Spin 2 Particle Theory, the Nonrelativistic Approximation

Nonlinear Phenomena in Complex Systems, vol. 28, no. 3 (2025), pp. 220 - 241 Spin 2 Particle Theory, the Nonrelativistic Approximation A. V. Ivashkevich,∗A. V. Bury,∗and V. M. Red’kov† B. I. Stepanov Institute of Physics of NAS of Belarus, 68 Nezavisimosti Ave., 220072 Minsk, BELARUS E. M. Ovsiyuk‡ Mozyr State Pedagogical University named after I. P. Shamyakin V. V. Kisel§ Belarus State University of Informatics and Radio-Electronics (Received 15 July, 2025) The goal of the present paper is investigation of the nonrelativistic approximation in the 39component theory for a spin 2 particle. We apply explicit expressions for four main matrices Γawith dimension 39×39 in the relevant first order system of equations, written in Cartesian coordinates and in the presence of external an electromagnetic field. For distinguishing the large and small parts in the complete wave function, we use three projective operators constructed on the base of the minimal polynomial of the 7-th order for the matrix Γ0. The relevant large and small components are found in explicit form. Among them we have found independent variables; in particular, among the large components there exist only five independent ones. We have derived the nonrelativistic equation for 5-component wave function; in which term describing interaction of the magnetic moment of the spin 2 particle with the external magnetic field is separated. PACS numbers: 02.30.Gp, 02.40.Ky, 03.65Ge, 04.62.+v Keywords: spin 2 particle, the non-relativistic approximation, external electromagnetic fields, projective operators DOI: https://doi.org/10.5281/zenodo.17237053 Introduction After the investigation by Pauli and Fierz [1], [2], the theory of massive and massless fields with spin 2 has always attracted much attention [3]–[27]. Several key aspects and challenges of this theory have been explored over the years. Most of the studies were performed in the framework of 2-nd order differential equations. It is known that many specific difficulties may be avoided if from the very beginning we start with 1st order systems. Apparently, the first systematic study of the theory of spin 2 fields within that formalism was performed by F.I. Fedorov [4]. It ∗E-mail: [email protected] †E-mail: [email protected]net.by ‡E-mail: [email protected] §E-mail: vasiliy-[email protected] turns out that this description requires a field function with 39 independent components. This theory was re-discovered by Regee in [5]. When studying the massless spin-2 field in a curved space-time, additional difficulties appear. For instance, unexpected constraints on space-time geometry arise to insure the gauge symmetry of the theory. In particular, the Ricci tensor Rαβ and the Riemann tensor Rαβρσ must vanish [15]. To resolve this, a non-minimal interaction term involving the Riemann tensor can be introduced into the basic equations [20], allowing the constraints to be reduced to Rαβ = 0. Another area of interest has been the problem of anomalous solutions in spin-2 theory [6, 7, 10]. A technical alternative for studying spin2 fields, both massive and massless, involves formulating first-order systems. This approach, based on the Gel’fand–Yaglom formalism [3], 220 Spin 2 Particle Theory, the Nonrelativistic Approximation 221 was first explored by Fedorov [4] and Regge [5]. Their papers demonstrated that a spin-2 particle requires a 39-component set of tensors for its description, it includes Φ,Φk,Φ(mn),Φ[mn]k. This formalism allows for exploration of new physical questions related to degrees of freedom. For instance, for the massless case, the 39-component matrix equation was solved in Minkowski space-time in [24], [25] using spherical and cylindrical coordinates. Six linearly independent solutions were found. By applying the Pauli–Fierz approach, adjusted to the tetrad formalism, the gauge solutions were constructed using exact solutions for the massless spin-1 field. This yielded four independent gauge solutions and two gauge-free solutions for the spin-2 field, as expected from physical reasoning. Additionally, F.I. Fedorov initiated a more general theory for the spin-2 particle based on a 50-component set of tensors. This theory, in the presence of external electromagnetic fields, describes a spin-2 particle with an anomalous magnetic moment4 see in [12, 13, 21–23]. One notable aspect of this theory is its allowance for a new massless limit for the spin-2 field [21]. This is particularly significant because the minimal Pauli–Fierz theory does not possess gauge symmetry in curved space-times with Rαβ = 0. However, the generalized theory exhibits gauge symmetry under these conditions. In the present paper, we will investigate the non-relativistic approximation in the basic 39component theory. Section 1 introduces the basic definitions and notations, including the structure of the 39component matrix equation. Explicit expressions for the four key matrices Γaof the equation, derived in [19], are assumed to be known. The system is formulated in Cartesian coordinates in the presence of external electromagnetic fields. The non-relativistic approximation is performed by distinguishing between large and small components of the wave function using three projective operators derived from the seventh-order minimal polynomial for the 39 ×39 matrix Γ0. The explicit structure of these components is determined; in Supplement A, we establish independent variables among large and small components. In section 2, we perform the procedure of the non-relativistic approximation. The basic point consists in decomposition of the components of the complete wave function in large and small constituents and splitting all 39 equations in equations of different orders of smallness. At this we should separate the rest energy. Besides, when performing the nonrelativistic approximation, we should assume the presence of terms of different smallness order; this permits in each equation to distinguish large and small terms. As the result, we derive the nonrelativistic equation for the 5-component wave function, in which the term describing interaction of the magnetic moment of the spin 2 particle with external magnetic field is separated. This additional term is constructed with the use of the spin matrices and the components of the magnetic field. 1. Basic equation for a spin 2 particle, projective operators We start with the matrix equation in Minkowski space [19, 24, 25] (ΓaDa−M)Ψ(x)=0,Γa=         0Ga0 0 1 2∆a0−1 3Ka0 0 Λa01 2Ba 0 0 Fa0         ,Ψ(x) =         Φ(x) Φl(x) Φ(mn)(x) Φ[mn]l(x)         , Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 222 A. V. Ivashkevich, A. V. Bury, E. M. Ovsiyuk, V. V. Kisel, and V. M. Red’kov where Da=∂a+ieAa; we use the block matrices G1,∆a, Ka, Ba, Fa(see in [19]); they dimensions are determined by the block columns in the complete field function; the field function consists of scalar, vector Φk(x), symmetric tensor Φ(mn)(x)and 3-rank tensor Φ[mn]l(x); it may be presented as the 39dimension column Ψ = {Φ; Φl;~ f, ~c, ~ d, f0;ϕ0, ϕ1, ϕ2, ϕ3}={H;H1;H2;H3}.(1.1) According to general theory, in order to perform the nonrelativistic approximation we should work with the matrix Γ0= Γ0= Γ (its blocks are given below; we indicate their dimensions) G1×4= +1000 ,∆4×1=         1 0 0 0         , K4×10 =         0000000001 0000001000 0000000100 0000000010         ,Λ10×4=                     1 2000 1 2000 1 2000 0000 0000 0000 0100 0010 0001 3 2000                     , B10×24 =                     . . . . . . 3/2. . . . . . −1/2. . . . . . −1/2... . . . . . . −1/2. . . . . . 3/2. . . . . . −1/2... . . . . . . −1/2. . . . . . −1/2. . . . . . 3/2... . . . . . . . . . . . . . . 1. . . . 1.... . . . . . . . . 1... . . ....1. .... . . . . . . . 1....1. .... . . .... 1. .... . . .... . . .... . . .... .1.... . . .... . . .... . . .... . . 1... . . .... . . .... . . .... . . . . . . 1/2. . . . . . 1/2. . . . . . 1/2...                     , Нелинейные явления в сложных системах Т. 28, № 3, 2025 Spin 2 Particle Theory, the Nonrelativistic Approximation 223 F24×10 =                                                  ...... 2/3... ...... . 2/3. . . . . . . . . . 2/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. . . . . . . . −1/3 .....1.... ....1..... . . . . . . . . . . . . . . . . . . −1/3. ...... . 1/3. . .....1.... .1. . . . . . . −1/3 ...1. . . . . . . . . . . . . . 1/3. . . . . . . . . . . ......−1/3... ....1..... ...1. . . . . . . . 1. . . . . . −1/3 ...... . −1/3. . ...... 1/3... . . . . . . . . . .                                                  . The matrix Γsatisfies the minimal equation (this is verified by direct calculation) Γ7−Γ5= 0; which permits us to introduce 3 projective operators P+=1 2Γ5(Γ + I) = P1, P−=1 2Γ5(Γ −I) = P2, P0=I−Γ6=P3(1.2) with the needed properties P2 i=Pi, P++P−+P0=I, i = 1,2,3.The components of the complete wave function are listed in accordance with the following notations (also see (1.1)) H3= Φ[mn]l=⇒             Φ[01]l Φ[02]l Φ[03]l Φ[23]l Φ[31]l Φ[12]l             =             E10 E20 E30 B10 B20 B30             =ϕ0,             E11 E21 E31 B11 B21 B31             =ϕ1,             E12 E22 E32 B12 B22 B32             =ϕ2,             E13 E23 E3k B13 B23 B33             =ϕ3. We can find explicit form of three projective operators. Acting by these operators on the complete wave functions, we obtain the structure of three projective constituents and their sum (Ψ++ Ψ−+ Ψ0)=Ψ(we introduce notations for the variables L... referring to large components; the notations S... and s... for variables referring to small components: In Supplement A, it is shown that among components there exist linear relations, they Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 224 A. V. Ivashkevich, A. V. Bury, E. M. Ovsiyuk, V. V. Kisel, and V. M. Red’kov permits us to present the complete wave functions in terms of only independent large and small components. 2. The non-relativistic procedure We should separate the rest energy by formal change (where Mis real valued and positive mass parameter)) D0=⇒(D0−iM). Besides, when performing the nonrelativistic approximation, we should assume the presence of terms of different smallness order): L∼1, S ∼x, s ∼x, 1 MDi∼x, D0 M∼x2, this permits in each of 39 equations distinguish large and small terms. They may be divided into three groups. The group Iconsists of the constraints among si: s2−s1= 0,(s1 2−s12 3)−s2= 0, −i1 4(−2s19 −S11 −S15) +3 4(s19 −S11 −S15)+ s2 2+i(s6+S11 +S15)=0, −i1 4(−2s19+S11)+3 4(s19+S11)+s2 2+i(s6−S11)=0, −i1 4(2s19+S15)+3 4(s19+S15)+s2 2+i(s6−S15)=0, S14 = 0, S13 = 0, S10 = 0, −i(1 4s19 +1 4s19 +1 4s19 +3s2 2) + is12 = 0, s16 = 0, s17 = 0, s18 = 0, −i(s6−s12 3+S11 +S15) + i(s19 −S11 −S15)=0, +iS10 +i(s20 +S10)=0, iS13 +i(s21 +S13) = 0, iS10 +i(−s20 +S10)=0, −i(s6−s12 3−S11) + i(s19 +S11)=0, iS14 +i(s27 +S14) = 0, iS13 +i(−s21 +S13)=0, iS14 +i(−s27 +S14)=0, −i(s6−s12 3−S15) + i(s19 +S15)=0, II permits to express small variables through the terms of the form DiLn: D1 M 1 3(−L11 −L15) + 1 3 D2 ML10 +1 3 D3 ML13 +is9 3+is3= 0, D2 M 1 3L11 +1 3 D1 ML10 +1 3 D3 ML14 +is10 3+is4= 0, D3 M 1 3L15 +1 3 D1 ML13 +1 3 D2 ML14 +is11 3+is5= 0, D2 2ML10 +D3 2ML13 +D1 2M(−L11 −L15) −i(s3+s13 2) + is9= 0, D1 2ML10 +D2 2ML11 +D3 2ML14 −i(s4+s14 2) + is10 = 0, D1 M 1 2L13 +D2 M 1 2L14 +D3 M 1 2L15 −i(s5+s15 2) + is11 = 0; D1 3M(−L11 −L15) + D2 3ML10 Нелинейные явления в сложных системах Т. 28, № 3, 2025 Spin 2 Particle Theory, the Nonrelativistic Approximation 225 +D3 3ML13 −2is9 3+is13 = 0; D2 3ML11 +D1 3ML10 +D3 3ML14 −2is10 3+is14 = 0, D3 3ML15 +D1 3ML13 +D2 3ML14 −2is11 3+is15 = 0, −D3 ML10 +D2 ML13 +is22 = 0, D3 M(1 3L15 −L11 −L15)−2 3MD1L13 +1 3MD2L14 +1 3is11 +is23 = 0, 1 3MD2(2L11 + 3L15) + 2 3MD1L10 −1 3MD3L14 −is10 3+is24 = 0, 1 3MD3(−3L11 −L15)−1 3MD1L13 +2 3MD2L14 −is11 3+is28 = 0, D3 ML10 −D1 ML14 +is29 = 0, D1 M(1 3(−L11 −L15) + L11) −2D2 3ML10 +D3 3ML13 +is9 3+is30 = 0, D2 M(1 3L11 +L15) + 1 3MD1L10 −2 3MD3L14 +1 3is10 +is34 = 0, D1 3M(L11 −2L15)−D2 3ML10 +D3 3ML13 −is9 3+is35 = 0, −D2 ML13 +D1 ML14 +is36 = 0, The group III contains the terms with the structure D0 M: D0 M(−L11 −L15) +D3 M(−s5 2+s15 4−3s23 4−s28 4) +D2 M(−s4 2+s14 4+3s24 4+s34 4) +D1 M(3s3 2+s13 4+s30 4−s35 4)=0, D0 ML11 +D3 M(−s5 2+s15 4+s23 4+3s28 4) +D2 M(3s4 2+s14 4−s24 4+s34 4) +D1 M(−s3 2+s13 4−3s30 4−s35 4)=0, D0 ML15 +D3 M(3s5 2+s15 4+s23 4−s28 4) +D2 M(−s4 2+s14 4−s24 4−3s34 4) +D1 M(−s3 2+s13 4+s30 4+3s35 4)=0, D0 M(1 2L14 +1 2L14) + D2 M(s5−s28 2) +D3 M(s4+s34 2) + D1 M(s29 2−s36 2)=0, D0 M(1 2L13 +1 2L13) + D1 M(s5+s23 2) +D3 M(s3−s35 2) + D2 M(s36 2−s22 2)=0, D0 M(1 2L10 +1 2L10) + D1 M(s4−s24 2) +D3 M(s22 2−s29 2) + D2 M(s3+s30 2)=0, D0 M(−L11 −L15)−2 3 D1 Ms9+D2 M s10 3+D3 M s11 3= 0, D0 ML10 −D2 Ms9= 0,24 D0 ML13 −D3 Ms9= 0, D0 ML10 −D1 Ms10 = 0, D0 ML11 +D1 M s9 3−D2 M 2s10 3+D3 M s11 3= 0, D0 ML14 −D3 Ms10 = 0,D0 ML13 −D1 Ms11 = 0, D0 ML14 −D2 Ms11 = 0, D0 ML15 +D1 M s9 3+D2 M s10 3−D3 M 2s11 3= 0. We will ignore all equations from the first groups because they are not needed to derive equations with the non-relativistic structure. First we resolve equations of the group II with respect to small variables: s3= 0, s4= 0, s5= 0, Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 226 A. V. Ivashkevich, A. V. Bury, E. M. Ovsiyuk, V. V. Kisel, and V. M. Red’kov s9=i(D2L10 +D3L13 −D1(L11 +L15)) M, s10 =i(D1L10 +D2L11 +D3L14) M, s11 =i(D1L13 +D2L14 +D3L15) M, s13 =i(D2L10 +D3L13 −D1(L11 +L15)) M, s14 =i(D1L10 +D2L11 +D3L14) M, s15 =i(D1L13 +D2L14 +D3L15) M, s22 =i(D2L13 −D3L10) M, s23 =−i(D1L13 +D3(L11 +L15)) M, s24 =i(D1L10 +D2(L11 +L15)) M, s28 =i(D2L14 −D3L11) M, s29 =i(D3L10 −D1L14) M, s30 =i(D1L11 −D2L10) M, s34 =i(D2L15 −D3L14) M, s35 =i(2D3L13 −3D1L15) 3M, s36 =i(D1L14 −D2L13) M. Then we substitute these expressions into equations from the group III; this leads to (we follow the order of the derivatives, and recollect the terms with respect to the large variables): iD2D1L10 M2+(i(D2D2+D3D3)−MD0)L11 M2 +i(12D3D1+D1D3)L13 12M2 +(i(D2D2+D3D3)−MD0)L15 M2= 0, iD1D2L10 M2+(MD0−i(D1D1+D3D3)) L11 M2 +iD1D3L13 12M2+iD3D2L14 M2= 0, 3iD1D3L13 4M2+iD2D3L14 M2 + + (MD0−i(D1D1+D2D2)) L15 M2= 0, iD1D3L10 2M2+iD2D3L11 2M2+iD1D2L13 2M2 +(2MD0−i(2D1D1+D2D2+D3D3)) L14 2M2 +iD3D2L15 2M2= 0, iD2D3L10 2M2−iD1D3L11 2M2 +(6MD0−i(3D1D1+ 6D2D2+ 2D3D3)) L13 6M2 +iD2D1L14 2M2+i(D3D1−D1D3)L15 2M2= 0, (2MD0−i(D1D1+D2D2+ 2D3D3)) L10 2M2 +i(D2D1−D1D2)L11 2M2 +iD3D2L13 2M2+iD3D1L14 2M2−iD1D2L15 2M2= 0, i(D2D1−2D1D2)L10 3M2 +(i(2D1D1+D2D2)−3MD0)L11 3M2 +i(D3D1−2D1D3)L13 3M2 +i(D3D2+D2D3)L14 3M2 +(i(2D1D1+D3D3)−3MD0)L15 3M2= 0, (MD0−iD2D2)L10 M2+iD2D1L11 M2 −iD2D3L13 M2+iD2D1L15 M2= 0, −iD3D2L10 M2+iD3D1L11 M2 +(MD0−iD3D3)L13 M2+iD3D1L15 M2= 0, (MD0−iD1D1)L10 M2−iD1D2L11 M2−iD1D3L14 M2= 0, i(D1D2−2D2D1)L10 3M2 +(3MD0−i(D1D1+ 2D2D2)) L11 3M2 +i(D3D1+D1D3)L13 3M2 +i(D3D2−2D2D3)L14 3M2 Нелинейные явления в сложных системах Т. 28, № 3, 2025 Spin 2 Particle Theory, the Nonrelativistic Approximation 227 −i(D1D1−D3D3)L15 3M2= 0, −iD3D1L10 M2−iD3D2L11 M2+(MD0−iD3D3)L14 M2= 0, (MD0−iD1D1)L13 M2 −iD1D2L14 M2−iD1D3L15 M2= 0, −iD2D1L13 M2+(MD0−iD2D2)L14 M2−iD2D3L15 M2= 0, i(D2D1+D1D2)L10 3M2 −i(D1D1−D2D2)L11 3M2+i(D1D3−2D3D1)L13 3M2 +i(D2D3−2D3D2)L14 3M2 +(3MD0−i(D1D1+ 2D3D3)) L15 3M2= 0. Let us separate 15 equations with the nonrelativistic structure (it is convenient to numerate them)): with the use of more short notations L10 =L1, L11 =L2, L13 =L3, L14 =L4, L15 =L5,(2.1) 1) −2D2D1L1M−2D2 2L2M−2D2 3L2M−2D3D1L3M −1 6D1D3L3M−2D2 2L5M−2D2 3L5M−2iD0L2M2−2iD0L5M2= 0, 2) −2D1D2L1M+ 2D2 1L2M+ 2D2 3L2M−1 6D1D3L3M−2D3D2L4M+ 2iD0L2M2= 0, 3) −3 2D1D3L3M−2D2D3L4M+ 2D2 1L5M+ 2D2 2L5M+ 2iD0L5M2= 0, 4) −D1D3L1M−D2D3L2M−D1D2L3M+ 2D2 1L4M+D2 2L4M+D2 3L4M−D3D2L5M+ 2iD0L4M2= 0, 5) −D2D3L1M+D1D3L2M+D2 1L3M+ 2D2 2L3M +2 3D2 3L3M−D2D1L4M−D3D1L5M+D1D3L5M+ 2iD0L3M2= 0, 6) D2 1L1M+D2 2L1M+ 2D2 3L1M−D2D1L2M+D1D2L2M−D3D2L3M −D3D1L4M+D1D2L5M+ 2iD0L1M2= 0, 7) −2 3D2D1L1M+4 3D1D2L1M−4 3D2 1L2M−2 3D2 2L2M−2 3D3D1L3M+4 3D1D3L3M−2 3D3D2L4M −2 3D2D3L4M−4 3D2 1L5M−2 3D2 3L5M−2iD0L2M2−2iD0L5M2= 0, 8) 2D2 2L1M−2D2D1L2M+ 2D2D3L3M−2D2D1L5M+ 2iD0L1M2= 0, 9) 2D3D2L1M−2D3D1L2M+ 2D2 3L3M−2D3D1L5M+ 2iD0L3M2= 0, 10) 2D2 1L1M+ 2D1D2L2M+ 2D1D3L4M+ 2iD0L1M2= 0, 11) (D0M−1 3i(D2 1+ 2D2 2))L2+1 3i(D1D2−2D2D1)L1 +1 3i(D3D1+D1D3)L3+1 3i(D3D2−2D2D3)L4−1 3i(D2 1−D2 3)L5= 0, 11) 4 3D2D1L1M−2 3D1D2L1M+2 3D2 1L2M+4 3D2 2L2M−2 3D3D1L3M−2 3D1D3L3M Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 228 A. V. Ivashkevich, A. V. Bury, E. M. Ovsiyuk, V. V. Kisel, and V. M. Red’kov −2 3D3D2L4M+4 3D2D3L4M+2 3D2 1L5M−2 3D2 3L5M+ 2iD0L2M2= 0, 12) 2D3D1L1M+ 2D3D2L2M+ 2D2 3L4M+ 2iD0L4M2= 0, 13) 2D2 1L3M+ 2D1D2L4M+ 2D1D3L5M+ 2iD0L3M2= 0, 14) 2D2D1L3M+ 2D2 2L4M+ 2D2D3L5M+ 2iD0L4M2= 0, 15) −2 3D2D1L1M−2 3D1D2L1M+2 3D2 1L2M−2 3D2 2L2M+4 3D3D1L3M−2 3D1D3L3M +4 3D3D2L4M−2 3D2D3L4M+2 3D2 1L5M+4 3D2 3L5M+ 2iD0L5M2= 0. Further, let us take into account the following identities (note that Fij =∂iAj−∂jAi) DiDj=1 2(DiDj+DjDi) + 1 2(DiDj−DjDi) = 1 2(DiDj+DjDi) + ieFij ≡D(ij)+ieFij;(2.2) so in all equations should arise only the following terms 2iMD0Lk, D(11) =D2 1Lk, D(22) =D2 2Lk, D(33) =D2 3Lk, D(23)Lk, D(32)Lk, D(12)Lk, F23Lk, F31Lk, F12Lk.(2.3) In this way, we obtain (below we will omit parentheses in D(ij)) 1) −2iD0L2M2−2iD0L5M2−2D12L1M−2D22L2M−2D33L2M −13 6D31L3M−2D22L5M−2D33L5M+ 2ieF12L1M−11 6ieF31L3M= 0, 2) 2iD0L2M2−2D12L1M+ 2D11L2M+ 2D33L2M−1 6D31L3M−2D23L4M−2ieF12L1M +1 6ieF31L3M+ 2ieF23L4M= 0, 3) 2iD0L5M2−3 2D31L3M−2D23L4M+ 2D11L5M+ 2D22L5M+3 2ieF31L3M−2ieF23L4M= 0, 4) 2iD0L4M2−D31L1M−D23L2M−D12L3M+ 2D11L4M+D22L4M+D33L4M−D23L5M +ieF31L1M−ieF23L2M−ieF12L3M+ieF23L5M= 0, 5) 2iD0L3M2−D23L1M+D31L2M+D11L3M+ 2D22L3M+2 3D33L3M−D12L4M −ieF23L1M−ieF31L2M+ieF12L4M−2ieF31L5M= 0, 6) 2iD0L1M2+D11L1M+D22L1M+ 2D33L1M−D23L3M−D31L4M+D12L5M +2ieF12L2M+ieF23L3M−ieF31L4M+ieF12L5M= 0, 7) −2iD0L2M2−2iD0L5M2+2 3D12L1M−4 3D11L2M−2 3D22L2M+2 3D31L3M−4 3D23L4M −4 3D11L5M−2 3D33L5M+ 2ieF12L1M−2ieF31L3M= 0, 8) 2iD0L1M2+ 2D22L1M−2D12L2M+ 2D23L3M−2D12L5M +2ieF12L2M+ 2ieF23L3M+ 2ieF12L5M= 0, 9) 2iD0L3M2+2D23L1M−2D31L2M+2D33L3M−2D31L5M−2ieF23L1M−2ieF31L2M−2ieF31L5M= 0, Нелинейные явления в сложных системах Т. 28, № 3, 2025 Spin 2 Particle Theory, the Nonrelativistic Approximation 235 Appendix A. Independent large and small variables Ψ+=                                                                               0 0 0 0 0 1 6(2E11 −E22 −E33 + 2f1−f2−f3) 1 6(−E11 + 2E22 −E33 −f1+ 2f2−f3) 1 6(−E11 −E22 + 2E33 −f1−f2+ 2f3) 1 4(2c1+E23 +E32) 1 4(2c2+E13 +E31) 1 4(2c3+E12 +E21) 0 0 0 0 0 0 0 0 0 0 1 6(2E11 −E22 −E33 + 2f1−f2−f3) 1 4(2c3+E12 +E21) 1 4(2c2+E13 +E31) 0 0 0 1 4(2c3+E12 +E21) 1 6(−E11 + 2E22 −E33 −f1+ 2f2−f3) 1 4(2c1+E23 +E32) 0 0 0 1 4(2c2+E13 +E31) 1 4(2c1+E23 +E32) 1 6(−E11 −E22 + 2E33 −f1−f2+ 2f3) 0 0 0                                                                               =                                                                               0 0 0 0 0 L1 L2 L3 L4 L5 L6 0 0 0 0 0 0 0 0 0 0 L7 L8 L9 0 0 0 L10 L11 L12 0 0 0 L13 L14 L15 0 0 0                                                                               , Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 236 A. V. Ivashkevich, A. V. Bury, E. M. Ovsiyuk, V. V. Kisel, and V. M. Red’kov Ψ−=                                                                               0 0 0 0 0 1 6(−2E11 +E22 +E33 + 2f1−f2−f3) 1 6(E11 −2E22 +E33 −f1+ 2f2−f3) 1 6(E11 +E22 −2E33 −f1−f2+ 2f3) 1 4(2c1−E23 −E32) 1 4(2c2−E13 −E31) 1 4(2c3−E12 −E21) 0 0 0 0 0 0 0 0 0 0 1 6(2E11 −E22 −E33 −2f1+f2+f3) 1 4(−2c3+E12 +E21) 1 4(−2c2+E13 +E31) 0 0 0 1 4(−2c3+E12 +E21) 1 6(−E11 + 2E22 −E33 +f1−2f2+f3) 1 4(−2c1+E23 +E32) 0 0 0 1 4(−2c2+E13 +E31) 1 4(−2c1+E23 +E32) 1 6(−E11 −E22 + 2E33 +f1+f2−2f3) 0 0 0                                                                               =                                                                               0 0 0 0 0 S1 S2 S3 S4 S5 S6 0 0 0 0 0 0 0 0 0 0 S7 S8 S9 0 0 0 S10 S11 S12 0 0 0 S13 S14 S15 0 0 0                                                                               , Нелинейные явления в сложных системах Т. 28, № 3, 2025 Spin 2 Particle Theory, the Nonrelativistic Approximation 237 Ψ0=P3Ψ =                                                                               Φ Φ0 Φ1 Φ2 Φ3 1 3(f1+f2+f3) 1 3(f1+f2+f3) 1 3(f1+f2+f3) 0 0 0 d1 d2 d3 f0 E10 E20 E30 B10 B20 B30 1 3(E11 +E22 +E33) 1 2(E21 −E12) 1 2(E31 −E13) B11 B21 B31 1 2(E12 −E21) 1 3(E11 +E22 +E33) 1 2(E32 −E23) B12 B22 B32 1 2(E13 −E31) 1 2(E23 −E32) 1 3(E11 +E22 +E33) B13 B23 B33                                                                               =                                                                               s1 s2 s3 s4 s5 s6 s7 s8 0 0 0 s9 s10 s11 s12 s13 s14 s15 s16 s17 s18 s19 s20 s21 s22 s23 s24 s25 s26 s27 s28 s29 s30 s31 s32 s33 s34 s35 s36                                                                               ,Ψ =                                                                               s1 s2 s3 s4 s5 −L11 −L15 +S11 +S15 +s6 L11 −S11 +s6 L15 −S15 +s6 L14 −S14 L13 −S13 L10 −S10 s9 s10 s11 s12 s13 s14 s15 s16 s17 s18 −L11 −L15 −S11 −S15 +s19 L10 +S10 +s20 L13 +S13 +s21 s22 s23 s24 L10 +S10 −s20 L11 +S11 +s19 L14 +S14 +s27 s28 s29 s30 L13 +S13 −s21 L14 +S14 −s27 L15 +S15 +s19 s34 s35 s36                                                                               . It is convenient to write down expressions for separate components: Ψ+, L1=1 6(2E11 −E22 −E33 + 2f1−f2−f3), L2=1 6(−E11 + 2E22 −E33 −f1+ 2f2−f3), L3=1 6(−E11 −E22 + 2E33 −f1−f2+ 2f3), L4=1 4(2c1+E23 +E32), L5=1 4(2c2+E13 +E31), L6=1 4(2c3+E12 +E21), L7=1 6(2E11 −E22 −E33 + 2f1−f2−f3), L8=1 4(2c3+E12 +E21), Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 238 A. V. Ivashkevich, A. V. Bury, E. M. Ovsiyuk, V. V. Kisel, and V. M. Red’kov L9=1 4(2c2+E13 +E31), L10 =1 4(2c3+E12 +E21), L11 =1 6(−E11 + 2E22 −E33 −f1+ 2f2−f3), L12 =1 4(2c1+E23 +E32), L13 =1 4(2c2+E13 +E31), L14 =1 4(2c1+E23 +E32), L15 =1 6(−E11 −E22 + 2E33 −f1−f2+ 2f3) ; Ψ−, S1=1 6(−2E11 +E22 +E33 + 2f1−f2−f3), S2=1 6(E11 −2E22 +E33 −f1+ 2f2−f3), S3=1 6(E11 +E22 −2E33 −f1−f2+ 2f3), S4=1 4(2c1−E23 −E32), S5=1 4(2c2−E13 −E31), S6=1 4(2c3−E12 −E21), S7=1 6(2E11 −E22 −E33 −2f1+f2+f3), S8=1 4(−2c3+E12 +E21), S9=1 4(−2c2+E13 +E31), S10 =1 4(−2c3+E12 +E21), S11 =1 6(−E11 + 2E22 −E33 +f1−2f2+f3), S12 =1 4(−2c1+E23 +E32), S13 =1 4(−2c2+E13 +E31), S14 =1 4(−2c1+E23 +E32), S15 =1 6(−E11 −E22 + 2E33 +f1+f2−2f3) ; Ψ0, s1= Φ, s2= Φ0, s3= Φ1, s4= Φ2, s5= Φ3, s6=1 3(f1+f2+f3), s7=1 3(f1+f2+f3), s8=1 3(f1+f2+f3), s9=d1, s10 =d2, s11 =d3, s12 =f0, s13 =E10, s14 =E20, s15 =E30, s16 =B10, s17 =B20, s18 =B30, s19 =1 3(E11 +E22 +E33), s20 =1 2(E21 −E12), s21 =1 2(E31 −E13), s22 =B11, s23 =B21, s24 =B31, s25 =1 2(E12 −E21), s26 =1 3(E11 +E22 +E33), s27 =1 2(E32 −E23), s28 =B12, s29 =B22, s30 =B32, s31 =1 2(E13 −E31), s32 =1 2(E23 −E32), s33 =1 3(E11 +E22 +E33), s34 =B13, s35 =B23, s36 =B33. We can find independent variables in all three sets. First consider the large components Li L1=1 6(2E11 −E22 −E33 + 2f1−f2−f3), L2=1 6(−E11 + 2E22 −E33 −f1+ 2f2−f3), L3=1 6(−E11 −E22 + 2E33 −f1−f2+ 2f3), L4=1 4(2c1+E23 +E32), Нелинейные явления в сложных системах Т. 28, № 3, 2025 Spin 2 Particle Theory, the Nonrelativistic Approximation 239 L5=1 4(2c2+E13 +E31), L6=1 4(2c3+E12 +E21), L7=1 6(2E11 −E22 −E33 + 2f1−f2−f3), L8=1 4(2c3+E12 +E21), L9=1 4(2c2+E13 +E31), L10 =1 4(2c3+E12 +E21), L11 =1 6(−E11 + 2E22 −E33 −f1+ 2f2−f3), L12 =1 4(2c1+E23 +E32), L13 =1 4(2c2+E13 +E31), L14 =1 4(2c1+E23 +E32), L15 =1 6(−E11 −E22 + 2E33 −f1−f2+ 2f3). Making up the matrix of the system, we find its rank, it turns out to be equal to 5. Eliminating the rows 1,...,9 and 12, we get the matrix with the same rank; the variables L10, L11, L13, L14, L15 can be taken as independent: let us use the notations L10 =B1, L11 =B2, L13 =B3, L14 =B4, L15 =B5;then we get L1=L7=−L11 −L15 =−B2−B5, L2=L11 =B2, L3=L15 =B5, L4=L14 =B4, L5=L9=L13 =B3, L6=L8=L10 =B1, L12 =L4=L14 =B4. Similarly, for small components Siwe get S1=1 6(−2E11 +E22 +E33 + 2f1−f2−f3), S2=1 6(E11 −2E22 +E33 −f1+ 2f2−f3), S3=1 6(E11 +E22 −2E33 −f1−f2+ 2f3), S4=1 4(2c1−E23 −E32), S5=1 4(2c2−E13 −E31), S6=1 4(2c3−E12 −E21), S7=1 6(2E11 −E22 −E33 −2f1+f2+f3), S8=1 4(−2c3+E12 +E21), S9=1 4(−2c2+E13 +E31), S10 =1 4(−2c3+E12 +E21), S12 =1 4(−2c1+E23 +E32), S11 =1 6(−E11 + 2E22 −E33 +f1−2f2+f3), S13 =1 4(−2c2+E13 +E31), S14 =1 4(−2c1+E23 +E32), S15 =1 6(−E11 −E22 + 2E33 +f1+f2−2f3). We make up the matrix of the system, its rank equals 5. Eliminating the rows 1− −9,12, we get the matrix with the same rank. The variables S10, S11, S13, S14, S15 are taken as independent. We find expressions for remaining components through independent ones S1=S11 +S15, S2=−S11, S3=−S15, S4=−S14, S5=−S13, S6=−S10, S7=−S1=−S11 −S15, S8=−S6=S10, S9=−S5=S13, S12 =−S4=S14. Now consider the small components si: s1= Φ, s2= Φ0, s3= Φ1, s4= Φ2, s5= Φ3, Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 240 A. V. Ivashkevich, A. V. Bury, E. M. Ovsiyuk, V. V. Kisel, and V. M. Red’kov s6=1 3(f1+f2+f3), s7=1 3(f1+f2+f3), s8=1 3(f1+f2+f3), s9=d1, s10 =d2, s11 =d3, s12 =f0, s13 =E10, s14 =E20, s15 =E30, s16 =B10, s17 =B20, s18 =B30, s19 =1 3(E11 +E22 +E33), s20 =1 2(E21 −E12), s21 =1 2(E31 −E13), s22 =B11, s23 =B21, s24 =B31, s25 =1 2(E12 −E21), s26 =1 3(E11 +E22 +E33), s27 =1 2(E32 −E23), s28 =B12, s29 =B22, s30 =B32, s31 =1 2(E13 −E31), s32 =1 2(E23 −E32), s33 =1 3(E11 +E22 +E33), s34 =B13, s35 =B23, s36 =B33. here we have 29 independent variables; besides there exist relations s6=s7=s8, s19 =s26 =s33, s20 =−s25, s21 =−s31, s27 =−s32. Let us collect all constraints together: Ψ+, L10, L11, L13, L14, L15 L1=−L11 −L15, L2=L11, L3=L15, L4=L14, L5=L13, L6=L10, L7=−L11 −L15, L8=L10, L9=L13, L12 =L14; Ψ−, S10, S11, S13, S14, S15 S1=S11 +S15, S2=−S11, S3=−S15, S4=−S14, S5=−S13, S6=−S10, S7=−S11 −S15, S8=S10, S9=S13, S12 =S14; Ψ0, s1, s2, s3, s4, s5, s6, s9, s10, s11, s12, s13, s14, s15, s16, s17, s18, s19, s20, s21, s22, s23, s24, s27, s28, s29, s30, s34, s35, s36, s7=s6, s8=s6, s25 =−s20, s26 =s19, s31 =−s21, s32 =−s27, s33 =s19. References [1] W. Pauli, M. Fierz, ¨ Uber relativistische Feldleichungen von Teilchen mit beliebigem Spin im elektromagnetishen Feld, Helv. Phys. Acta, 12, 297-300 (1939). [2] M. Fierz, W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. London, A173, 211-232 1939). [3] I.M. Gel’fand, A.M. Yaglom, General relativistically invariant equations and infinitedimensional representations of the Lorentz group, Journal of Experimental and Theoretical Physics, 18(8), 703-733 (1948). [4] F.I. Fedorov, To the theory of particles with spin Нелинейные явления в сложных системах Т. 28, № 3, 2025 Spin 2 Particle Theory, the Nonrelativistic Approximation 241 2, Proceedings of Belarus State University, Ser. Phys.-Math, 12, 156-173 (1951). [5] T. Regge, On properties of the particle with spin 2. Nuovo Cimento, 5(2), 325-326 (1957). [6] G. Velo, D. Zwanziger, Noncausality and other defects of interaction Lagrangians for particles with spin one and higher, Phys. Rev., 188(5), 2218–2222 (1969). [7] G. Velo, Anomalous behavior of a massive spin two charged particle in an external electromagnetic field, Nucl. Phys., B43, 389-401 (1972). [8] C.R. Hagen, Minimal electromagnetic coupling of spin-two fields, Phys. Rev., D6(4), 984-987 (1972). [9] J.F. Reilly, Minimal coupling in spin-2 field, Nucl. Phys., B1, 356-364 (1974.). [10] P.M. Mathews, M. Seetharaman, J. Prabhakaran, Inconsistencies in the symmetric tensor field description of spin-2 particles in an external homogeneous magnetic field, Phys. Rev., D14, 1021-1031(1976). [11] W. Cox, First-order formulation of massive spin2 field theories, J. Phys., A15, 253-268 (1982). [12] V.V. Kisel, On relativistic wave equations for a spin 2 particle, Proceedings of the National Academy of Sciences of Belarus, Physics and Mathematics Series, 5, 94-99 (1986). [13] A.A. Bogush, V.V. Kisel, On description of anomalous magnetic moment of a massive particle with spin 2 in the theory of relativistic wave equations, Izvestia Vuzov. SSSR. Fizika, 31(3), 11-16 (1988). [14] I.L. Buchbinder, D.M. Gitman, V.A. Krykhtin, V.D. Pershin, Equations of motion for massive spin 2 field coupled to gravity, Nucl. Phys., B584, 615-640 (2000). [15] V.M. Red’kov, N.G. Tokarevskaya, V.V. Kisel, Graviton in a curved space-time background and gauge symmetry, Nonlinear Phenomena in Complex Systems, 6(3), 772-778 (2003). [16] A.A. Bogush, V.V. Kisel, N.G. Tokarevskaya, V.M. Red’kov, On equations for a spin 2 particle in external electromagnetic and gravitational fields, Proceedings of the National Academy of Sciences of Belarus, Physics and Mathematics Series, 1, 62-67 (2003). [17] L. Bernard, C. Deffayet, A. Schmidt-May and M. von Strauss, Linear spin-2 fields in most general backgrounds, Phys. Rev., D93. 084020 (2016). [18] Charles Mazuet,a Mikhail S. Volkov, Massive spin-2 field in arbitrary spacetimes, the detailed derivation, Phys. Rev., D96, 124023 (2017). [19] A. Ivashkevich, A. Buryy, E. Ovsiyuk, V. Balan, V. Kisel, V. Red’kov, On the matrix equation for a spin 2 particle in pseudo-Riemannian spacetime, tetrad method, Proceedings of Balkan Society of Geometers, 28, 1-23 (2021). [20] V.M. Red’kov, N.G. Tokarevskaya, V.V. Kisel, Graviton in a curved space-time background and gauge symmetry, Nonlinear Phenomena in Complex Systems, 6(3), 772-778 (2003). [21] I.G. Dudko, O.A. Semenyuk, V.V Kisel, V.M. Red’kov, Spin 2 particle with anomalous magnetic moment in Riemann space-time, restriction to massless case, gauge symmetry, Nonlinear Phenomena in Complex Systems, 25(3), 286-296 (2022). [22] A.V. Ivashkevich, A.V. Bury, V.M. Red’kov, V.V. Kisel, On new form of the 50-component theory for spin 2 particle with anomalous magnetic moment in the basis of tensors of 2nd and 3-rd ranks, Nonlinear Dynamics and Applications, 29, 289-330 (2023). [23] A.V. Ivashkevich, V.M. Red’kov, A.M.Ishkhanyan, Massless spin 2 field in 50-component approach, exact solutions with cylindrical symmetry, eliminating the gauge degrees of freedom, Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 60(2), 132-145 (2024). [24] A.V. Ivashkevich, A.V. Chichurin, A.M. Ishkhanyan, V.M. Red’kov, Massless spin 2 field solutions with spherical symmetry: eliminating gauge degrees of freedom, Modern Physics Letters, A39, Paper 2450065 [22 pages] (2024). [25] A.V. Ivashkevich, A.V. Bury, V.M. Red’kov, E.M. Ovsiyk, V.V. Kisel, Solutions with cylindrical symmetry for graviton field in linear approximation, the gauge degrees of freedom, Nonlinear Phenomena in Complex Systems, 27(3), 256-277 2024. [26] A.V. Ivashkevich, V.M. Red’kov, A.M. Ishkhanyan, Nonrelativistic approximation in the theory of a spin 2 particle with anomalous magnetic moment Axioms, 14(1), 35 (2025). [27] Ivashkevich, Alina. A. Ivashkevich, V. Red’kov, A. Ishkhanyan, Spin-2 particle in Coulomb field: non-relativistic approximation, Symmetry, 17(7), 1075 [29 pages] (2025). Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025