On the relation between short-range forces and the
concept of neutrality in Hidrino and Widom-Larsen theories
Jacques Chauveheid
Abstract
A unified theory [1] details the relation between the
strong-nuclear force and nuclear reactions, nuclear fusion included. The
asymmetry between electron and nucleon sustains the primary concept of
electrical neutrality through a weak nuclear force mechanism.
Key-words: strong-nuclear force,
weak interaction, two-body problem, Newton's third principle, neutron stars.
I. INTRODUCTION
A.
Preliminary remarks
If quantum mechanics can
provide quantitative expressions of forces in conformity with the work of Erhenfest and the principle
of correspondence [2], recognized
quantitative expressions for nuclear and weak forces do not currently exist [3]. In addition, the four basic forces do not depend on temperature, since measured in
vacuum between particles.
In one of his books [4],
Abraham Pais recalled a comment by Rutherford during the 1914-1919 period: "the Coulomb forces dominate if
v (speed of alpha particles) is sufficiently small", evidencing by these
words the velocity-dependence of the strong-nuclear force. However, since
Rutherford did not apparently refer to temperature, optimal conditions for nuclear
fusion do not necessarily arise in disordered configurations characterized by
extremely high temperatures, such as those encountered in stars like the sun.
Even compared with galaxy formation, hot fusion in many stars seems the slowest
and most inefficient physical phenomenon in the universe, because the sun's ten
billion year lifetime has an order of magnitude similar to the age of the
universe, this circumstance having been highly beneficial for the life on
earth.
Although not based on equations, Rutherford’s conclusion constitutes the
essence of the “cold” approach to nuclear fusion and reactions starting from
moderate energy levels, instead of extreme temperatures hardly controlling with
precision the physical parameters ruling nuclear phenomena, besides the great
difficulty of chasing particle targets moving at incredibly high velocities. In
this view, a better theoretical understanding of these parameters will help
nuclear technologies.
B. Theoretical antecedents
Eddington mentioned the concept of asymmetric affine connection in
1921 and pointed out applications in microphysics, but he did not pursue this
idea [5]. 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 [5] in which he recommended the use of the differential formalism
he developed, but Einstein did not follow Cartan's advice.
Between 1944 and 1950, J. Mariani published four papers dealing with
astrophysical magnetism [6] and introduced
an "ansatz" structurally similar to that used in the present theory.
The German word "ansatz", used by Ernst Schmutzer (correspondence),
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 being trivial.
The organization of the paper is the following: 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 the short-range force between charged
particles, first referred to as strong-nuclear between nucleons. Section V is
on Yukawa and complexity. Section VI details the short-range forces in both
systems electron-proton and electron-neutron, evidencing a weak nuclear
mechanism in LENR technologies.
When not stated otherwise, mathematical conventions are those of reference
[1].
II. THE THEORY AND ITS
METHOD
Following Einstein's program, the field Lagrangian of this theory [1] is essentially gravitational and
electromagnetic, with five fields and a new constant. This Lagrangian retrieves
Einstein's equation for gravitation, and Maxwell's linear electromagnetism in a
first approximation. The torsion structure of this Lagrangian is part of an
extended (4-dimensional) Einstein-Hilbert Lagrangian, whose full affine
connection includes the torsion T schematically introduced by the ansatz T
= FJ (without indices), where F is the
electromagnetic field and J is the electric current density. Torsion produces
three quadratic Lagrangian terms evidencing quadratic electromagnetic couplings
subsequently describing short-range forces, besides magnetic moments coupled
with electromagnetism.
A second Lagrangian, in simpler reduced form for motion, produces
the forces acting on massive matter, through coordinate variation. These forces
include short-range forces Interpreted as strong-nuclear and weak, besides spin
forces [7] 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 [2].
Due to the local character of physical forces, implied by the
presence of the current J and the non-vanishing of 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 conclusion, these
three Lagrangians describe the fields inside matter and the forces acting on
this massive matter, all massive particles being electrically charged.
Moreover, these forces are naturally extended in vacuum. In relation with this,
physics appears quite far from a unique theoretical model, besides the Standard
Model and a long list covering classical mechanics, thermodynamics, large
number of quantum approaches, etc...
III. THE EQUATIONS
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 [8] in a curved 4-space in general
relativity:
S = ∫- mcds = ∫pkdxk, (1a)
where pk is
the 4-momentum defined by
pk ≡ mcdxk/ds
; (ds2 ≡ - dxkdxk). (1b)
For other interactions,
gravity is switched off [5] and the
formalism of special relativity is used. Such interactions are derivable from
the action [8]
S = ∫(pk + eNk)dxk, (2)
where e is the electric
charge and Nk 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(ds2) =
2ds(δds) = - 2dxkδdxk. (3)
Eq. (3) implies
(δpk)dxk
≡ 0, (4)
obtained by replacing pk
with its expression (1b). For the variation of (2), one uses δdxk =
dδxk and δNk = ∂iNkδxi,
integrating by parts according to the known procedure [8] and finds
δS = │(pk
+ eNk)δxk │ -
∫ [dpk - e(∂kNi
- ∂iNk)dxi]δxk (5)
(from points A fo B).
In accordance with usual
limit conditions, the null-variation of S (δS = 0) implies the equation of
motion
Fk ≡ dpk/dt
= e(∂k Nn - ∂nNk)dxn/dt, (6)
where Fk is
the generalization of the Newtonian force in special relativity. Since dt is
not an invariant, Fk 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 to
delimit 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 + Ta,bc(Ta,bc
+ ΦJ(aFbc))] + AiJi
- (1/4μo)FikFik
+ αΦJiJi, (7)
where gab is
the symmetric metric tensor with g ≡ - det(gab). Ta,bc is
the torsion tensor and parentheses around three indices mean their cyclic
permutation. Jk is the electric current density, Ak is
the 4-vector potential, Fik is the electromagnetic tensor. R is
Riemann's scalar, K is Einstein's constant of gravitation and μo 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 [9], only the relation between torsion and physical fields is
meaningful regarding the relation between geometry and physics.
In line with the 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 Ji containing the 4-velocity c(dxi/ds)
for coordinate variation. For motion, one therefore discards the terms not
containing Ji.
In special relativity
(gravitation switched off [5]), this
procedure thus leads to the second reduced Lagrangian scalar L for motion,
given by
L ≡ (1/2K)Ta,bcΦJ(aFbc)
+ AiJi + αΦJiJi, (8)
where
Ta,bc = -
(Φ/2)J(aFbc) (9)
is the equation-definition
for torsion [1]. One details
Jk ≡ ρouk
= ρoc(dxk/ds) = = ρ(dxk/dt) (10)
(uk is the
4-velocity), where ρo is the rest electric charge density and ρ is
this charge density in the referential of the observer, so that (10) implies
ρ = ρo / (1 -
v2/c2)1/2. (11)
Using (9), one puts (8)
in the form
L = NiJi
+ αΦJiJi,
(12)
with
Na ≡ Aa
- (3Φ2 / 4K)J(aFik)Fik, (13)
due to the substitution
of torsion by the right member of (9) and the identity
J(aFbc).J(aFbc)
= 3JaFbc J(aFbc). (14)
One first shows that the
second "nuclear" term including JiJi in the right member of (12) produces
the line-integral (1a) after 4-volume integration, according to the mass
condition [1]
m = αΦρoe, (15)
in which the Φ-field
plays a key-role in next equations (16) to (18). Φ originally came from
solutions of Einstein's equation for gravitation in a static, spherically
symmetric space-time inside matter [1].
Such solutions would not exist if αΦ were a 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 in relation with
their common radius.
Using (15) and the
notation uk ≡ Jk/ρo for relativist 4-velocity,
one writes
∫dt(αΦJkJk)d3x = ∫dt(αΦρoukρ)(dxk/dt)d3x
= ∫(m/e)ukρdxkd3x
= (1/e)∫pkρd3xdxk
= ∫pkdxk , (16)
where Φρo
becomes a constant at the particle level, in relation with m and e becoming the
respective mass and electric charge of a particle. NiJi
in (12) then gives
∫dt(NkJk)d3x = ∫dt(Nkρdxk/dt)d3x
= ∫eNkdxk, (17)
and gets
S ≡ ∫(L.d3x)dt = ∫(pk + eNk)dxk. (18)
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. One therefore interprets
ρo as the average value of rest charge density.
IV. THE SHORT-RANGE
FORCE AT THE ELECTRO-NUCLEAR APPROXIMATION
To evidence short-range
forces, one discards the Lorentz force engendered by Aa in the right
member of (13), whose second term will produce an attractive force in 1/r5,
of intensity proportional to the square of non-relativistic momentum multiplied
by the particle volume at rest (see further). This short-range force is
distance-dependent [10, 11], in
opposition to spin forces of greater local character.
One now calculates the
components fa 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 [2]. Accordingly, this short-range force of electro-nuclear
character reads
fa ≡ dpa/dt
= e(∂aQb - ∂bQa)dxb/dt, (19)
with
Qa = - (3Φ2
/ 4K)J(aFik)Fik, (20)
from (13), thus without
Aa implying the Lorentz force. Using (15), Eq. (20) then becomes
Qa = - (3m2
/ 4Kα2e2ρo2)J(aFik)Fik. (21)
Due to the existence of
quarks, the simplest 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 xo
= ct in the approximation of point-like protons, the components Ex
and Ey of the electric field produced by the proton at rest are
Ex
= cFxo = ex/r3 ; Ey
= cFyo = ey/r3
(22)
(r2 ≡ x2
+ y2 ; vy ≡ v), at
the approximation of Maxwell's electric field in vacuum.
The electric current
density reads
Jy = ρovy
/ Γ ; Jx = 0, (23)
where Γ ≡ (1 - v2/c2)1/2. (24)
One calculates
J(xFik)Fik
= 2JyFxoFyo ; J(yFik)Fik = - 2Jy(Fxo)2, (25)
and finds
Qx = (-3m2v
/ 2Kα2ρoΓc2)(xy / r6), (26a)
Qy = (3m2v
/ 2Kα2ρoΓc2)(x2 / r6), (26b)
Qo = 0. (27).
The components of fa
then read
fx
= e(∂xQy - ∂yQx)v, (28a)
fy = 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
fx = (-9m2v2Vsinδ
/ 2Kα2c2Γ)r-5 (29)
(fy = 0) where V ≡ e/ρo
is the proton volume, defined before in the approximation of a constant rest
electric charge density ρo.
In the more general case of two
particles with respective charges e1 and e2, the particle
1 being at rest, (29) goes over into
fx = -[9(m2)2(v2)2V2sinδ / 2Kα2c2Γ ](e1/e2)2r-5, (30)
where m2, v2,
V2, e2 are the respective mass, velocity, volume at rest
(V2 ≡ e2/ρo), and electric charge of the
moving particle, e1 being the
charge of the particle at rest, r being the distance between them. 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 eventual quark
structure also implies that (e1)2 and (e2)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 [12]. Moreover, Eq. (30) applies to all
massive particles, here built on
electricity (Mie's idea, see below).
These sums of squared
quark charges also relate to the equation e = Cr, unnumbered formula between
Eqs. (45) and (46) in ref. [1] where
e is the charge, C is a constant and r is the radius of a fundamental charged
particle such as a quark, which presents another relation with the words "neutron mean squared intrinsic charge
radius" in ref. [13]. The 1/ r5 dependence of the strong interaction came
out in 1926 after unsuccessful attempts
with 1/ r2 and 1/ r4 [4,
7], reference in which A. Pais recalls the discovery of a non-central
component of the nuclear force, discovered by Schwinger and Bethe in 1939 [4]. This non-central character was
confirmed in the forties [4].
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 [2],
Mariani's ansatz toward field unification 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 (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]. On the other hand, the quantum
treatment of strong-weak couplings has been done regarding magnetic moments [1], but the procedure has to be applied
to short-range forces, strong-nuclear and weak, because quantum theory is
over-imposed on the classical structure.
V. YUKAWA AND COMPLEXITY
Since perpendicular to
velocity according to Eqs. (28), the short-range force defined by Eq. (30) 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]. A complex
and chaotic kinematics, together with a rather unexpected dynamics characterize
therefore strong-nuclear forces manifesting a tendency toward unpredictability
and instability [4], besides
radioactivity opposing nuclear stability.
Apart from this
stability issue, nuclear forces are velocity-dependent and complex [4], so that Lev Landau's suggested to
limit the study of the strong-nuclear force to binary nuclear interactions
("two by two" [7]). 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" [4]. 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. (28). 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 an invariant
according to a definite arrow of time.
Within a
non-relativistic approximation (Γ= 1), one makes (e1)2 =
(e2)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 mv2/(R/2) for motion around the center of mass and
assuming a circular motion. The factor v2 then simplifies in both
members, which gives
R4 ≈ 9mV /
4Kα2c2,
(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 fermis and allows the calculation of the
constant α according to a nucleon radius of 0.7 fermis (approximation).
VI.
THE WEAK NUCLEAR FORCE IN THE SYSTEM ELECTRON-NUCLEON
A. Electrons and
nucleons
In ref. [7], 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).
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 [2], which would
bring new elements, besides reassuring the calculation of the important tunnel
effect within an appropriate framework.
In this classical
approach, the strong-nuclear force does not produce energy to realize nuclear
fusion in a first approximation, This situation looks deceiving but leads to an
apparently positive conclusion. 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 might therefore play a
significant role, if theoretically validated. In addition, LENR experiments
already produce excess heat, which establishes the experimental foundation for
future developments, starting from simple configurations, to improve in a
second stage.
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 of fields around it. The
phenomenon of orbital electron capture from
atomic levels K, L, M exists in neutron stars [14],
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 (e1)2
= (e2)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 / 2Kα2c2r5, (32)
yields
(me)2(ve)2Ve
= (mp)2(vp)2Vp, (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
produces
meve
= mpvp(Vp / Ve)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 gives meve in the left member
of Eq. (34). The time-derivative of both members then leads to
[m(dv/dt)]e =
[m(dv/dt)]p(Vp / Ve)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)
so that Eq. (35) reduces to
Vp = Ve, (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
(me)2(ve)2Ve(2/3) =
(mn)2(vn)2Vn(3/2). (38)
In
relation with Eq. (36), one writes
meve
= mnvn,
(39)
for
equality of absolute values of momenta, so that Eq. (38) reduces to
Vn
= (4/9)Ve. (40)
implying Vn =
(4/9)Vp in relation with Eq. (37). 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” in ref. [7]).
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 its 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 [15]
which brings forth a neutral deuteron in conformity with Eq. (38).
The spherical symmetry, fundamental in Bohr's model of the hydrogen
atom and Schrödingers's equation in atomic theory, besides the statistical
interpretation of the wave-function, seems typical in relation with a classically
defined “weak nuclear” force mechanism in Eq. (33), 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 (e1/e2)2
previously 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. [14] indicates
that “β-decay applies to all nuclear
reactions implying neutrinos or anti-neutrinos” summarized by
p
+ e- ↔
n + νe, (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 includes this electron capture
producing a neutron, subsequently fusing with a nucleus since being essentially
acted upon by the strong-nuclear force 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, as in the sun where the presence of electrons
in momentary bound systems electron-proton could play a role in the fusion of
hydrogen [16], but this is another story. Obviously, the differentiation
between short-range forces in the weak
nuclear scenario of electron capture by a proton [14] supports the low
energy approach to nuclear fusion.
B. Numbers
From Eq. (30), the coupling constant of the weak force
between two electrons is (me)2, which constitutes a first
number. According to the approximation mn ≈ mp,
the strong nuclear force between two nucleons is characterized by the coupling
constant (mp)2. Based on mp = 1836 me,
the coupling constant of the strong-nuclear force between nucleons is 3.37 x 106
times greater than the coupling constant of the weak force between electrons,
this the second number.
In the bound system electron-nucleon, the weak nuclear
mechanism includes the two coupling constants (mp)2 and
(me)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 106 because (me)2
is negligible in regard to (mp)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 force intensities. 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.
REFERENCES
[1] J. Chauveheid:
A Proposal for a Quadratic
Electromagnetic Coupling Based on the Underlying Philosophy of Einstein-Maxwell
Theory, Physics Essays, 10,
474 (1997).
[2] W.
Heisenberg: Principes Physiques de la Théorie des Quanta (Gauthier-Villars,
Paris 1957), pp. 29, 101, 104.
[3] Sunil Kumar Singh in
[4] A. Pais: Inward Bound:
of Matter and Forces in the Physical World (Oxford University Press, 1986),
pp. 124, 238, 240, 403, 417, 441, 481, 483.
[5] F. Hehl, P. von der Heyde, G. Kerlick, J. Nester: Rev. Mod. Phys. 48,
3 (July 1976).
[6] M.-A. Tonnelat: Les Théories Unitaires de l'Électromagnétisme et de la Gravitation
(Gauthiers-Villars, Paris 1965), pp. xviii, xix, 220, 283.
[7] L. Landau and E.
Lifshitz: Mécanique Quantique
(Mir, Moscow 1967), pp. 10, 511, 517, 519.
[8] L. Landau and E. Lifshitz: The Classical Theory
of Fields (Addison-Wesley, Reading, MA 1961), pp. 56, 73, 268.
[9] W. Yourgrau and S. Mandelstam: Variational
Principles in Dynamics and Quantum Theory (Dover, NY 1979), p. 170.
[10] N. Koshkin and M. Shirkévich:
Manual
de Física Elemental (Mir,
Moscow 1975), p. 220.
[11] K. Sholkin: Física del Micromundo (Mir, Moscow
1972), pp. 82, 94, 107.
[12]
[13]
[14]
[15]
[16] S. Weinberg: Gravitation and Cosmology (John Wiley & Sons, NY, 1972), pp. 546,
556.
No hay comentarios:
Publicar un comentario