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W.I. Fushchych
Scientific Works



Volume 5
1993–1995




Editor
Vyacheslav Boyko




Kyiv 2003
W.I. Fushchych, Scientific Works 2003, Vol. 5, 1–4.

Fundamental constants
of nucleon-meson dynamics
O. BEDRIJ, W.I. FUSHCHYCH
i ii
. i: -
i “i” () c -i i iii i
i i.

In this paper, we are proposing an approach to calculate fundamental physical
constants that characterize nucleon-meson dynamics. The approach is based on the
referenced papers [1, 2], and on the premise that fundamental constants are reducible
to mathematical relations and operations, which can be used to predict, define and
calculate other fundamental “natural” unit systems (quanta).
At the present, we have, when compared to available data on quantum electrody-
namics (electron-photon dynamics), very limited experimental fundamental constant
data for the proton and the neutron. Such constants as the neutron or proton radius,
or the Rydberg constant are not adequately defined in nucleon-meson dynamics.
From experiment, we know the mass and the charge of proton and neutron. Other
physical characteristics such as nuclear magneton. Compton wavelength of the proton
and the neutron are derived quantities, that incorporate and c constants in the
relations. It is presently assumed in physics that electrodynamic constants of and c
are applicable to characterization of nucleon-meson dynamics. Our calculations show
that constants and c for nucleon-meson dynamics are different from the same
constants in quantum electrodynamics. This is natural, because the electron emits
a photon, while the nucleon emits a meson.
We propose that standard formulas for fundamental characteristics of proton and
neutron can be modified to represent the nucleon-meson constants and not electrody-
namic constants. Below we show the proposed modifications (Definitions of Quantities
are shown in [2]):

Standard Proposed
Relationships Relationships
?p = /mp c ?p =
Compton Wavelength of proton p mp vp

?n = /mn c ?n =
Compton Wavelength of neutron n /mn vn

µp = q /2mp c µp = q p
Proton magneton p /2mp vp

µn = q /2mn c µn = q n
Neutron magneton n /2mn vn

rp = =
Proton radius p ?f p mp vp , p

rn = =
Neutron radius n ?f n /mn vn , n


where vp and vn are velocities of mesons which are emitted by proton and neutron.
In our approach, we assume that:
ii , 1993, 5, . 62–64.
2 O. Bedrij, W.I. Fushchych

1. The physical relationships between quantities are the same for all inertial frames
of reference.
2. The scale-symmetry is a fundamental concept in all of physics, including the
photon, electron, meson, proton, neutron, etc.: that is, the scale-invariance of the
physical relationships between quantities with respect to the scale group.
3. Physical quantities have a fundamental relationship to, an equilibrium frame of
reference and that the equilibrium frame of reference is scale invariant [2].
When we consider that the laws of physics are invariant in all inertial frames
of reference, and that the scale-symmetry is a fundamental aspect of physical re-
lationships and constants, constant values that deal with quantum electrodynamics
(constants that satisfy physical relationships for electron mass, photon, Compton
wavelength, etc., ([2] — Table 1), are not applicable for the proton or neutron, which
have different masses and hence, different scales of reference.
Earlier [2] we stated that:
1 = q1 1 q2 2 q3 3 · · · qs s / pj1 pj2 2pj3 · · · pjz
xxx x
(1)
z
1 3

or,
(2)
1 = Y /KX,
where
Y ? q1 1 q2 2 q3 3 · · · pjz ,
xxx
(3)
z


1/KX ? pj1 pj2 pj3 · · · pjz , (4)
z
3

?1
(qs )0 = 1, (qs )0 = 1,
q1 , q2 , q3 , . . . , qs , p1 , p2 , q3 , . . . , pz are quantities,
x1 , x2 , x3 , . . . , xs , j1 , j2 , j3 , . . . , jz are real numbers,
K is the slope for line Y = KX,
j, s, x, z = 1, 2, 3, . . . .

We require that the equations (1) and (2) are scale invariant. That is the equa-
tions (1) and (2) are invariant with respect to the following transformations:
q1 > q1 = aq1 , q2 > q2 = aq2 , q 3 > q3 = aq3 , (5)
...,

p1 > p1 = ap1 , p2 > p2 = ap2 , p3 > p3 = ap3 , (6)
...,

where “a” is a scale transformation parameter, and all physical quantities (qs and pz )
have to be subjected to transformation. Hence, based on equations (1) and (2), it
follows that “1” is always invariant with respect to scale transformations (5) and (6).
Thus, electron, proton, and neutron constants are on the lines:
where Ke is the slope for electron line, (7)
1 = Y /Ke X,

where Kp is the slope for proton line, (8)
1 = Y /Kp X,

where Kn is the slope for neutron line, (9)
1 = Y /Kn X,
Fundamental constants of nucleon-meson dynamics 3

Table 1. Fundamental Constants of Proton Dynamics
Symbols Constants Relationships of Quantities
1, 075827 · 10?36 Vop = m/d
Vop
2, 667688 · 10?30 hp = W/f
hp
1, 672623 · 10?27 mp = F/Y
mp
1, 440869 · 10?22 Cp = q/V
Sp
5, 635247 · 10?18 Lp = ?/i
Lp
3, 491143 · 10?15 ?p = F/H
?p
1, 024662 · 10?12 Sp = V /E
Sp
2, 162829 · 10?12 Wp = P t
Wp
4, 435318 · 10?11 ?p = v/f
?p
1, 155117 · 10?2 ?f p = S/2?
?f p
1, 000000 · 100
1 1 = GR
1, 504171 · 106 3
R?p = ?f p /S
R?p
1, 681364 · 107 Dp = q/A
Dp
3, 595937 · 107 Vp = H/D
Vp
3, 325110 · 109 Bp = E/v
Bp
6, 046079 · 1014 Hp = i/S
Hp
1, 195689 · 1017 Ep = V /S
Ep
8, 107560 · 1017 fp = W/h
fp
3, 509387 · 1019 ?p = (?)1/2
?p

Table 2. Fundamental Constants of Neutron Dynamics
Symbols Constants Relationships of Quantities
1, 077819 · 10?36 Von = m/d
Von
2, 671749 · 10?30 hn = W/f
hn
1, 674929 · 10?27 mn = F/Y
mn
1, 442489 · 10?22 Cn = q/V
Cn
5, 640249 · 10?18 Ln = ?/i
Ln
3, 493739 · 10?15 ?n = F/H
?n
1, 025295 · 10?12 Sn = V /E
Sn
2, 164127 · 10?12 Wn = P t
Wn
4, 437681 · 10?11 ?n = v/f
?n
1, 155214 · 10?2 ?f n = S/2
?f n
1, 000000 · 100
1 1 = GR
1, 503623 · 105 3
R?n = ?f p /s
R?n
1, 680739 · 107 Dn = q/A
Dn
3, 594539 · 107 Vn = H/D
Vn
3, 323482 · 109 Bn = E/v
Bn
6, 041484 · 1014 Hn = i/S
Hn
1, 194639 · 1017 En = V /S
En
8, 100040 · 1017 fn = W/h
fn
3, 505861 · 1019 ?n = (?)1/2
?n

The equations (7)–(9) are straight lines in the X ? Y plane that go through the
Absolute frame of reference of 1. Therefore, all electron, proton, and neutron constants
are located on straight lines that have fixed slopes of Ke , Kp , and Kn , and a common
hidden Absolute frame of reference of 10? or 1. Note, because the lines with slopes
Ke , Kp , and Kn go through the center of equilibrium, it requires only one constant
4 O. Bedrij, W.I. Fushchych

and the Absolute frame of reference of 1 to compute another set of constants for a new
particle.
We computed constants that characterize proton and neutron, by raising electrons
constant values ([2] — Table 1) to a power of the difference between the masses of
the proton (and neutron) and the electron [ln mp /me = 0, 89135 and ln mn / ln me =
0, 89133]. Some of the calculations are listed in the Tables 1 and 2.

1. Bedrij O., Fundamental constants in quantum electrodynamics, Dopovidi Ukrainian Academy of
Sciences, 1993, 3, 40–45.
2. Bedrij O., Scale invariance, unifying principle order and sequence of physical quantities and funda-
mental constants, Dopovidi Ukrainian Academy of Sciences, 1993, 4, 67–73.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 5–8.

On maximal subalgebras of the rank n ? 1
of the conformal algebra AC(1, n)
A.F. BARANNYK, W.I. FUSHCHYCH
ii i n ? 1 AC(1, n), i
?
aepi AP (1, n).

Consider the multidimensional eikonal equation
2 2 2
?u ?u ?u
? ? ··· ? (1)
= 1,
?x0 ?x1 ?xn?1
where u = u(x) is a scalar function of the variable x = (x0 , x1 , . . . , xn?1 ), n ? 2.
In [1] it was established that the Lie algebra AC(1, n) of the group C(1, n) of the
Minkowski R1,n space with the metric x2 ? x2 ? · · · ? x2 , where xn = u, is a maximal
n
0 1
algebra of the equation (1) invariance. The basis of the algebra AC(1, n) is formed by
such vector fields as:
P? = ?? , J?? = g ?? x? ?? ? g ?? x? ?? , D = ?x? ?? ,
K? = ?2(g ?? x? )D ? (g ?? x? x? )?? ,

where g00 = ?g11 = · · · = ?gnn = 1, g?? = 0, when ? = ? (?, ?, ? = 0, 1, . . . , n).
The algebra AC(1, n) contains the Poincar? algebra AP (1, n) which is generated by

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