научная статья по теме A SYMMETRY REDUCTION SCHEME OF THE DIRAC ALGEBRA WITHOUT DIMENSIONAL DEFECTS Физика

Текст научной статьи на тему «A SYMMETRY REDUCTION SCHEME OF THE DIRAC ALGEBRA WITHOUT DIMENSIONAL DEFECTS»

ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 2, с. 297-302

= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

A SYMMETRY REDUCTION SCHEME OF THE DIRAC ALGEBRA WITHOUT DIMENSIONAL DEFECTS

©2010 R. Dahm*

Zentrum fiir Datenverarbeitung, Universityt Mainz, Germany; Beratung fiir IS, Mainz, Germany

Received April 17,2009

In relating the Dirac algebra to homogeneous coordinates of a projective geometry, we present a simple geometric scheme which allows to identify various Lie algebras and Lie groups well-known from classical physics as well as from quantum field theory. We introduce a 1-point-compactification and quaternionic Mobius transformations, and we use SU*(4) and a symmetry reduction scheme without dimensional defects to identify transformations and particle representations thoroughly. As such, two subsequent nonlinear a models SU*(4)/USp(4) and USp(4)/SU(2) x U(1) emerge as well as a possible double coset decomposition of SU*(4) with respect to SU(2) x U(1). Whereas the first model leads to equivalence classes of hyperbolic manifolds and naturally introduces coordinates and velocities, the second coset model leads to a Hermitian symmetric (vector) space (Kahlerian space) of real dimension 6, i.e., to a 3-dimensional complex space with a global symplectic and a local SU(2) x U(1) symmetry which allows to identify the (local) gauge group of electroweak interactions as well as under certain assumptions it admits compact SU(3) transformations as automorphisms of this 3-dimensional (hyper)complex vector space. In the limit of low energies, this geometric SU*(4) scheme naturally yields the (compact) group SU(4) to describe "chiral symmetry" and conserved isospin of hadrons as well as the low-dimensional hadron representations. Last not least, with respect to some of the SU*(4) generators we find a multiplication table which (up to signs) is identical with the octonions represented in the Fano plane.

1. INTRODUCTION

In [1] we have summarized a classification and representation scheme which extends (compact) spin ® isospin symmetries [2, 3] of hadrons, and we have identified some of their spinorial representations, moreover, we've described relativistic transformations and space—time symmetries via the group Sl(2, H), respectively, its complex representation SU*(4) and certain substructures. Here, we focus on aspects of this symmetry approach mostly related to its two spontaneously broken symmetry modes, SU*(4)/USp(4) and USp(4)/U(2) (respectively, USp(4)/SU(2) x U(1)), as well as to some properties of the underlying Lie algebra s-u*(4) while a more detailed overview, a discussion of the physical aspects/motivations and more detailed calculcations will be published soon [4].

As such, we've organized this paper as follows: Based on the Lie algebra s-u*(4), we'll identify the group generators of each of the two spontaneously broken symmetry models SU*(4)/USp(4) (Section 2) and USp(4)/SU(2) x U(1) (Section 3),

E-mail: dahm@uni-mainz.de, dahm@bf- is .de

and we'll discuss some of their aspects and consequences. Especially in the first case, i.e., the symmetry breakdown of SU*(4) to USp(4), we discuss the relations to a general relativistic description based on the hyperbolic spaces H3, H4, and H5. We describe some of their properties and symmetries by using Lie triple systems, and we thus introduce isometry groups and geodesics very naturally. The second decomposition leads to a natural complex structure and in addition it emphasizes one of the 15 s-u*(4) generators which appears in the identification of isospin transformations as well as in its role as U(1) generator of the four U(2), respectively, SU(2) x U(1) generators and as a (hyper)complex unit with respect to the Hermitian symmetric space USp(4)/U(2).

In the last Section 4, by taking up another position on the s-u*(4) generators, we derive a 6-dimensional algebra isomorphic to the Lie algebra of the Lorentz group and some interesting multiplication properties of the SU*(4) generators which are very similar to the octonionic multiplication table represented in the Fano plane. For a more detailed discussion of physical identifications and motivation as well as for an extended overview on the various representations of the symmetry reduction scheme we refer to the upcoming papers [4].

2. SU*(4)/USp(4)

As emphasized in the publications mentioned above, our initial motivation resulted from hadronic interactions described by chiral SU(2) x SU(2) symmetry. There, as a consequence of low-energy hadron physics, we've generalized chiral pion transformations towards general quaternionic Mobius transformations and we've introduced homogeneous quaternionic coordinates q1 and q2, respectively, the variable

q

qiq- 1,

q —► q' = fA(q) =

(1)

aq + b cq + d

:= (aq + b)(cq + d)

where a, b,c,d e H. As a generalization of the Rie-mannian sphere from C to H, which is well defined and unique as the quaternions constitute a division algebra with unit element, we obtain the "simple" geometry in Fig. 1 which (in terms of homogeneous coordinates) leads to the transformation group Sl(2,

H) and, by restricting the (compactified) sphere S4 to "rotations around the -axis", to the compact group U(2, H). Then, when representing the quaternions by 2 x 2 complex matrices, we may as well discuss the (complex) groups SU*(4) and USp(4), respectively, and we may use an operator representation Qap, 0 < a,3 < 3, to denote the 4 x 4 complex representations of the generators and group elements, where the first index a denotes the 2 x 2 block structure, whereas the second index 3 denotes the 2 x 2 content of the block (see, e.g., [2]). We've called these operators "qquaternions" due to their twofold quaternionic structure. While global SU*(4) transformations can be related to the Dirac algebra {1,75,7^,757^, }, to quantum field theory (QFT) and (at low energies) to SU(4) hadron descriptions [3,

I], the first reduction step comprizes equivalence classes of the noncompact group SU*(4) with respect to its maximal compact subgroup USp(4). This corresponds to the Cartan decomposition of the Lie algebra su*(4) where we find a direct (reductive) decomposition of the 15-dimensional Lie algebra su*(4) according to h © p, where dim h = dim usp(4) = 10 and dim p = 5. For the operator subsets h and p we find the relations

su*(4) = h ® p, su(4) = h ® ip, [h,h] С h, [h,p] С p, [p,p] С h.

(2)

Due to [h,p] Q p, the space p admits to represent the (compact) symplectic symmetry transformations generated by h on p, and at first glance we find an USp(4) stability group. The related nonlinear a model SU*(4)/USp(4) is an irreducible Riemannian globally symmetric space of type AII [5, 6] of rank 1

and dimension 5. This model has a global SU*(4) symmetry and a "local" symplectic symmetry.

In order to introduce coordinates, respectively, to use differential geometry, one can map the 5-dimensional difference vector space p of su*(4) via the exponential mapping to a 5-dimensional hyperboloid H5 in R6 which may be parametrized by six real coordinates (X0,Xa), 1 < a < 5, with the additional (hyperbolic) constraint X$ — X1 = = const2. Note, however, that because we can calculate exp V = exp xaPa, Pa e p, respectively, instead of only V we can calculate also parametrized (linear) paths in the Lie algebra su*(4) and with respect to the difference space p exactly and as a closed expression in terms of hyperbolic functions,

exp V = cosh\\x\\■ 1 + sinh ||ж||

l|x||

Va

(3)

we naturally obtain hyperbolic representations of the (coset) coordinates X0 and Xa in terms of cosh x and sinhx which automatically fulfil the hyperbolic constraint on the manifold given above [4]. Therefore, the definition and a comparison of equivalence classes naturally introduces "physical" space—time coordinates and velocities (boosts), and this immediately requires to define space—time transformations just in order to compare (generalized) "locations" on the hyperboloids in R6 and (covariant) objects defined at the various points in R6. Mapping points and paths from the Lie algebra exponentially to the manifold, we automatically obtain points, lines, and ordering relations as well as distances, vectors and tensors, which on the one hand serve to discuss projective geometry and duality, on the other hand one can use differential geometry and discuss metrical properties and tensor analysis. Here, however, we benefit from further substructures in p, especially from differen-tiable structures, which we are going to discuss later in terms of Lie triple systems. Because SU*(4) acts transitively on SU*(4)/USp(4) ^ H5 and USp(4) is a 10-dimensional stability group of the Riemannian space, such an approach to the geometry can study the nonlinear a model, its differential geometry, the Riemannian structure, the Laplace—Beltrami operator, the affine connection, etc. in more detail. Some of these calculations can be found in [4], for some of the underlying definitions and considerations concerning coordinates and measures of Hn, representations of the Laplace—Beltrami operators on hyperboloids and (related) spheres, the metric, etc. we refer the reader directly to [7]. Moreover, we can identify Killing vectors or formulate a Kaluza—Klein theory where we identify three space—time operators and two (complexified/noncompact) "isospin" operators in p (see [4] and si in the enumeration below),

x

a

A SYMMETRY REDUCTION SCHEME

N

299

H ^^

p

Fig. 1. Quaternionic "sphere" and projection coordinates.

however, we benefit from overall and fixed definitions of phases, operators, coordinates, metrics, etc. because we've started from the (projective) quaternionic geometry (see Fig. 1) and the Lie group Sl(2, H), respectively, SU*(4) by acting on homogeneous quaternionic coordinates.

Here, however, we want to look for further structures and information admitted and/or provided by the vector space p itself. The (physical) symmetry transformations described so far in terms of

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