Hurwitz transformation

The general framework for defining and studying the Hurwitz transformations is that of the Cayley–Dickson algebras (cf. also Cayley–Dickson algebra). Familiar examples of Cayley–Dickson algebras are:

and , the algebras of ordinary and hyperbolic complex numbers, respectively;

and , the algebras of ordinary (or Hamilton) and hyperbolic quaternions, respectively;

and , the algebras of ordinary (or Cayley) and hyperbolic octonions, respectively. Higher-dimensional Cayley–Dickson algebras over the real field are denoted by , where is a -tuple with (). Such a -dimensional algebra , with , may be constructed from an -dimensional Cayley–Dickson algebra by a "doubling" process [a1], [a2], [a3]. (This process generalizes .) For fixed , the algebra with corresponds to ordinary (or elliptic) hypercomplex numbers (cf. also Hypercomplex number), while the other algebras with correspond to hyperbolic hypercomplex numbers. The Cayley–Dickson algebra is referred to as normed or pseudo-normed according to whether the metric

is Euclidean or pseudo-Euclidean (cf. also Euclidean space; Pseudo-Euclidean space). For each algebra there exist anti-involutions : , (the mapping satisfies and ). One of the anti-involutions is the mapping

the remaining anti-involutions correspond to anti-involutions of type on the various -dimensional Cayley–Dickson subalgebras of .

An element has real components ; these define a vector in and can thus be associated with a column vector . From the product of two elements and , an (generalized) Hurwitz matrix is defined via .

The application

(a1)

where with (), defines a mapping , called Hurwitz transformation and denoted by . The row vector consists of:

i) total differentials, leading to a vector ;

ii) one-forms (when ) which are not total differentials and which are taken to be equal to zero to account for the non-bijectivity of the mapping : . The integer , , depends on . The various possibilities for are:

1) is the unit -matrix;

2) is such that () and thus corresponds to the anti-involution of ( if and only if );

3) corresponds to one of the remaining anti-involutions of ;

4) is a matrix not listed in the other cases. Equation (a1) defines the components of as quadratic functions of the components of . For , the vector may also be generated from the product that produces a column vector, with vanishing entries and non-vanishing entries, corresponding to .

Another type of Hurwitz transformation, denoted by , is formally obtained by replacing in (a1) or in by with . This leads to non-quadratic transformations [a4].

The cases deserve special attention, since they correspond to the Hurwitz factorization problem (the situations addressed in [a5] concern for , for and for ). In these cases, the -matrix satisfies

and may be written in terms of elements of a Clifford algebra of degree . As a consequence, the factorization property

for and fixed, is satisfied by for .

The geometric and group-theoretical properties of the transformations for , and are well known. From the geometrical point of view, they correspond to fibrations on spheres and hyperboloids [a3], [a6]. From the point of view of group theory, they are associated to Lie algebras under constraints [a7].

Some typical examples of are as follows. The case , where is the unit -matrix, corresponds to the Levi-Civita transformation used in the restricted three-body problem of classical mechanics [a8]. The case , where , corresponds to the Kustaanheimo–Stiefel transformation used in the regularization of the Kepler problem [a9] and associated to the Hopf fibration of fibre [a10]. The case , where , corresponds to the Fock (stereographic) projection used in the quantum mechanical problem of the hydrogen atom [a11]. More generally, Hurwitz transformations are useful in number theory and in theoretical physics (classical and quantum mechanics, quantum field theory, local gauge symmetries). In particular, they can be useful for transforming a dynamical system in into a dynamical system in subject to constraints. (Under such a transformation, the coupling constant of one system is exchanged with the energy of the other.)

References

[a1]	L.E. Dickson, "On quaternions and their generalization and the history of the eight square theorem" Ann. of Math. , 20 (1919) pp. 155
[a2]	G.P. Wene, "A construction relating Clifford algebras and Cayley–Dickson algebras" J. Math. Phys. , 25 (1984) pp. 2351
[a3]	D. Lambert, M. Kibler, "An algebraic and geometric approach to non-bijective quadratic transformations" J. Phys. A: Math. Gen. , 21 (1988) pp. 307
[a4]	M. Kibler, P. Labastie, "Transformations generalizing the Levi-Civita, Kustaanheimo–Stiefel and Fock transformations" Y. Saint-Aubin (ed.) L. Vinet (ed.) , Group Theoretical Methods in Physics , World Sci. (1989)
[a5]	A. Hurwitz, "Über die Komposition der quadratischen Formen von beliebig vielen Variablen" Nachr. K. Gesellschaft Wissenschaft. Göttingen (1898) pp. 309
[a6]	I.V. Polubarinov, "On the application of Hopf fiber bundles in quantum theory" , Report E2-84-607 , JINR: Dubna (Russia) (1984)
[a7]	M. Kibler, P. Winternitz, "Lie algebras under constraints and non-bijective transformations" J. Phys. A: Math. Gen. , 21 (1988) pp. 1787
[a8]	T. Levi-Civita, "Sur la régularisation du problème des trois corps" Acta Math. , 42 (1918) pp. 99
[a9]	P. Kustaanheimo, E. Stiefel, "Perturbation theory of Kepler motion based on spinor regularization" J. Reine Angew. Math. , 218 (1965) pp. 204
[a10]	H. Hopf, "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" Math. Ann. , 104 (1931) pp. 637
[a11]	V. Fock, "Zur Theorie des Wasserstoffatoms" Z. Phys. , 98 (1935) pp. 145