members by
9 / 2Kα

^{2}c^{2}r^{5}, (32)
yields

(m

_{e})^{2}(v_{e})^{2}V_{e}= (m_{p})^{2}(v_{p})^{2}V_{p,}(33)
with sinδ =
1 for both forces in opposition, in relation with round orbits of particle
centers (analogy with Bohr's 1913 model). The square root of both members gives

m

_{e}v_{e}= m_{p}v_{p}(V_{p }/_{ }V_{e})^{1/2}. (34)
In Eq. (30),
the attractive short-range force is proportional to the squared
non-relativistic momentum of a particle, now retrieved in the left member of
Eq. (33), whose square root produces m

_{e}v_{e}in the left member of Eq. (34). The time-derivative of both members then reads
[m(dv/dt)]

_{e}= [m(dv/dt)]_{p}(V_{p }/_{ }V_{e})^{1/2}. (35)_{ }
whose left member is the force dp/dt figuring in Eq. (19), acting
now on the electron. For consistency, therefore not only in relation with
Newton's third principle, one writes

[m(dv/dt)]

_{e}= [m(dv/dt)]_{p}, (36)
implying

V

_{p}= V_{e .}(37)
The electron and the proton have therefore the same volume in this
theory.

One repeats the same procedure, by adapting Eq. (33) to the bound
system electron-neutron, taking into account the sum of squared quark charges
worth 2/3 for the neutron, which gives

(m

_{e})^{2}(v_{e})^{2}V_{e}(2/3) = (m_{n})^{2}(v_{n})^{2}V_{n}(3/2). (38)
In relation with Eq. (36), one writes

m

_{e}v_{e}= m_{n}v_{n}, (39)
for equality of absolute values of momenta, so that Eq. (38)
reduces to

V

_{n}= (4/9)V_{e}. (40)
This result would be
unphysical if the proton and neutron volumes need to be equal for defining
a unique Yukawa length in Eq. (31). However, the imperfect equivalence between
(p-p) and (n-n) interactions is an established fact (“very similar”

**[5]**).**In this view, the 18 % difference for R in Eq. (31) might be physical, and not important enough to discard this type of capture, in spite of apparent lack of observational data. All this seems good news for the possibility of electron capture by a neutron, part of the Widom-Larsen theory****[17, 18]**.
Spherical symmetry, fundamental in Bohr's model of the hydrogen atom
and Schrödinger's equation in atomic theory
seems typical in relation with a “weak
nuclear force”

**[16]**in Eq. (33), here referring to two distinct forces of equal intensity in the system electron-proton. The weak force acts on the electron with the same intensity as if the source were another electron, instead of a proton, and the second strong-nuclear force acts on the proton with the same intensity as if the source were another proton, instead of an electron. Short-range forces work this way because proportional to the squared mass of the particle acted upon, multiplied by the factor (e_{1}**/**e_{2})^{2}defined in relation with quarks.
Here,
the bound system electron-proton represents the electron capture through two
forces, strong-nuclear and weak, system literally characterizing a “weak
nuclear” mechanism of forces, according to which proton and electron react. In
relation with this, ref.

**[16]**indicates “β-decay applies to all nuclear reactions implying neutrinos or anti-neutrinos” summarized by
p
+ e

**↔ n + ν**^{-}_{e}, (41)
adding
that “all these reactions are ruled by the weak nuclear force”, represented
here by two forces, strong-nuclear and weak of equal intensity, which are
Newton's inevitable action and reaction. The left member of (41) presents the
electron capture explaining the neutralization of the Coulomb
barrier (including eventual shielding during definite time intervals before
neutron formation ?). The Widom-Larsen theory

**[17, 18]**comprehends this electron capture producing a neutron subsequently fusing with a nucleus, since being acted upon by the conjunction of all nuclear forces between nucleons.
In
conclusion, the bound system electron-proton with two distinct forces, weak and
strong-nuclear of equal intensities, contrasts with the scenario of two
strong-nuclear forces representing action and reaction in the usual outline of
nuclear fusion between protons. This, as in the sun where the presence of
electrons in momentary bound systems electron-proton could play a role in the
fusion of hydrogen

**[19]**, but this is another story. Obviously, the differentiation between short-range forces in the weak nuclear scenario of electron capture by a proton supports the low energy approach to nuclear fusion.**B. Numbers**

From Eq. (30), the coupling constant of the weak force
between two electrons is (m

_{e})^{2}, which constitutes a first number. According to the approximation m_{n}≈ m_{p}, the strong nuclear force between two nucleons is characterized by the coupling constant (m_{p})^{2}. Based on m_{p}= 1836 m_{e}, the coupling constant of the strong-nuclear force between nucleons is 3.37 x 10^{6}times greater than the coupling constant of the weak force between electrons, this is the second number.
In the bound system electron-nucleon, the weak nuclear
mechanism includes the two coupling constants (m

_{p})^{2 }and (m_{e})^{2}because two distinct forces are present. Since the weak force acts on the electron and the strong-nuclear force acts on the nucleon, the mean value of these two coupling constants would be half the second number above, so roughly 1.69 x 10^{6}because (m_{e})^{2}is negligible in regard to (m_{p})^{2}. However, this third number is unrelated to relative intensities of these two forces in a bound system where the equality between action and reaction implies the equality of both forces. In relation with Eq. (30), the non-central short-range force is velocity-dependent, so that relative force intensities are relative values of squared non-relativistic momenta, in opposition to relative values of coupling constants.

__COMMENTS__
Einstein's program, formulated at Princeton in
1932, recommends the retrieval of general relativity and Maxwell's
electromagnetism, as a step toward incorporating other theories, even
explaining quantum mechanics

**[20]**, which is only feasible from a Lagrangian containing a quantum field**[1]**. In vacuum, the present theory retrieves general relativity and Maxwell's theory in torsion-free situations. In matter, one gets the usual energy-tensor in the right member of Einstein's equation, standard procedure in classical unification. In addition, generalized Maxwell's equations inside matter include additional couplings of torsion with the electromagnetic field, due to the non-vanishing of torsion in massive matter.
Mathematically speaking, the present theory
follows Cartan's geometric program based on torsion

**[3]**. Moreover, usable physical results, chiefly for technologies concerning new energy sources, hardly come from equations only since equations mean little without their solutions. One thus needs to calculate these solutions to uncover crucial relations and conditions that would remain undisclosed in the absence of Einstein's gravitational solutions.
Einstein's equation appears indispensable
because gravitation unites all forms of mass-energy, including massless
particles. In this view, the Lagrangian formulation provided eventually workable
solutions of Einstein's equation, in a static spherically symmetric
gravitational field, which led to the mass condition (15) and the classical
scalar Φ-field of Higgs essence.

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*A Proposal for a Quadratic Electromagnetic Coupling Based on the Underlying Philosophy of Einstein-Maxwell Theory*, Physics Essays,

__10__, 474 (1997).

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*Phys.*

__48__, 3 (July 1976).

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*Les Théories Unitaires de l'Électromagnétisme et de la Gravitation*(Gauthiers-Villars, Paris 1965), pp. xviii, xix, 220, 283.

**[5]**L. Landau and E. Lifshitz

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*Mécanique Quantique*(Mir, Moscow 1967), pp. 10, 511, 517, 519.

**[6]**W. Heisenberg

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*Principes Physiques de la Théorie des Quanta*(Gauthier-Villars, Paris 1957), pp. 29, 101, 104.

**[7]**L. Landau and E. Lifshitz

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**[8]**W. Yourgrau and S. Mandelstam

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*Variational Principles in Dynamics and Quantum Theory*(Dover, NY 1979), p. 170.

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*On the Connection between Classical and Quantum Fields*, Physics Essays,

__12__, 146 (1999).

**[10]**Sunil Kumar Singh in

**[11]**N. Koshkin and M. Shirkévich

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*Manual de Física Elemental*(Mir, Moscow 1975), p. 220.

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**[14]**

**[15]**A. Pais

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*Inward Bound*(Oxford University Press, 1986), pp. 124, 238, 240, 403, 417, 441, 481, 483.

**:**of Matter and Forces in the Physical World**[16]**

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**[18]**

**[19]**S. Weinberg:

*Gravitation and Cosmology*(John Wiley & Sons, NY 1972), pp. 546, 556.

**[20]**A. Pais

**:**”

*Subtle is the Lord..."*(Oxford University Press, 1982).

**:**The Science and the Life of Albert Einstein
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