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Generalized Weierstrass representation for surfaces in terms of Dirac-Hestenes spinor field

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Generalized Weierstrass representation for surfaces in terms of Dirac-Hestenes spinor field
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    a  r   X   i  v  :  m  a   t   h   /   9   8   0   7   1   5   2  v   5   [  m  a   t   h .   D   G   ]   2   S  e  p   2   0   0   1 Generalized Weierstrass representation forsurfaces in terms of Dirac-Hestenesspinor field Vadim V. Varlamov Computer Division, Siberia State University of Industry,Novokuznetsk 654007, Russia  Abstract A representation of generalized Weierstrass formulae for an immer-sion of generic surfaces into a 4-dimensional complex space in terms of spinors treated as minimal left ideals of Clifford algebras is proposed.The relation between integrable deformations of surfaces via mVN-hierarchy and integrable deformations of spinor fields on the surfaceis also discussed. Mathematics Subject Classification (1991): 53A05, 53A10, 15A66.Key words: Weierstrass representation, Clifford algebras, spinors.  1 Introduction The theory of integrable deformations and immersions of surfaces due its aclose relationship with the theory of integrable systems at present time is arapid developing area of mathematical physics. One of the most powerfulmethods in this area is a Weierstrass representation for minimal surfaces [1],the generalization of which onto a case of generic surfaces was proposed byKonopelchenko in 1993 [2, 3] served as a basis for the following investigations. So, the generalized Weierstrass formulae for conformal immersion of surfacesinto 3-dimensional Euclidean space are used for the study of the basic quan-tities related to 2D gravity, such as Polyakov extrinsic action, Nambu-Gotoaction, geometric action and Euler characteristic [4]. This method is alsointensively used for the study of constant mean curvature surfaces, Willmoresurfaces, surfaces of revolution and in many other problems related withdifferential geometry [5]-[14]. A further generalization of Weierstrass rep- resentation onto a case of multidimensional Riemann spaces, in particularonto a case of 4-dimensional space with signature (+ , + , + , − ) (Minkowskispace-time) has been proposed in the recent paper [15].In the present paper we consider a relation between a Weierstrass repre-sentation in a 4-dimensional complex space  C 4 and a Dirac-Hestenes spinorfield which is defined in Minkowski space-time  R 1 , 3 . Dirac-Hestenes spinorswere srcinally introduced in [16, 17] for the formulation of a Dirac theory of  electron with the usage of the space-time algebra  Cℓ 1 , 3  [18] in  R 1 , 3 (see also[19]). On the other hand, there is a very graceful formulation [20]-[23] of the Dirac-Hestenes theory in terms of modern interpretation of spinors as min-imal left ideals of Clifford algebras [24, 25], a brief review of which we give in Section 2. In Section 3 after a short historical introduction, generalizedWeierstrass formulae in  C 4 are rewritten in a spinor representation type form(matrix representation of a biquaternion algebra  C 2  ∼ = M 2 ( C )) and are iden-tified with the Dirac-Hestenes spinors, the matrix representation of which isalso isomorphic to M 2 ( C ). It allows to use a well-known relation betweenDirac-Hestenes and Dirac spinors [23, 26] (see also [27]) to establish a rela- tion between Weierstrass-Konopelchenko coordinates for surfaces immersedinto  C 4 and Dirac spinors. Integrable deformations of surfaces defined bya modified Veselov-Novikov equation and their relation with integrable de-formations of Dirac field on surface are considered at the end of the Section3.1  2 Spinors as minimal left ideals of Cliffordalgebras Let us consider a Clifford algebra  Cℓ  p,q ( V,Q ) over a field  K  of characteristic0 ( K  =  R ,  K  = Ω =  R ⊕ R ,  K  =  C ), where  V   is a vector space endowedwith a nondegenerate quadratic form Q  =  x 21  +  ...  +  x 2  p − ... − x 2  p + q . The algebra  Cℓ  p,q  is naturally  Z 2 -graded. Let  Cℓ +  p,q  (resp.  Cℓ −  p,q ) be a setconsisting of all even (resp. odd) elements of algebra  Cℓ  p,q . The set  Cℓ +  p,q  is asubalgebra of   Cℓ  p,q . It is obvious that  Cℓ  p,q  =  Cℓ +  p,q ⊕ Cℓ −  p,q .When  n  is odd, a volume element  ω  =  e 12 ...p + q  commutes with all elementsof algebra  Cℓ  p,q  and therefore belongs to a center of   Cℓ  p,q . Thus, in the caseof   n  is odd we have for a center Z  p,q  =  R ⊕ i R  if   ω 2 = − 1; R ⊕ e R  if   ω 2 = +1 , (1)where  e  is a double unit. In the case of   n  is even the center of   Cℓ  p,q  consiststhe unit of algebra.Let  R  p,q  =  Cℓ  p,q ( R  p,q ,Q ) be a real Clifford algebra ( V   =  R  p,q is areal space). Analogously, in the case of a complex space we have  C  p,q  = Cℓ  p,q ( C  p,q ,Q ). Moreover, it is obvious that  C  p,q  ∼ =  C n , where  n  =  p + q  . Fur-ther, let us consider the following most important in physics Clifford algebrasand their isomorhisms to matrix algebras:quaternions  R 0 , 2  = IHbiquaternions  C 2  =  R 3 , 0  ∼ = M 2 ( C )space-time algebra  R 1 , 3  ∼ = M 2 (IH)Dirac algebra  C 4  =  R 4 , 1  ∼ = M 4 ( C ) ∼ = M 2 ( C 2 )The identity  C 2  =  R 3 , 0  for a biquaternion algebra known in physics as aPauli algebra is immediately obtained from definition of the center of thealgebra  Cℓ  p,q  (1). Namely, for  R 3 , 0  we have a volume element  ω  =  e 123  ∈ Z 3 , 0  =  R ⊕ i R , since  ω 2 = − 1. The identity  C 4  =  R 4 , 1  is analogously proved.The isomorphism  R 4 , 1  ∼ = M 2 ( C 2 ) is a consequence of an algebraic modulo 2periodicity of complex Clifford algebras:  C 4  ∼ =  C 2 ⊗ C 2  ∼ =  C 2 ⊗ M 2 ( C )  ∼ =M 2 ( C 2 ) [28, 29, 30]. 2  The left (resp. right) ideal of algebra  Cℓ  p,q  is defined by the expression Cℓ  p,q e  (resp.  eCℓ  p,q ), where  e  is an idempotent satisfying the condition  e 2 =  e .Analogously, a minimal left (resp. right) ideal is a set of type  I   p,q  =  Cℓ  p,q e  pq (resp.  e  pq Cℓ  p,q ), where  e  pq  is a primitive idempotent, i.e.,  e 2  pq  =  e  pq  and  e  pq cannot be represented as a sum of two orthogonal idempotents, i.e.,  e  pq   = f   pq  + g  pq , where  f   pq g  pq  =  g  pq f   pq  = 0 , f  2  pq  =  f   pq , g 2  pq  =  g  pq . In the general casea primitive idempotent has a form [20] e  pq  = 12(1 + e α 1 )12(1 + e α 2 ) ...  12(1 + e α k ) ,  (2)where  e α 1 ,... , e α k  are commuting elements of the canonical basis of   Cℓ  p,q such that ( e α i ) 2 = 1 ,  ( i  = 1 , 2 ,... ,k ). The values of   k  are defined by aformula k  =  q  − r q −  p ,  (3)where  r i  are the Radon-Hurwitz numbers, values of which form a cycle of theperiod 8 : r i +8  =  r i  + 4 .  (4)The values of all  r i  are i  0 1 2 3 4 5 6 7 r i  0 1 2 2 3 3 3 3For example, let consider a minimal left ideal of the space-time algebra  R 1 , 3 .The Radon-Hurwitz number for algebra  R 1 , 3  is equal to  r q −  p  =  r 2  = 2, andtherefore from (3) we have  k  = 1. The primitive idempotent of   R 1 , 3  has aform e 13  = 12(1 + e 0 ) , or  e 13  =  12 (1+Γ 0 ), where Γ 0  is a matrix representation of the unit  e 0  ∈ R 1 , 3 .Thus, a minimal left ideal of   R 1 , 3  is defined by the following expression I  1 , 3  =  R 1 , 3 12(1 + Γ 0 ) .  (5)Analogously, for the Dirac algebra  R 4 , 1  on using the recurrence formula (4)we obtain  k  = 1 − r − 3  = 1 − ( r 5 − 4) = 2, and a primitive idempotent of   R 4 , 1 may be defined as follows e 41  = 12(1 + Γ 0 )12(1 +  i Γ 12 ) ,  (6)3  where Γ 12  = Γ 1 Γ 2  and Γ i  ( i  = 0 , 1 , 2 , 3) are matrix representations of theunits of   R 4 , 1  =  C 4 :Γ 0  =  1 0 0 00 1 0 00 0  − 1 00 0 0  − 1  ,  Γ 1  =  0 0 0 10 0 1 00  − 1 0 0 − 1 0 0 0  , Γ 2  =  0 0 0  − i 0 0  i  00  i  0 0 − i  0 0 0  ,  Γ 3  =  0 0 1 00 0 0  − 1 − 1 0 0 00 1 0 0  . Further, for a minimal left ideal of Dirac algebra  I  4 , 1  =  R 4 , 112 (1+Γ 0 ) 12 (1+ i Γ 12 ) using the isomorphisms  R 4 , 1  =  C 4  =  C ⊗ R 1 , 3  ∼ = M 2 ( C 2 ) ,  R +4 , 1 ∼ = R 1 , 3  ∼ = M 2 (IH) and also an identity  R 1 , 3 e 13  =  R +1 , 3 e 13  [22, 23] we have the following expression [27]: I  4 , 1  =  R 4 , 1 e 41  = ( C ⊗ R 1 , 3 ) e 41  ∼ =  R +4 , 1 e 41  ∼ =  R 1 , 3 e 41  = R 1 , 3 e 13 12(1 +  i Γ 12 ) =  R +1 , 3 e 13 12(1 +  i Γ 12 ) .  (7)Let Φ  ∈  R 4 , 1  ∼ = M 4 ( C ) be a Dirac spinor and  φ  ∈  R +1 , 3 ∼ =  R 3 , 0  =  C 2 be a Dirac-Hestenes spinor. Then from (7) the relation immediately followsbetween spinors Φ and  φ :Φ =  φ 12(1 + Γ 0 )12(1 +  i Γ 12 ) .  (8)Since  φ ∈ R +1 , 3 ∼ =  R 3 , 0 , then the Dirac-Hestenes spinor can be represented bya biquaternion number φ  =  a 0 +  a 01 Γ 01  +  a 02 Γ 02  +  a 03 Γ 03  +  a 12 Γ 12  +  a 13 Γ 13  +  a 23 Γ 23  +  a 0123 Γ 0123 . (9)Or in the matrix representation φ  =  φ 1  − φ ∗ 2  φ 3  φ ∗ 4 φ 2  φ ∗ 1  φ 4  − φ ∗ 3 φ 3  φ ∗ 4  φ 1  − φ ∗ 2 φ 4  − φ ∗ 3  φ 2  φ ∗ 1  , φ i  ∈ C ,  (10)4
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