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Two-Variable Relativistic Tensor Harmonics

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Two-Variable Relativistic Tensor Harmonics ^ сообщения объединенного института ядерных исследований дубна Е2 - 12936 M.Gmitro, A.A.Ovchinnikova TWO-VARIABLE RELATIVISTIC TENSOR HARMONICS 1979 ч л i Е2 - 12936 M.Gmitro, A.A.Ovchinnikova TWO-VARIABLE RELATIVISTIC TENSOR HARMONICS F2 - !2Q4h Гмитро М., Овчинникова А.А. '** *****v Релятивистские тензорные гармоники в случае двух переменных Рассмотрено обобщение релятивистских сферических тензоров на слу­ чай двух переменных. В сферическом и спиральном базисах а системе от­ счета Брейта построены лоренцесские коварианты, зависящие от трех пе­ ременных. Изучены ид свойства: четность, ортонормальность на сфере, вложенной в трехмерное Евклидово пространство. Получено выражение для скалярного произведения в четырехмерном пространстве и условие незави симости релятивистских тензорных гармоник. Переход к нерелятивистским коеариантам является очевидным. Построенные релятивистские тензоры могут быть полезны при изучении реакций с числом частиц в конечном состоянии больше деух, Работа выполнена в Лаборатории теоретической физики ОИЯИ. Сообщение Объединенного института ядерных исследований, Дубна 1979 F2 • 12Q46 Gmitro «., Ovchi,:r.ikova A.A. *** '*™« Two-Variable Relatrvistic Tensor Harmonics Three bases in the Hilbert space of tensor fields on the unit spheres associated with two independent vectors are discussed: the tensor spherical harmonics and the symmetric and unsymmetric tensor helicity harmonics. Under the conditions which we specify they form complete sets of independent Lorentz covariants which may serve the purpose of the analysis of reactions with several particles in the final state. The investigation has been performed at the Laboratory of Theoretical Physics, JINR. Communication of the Joint institute for Nuclear Research. Dubno 1979 ) 1979 Объединенный институт ядерных исследований Дубна 1. Introduction Recently, a new interest in the intermediate-energy (=к10 MeV) lepton- and meson-induced processes on nucleons and nuclei has arisen in connection with the increasing flow of the high- quality data coming from the SIN, TRIUMF and LAMPF meson faci­ lities. The Lorentz covariants written in the Cartesian basis have been the traditional theoretical tool to analyse such reac­ tions. Though manifestly covariant, they frequently do not fit the purposes of the physical investigations, e.g.,already the basic problem of selecting the independent covariants may prove to be practically insoluble in many cases. It is our experience that the tensor harmonics in the sphe­ rical and helicity bases constitute a convenient, highly flexible framework for the construction of the Lorentz covariants: The most important mathematical properties of the tensor harmonics follow directly from the well-known formulas of the angular-mo­ mentum algebra, the construction of the independent sets of co- variants is straightforward. Orthogonality properties of the tensor bases make the calculations of rates and other physical quantities much easier than with the cumbersome Cartesian tech- 3 niques. Besidea these technical advantages two gratifying proper­ ties of the new formalism constitute its main merit and should be mentioned. First, the relativistic tensor harmonics allow a natu­ ral unification of the theoretical treatment of a big class of dif­ ferent physical processes. Second, the formalism, though fully equivalent to the covariant Cartesian expressions is actually very much similar to the familiar nonrelativistlc multipole-expan- aion formulae. Therefore, the physical results may always be easi­ ly interpreted by a direct extrapolation to the domain of classi­ cal nuclear physics. Using the tensor harmonics we need not to perform any "nonrelativistic reduction" which is normally done, e.g., via the Foldy-Wouthuysen transformation. Namely this is the step which frequently makes the treatment of physical processes unwieldy and brings in the approximations which are usually dif­ ficult to control. The trick hare is indeed in choosing the ap­ propriate referenoe frame.It is the Breit frame which being fully appropriate physically, gives simultaneously an enormous ampli­ fication of the formulas. The formalism of the relativistic s-th order tensor spheri­ cal harmonica has been presented recently by Daumena and llinnaert . For the corresponding analysis performed in the helicity ba- /2/ sis we refer the reader to the paper by Akyeampong ' '. As a matter of fact the method was first introduced by Stech and Schu- lke who have considered, however, only the specific case of nuclear beta-decay. Recently, Delorme ' ' presented the applica­ tion of the relativistic spherical tensor harmonics in the context of the so-called elementary-particle theory of nuclear currents. The treatment in Refe. is always limited to the one-variable harmonica which correspond to the case of binary reactions. Here we shall present our results concerning the two-vari­ able Lorentz-covariant tensor harmonics in the spherical and he­ licity bases. It will be shown below that they provide actually the most general description of the multi-variable tensor fields, which may be needed in the analysis of any reaction of the type atb—• 1»г«- •• и. In Sec. 2 we define the spherical tetrads and build up the second order tensor spherical basis. Section 3 is devoted to the (scalar) spherical harmonics in two variables. There we display the reduction formula which permits an еаву elimination of those 4 harmonics which can be expressed as linear combinations (with sca­ lar coefficients) of the harmonlcB which form the basic set. In Sec. 4 the second order tensor spherical harmonics in two vari­ ables are introduced and their most important properties are listed. In Sec, 5 two different forms of tensor harmonics in the helieity bases are deduced from the two-variable tensor spherical harmonics constructed in the preceding section. Finally, in Sec. 6 we indi­ cate, using a particular example of the reaction with three par­ ticles in the final state, how the formalism of the present paper may be applied and indicate some of its merits in comparison with the Cartesian expressions. 2. Tensor Spherical Basis First we have to introduce a set of orthogonal 4-vectors on which to define the projections of the tensor fields. Following Daumens and Hinnaert ' ' we choose three враее-like vectors e" (n = *1,0) on the unit sphere S (e) embedded in the sub- space £ fe) orthogonal to the time-like vector e~ =S^ . The complex vectors e"1 satisfy the following conditions f ^r = (-'r\ ; , (1, п е; (ерГ^гЛп.. (2) We иве the Paull metrics (i.e., a^ =a6^ а"& - Q0bo ) and the usual summation convention for repeated Greek indices ( ix - 0,1,2,3). Note that three vectors e" ( *) = *1,0) form the usual 3-dimensional spherical basis. The spherical components of an arbitrary vector Q„ in the basis Just introduced are given from the decomposition CL-Ia*"e£* + aMe"* . (3) This means 00 _ JX> _'»• _ •='•e " (4) а"- «ле™, а™ = °л r • The construction of the tensor spherical bases of an arbit­ rary order has bnen discussed in detail by Daumens and Hinnaert '' . We shall deal with the 2nd order tensor basis 5 п,л2 only, where the symbol 1---J denotes the Clebsch-Gordan coeffi­ cient. The parity and orthogonality properties of basis (5) read as followsj M^.M'-"'^, (6) ^"•'Г'^'ЧЛЛД". (7) 3. Spherical Harmonics in Two Variables The relativietic spherical harmonica may be introduced by taking the projections of an arbitrary unit 4-vector U^ on the basis #1" • The constructions aa performed,e.g.,in Ref. holds for the purely space-like vectors (i.e. ue =0 ) only. This condition, however, is not too restrictive for the applica­ tions, since it can always be met if one works in the rest (Breit) system which is fully appropriate in major phyaical situations. Indeed, taking the definition of the Breit system аа Q^-(0,iQ0) l and choosing the spherical basis in such a way that e" *Qh/j-Q > we may always instead of a^ consider the vector which is orthogonal to е., : and then define the unit 4-vector U^ = a f\f £г , in this way since the time component of U^ vanish, the spherical har­ monics defined in (8) on S (e) become identical with /5/ the usual spherical harmonics as defined, e.g., in Ref. In the simplest oase t = / we have 6 Proceeding to the case of two variables U, and v we construct as usual the objects, which transform according to the irreducible representations of the rotation group! They form a complete orthogonal basis. We would like to keep only independent terms of this infinite-dimensional basie. By indepen­ dent we mean such terms which cannot be expressed through the re­ maining onee as linear combinations with scalar (i.e., depending on U , V<L , and uv and their powers) coefficients. To separate the independent harmonics we apply the identity (13) l J цг '< '• which represents actually the relation between the two coupling schemes of the four momenta e,t, s, and t' re-written using the expansion Y„>)Y,'(u)=I«6U) OYi», ' „ '•-< (14) where ц=/2о+7, and (••;( stands for the well-known 93 sym­ bol. Using (13), any harmonies |(',4') *».| with 4t4>l*i can be expressed through the harmonics J(f,^)''n} with P, • (2 < f, <• ^ , It is easy to see that this proaeas can be repeated as many times as necessary in order to reach the harmonics {«,«,)?».} with t,*ti*t, (15a) 1 ( Ы <>>* - (,5Ь) The construction just desoribed leaves us with 11<-4 orthogonal terms of the form (12), which are independent in the above sense. They cannot be further reduced in view of the triangular condi­ tion Т,*Тг -Г. 7 Concerning the case of the spherical harmonica in v vari­ ables it should be noted, that further generalization of (12) is actually not needed,In physical applications which we have in mind we always deal with the 4-dimensional Minkowski space. In this space there are only four independent vec< re, that is the time-like vector Qr = (°,-4i) and three space-like vectors, e.g., u v Anv Uf.
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