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 s p i n o r r e p r e s e n t a t i o n f o r the K-S t r a n s f o r m a t i o n
by means o f t h e Cartan s p i n o r theory.
i s derived
INTRODUCTION
I t i s w e l l known t h a t a r e g u l a r i z a t i o n o f the Kepler motion i n
the t r i d i m e n s i o n a l space R 3 i s developed by u s i n g a simple mapping o f a
.
simple mapping i s known
four- dimensional space R4 o n t o R ~ This
Kustaanheimo-Stiefel ( K - S ) t r a n s f o r m a t i o n ' .
as the
I n R4, the equations o f the
Kepler motion a r e l i n e a r d i f f e r e n t i a l equations w i t h c o n s t a n t
coef-
f i c i e n t s and remain completely r e g u l a r a t the c e n t e r o f a t t r a c t i o n .
These a r e equations o f simple harmonic o s c i l l a t o r motions.
The K- S t r a n s f o r m a t i o n i s n o t a complete g e n e r a l i z a t i o n o f the
two-dimensional ~ e v i - c i v i t a 2transformation,
b u t o n l y a mapping
of
R'
o n t o R 3 having the d e s i r e d behaviour i n o r d e r t o g e t the r e g u l a r i z a t i o n
o f the Kepler motion.
I n t h i s paper we use t h e Cartan s p i n o r ~ h e o irn~o r~d e r t o g e t
a s p i n o r r e p r e s e n t a t i o n f o r t h e K-S t r a n s f o r m a t i o n and reduce t h e equat i o n s o f t h e Kepler motion i n a s p i n o r d i f f e r e n t i a l equation.
1. THE K S TRANSFORMATION
The K- S transformat i o n r e l a t e s a v e c t o r
-+
a v e c t o r u E (u, ,u2,u3,u4) €
where A(;)
i s a 4x4 m a t r i x
r (x, ,x2 ,o,)
by t h e f o l l o w i n g r e l a t i o n
€ R3 to
Revista Brasileira de Física, Vol. 17, n? 2, 1987
The main p r o p e r t i e s o f t h i s t r a n s f o r m a t i o n are: c01 umn o r 1i ne
T
v e c t o r s o f A(u) a r e orthogonal t o each o t h e r ; A
($)A@)
the
=
3
1 ength r o f t h e posi t i o n v e c t o r x € R3 i s g i v e n by
One t r o u b l e o f t h i s t r a n s f o r m a t i o n i s t h a t i t i s n o t
one.
-b
to
I f two v e c t o r s u and v E R' a r e r e l a t e d by
O
-
u4
sen O
= u l sen O
+
U'
cos
vi = u l cos
V,
with arbitrary
4,
cb +
u 3
sen O
v 3 =-u2sencb +
UJ
COSO
v 2 = u2 cos
(1.4)
O
they a r e mapped o n t o t h e same v e c t o r o f R 3 . The image
o f a p o i n t i n R3 i s a c i r c l e o f r a d i u s
R'
one
3
i n the
parametric
space
.
Kustaanheimo and S t i e f e l showed t h a t w i t h
R4 p a r a m e t r i z a t i o n
we can analyse t h e map o f a p o l a r i z e d R2 plane o n t o R ~ .I f two
8 and 3 a r e
vectors
any p a i r o f v e c t o r s of R2 they s a t i s f y t h e f o l l o w i n g b i l i n e a r
relation
Thls R2 plane i s conforma1 l y mapped o n t o a plane o f R3 and the
mapping i s a o f L e v i - C i v i t a ' s type: distances from o r i g i n a r e s q u a r e d
and angles a t t h e o r i g i n a r e doubled. A g i v e n p o i n t i n R2 has
a
point
image i n R 3 . The p o s i t i o n v e c t o r i n R3 i s g i v e n i n a p a r t i c u l a r o r t h o g elernents o f
onal basis, where t h e v e c t o r s o f t h i s b a s i s a r e t h e column
t h e Cayley m a t r i x : t h e well-known Cayley p a r a m e t r i za t i o n o f t h e r o t a t i o n s i n t h e 3-dimensional space.
-f
3
I t i s important t o n o t e t h a t i f t w o v e c t o r s u a n d V E R'
s a t i s f y t h e b i l inear r e l a t i o n s
eq. (1.5),
3 +
-+ 3
then A ( u ) v = A(V)U.
These p r o p e r t i e s o f t h e K- S t r a n s f o r m a t i o n have one fundament a l r o l e i n the quantum appl i c a t i o n o f t h e K-S t r a n s f o r m a t i o n
3
in
the
3
Coulomb problem: the o p e r a t o r s associated t o u and V a r e quantumcanonic a l conjugates and t h e o p e r a t o r associated t o t h e b i l i n e a r r e l a t i o n determines a c o n s t r a i n t c o n d i t i o n i n the d e t e r m i n a t i o n o f
the wave func-
Revista Brasileira de Física, Vol. 17, n? 2, 1987
4
t i on .
By using t h i s transformation and a time transformationdo=dt/r,
Kustaanheimo-Stiefel showed t h a t t h e equation o f t h e Kepler motion i
n
can be w r i t t e n i n t h e form
w i t h w2 =M/4ao, whereM i s t h e product o f
tationa
constant and a,
The eqs. (1.6)
the
and t h e gravi-
i s a semi-major a x i s o f t h e o r b i t .
a r e 1 inear d i f f e r e n t i a l equat ions wi t h constant
coef f i c ents. The image-point moves as i f i t were
origin
mass
y an e l a s t i c s t r i n g o f r i g i d i t y
,
connected w i t h t h e
w2. I t s path i s a c o n i c a l sec-
t i o n centered a t t h e o r i g i n .
2. SPINOR REPRESENTATION FOR THE K S TRANSFORMATION
I n order t o g e t a s p i n o r representation f o r t h e K-S transforma t i o n we introduce a nu1 1 f o u r - v e c t o r xa i n a Minkowski space with m e t r i c
a
(+I,-1 ,-1 , - I ) . The components o f x a r e
By t h e Cartan theory o f s p i n o r
3
we can associate t o
complex m a t r i x
o r equivalently,
i n a tensor form,
1
T~~ =
where t h e T~~
tensors a r e
2
.c
'&B
xa a 2 x 2
~
~
Revista Brasileira de Física, Vol. 17, n? 2, 1987
and have t h e f o l l o w i n g s p r o p e r t i e s
and
By eq.
(2.2),
a
the components o f t h e f o u r - v e c t o r x
are
a i s "given by t h e determinant o f TAB wh i c h i s nu1 1 .
and t h e l e n g t h o f x
a
The f o u r - v e c t o r x
i s associated w i t h a s i n g u l a r m a t r i x ,
Cartan s p i n o r theory t h e r e e x i s t two complex ( 1 , l )
hence by
s p i n o r s $A
such TAB = $ A $ ~ ,where the bar means complex conjugate.
and
Therefore
the
JiB
we
can w r i t e
and by eq. (2.2)
,
we have
i components o f t h e f o u r - v e c t o r xa a r e the components
If the x
o f a v e c t o r i n R ~ then
,
we can i d e n t i f y eq. (2.6) wi t h t h e s p i n o r transf o r m a t i o n o f ~ustaanheimo'.
I n o r d e r t o g e t t h e K-S t r a n s f o r m a t i o n we make a l i n e a r corre-
+
spondence between a v e c t o r u i n the
spinor (1,l)
qA.
parametric space and t h e complex
We d e f i n e o u r complex s p i n o r
+A
i n terms o f f o u r r e a l
Revista Brasileira de Física, Vol. 17, no 2, 1987
This correspondence i s l i n e a r
and t h e l e n g t h o f t h e v e c t o r i s equal t o t h e square r o o t o f t h e norm o f
t h e associated s p i n o r .
.
By eq (2.4)
,
we have
which a r e t h e r e l a t i o n s between R 3 and R' obtained
by
t h e K-S t r a n s -
format ion.
The f a c t t h a t t h e K-S t r a n s f o r m a t i o n i s n o t one t o one can be
e a s i l y reproduced i n t h e s p i n o r space by a simple gauge transformation.
3
I n f a c t , i f two v e c t o r s u and
associated t o
3
u
3
v € R4 a r e r e l a t e d by eq.(1.41, t h e spinor
i s given by
3
The s p i n o r gauge t r a n s f o r m a t i o n $A+$A
(2.10)
ei( shows t h a t
the
K-S
trans-
formation i s n o t one t o one.
We need t o show t h a t wi t h eq. (2.4) we can reproduce t h e map o f
a p o l a r i z e d R' plane o n t o t h e R 3 and t h e mapping
t Y Pe
3
Let u and
-+
v
i s of Levi- Civita's
be two orthogonal u n i t - v e c t o r s i n R4 s p a n n i n g
plane R* through the o r i g i n
a
Revista Brasileira de Fisica, Vol. 17, no 2, 1987
and b u i l d i n g a cartesian coordinate-system i n R ~ They
.
a l s o satisfy
the b i l i n e a r r e l a t i o n given by eq.(1.5)
I f w i s the
p o l a r i z a t i o n angle, then t h i s system o f two 1 i-%
v
near equations i n the components o f
V1 =
U2C0s O
+
U3
has i t s s o l u t i o n i n the form
sen o
v 4 = -u2 senw + u , cosw
Consider a given p o i n t i n
0 w i t h respect t o the basis
V,
= ul cos w
+
V,
= u , senw
- u,
sen w
74.,
(2.13)
cosw
having polar coordinates
(;,$I i n
p
and
R2. The vector representing t h i s
p o i n t i s given by
-%
-%
p = p sen Ou
+
+
p cos 0 v
+
By the correspondente i n eq. (2.7) and eq. (2.8) the spinor $(p)
-+
associated t o p, i s given by
By using eqs. (2.4) and (2.13) we can see, a f t e r
some
straighforward
computation, t h a t the image o f t h i s p o i n t i n R3 i s given by
I x3j
+
p2 COS w
I
I
2(u,u3
sen 28
+
+
p2 sen w 2 (u,u,
I
- u2u,)'
-u2-u2+u2+u2
1
2
sen 20
u,u,)
3
4
2
(2.15)
Revista Brasileira de Física, Vol. 17, n? 2, 1987
o r , i n t h e Kustaanheimo- Stiefel a b b r e v i a t i o n ,
$
=
3
28
P 2 [COS
The two v e c t o r s a and
a + [-
cos w
+
cos w
+
C sen w l sen
3
c sen w a r e orthogonal
2e]
(2.16)
. This
f o l lows from
the Cayley p a r a m e t r i z a t i o n o f the r o t a t i o n s i n the 3-dimensional space.
As t h e x O component i s t h e l e n g t h o f t h e image p o i n t i n R ~ , we
have
t h a t t h e mapping i s o f L e v i - C i v i t a ' s type.
We n o t e t h a t t h e s p i n o r t r a n s f o r m a t i o n g i v e n by eq.(2.4)
and
S
3
3
a r e two v e c t o r s i n R~ and +(u) and + ( v ) a r e
s p i n o r s i n the s p i n o r i a l space, then by eq.
3
-+
(2.4)
3
the c o r r e s p o n d i ng
we can see t h a t Re(lço)
given
i s the s c a l a r product (u,v) and - l m ( x 0 ) i s t h e b i l inear r e l a t i o n
i n eq. (1.5) ; Re(;ci)
con-
If u
t a i n s a l s o t h e e i g h t s i g n i f i c a n t r e a l s c a l a r s o f t h e K-S theory.
-+
3
a r e t h e f i r s t t h r e e components o f ~ ( u ) and
v
3
+
lm(xi)
a r e the t h r e e components o f t h e skew-symnetric cross product uxv.
With t h i s f i n a l v e r i f i c a t i o n we can conclude t h a t w i t h a l i n e a r
correspondence between a v e c t o r i n R'
and a complex
f ined i n terms o f f o u r r e a l parameters by eq.
(2.7),
spi nor
the
( 1 ,I) de-
K-S
formations a r e completely described by t h e s p i n o r t r a n s f o r m a t i o n
transgiven
by eq. ( 2 . 4 ) .
Using t h i s p a r t i c u l a r s p i n o r r e p r e s e n t a t i o n i n eq.(1.6)wehave
t h a t t h e f o u r d i f f e r e n t i a l equations f o r the Kepler motion can be transformed i n t o a two component s p i n o r d i f f e r e n t i a l equation
where the dots means d i f f e r e n t i a t i o n w i t h respect a õ.
José Bel l a n d i would 1 i k e t o thank FAPESP f o r f i n a n t i a l support
and A. Di Giacomo f o r h o s p i t a l i t y a t the I s t i t u t o d i F i s i c a - U n i v e r s i t a
d i Pisa where p a r t o f t h i s work was done.
REFERENCES
1 . P. Kustaanheimo,
E. S t i e f e l , J. Reine angew. Math. 218, 204 ( 1 9 6 5 ) .
2. L e v i - C i v i t a , Opere Mathematiche
6, 1 1 1 (1973).
Revista Brasileira de Física, Vol. 17. n? 2, 1987
3 . E.Cartan, Thhe Theoory of Spinors, Hermann, P a r i s ( 1 9 6 6 ) .
4 . M.Boi teux, Physica 65, 381 (1973).
5. P.Kustaanheimo,
P u b l i c a t i o n Astronom. Obs.
3, 102 (1965) H e l s i n k i .
Resumo
Deriva- se uma representa ç ão s p i n o r i a l para a transformação K-S
usando-se a t e o r i a dos spinores de Cartan.