ebook img

Scientific Works PDF

521 Pages·2003·3.141 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Scientific Works

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 1. Fushchych W.I, Shtelen W.M., The symmetry and some exact solutions of the relativistic eikonal equation,Lett.NuovoCimento,1982,34,498–502. 2. GrundlandA.M.,HarnadL.,WinternitzP.,Symmetryreductionfornonlinearrelativisticallyinvari- antequations,J.Math.Phys.,1984,25,№4,791–806. 3. ФущичВ.И.,БаранникЛ.Ф.,Максимальныеподалгебрырангаn−1алгебрыAP(1,n)иреду- кциянелинейныхволновыхуравнений.I,Укр.мат.журн.,1990,42,№11,1260–1256. 4. ФущичВ.И.,БаранникЛ.Ф.,Максимальныеподалгебрырангаn−1алгебрыAP(1,n)иреду- кциянелинейныхволновыхуравнений.II,Укр.мат.журн.,1990,42,№12,1693–1700. 5. БаранникА.Ф.,БаранникЛ.Ф.,ФущичВ.И.,Редукциямногомерногопуанкаре-инвариантного нелинейногоуравнениякдвумернымуравнениям,Укр.мат.журн.,1991,43,№10,1314–1323.

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.