The Kustaanheimo-Stiefel Transformation in a Spinor Representation

Transcription

1 Revista Brasileira de Fkica, Vol. 17, n? 2, 1987 The Kustaanheimo-Stiefel Transformation in a Spinor Representation J. BELLANDI FILHO and M.L.T. MENON Instituto de Fkíca, Universidade Estadual de Campinas, Caixa Postal 6165, Campinas, 13083, Sf, Brasil Recebido em 29 de agosto de 1986 Abstract A spinor representation for the K-S transformation is derived by means of the Cartan spinor theory. INTRODUCTION It is well known that a regularization of the Kepler motion in the tridimensional space R3 is developed by using a simple mapping of a four-dimensional space R4 onto R ~ This. simple mapping is known Kustaanheimo-Stiefel (K-S) transformation'. as the In R4, the equations of the Kepler motion are linear differential equations wi th constant coef- ficients and remain completely regular at the center of attract ion. These are equations of simple harmonic oscillator motions. The K-S transformation is not a complete generalization of the two-dimensional ~evi-civita2 transformation, but only a mapping of R' onto R3 having the desired behaviour in order to get the regularization of the Kepler motion. In this paper we use the Cartan spinor ~ h e o in r ~ order ~ to get a spinor representation for the K-S transformation and reduce the equa- tions of the Kepler motion in a spinor differential equation. 1. THE KS TRANSFORMATION The K-S transformat ion relates a vector r (x,,x2,o,) R3 to -+ a vector u E (u,,u2,u3,u4) by the following relation where A(;) is a 4x4 matrix

2 Revista Brasileira de Física, Vol. 17, n? 2, 1987 The main propert ies of this transformation are: c01 umn or 1 i ne T vectors of A(u) are orthogonal to each other; A ($)A@) = the 3 1 ength r of the posi t ion vector x R3 is given by One trouble of this transformation is that it is not one to -b 3 one. If two vectors u and v E R' are related by vi = ul cos O - u4 sen O v2 = u2 cos cb + u3 sen O V, = ul sen O + U' cos O v3 =-u2sencb + UJ COSO (1.4) with arbitrary 4, they are mapped onto the same vector of R3. The image of a point in R3 is a circle of radius in the parametric space R'. Kustaanheimo and Stiefel showed that with R4 parametrization we can analyse the map of a polarized R2 plane onto R ~. If two vectors 8 and 3 are any pair of vectors of R2 they satisfy the following bilinear relation Thls R2 plane is conforma1 l y mapped onto a plane of R3 and the mapping is a of Levi-Civita's type: distances from origin are squared and angles at the origin are doubled. A given point in R2 has a point image in R3. The position vector in R3 is given in a particular orthog- onal basis, where the vectors of this basis are the column elernents of the Cayley matrix: the well-known Cayley parametr i za t ion of the ro- tations in the 3-dimensional space. 3 -f It is important to note that if two vectors u and V E R' satisfy the bil inear relations eq. (1.5), then A(u)v = A(V)U. These properties of the K-S transformation have one fundamen- tal role in the quantum appl ication of the K-S transformation in the 3 3 Coulomb problem: the operators associated to u and V are quantumcanoni- cal conjugates and the operator associated to the bilinear relation de- termines a constraint condition in the determination of the wave func-

3 Revista Brasileira de Física, Vol. 17, n? 2, 1987 t ion 4. By using this transformation and a time transformationdo=dt/r, Kustaanheimo-Stiefel showed that the equation of the Kepler motion i n ~ ~ can be written in the form with w2 =M/4ao, wherem is the product of the mass and the gravitationa constant and a, is a semi-major axis of the orbit. The eqs. (1.6) are 1 inear differential equat ions wi th constant coef f i c ents. The image-point moves as if it were connected with the origin y an elastic string of rigidity, w2. Its path is a conical section centered at the origin. 2. SPINOR REPRESENTATION FOR THE KS TRANSFORMATION In order to get a spinor representation for the K-S transformation we introduce a nu1 1 four-vector xa in a Minkowski space with metric a (+I,-1,-1,-I). The components of x are complex matrix By the Cartan theory of spinor 3 we can associate to xa a 2 x 2 or equivalently, in a tensor form, 1 c. T~~ = 2 '&B where the T~~ tensors are

4 Revista Brasileira de Física, Vol. 17, n? 2, 1987 and have the followings properties and a By eq. (2.2), the components of the four-vector x are and the length of x a is "given by the determinant of TAB wh i ch is nu1 1. a The four-vector x is associated with a singular matrix, hence by the Cartan spinor theory there exist two complex (1,l) spinors $A and JiB such TAB = $A$~, where the bar means complex conjugate. Therefore we can wri te and by eq. (2.2), we have i a If the x components of the four-vector x are the components of a vector in R ~ then, we can identify eq. (2.6) wi th the spinor trans- formation of ~ustaanheimo'. In order to get the K-S transformation we make a linear corre- + spondence between a vector u in the parametric space and the complex spinor (1,l) qa. We define our complex spinor + in terms of four real A

5 Revista Brasileira de Física, Vol. 17, no 2, 1987 This correspondence is linear and the length of the vector is equal to the square root of the norm of the associated spinor. By eq. (2.4), we have which are the relations between R3 and R' obtained by the K-S trans- format ion. The fact that the K-S transformation is not one to one can be easily reproduced in the spinor space by a simple gauge transformation. 3 3 In fact, if two vectors u and v R4 are related by eq.(1.41, 3 associated to u is given by the spinor 3 (2.10) The spinor gauge transformation $A+$A ei( shows that the K-S trans- formation is not one to one. We need to show that wi th eq. (2.4) we can reproduce the map of a polarized R' plane onto ther3 and the mapping ty Pe is of Levi-Civita's 3 -+ Let u and v be two orthogonal unit-vectors in R4 spann ing a plane R* through the origin

6 Revista Brasileira de Fisica, Vol. 17, no 2, 1987 and building a cartesian coordinate-system in R ~ They. also satisfy the bilinear relation given by eq.(1.5) If w is the polarization angle, then this system of two 1 i- -% near equations in the components of v has its solution in the form V1 = U2C0s O + U3 sen o V, = ul cos w + 7.4, sen w (2.13) v4 = -u2 senw +u, cosw V, =u, senw - u, cosw Consider a given point in having polar coordinates p and 0 with respect to the basis (;,$I in R2. The vector representing this point is given by -% -% + p = p sen Ou + p cos 0 v + By the correspondente in eq. (2.7) and eq. (2.8) the spinor $(p) -+ associated to p, is given by By using eqs. (2.4) and (2.13) we can see, after some computation, that the image o f this point in R3 is given by straighforward I x3j + p2 COS w I sen 28 + p2 sen w I 2(u,u3 - u2u,)' I 2 (u,u, + u,u,) -u2-u2+u2+u sen 20 (2.15)

7 Revista Brasileira de Física, Vol. 17, n? 2, 1987 or, in the Kustaanheimo-Stiefel abbreviation, $ = P2 [COS 28 a + [- cos w + C sen w l sen 2e] (2.16) 3 3 The two vectors a and cos w + c sen w are orthogonal. This fol lows from the Cayley parametrization of the rotations in the 3-dimensional space. As the x O component is the length of the image point in R ~, we have that the mapping is of Levi-Civita's type. We note that the spinor transformation given by eq.(2.4) tains also the eight significant real scalars of the K-S theory. 3 3 and S are two vectors in R~ and +(u) and +(v) are con- 3 If u the correspondi ng spinors in the spinorial space, then by eq. (2.4) we can see that Re(lço) 3 -+ is the scalar product (u,v) and -lm(x0) is the bil inear relation given -+ 3 in eq. (1.5) ; Re(;ci) are the f i rst three components of ~(u)v and lm(xi) 3 + are the three components of the skew-symnetric cross product uxv. With this final verification we can conclude that withalinear correspondence between a vector in R' and a complex spi nor ( 1,I ) de- f ined in terms of four real parameters by eq. (2.7), the K-S trans- formations are completely described by the spinor transformation by eq. (2.4). given Using this particular spinor representation in eq.(1.6)wehave that the four differential equations for the Kepler motion can be trans- formed into a two component spinor differential equation where the dots means differentiation with respect a õ. José Bel landi would 1 ike to thank FAPESP for f inantial support and A. Di Giacomo for hospitality at the Istituto di Fisica- Universita di Pisa where part of this work was done. REFERENCES 1. P. Kustaanheimo, E. Stiefel, J. Reine angew. Math. 218, 204 (1965). 2. Levi-Civita, Opere Mathematiche 6, 111 (1973).

8 Revista Brasileira de Física, Vol. 17. n? 2, E.Cartan, Thhe Theoory of Spinors, Hermann, Paris (1966). 4. M.Boi teux, Physica 65, 381 (1973). 5. P.Kustaanheimo, Publication Astronom. Obs. 3, 102 (1965) Helsinki. Resumo Deriva-se uma representação spinorial para a transformação K-S usando-se a teoria dos spinores de Cartan.