Scientific Works

521 Pages·2003·3.141 MB·English

by  Fushchych W.I., Boyko V.M. (ed.)

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
Запропоновано новий феноменологiчний пiдхiд для обчислення констант протону та
нейтрону. В основу роботи покладено нестандартну iдею: стала Планка (cid:1) та швид-
кiсть “свiтла” (мезону) c в нуклон-мезоннiй динамицi вiдмiннi вiд цих же констант
в квантовiй електродинамицi.
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 (cid:1) and c constants in the
relations. It is presently assumed in physics that electrodynamic constants of (cid:1) and c
are applicable to characterization of nucleon-meson dynamics. Our calculations show
that constants (cid:1) 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-
namicconstants.Belowweshowtheproposedmodifications(DefinitionsofQuantities
are shown in [2]):
Standard Proposed
Relationships Relationships
Compton Wavelength of proton λp =(cid:1)/mpc λp =(cid:1)pmpvp
Compton Wavelength of neutron λn =(cid:1)/mnc λn =(cid:1)n/mnvn
Proton magneton µp =q(cid:1)/2mpc µp =qp(cid:1)p/2mpvp
Neutron magneton µn =q(cid:1)/2mnc µn =qn(cid:1)n/2mnvn
Proton radius rp =(cid:1)pαfpmpvp, (cid:1)p (cid:1)=(cid:1)
Neutron radius rn =(cid:1)nαfn/mnvn, (cid:1)n (cid:1)=(cid:1)
where v and v are velocities of mesons which are emitted by proton and neutron.
p n
In our approach, we assume that:
ДоповiдiАНУкраїни,1993,№5,С.62–64.
2 O. Bedrij, W.I. Fushchych
1.The physicalrelationshipsbetweenquantities arethe samefor 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:
(cid:1) (cid:2) (cid:1) (cid:2)
1= qx1qx2qx3···qxs / pj1pj22pj3···pjz (1)
1 2 3 s 1 3 z
or,
(cid:1)
1=Y /KX, (2)
where
(cid:1) (cid:2)
Y(cid:1) ≡ qx1qx2qx3···pjz , (3)
1 2 3 z
(cid:1) (cid:2)
1/KX ≡ pj1pj2pj3···pjz , (4)
3 z
(q )0 =1, (q−1)0 =1,
s s
q1(cid:1),q2(cid:1),q3(cid:1),...,qs(cid:1),p1(cid:1),p2(cid:1),q3(cid:1),...,pz(cid:1) are quantities,
x1(cid:1),x2(cid:1),x3(cid:1),...,xs(cid:1),j1(cid:1),j2(cid:1),j3(cid:1),...,jz(cid:1) are real numbers,
K is the slope for line Y(cid:1) =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:
q →q(cid:1) =aq , q →q(cid:1) =aq , q3 →q(cid:1) =aq , ..., (5)
1 1 1 2 2 2 3 3
p →p(cid:1) =ap , p →p(cid:1) =ap , p3 →p(cid:1) =ap , ..., (6)
1 1 1 2 2 2 3 3
where “a” is a scale transformation parameter, and all physical quantities (q and p )
s z
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:
(cid:1)
1=Y /K X, where K is the slope for electron line, (7)
e e
(cid:1)
1=Y /K X, where K is the slope for proton line, (8)
p p
(cid:1)
1=Y /K X, where K is the slope for neutron line, (9)
n n
Fundamental constants of nucleon-meson dynamics 3
Table 1. Fundamental Constants of Proton Dynamics
Symbols Constants Relationships of Quantities
Vop 1,075827·10−36 Vop =m/d
hp 2,667688·10−30 hp =W/f
mp 1,672623·10−27 mp =F/Y
Sp 1,440869·10−22 Cp =q/V
Lp 5,635247·10−18 Lp =φ/i
φp 3,491143·10−15 φp =F/H
Sp 1,024662·10−12 Sp =V/E
Wp 2,162829·10−12 Wp =Pt
λp 4,435318·10−11 λp =v/f
αfp 1,155117·10−2 αfp =S/2λ
1 1,000000·100 1=GR
R∞p 1,504171·106 R∞p =αf3p/S
Dp 1,681364·107 Dp =q/A
Vp 3,595937·107 Vp =H/D
Bp 3,325110·109 Bp =E/v
Hp 6,046079·1014 Hp =i/S
Ep 1,195689·1017 Ep =V/S
fp 8,107560·1017 fp =W/h
ωp 3,509387·1019 ωp =(α)1/2
Table 2. Fundamental Constants of Neutron Dynamics
Symbols Constants Relationships of Quantities
Von 1,077819·10−36 Von =m/d
hn 2,671749·10−30 hn =W/f
mn 1,674929·10−27 mn =F/Y
Cn 1,442489·10−22 Cn =q/V
Ln 5,640249·10−18 Ln =φ/i
φn 3,493739·10−15 φn =F/H
Sn 1,025295·10−12 Sn =V/E
Wn 2,164127·10−12 Wn =Pt
λn 4,437681·10−11 λn =v/f
αfn 1,155214·10−2 αfn =S/2
1 1,000000·100 1=GR
R∞n 1,503623·105 R∞n =αf3p/s
Dn 1,680739·107 Dn =q/A
Vn 3,594539·107 Vn =H/D
Bn 3,323482·109 Bn =E/v
Hn 6,041484·1014 Hn =i/S
En 1,194639·1017 En =V/S
fn 8,100040·1017 fn =W/h
ωn 3,505861·1019 ωn =(α)1/2
The equations (7)–(9) are straight lines in the X −Y(cid:1) plane that go through the
Absoluteframeofreferenceof1.Therefore,allelectron,proton,andneutronconstants
are located on straight lines that have fixed slopes of K , K , and K , and a common
e p n
hidden Absolute frame of reference of 10◦ or 1. Note, because the lines with slopes
K , K , and K go through the center of equilibrium, it requires only one constant
e p n
4 O. Bedrij, W.I. Fushchych
andtheAbsoluteframeofreferenceof1tocomputeanothersetofconstantsforanew
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 [lnm /m = 0,89135 and lnm /lnm =
p e n e
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. BedrijO.,Scaleinvariance,unifyingprincipleorderandsequenceofphysicalquantitiesandfunda-
mentalconstants,DopovidiUkrainianAcademyofSciences,1993,№4,67–73.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 5–8.
n − 1
On maximal subalgebras of the rank
AC(1, n)
of the conformal algebra
A.F. BARANNYK, W.I. FUSHCHYCH
Проведено класифiкацiю максимальних пiдалгебр рангу n−1 алгебри AC(1,n), якi
належать aлгeбpi AP˜(1,n).
Consider the multidimensional eikonal equation
(cid:3) (cid:4) (cid:3) (cid:4) (cid:3) (cid:4)
∂u 2 ∂u 2 ∂u 2
− −···− =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 R space with the metric x2−x2−···−x2, where x =u, is a maximal
1,n 0 1 n n
algebra of the equation (1) invariance. The basis of the algebra AC(1,n) is formed by
such vector fields as:
P =∂ , J =gαγx ∂ −gβγx ∂ , D =−xα∂ ,
α α αβ γ β γ α α
K =−2(gαβx )D−(gβγx x )∂ ,
α β β γ α
where g = −g = ··· = −g = 1, g = 0, when α (cid:4)= β (α,β,γ = 0,1,...,n).
00 11 nn αβ
The algebra AC(1,n) contains the Poincar´e algebra AP(1,n) which is generated by
vector fields P , J and the extended Poincar´e algebra AP˜(1,n)=AP(1,n)+⊃ (cid:6)D(cid:7).
α αβ
In order to reduce the equation (1) by subalgebras of the algebra AC(1,n), it
is necessary to describe all C(1,n)-nonequivalent subalgebras of this algebra. The
subalgebras K and K of the algebra AC(1,n) are called as C(1,n)-equivalent ones
1 2
if they have the same invariants with respect to C(1,n)-conjugation. Among C(1,n)-
equivalent algebras there exists one (maximal) subalgebra containing all the other
subalgebras. The maximal subalgebras K and K of the algebra AC(1,n) are equi-
1 2
valent if and only if K and K are C(1,n)-conjugated.
1 2
The maximal subalgebras of the rank n of the algebra AP(1,n) with respect to
P(1,n)-conjugation are described in [2]. The maximal subalgebras of the rank n of
the algebra AP˜(1,n) with respect to P˜(1,n)-conjugation are described in [3, 4].
The present article is a continuation of researches which were realized in [3, 4].
The full classification of the maximal subalgebras of the rank n−1 of the algebra
AC(1,n) which are contained in the algebra AP˜(1,n) has been carried out in the
present article. Ansatzes corresponding to these subalgebras reduce the equation (1)
to ordinary differential equations.
We will use the notations:
1
M =P +P , T = (P −P ), G =J −J , a=1,...,n−1,
0 n 2 0 n a 0n an
AO[r,s]=(cid:6)J |a,b=r,...,s(cid:7), r ≤s,
ab
ДоповiдiАНУкраїни,1993,№6,С.38–41.
6 A.F. Barannyk, W.I. Fushchych
AE[r,s]=(cid:6)P ,...,P (cid:7)+⊃ AO[r,s], r ≤s,
r s
AE [r,s]=(cid:6)G ,...,G (cid:7)+⊃ AO[r,s], r ≤s.
1 r s
If s>r then AO[r,s]=0, AE[r,s]=0 by definition.
Let
Φ(r,s,γ)=(cid:6)G +γP ,...,G +γP (cid:7)+⊃ AO[r,s], r,s∈N, r ≤s, γ ∈R.
r r s s
Let
Γ =U+⊃ F,
d,q
where F is the diagonal of AO[1,d]⊕AO[d+1,2d]⊕···⊕AO[(q−1)d+1,qd], and
U is the Abelian algebra which has the basis
G1+γ1P1+λ1P(q−1)d+1, ..., Gd+γ1Pd+λ1Pqd,
Gd+1+γ2Pd+1+λ2P(q−1)d+1, ..., G2d+γ2P2d+λ2Pqd,
·························································
G(q−2)d+1+γq−1P(q−2)d+1+λq−1P(q−1)d+1, ..., G(q−1)d+
+γq−1P(q−1)d+λq−1Pqd,
where 0≤γ1 <γ2 <···<γq−1, λ1 >0, λ2 >0, ..., λq−1 >0.
Resultsofthework[5]reducetheproblemconstructinginvariantsofanysubalgeb-
ra of the algebra AP˜(1,n) to the problem of constructing invariants of the irreducible
subalgebras of the orthogonal algebra AO(k) for all k ≤ n. The latter problem has
no solution in quadratures. Therefore, we shall restrict ourself considering of such
subalgebras of the algebra AP˜(1,n) which projections onto AO[1,n] are subdirect
sums on the algebras AO[r,s]. Moreover, to find real solutions of the equation (1) it
is necessary to exclude from consideration such subalgebras of the algebra AP(1,n)
which with respect to equivalence contain P +P or P . Therefore we prove the
0 n 0
following theorems.
Theorem 1. Let L be the maximal subalgebra of the rank n − 1 of the algebra
AP(1,n). Then L is C(1,n)-conjugated with one of the following algebras:
1) L =AE[1,n−1];
1
2) L =AO[1,m]⊕AE[m+1,n], m=1,...,n, n≥2;
2
3) L =AE [1,m]⊕AE[m+1,n−1], m=1,...,n−1, n≥2;
3 1
4) L =AO[1,m]⊕AE[m+1,n−1]⊕(cid:6)J (cid:7), m=1,...,n−1, n≥3;
4 0n
5) L =AO[0,m]⊕AE[m+1,n−1], m=2,...,n−1, n≥3;
5
6)L =AO[0,m]⊕AO[m+1,q]⊕AE[q+1,n−1],m=2,...,n−1,q =m+1,...,n,
6
n≥3;
7) L =(cid:6)G +P −P (cid:7)⊕AE[2,n−1], n≥2;
7 1 0 n
8)L8 =Φ(d0+1,d1,γ1)⊕···⊕Φ(dt−1+1,m,γt)⊕AE[m+1,n−1],m=1,...,n−1,
n≥3;
9) L =(cid:6)J +P (cid:7)⊕AE[2,n−1], n≥2;
9 0n 1
10) L =(AE [1,m]⊕(cid:6)J +P (cid:7))⊕AE[m+2,n−1], m=1,...,n−2, n≥3;
10 1 0n m+1
11) L =(cid:6)J +P (cid:7)⊕AE[3,n], n≥2.
11 12 0
Theorem 2. Let L be the maximal subalgebra of the rank n − 1 of the algebra
AP˜(1,n) which has a nonzero projection onto (cid:6)D(cid:7). Then L is C(1,n)-conjugated
with one of the following algebras:
On maximal subalgebras of the rank n−1 of the conformal algebra AC(1,n) 7
1) L = (AO[0,d]⊕AO[d+1,m]⊕AO[m+1,q]⊕AE[q +1,n])+⊃ (cid:6)D(cid:7), d =
1
2,...,n−2, m=d+1,...,n−2, q =m+1,...,n−1, 2n≤d+q, n≥4;
2) L2 =(AO[0,m]⊕AE[m+1,n−2])+⊃ (cid:6)D+αJn−1,n(cid:7), m=2,...,n−2, n≥4,
α>0;
3) L = (AO[1,m] ⊕ AO[m + 1,q] ⊕ AE[q + 1,n])+⊃ (cid:6)D(cid:7), m = 2,...,n − 2,
3
q =m+2,...,n, 2m≤q, n≥2;
4) L4 = (AE1[1,m]⊕AE[m+1,n−3])+⊃ (cid:6)Jn−2,n−1 +cJ0n,D +αJ0n(cid:7), m =
1,...,n−3, n≥4, c>0, α≥0;
5) L =(AO[1,m]⊕AO[m+1,q]⊕AE[q+1,n−1])+⊃ (cid:6)D,J (cid:7), m=1,...,n−2,
5 0n
q =m+1,...,n−1, 2m≤q, n≥3;
6) L =AE[3,n−1]+⊃ (cid:6)J +cJ ,D+αJ (cid:7), c>0, α≥0, n≥3;
6 12 0n 0n
7)L =(AE [1,d]⊕AO[d+1,m]⊕AE[m+1,n−1])+⊃ (cid:6)D+αJ (cid:7),d=1,...,n−2,
7 1 0n
m=d+1,...,n−1, n≥3, α≥0;
8) L = (AO[1,m]⊕AE[m+1,n−1])+⊃ (cid:6)D+αJ (cid:7), m = 1,...,n−1, n ≥ 2,
8 0n
α≥0;
9) L =((cid:6)G +2T(cid:7)⊕AO[2,m]⊕AE[m+1,n−1])+⊃ (cid:6)2D−J (cid:7), m=2,...,n−1,
9 1 0n
n≥3;
10) L = (AE [1,d] ⊕ AO[d + 1,m] ⊕ AE[m + 1,n − 1])+⊃ (cid:6)D + J + M(cid:7),
10 1 0n
d=1,...,n−2, m=d+1,...,n−1, n≥3;
11) L = (AO[1,m]⊕AE[m+1,n−1])+⊃ (cid:6)D+J +M(cid:7), m = 1,...,n−1,
11 0n
n≥2;
12) L12 = (AE1[1,m]⊕AE[m+1,n−3])+⊃ (cid:6)Jn−2,n−1 +αM,D +J0n +M(cid:7),
m=1,...,n−3, n≥4, α≥0;
13) L13 = (AE1[1,m] ⊕ AE[m + 1,n − 3])+⊃ (cid:6)Jn−2,n−1 + M,D + J0n(cid:7), m =
1,...,n−3, n>4;
14) L =AE[3,n−1]+⊃ (cid:6)J +αM,D+J +M(cid:7), n≥3, α≥0;
14 12 0n
15) L =AE[3,n−1]+⊃ (cid:6)J +M,D+J (cid:7), n≤3;
15 12 0n
16) L =(Γ ⊕AE[dq+1,n−1]+⊃ (cid:6)D−J (cid:7), d≥2, n≥5;
16 d,q 0n
17) L17 =(Φ(d0+1,d1,γ1)⊕Φ(d1+1,d2,γ2)⊕···⊕Φ(dt−1+1,dt,γt)⊕AO[dt+
1,m]⊕AE[m+1,n−1])+⊃ (cid:6)D−J (cid:7), where d = 0, γ < γ < ··· < γ , t > 1,
0n 0 1 2 t
m=1,...,n−2, n≥3;
18) L18 = (Γd,q ⊕Φ(l0 +1,l1,µ1)⊕Φ(l1 +1,l2,µ2)⊕···⊕Φ(lt−1 +1,lt,µt)⊕
AE[l +1,n−1])+⊃ (cid:6)D−J (cid:7), where µ <µ <···<µ , t≥1, l =dq.
t 0n 1 2 t 0
L –L and L –L of the theorems 1 and 2 respectively and to carry out a
1 11 1 18
reduction of the equation (1). Consider, for example, the subalgebra L . The ansatz
17
(cid:5) (cid:9)
(cid:6)t (cid:7) (cid:8)
1
u2 = −(x +x ]+ x2 +···x2 ϕ(ω)−
0 m x −x +γ di−1+1 di
0 m i
i=1
−x2 −···−x2 , ω =x −x ,
dt+1 m−1 0 m
corresponds to this subalgebra. This ansatz reduces the equation (1) to equation
ϕϕ˙ −ϕ=0. Using the solution of this equation we find the following solution of the
equation (1):
(cid:5) (cid:9)
(cid:6)t (cid:7) (cid:8)
1
u2 = −(x +x ]+ x2 +···+x2 ×
0 m x −x +γ di−1+1 di
0 m i
i=1
×(x −x +C)−x2 −···−x2 .
0 m dt+1 m−1
8 A.F. Barannyk, W.I. Fushchych
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