A geometric model of gravitating mass formation and baryon mass spectrum
aa r X i v : . [ phy s i c s . g e n - ph ] A p r Gravitation & Cosmology,
Vol. 7 (2001), No. 1 (25), pp. 1–6 c (cid:13) A GEOMETRIC MODEL OF GRAVITATING MASS FORMATIONAND BARYON MASS SPECTRUM
Nikolay Popov and Petr Roshchin Computation Centre of the Russian Academy of SciencesReceived 31 May 2000
The operator of elementary particle mass spectrum is constructed on the basis of the postulate of a six-dimensionalspace-time manifold having a Riemann-Cartan space structure, proceeding only from its geometric properties. Specialsolutions found for this operator well describe the mass spectrum of the known quark triplet and hence of some baryonsupermultiplets.
1. Introduction
In [1], using a Riemann-Cartan six-dimensional man-ifold of signature ( − − − + ++), a purely geometricmodel has been successfully constructed, describing thegravitational field produced by a point source of timeaxis rotation in the temporal subspace. Thus one of theprincipal results of this study is that, according to theproposed geometric model, the gravitating mass con-centrated at a given point of space originates due torotation of the time axis associated with that point. Itis noted that the gravitational field itself produced bysuch rotation coincides with the Schwarzschild field inthe four-dimensional space-time submanifold, while thepoint mass turns out to be proportional to the squaredangular velocity of time axis rotation. These results,based on new geometrical notions of space and time,are rather qualitative than quantitative and do not allowone to explain the existing laws governing the elemen-tary particle mass spectrum. To solve such a problem,the formalism of quantum mechanics should be used.If the above geometric model is sufficiently realistic,one would expect that the elementary particle mass val-ues, for particles considered, as a first appoximation, aspoints, should represent eigenvalues of some invariantoperator, uniquely induced by the metric of the space-time manifold constructed in [1]. A significant part inthe mass spectrum formation should be played, accord-ing to the geometric model, by the operator componentthat acts in the three-dimensional temporal subspace.The present study is concerned with solving this prob-lem by establishing a strong relationship between thenew notions of physical space-time structure and theproperties of elementary particles.
2. Statement of the problem
Let a six-dimensional manifold M be equipped witha Riemann-Cartan space structure, which is character-ized by the metric g ij and the connection ∆ ij . Thetangential subspace at each point of M is the pseudo-Euclidean space R , of signature ( − − − + ++). It e-mail: [email protected] should be noted that for the subsequent constructionsit will be sufficient to know the metric g , which is equiv-alent to assigning to M the structure of a pseudo-Riemannian space. However, proceeding in the spiritof [1], to which we shall have to refer in what follows,we prefer to leave everything as it is. Let x , ..., x bepseudo-Euclidean coordinates in M , with respect towhich the following requirements are satisfied:1. The metric tensor field g ij ( x ) is continuously dif-ferentiable in the entire manifold M , except the line x = x = x = x = x = 0 , where the metric may bediscontinuous.2. The metric components g ij i, j = 1 , ..., x , x , x , while g ij , i, j = 5 , x , x only.3. g ij = g ji = 0 for i = 1 , , j = 4 , , g ij are invariant under or-thogonal transformations of the coordinates x , x , x and, separately, x , x .5. At infinity all the metric components tend to zero,except six, which have the following limiting values: g = g = g = − , g = g = g = 1 . Our task is to construct, on the manifold M witha given metric g , the most general form of a self-adjointdifferential operator of second order, ∆ , , which pos-sesses the following properties:1 ′ . The operator ∆ , is invariant under the group oftransformations O (3) × O (2);2 ′ . ∆ , takes at infinity the following form:∆ , = η ij ∂ i ∂ j , where η ij is the pseudo-Euclidean metric in the space R , (here and henceforth ∂ i ≡ ∂/∂ x i and, for any z , ∂ z ≡ ∂/∂z ). We must then investigate its spectrum andestablish a direct relationship of this spectrum with themass spectra of some elementary particles. Nikolay Popov and Petr Roshchin
3. Construction of the operator∆ , Let us introduce the linear element ds = ( g ij dx i dx j ) / Then the most general form of the squared linear ele-ment, according to [1] and taking into account the re-quirements 2–5, can be represented as ds = F ( r ) dx + M ( ρ )( dx + dx )+ N ( ρ )( x dx + x dx ) − G ( r )( dx + d x + d x ) − H ( r )( x d x + x d x + x dx ) , (1)where ρ = ( x + x ) / , r = ( x + x + x ) / . The functions
F, G, H, M and N have been foundin [1], the first three of them coinciding with simi-lar functions for the Schwarzschild metric in the four-dimensional manifold [2]. In the region r > ε , ρ > ε ,where ε , ε are arbitrary but small quantities, thesefunctions have the form F ( r ) = 1 − a ( r + a ) / , G ( r ) = ( r + a ) / r ,H ( r ) = ( r + a ) − / r − a ( r + a ) − / − ( r + a ) / r ,M ( ρ ) = ρ + β ρ , N ( ρ ) = 1 ρ + β − ρ + β ρ , (2)where a is the Schwarzschild radius and β is a constant.The most general form of a self-adjoint second-order dif-ferential operator which satisfies the properties (1 ′ , ′ )is as follows:∆ , = F ( r ) ∂ − X k =5 , i ∂ x k M ( ρ ) 1 i ∂ x k − (cid:16) i ∂ x x + 1 i ∂ x x (cid:17) N ( ρ ) (cid:16) x i ∂ x + x i ∂ x (cid:17) + X k =1 , , i ∂ x k G ( r ) 1 i ∂ x k + (cid:16) i ∂ x x + 1 i ∂ x x + 1 i ∂ x x (cid:17) H ( r ) × (cid:16) x i ∂ x + x i ∂ x + x i ∂ x (cid:17) , (3)provided that r > ε , ρ > ε . Formally, the oper-ator (3) is obtained from the squared linear element(1) by substituting the Hermitian operator (1 /i )( ∂/∂x j )for the differential dx j and appropriate regrouping ofthe operator cofactors. Before proceeding to determinethe range of definition, D (∆ , ), of the operator (3),let us present it in new, partially spherical and par-tially cylindrical coordinates r, θ, ϕ, ρ, ψ, t where x = r sin θ cos ϕ , x = r sin θ sin ϕ , x = r cos θ , x = t , x = ρ cos ψ , x = ρ sin ψ , Then x ∂ + x ∂ = ρ∂ ρ ,∂ + ∂ = ∂ ρ + 1 ρ ∂ ρ + 1 ρ ∂ ψ ,x ∂ + x ∂ + x ∂ = r∂ r ,∂ + ∂ + ∂ = ∂ r + 2 r ∂ r + 1 r Λ( θ, ϕ ) , whereΛ( θ, ϕ ) = ∂ θ + cos θ sin θ ∂ θ + 1sin θ ∂ ϕ , and the operator (3) will be represented in the new co-ordinates as follows:∆ , = F ( r ) ∂ t + ( M + ρ N ) (cid:16) ∂ ρ + 1 ρ ∂ ρ (cid:17) + ( M ′ + ρ N ′ + ρN ) ∂ ρ + Mρ ∂ ψ − ( G + r H ) (cid:16) ∂ r + 2 r ∂ r (cid:17) − ( G ′ + r H ′ ) ∂ r − Gr Λ( θ, ϕ ) . (4)The change from one coordinate system to another isachieved in a covariant manner. The operator (4) maybe written as a sum of two commutative operators,∆ , = ∆ (1)3 , + ∆ (2)3 , , where∆ (1)3 , = ( M + ρ N ) (cid:16) ∂ ρ + 1 ρ ∂ ρ (cid:17) + (cid:16) M ′ + ρ N ′ + ρN (cid:17) ∂ ρ + Mρ ∂ ψ , (5)∆ (2)3 , = F ( r ) ∂ t − ( G + r H ) (cid:16) ∂ r + 2 r ∂ r (cid:17) − (cid:16) G ′ + r H ′ (cid:17) ∂ r − Gr Λ( θ, ϕ ) . (6)The range of definition D (∆ (1)3 , ) of the operator (5) con-sists of doubly differentiable, quadratically integrablefunctions of the parameters ρ, ψ relative to the measure dµ = ρ dρ dψ which satisfy the condition Z ∞ | g | ρ − dρ < ∞ . (7)The condition (7) is sufficient to ensure that∆ (1)3 , g ∈ L ( R , µ ) for any g ∈ D (∆ (1)3 , ) . Taking into account that the operator ∆ (1)3 , in the form(5) is defined only for ρ > ε , let us confine ourselves toconsidering the action of the operator ∆ (1)3 , upon func-tions belonging to D (∆ (1)3 , ) and defined in the domain R \ O , where O = { ρ : ρ < ε } . The range of definition D (∆ (1)3 , ) of the operator (6)consists of twice differentiable and square-integrable Geometric Model of Gravitating Mass Formation and Baryon Mass Spectrum functions of the parameters r, θ, ϕ, t relative to themeasure dµ = r sin θ dr dθ dϕ dt , which satisfy thecondition Z ∞ | g | r − dr < ∞ . (8)Integration in the parameter t is done in an arbitrarycompact. The condition (8) is sufficient for the oper-ator ∆ (2)3 , to be essentially self-adjoint. Since the op-erator ∆ (2)3 , in the form (6) is only defined at r > ε ,let us restrict ourselves to considering its action uponthe functions constituting D (∆ (2)3 , ) and defined in thedomain R , \ O , where O = { r : r < ε } .
4. Eigenvalues of the operator ∆ ( ) , With Eq. (2), one can represent the operator (5) in theform∆ (1)3 , = ρ ρ + β ∂ ρ + (cid:16) ρ β ( ρ + β ) + ρ + β ρ (cid:17) ∂ ρ + ρ + β ρ ∂ ψ . (9)We will only be interested in positive eigenvalues of theoperator (9). It can be noted that the essential spec-trum of the formally self-adjoint operator (9) representsa negative semiaxis; therefore, all non-negative eigen-values belong to a discrete spectrum. Accordingly, theproblem of finding the eigenvalues reduces to that ofsolving the differential equation∆ (1)3 , g = λ g, g ∈ D (∆ (1)3 , ) . (10)We will seek the function g in (10) in the form g ( ρ, ψ ) = e imψ g ( ρ ) , m = 0 , ± , ± , . . . . (11)Substituting (11) into (10) and neglecting the terms con-taining the first derivative with coefficient β/ρ in powers2 and higher yields h ρ ρ + β (cid:16) ∂ ρ + 1 ρ ∂ ρ (cid:17) − ρ + β ρ m i g ( ρ ) = λ g ( ρ ) . (12)Let us introduce the new variable x = β /ρ , x ∈ (0 , β /ε ), then the relation (12) assumes the form (cid:20) x β ( x + 1) (cid:16) ∂ x + 1 x ∂ x (cid:17) − x ( x + 1) β m (cid:21) g ( x ) = λ g ( x ) , (13)and the function g should be square-integrable in theregion (0 , β /ε ) over the measure dµ = [ β / (2 x )] dx ,or, taking into account the condition (7), we arrive atthe requirement Z β ε − | g | dx < ∞ . For x > (cid:18) ∂ x + 1 x ∂ x − m − m x − m + β λ x − λ β x (cid:19) g ( x ) = 0 . (14)We will seek the function g ( x ) in (14) in the form g = x − / v , then for v we get the equation v ′′ − m v − m vx − m + β λ − x v − β λ x v = 0 . (15)
5. Finding a partial spectrum of∆ ( ) , Let as set λ = 0 in (15) and find the eigenfunctions andallowed values of the parameter m which correspond tothe eigenvalue λ = 0 of the operator ∆ (1)3 , . Considerthe equation v ′′ − m v − m vx + 1 − m x v = 0 . (16)We will seek its solution in the form v = e − mx/ ∞ X k = − C k x r +1+ k . (17)The lower limit of the sum in (17) is chosen on the ba-sis of the requirement of quadratic integrability of thefunction v x − / near zero. Substituting (17) to (16)and denoting y = ∞ P k = − C k x r +1+ k , we arrive at the re-lation y ′′ − my ′ − m x y + 1 − m x y = 0 . (18)From (18) we obtain an infinite set of recurrent relations: h ( r + k )( r + k + 1) + (1 − m ) i C k − m (cid:16) r + k + m (cid:17) C k − = 0 , k = 0 , , , ... , with a condition on r in the form r = − B m − , where B = 13 . Hence, C k = m (cid:16) m − B + k − (cid:17) k ( m + k ) C k − , k = 0 , , ... . (19)From (19) it follows that the sum ∞ P k = − C k +1 x r +1+ k should contain a finite number of terms to assure thatthe function v be square integrable. If m = 0 , then atsome m, k, B the equality k −
13 + m − B Nikolay Popov and Petr Roshchin should hold. The parameters λβ, m, k − , B admitthe following physical interpretation: let Q be thequark charge, B the baryon charge, I the isotopicspin, I a projection of the isotopic spin, S the particlestrangeness, then S = − λβ, Q = k − , I = | m | , I = −| m | , ..., | m | . It can be noted that the idea to consider the isospinas a spin of a particle in the three-dimensional time-space was first advanced in [3]. From (20) we obtain thefollowing relation: S = − I − B + 2 Q (21)for the case S = 0 . The relation (21) is nothing else butthe Nishijima formula for the baryon supermultiplet. If S = 0 and m = 1 / I = ± , Q = − , , which corresponds to the quark doublet ( u, d ).Let us consider Eq. (15) with m = 0 , then we obtainthe following relation: v ′′ + 1 − λ β x v − λ β x v = 0 . (22)We will seek its solution in the form v = e − λβ/ √ x y. (23)Substituting (23) into (22), we come to the following setof equations for y : y ′′ + 1 − λ β x y = 0 , y ′ − yx = 0 . (24)The system (24) has the solution y = cx / only if λβ = . This means that the Nishijima formula is true in thecase λβ = , I = 0 , B = , Q = − , whichcorresponds to the quark s .Thus we have found two eigenvalues 0 and 1 / (2 β )of the operator ∆ (1)3 , , which correspond to the quarktriplet ( u, d, s ). It can already be seen at this stage ofthe investigation that the s quark mass differs from themasses of u and d quarks.
6. Finding a partial spectrum of∆ ( ) , With Eq. (2), one can represent the operator (6) in theform∆ (2)3 , = f ∂ t − f r ∂ r − f r Λ( θ, ϕ ) − (cid:16) f r + 4 f r + f ′ r (cid:17) ∂ r , (25)where f = G/r + H/r , f = r G, f = F .Accordingly, the problem of finding the eigenvaluesreduces to that of solving the differential equation∆ (2)3 , g = ∆ g , (26) where g ∈ D (∆ (2)3 , ). We will seek the function g in (26)in the form g ( t, r, θ, ϕ ) = G ( t ) R l,n ( r ) Y l ( θ, ϕ ) (27)where Y l is an eigenfunction of the operator Λ( θ, ϕ ),corresponding to the eigenvalue − l ( l + 1), so thatΛ( θ, ϕ ) Y l ( θ, ϕ ) = − l ( l + 1) Y l ( θ, ϕ ) ; (28) G ( t ) is an eigenfunction of the formally self-adjoint op-erator − ∂ t , which has only an essential spectrum in thedomain (0 , ∞ ), so that ∂ t G ( t ) = α G ( t ) . (29)Taking into account the relations (25), (27), (28), (29),one can represent Eq. (26) in the form h f α − f r ∂ r − (cid:16) f r + 4 f r + f ′ r (cid:17) ∂ r + f r l ( l + 1) i R l,n = ∆ R l,n . (30)The essential spectrum of the formally self-adjointoperator in the left-hand side of Eq. (30) is representedby the range ( α , ∞ ), therefore all non-negative eigen-values of the operator ∆ (2)3 , restricted above by the value α are discrete. From the condition f r = 0 for r ∈ ( ε , ∞ ) it follows: (cid:20) ∂ r + (cid:16) r + f ′ f + 2 f r f (cid:17) ∂ r − f f r α − f f r l ( l + 1) + ∆ f r (cid:21) R l,n = 0 . (31)We will seek its solution in the form R l,n = exp (cid:20) − Z (cid:16) r + f ′ f + 2 f r f (cid:17) dr (cid:21) v l,n . (32)After substitution of (32), Eq. (31) may be rewritten inthe form v ′′ l,n − (cid:16) P + P ′ + Q (cid:17) v l,n = 0 , (33)where [4] P = 4 r + f ′ f + 2 f r f ; Q = f f r α + f f r l ( l +1) − ∆ f r . Taking into account that1 f r ∼ − ar + o (cid:16) a r (cid:17) , f f r ∼ − ar + a r + o (cid:16) a r (cid:17) ,f f r ∼ − ar + o (cid:16) a r (cid:17) , P + 12 P ′ ∼ o (cid:16) a r (cid:17) and neglecting the terms containing ar in powers 3 andhigher yields v ′′ l,n − ( α − ∆ ) v l,n + ar (2 α − ∆ ) v l,n − a α + l ( l + 1) r v l,n = 0 . (34) Geometric Model of Gravitating Mass Formation and Baryon Mass Spectrum We will seek the solution of Eq. (34) in the form v = exp[ − ( α − ∆ ) / r ] y l,n , (35)where y l,n = ∞ P k =0 C k r ν + k . Substituting (35) into (34),we come to the following relation for y l,n : y ′′ l,n − α − ∆ ) / y ′ l,n + (2 α − ∆ ) ar y l,n − l ( l + 1) + a α r y l,n = 0 . (36)Eq. (36) gives an infinite set of recurrent relations[( ν + k + 1)( ν + k ) − l ( l + 1) − a α )] C k +1 − [2( α − ∆ ) / ( ν + k ) − (2 α − ∆ ) a ] C k = 0 , where k = 0 , , , . . . and ν = ± p l ( l + 1) + 4 a α . (37)Hence, C k +1 = 2( α − ∆ ) / ( ν + k ) − (2 α − ∆ ) a ( ν + k + 1)( ν + k ) − l ( l + 1) − a α C k . (38)From (38) it follows that the sum ∞ P k =0 C k r ν + k shouldcontain a finite number of terms to assure that the func-tion v l,n be square-integrable in the region ( ε , ∞ ).Thus at some k = n the equality2( α − ∆ ) / ( ν + n ) − (cid:16) α − ∆ (cid:17) a = 0 . (39)should hold, and for all k < n the inequality( ν + k + 1)( ν + k ) − l ( l + 1) − a α = 0is valid. Solving Eq. (39) for ∆ , we obtain∆ , = 2 (cid:20) α − (cid:16) ν + na (cid:17) (cid:21) ± (cid:20) α − (cid:16) ν + na (cid:17) (cid:21) × (cid:20) − α α − [( ν + n ) /a ] (cid:21) / . (40)From (40) it follows (cid:16) ν + na (cid:17) > α (41)because 1 − α α − [( ν + n ) /a ] ≥ α and ∆ the followingrelations: α = ν ( ν − − l ( l + 1) a , ∆ = 2 a A (cid:20) | ν + n | A / − (cid:21) , where A = ( ν + n ) + l ( l + 1) − ν ( ν − l is the spinmoment of a particle, while the physical meaning of thequantum parameters ν and n remains obscure.
7. Partial spectrum of ∆ , andthe mass spectrum of the quarktriplet The obtained partial spectra of the operators ∆ (1)3 , and∆ (2)3 , lead to a partial spectrum of the operator ∆ , :∆ , g = ( λ + ∆ ) g, or in an expanded form∆ , g = (cid:20)(cid:16) Q − B − I β (cid:17) + 2 a A (cid:16) ν + nA / − (cid:17)(cid:21) g. (42)If Q = − , , B = , I = − , , , l = , thenthis set of quantum characteristics corresponds to thequark triplet d, u, s . From (42), for the mass doubletof the quarks ( d, u ) and the quark s we get the followingrelations: m d,u = (cid:26) a (cid:20) ( ν + n ) − ν ( n −
1) + 34 (cid:21) × (cid:20)(cid:16) − ν ( ν − − / ν + n ) (cid:17) − / − (cid:21)(cid:27) / , (43) m s = ( β − + m d,u ) / . (44)We note that the relations (43) and (44) are given inprovisional units of measurement h = 1 , c = 1 , e = 1 .Knowing the mass spectrum of the quark triplet, wecan readily find the mass spectrum of, e.g., the baryondecuplet, ignoring the spin-spin and spin-orbital inter-actions. The result satisfactorily describes the nature ofthe supermultiplet spectrum.
8. Conclusion
The very attempt to relate the geometric properties ofthe space-time continuum to the properties of elemen-tary particles, in particular, to the quark mass spec-trum appears, judging from the results that have beenobtained, to be promising. Apparently, the first theo-retical substantiation of the Nishijima formula for thequark triplet ( u, d, s ) on the basis of the geometricalproperties of space and time is not coincidental.The main achievement of the suggested work is ap-parently the construction of an unlimited self-adjoinedoperator, possessing a discrete spectrum adequately de-scribing the quark triplet mass spectrum.
References [1] N. Popov,