Spinor structures and nonlinear connections in vector

Spinor structures and nonlinear connections in vector bundles, generalized Lagrange and Finsler spaces Sergiu I. Vacaru Institute of Applied Physics, Academy of Sciences, 5 Academy Str., Chisinau-028, Republic of Moldova

~Received 11 January 1995; accepted for publication 21 June 1995! It is our purpose here to show that the spinor theory admits generalization for curved spaces with local anisotropy ~for example, for Finsler, Lagrange, and generalized Lagrange spaces!. © 1996 American Institute of Physics. @S00222488~95!02210-7#

I. INTRODUCTION

The space–times with local anisotropy have generated growing interest in theoretical and mathematical physics.1– 8 In different models of locally anisotropic space–times one considers nonlinear and linear connections and metric structures in vector bundles ~isotopic anisotropy! and tangent bundles ~space–time anisotropy! on locally isotropic space–times @~pseudo!-Riemannian, Einstein–Cartan, or more general types of curved spaces with torsion and nonmetricity#. It seems likely that locally anisotropic space–times ~la-spaces! make up a more convenient geometrical background for developing, in a self-consistent manner, classical and quantum statistical and field theories in dispersive media with radiational or turbulent and random processes. In this connection, the formulation of spinor theory on la-spaces presents substantial interest. Questions on spinors and la-space geometry were considered, for example, in the frame of Finsler bundles on space–time5 and of the spinor gauge field theory,9 but up to the present, we do not have a rigorous mathematical definition of spinors on la-spaces. The aim of this paper is to present a geometric study of the Clifford and spinor structures in vector and tangent bundles provided with nonlinear and linear connections and metric structures and to formulate the spinor theory for spaces with the most general anisotropy of metric called generalized Lagrange spaces1,2,10 GL-spaces. The geometry of vector bundles, endowed with mutually adapted nonlinear connection, distinguished connection and metric structures and the geometry of GL-spaces are briefly reviewed in Sect. II. Distinguished Clifford algebras are introduced in Sec. III. Then, in Sec. IV, we define Clifford bundles and spinor structures on vector bundles and GL-spaces. Almost complex spinor structures on GL-spaces are studied in Sec. V. A brief introduction into algebraic and geometric theory of distinguished spinors in vector bundles and GL-spaces is given in Sec. VI. Finally, the results presented in the paper are discussed in Sec. VII. II. NONLINEAR CONNECTIONS IN VECTOR BUNDLES AND GENERALIZED LAGRANGE SPACES

In this section we present for our further considerations the necessary definitions and basic results on vector bundles and spaces with local anisotropy.1,2,11,12 Let us introduce differentiable bundle spaces: the principal bundle, denoted as P 5~P,p,Gr,M ! where P and M are differentiable manifolds, map p : P→M is a differentiable surjection, and Gr is the structural group, the vector bundle, v -bundle, denoted as j 5(E,p,M )5(E,p,Gr,M ,F! where differentiable manifolds E and M are called, respectively, the total ~E5tot j! and base ~M 5bas j! spaces of v -bundle j, map p:E→M is a differentiable surjection, typical fiber F is a real vector space of dimension m, dim F5m, and as the structural group Gr of bundle j we consider the group of linear transforms of F, i.e., Gr5GL(m,R!.

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Sergiu I. Vacaru: Spinors in locally anisotropic spaces

509

For a base space M of dimension n, bundle j has E a space of dimension n1m. Local coordinates on j are denoted as u5(u a )5(x i ,y a ), (i51,2,...,n), (a51,2,...,m), where x5(x i ) are considered as local coordinates on M and y a as coordinates on fiber F x . Coordinate transforms on v -bundle j are defined as u a 5 ~ x i ,y a ! →u a 8 5 ~ x i 8 ,y a 8 ! ,

~1!

where x k 8 5 x k 8 (x k ), rank ( ] x k 8 / ] x k ) 5 n, y a 8 5M aa 8 ~ x ! y a , i M aa 8 ~ x !i P Gr, matrices M aa 8 have the property that for a superposition of coordinate transforms (x i ,y a )→(x i 8 ,y a 8 )→(x i 8 ,y a 8 ), M aa 9 ~ x 8 ! M aa 8 ~ x ! 5M aa 9 ~ x ! ,M aa 8 ~ x ! 5 d aa 8 . The concept of nonlinear connection, i.e., N-connection, was introduced in the frame of Finsler geometry; the definition of N-connection as a global geometric structure was first given in Ref. 13 ~see related topics in Refs. 14 and 15!. In Refs. 1, 2, 9, and 10 N-connection structures are studied in detail. Definition 1: A nonlinear connection in a vector bundle j is a distribution $ E u →H u E,T u E5H u E ^ V u E % on E defining a global decomposition, as a Whitney sum, into horizontal, HE, and vertical, VE, subbundles of the tangent bundle TE: ~2!

TE5HE ^ VE. To a N-connection one associates a covariant derivation ¹ X A5X i

H

J

]Aa 1N ai ~ x, A ! s a ]xi

~3!

on M , where s a are local linear independent sections of (E,p,M ), A5A a s a is a tensor field in E, and X5X i s i is a vector field on M decomposed on local basis s i . Differentiable functions N ai from ~3! written as functions on x i and y a , N ai (x, y) are called coefficients of the N-connection and satisfy these transformation laws under coordinate transforms ~1! and ~2!: a N i 88

] M aa 8 ~ x ! a ] x i8 a8 a 5M a N i 2 y . ]xi ]xi

Remark 1: Linear connections are particular cases of N-connections, when N ai (x, y) are parametized as N ai (x, y)5K a bi (x)X i y b ; functions K a bi (x) defined on M are called as Christoffel coefficients. In vector bundle j we can introduce a local frame basis adapted to the given N-connection,

d

S

D

d ] d ] a . i 5 ] i 2N i ~ x, y ! a ,X a 5 a5 dx ]y dy ]ya

~4!

X a 5du a 5 ~ dX i 5 d x i 5dx i , X a 5 d y a 5dy a 1N ai ~ x, y ! dx i ! .

~4a!

X a5

du

a5

X i5

The dual to ~4! basis is defined as

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Sergiu I. Vacaru: Spinors in locally anisotropic spaces

By using adapted bases ~4! and ~4a! one introduces algebra DT(E) of tensorial distinguished pr fields ~d-fields, d-tensors, d-objects! on j, T 5T qs which is equivalent to the tensorial algebra of the v -bundle p d :HE ^ VE→E, hereafter briefly denoted as j d . An element tPT pr qs, d-tensor field pr of type ( qs ) can be written in local form as i •••i a 1 •••a r ~ x, 1 q b 1 •••b s

t5t j1 ••• jp

y!

d dx

i 1 ^ ••• ^

d dx

j1 jq i p ^ dx ^ ••• ^ dx ^

] ] ^ d y b 1 ^ ••• ^ d y b s . a 1 ^ ••• ^ ]y ] y ar

In addition to d-tensors we can introduce d-objects with various group and coordinate transforms adapted to global splitting ~2!. For example, we define linear d-connections in this form. Definition 2: A linear d-connection on E is a linear connection D on E conserving under parallelism the global decomposition ~2! into horizontal and vertical sub-bundles of the tangent bundle TE. By using decompositions of N-adapted frames ~4! we define components of connection D, ˜ -derivations of X : G˜ a bg , as covariant D b ˜ X :5D ˜ X 5G˜ a X . D bg a g b Xg b Torsion T˜ a bg and curvature R˜ b a g d of connection G˜ a bg can be introduced in standard manner:1,2,10 T˜ ~ X g ,X b ! 5T˜ a bg X a , where T˜ a bg 5G˜ a bg 2G˜ a gb 1 v a bg ,

~5!

and, respectively, R˜ (X d ,X g ,X b )5R˜ b a g d X a , where R˜ b a g d 5X d G˜ a bg 2X g G˜ a b • d 1G˜ w bg G˜ a w d 2G˜ w b d G˜ a w g 1G˜ a b w v w g d .

~6!

In formulas ~5! and ~6! we have used nonholonomy coefficients w a bg of adapted frames, defined as @ X a ,X b # 5X a X b 2X b X a 5w g ab X g .

~7!

Let us consider v -bundle j 5(E, p, M ) with paracompact base M . Definition 3: The metric structure G on total space E of vector bundle j is defined as a second-order covariant, tensor field nondegenerate, and of constant signature. In the adapted frame metric G on E is expressed as G5G ab ~ u ! d u a ^ d u b 5g i j ~ x, y ! dx i ^ dx j 1q ab ~ x, y ! d y a ^ d y b .

~8!

Definition 4: Distinguished connection structure D on E is compatible with metric structure G on E if ˜ G 50. D a gd

~9!

In Lagrange and Finsler geometry the basic geometric constructions are realized on the tangent bundle (TM , t ,M ) In this case, an N-connection, with local coefficients N ij (x k ,y l ) is associated to a global Whitney sum decomposition: TTM 5HTM % VTM .

~10!

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Sergiu I. Vacaru: Spinors in locally anisotropic spaces

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Let consider a d-metric g i j (x, y) on M ~fundamental tensor on M ! as a second-order covariant and nondegenerate tensorial d-field on M . Definition 5 (see Refs. 1, 2, and 10): A pair M n 5(M , g i j (x, y)) is called a generalized Lagrange space ~GL-space!. Remark 2: Lagrange spaces L n are a particular case of GL-spaces: when the d-metric on M can be expressed as g i j ~ x,y ! 5

] 2L , ]yi ]y j

where L:TM →R ~R is the real number field! is a differentiable function, called a Lagrangian on M. Remark 3: We obtain a Finsler space (M ,L), also as a particular case, if L5L 2 , where L is a Finsler metric on M . For our purposes it is convenient to use Miron’s1,2 almost Hermitian model H 2n (M ,G,J) of a GL-space, denoted H 2n -space, a correspondingly defined lift of M n 5(M , g i j (x, y)) to TM , i

0 dj ) 0 j

which is almost compatible with complex structure J a b 5 ( 2 d i 2n

on TM , with J•J52I. In the

construction of H -spaces it is a very important fact that the N-connection on TM uniquely determines the metric structure G on TM , the H 2n -metric, G ~ u ! 5g i j ~ x,y ! dx i ^ dx j 1g i j ~ x,y ! d y i ^ d y j

~11!

@with components of d-metric g i j (x, y) defined from relations g i j 2N k i g k j 50# being compatible ˜ -connection, i.e., with the D ˜ G 50, D a bg

~12!

and with the almost complex structure, i.e., J a b J g d G b d 5G ag

and D a J g b 50.

The spinor formalism proposed in this paper will be formulated for v -bundles provided with an N-connection structure compatible with the corresponding d-connection and metric structures ~8! and satisfying metricity conditions ~9!. We point out that for GL-spaces, H 2n -metric ~11! satisfying metricity conditions ~12! is uniquely determined by the N-connection; we shall construct Clifford bundles and define spinor structures generated by this nonlinear connection structure. III. DISTINGUISHED CLIFFORD ALGEBRAS

The typical fiber of v -bundle j d , p d :HE % VE→E is a d-vector space F 5hF % v F , split into vertical v F and horizontal hF subspaces, with metric G(g, q) induced by v -bundle metric ~8! @or by H 2n -metric ~11! in the case when E5TM #. Clifford algebras ~see, for example, Refs. 16 –18! formulated for d-vector spaces will be called Clifford d-algebras. In this section we shall consider the main properties of Clifford d-algebras. The proof of theorems will be based on the technique developed in Ref. 16 correspondingly adapted to the distinguished character of spaces in consideration. Let k be a number field ~for our purposes k5R or k5C, R and C are, respectively, real and complex number fields! and define F as a d-vector space on k provided with nondegenerate symmetric quadratic form ~metric! G. Let C be an algebra on k ~not necessarily commutative! and j:F →C a homomorphism of underlying vector spaces such that j(u) 2 5G(u)•1 ~1 is the unity in algebra C and d-vector uPF !. We are interested in definition of the pair (C, j) satisfying the next

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Sergiu I. Vacaru: Spinors in locally anisotropic spaces

universality conditions. For every k-algebra A and arbitrary homomorphism w:F →A of the underlying d-vector spaces, such that ( w (u)) 2 →G(u)•1, there is a unique homomorphism of algebras c :C→A transforming the diagram

into a commutative one. The algebra solving this problem will be denoted as C~F ,G! @equivalently as C(G) or C~F !# and called as Clifford d-algebra associated with pair ~F , G!. Theorem 1: The above-presented diagram has a unique solution (C, j) up to isomorphism. Proof: ~We adapt for d-algebras that of Ref. 16, p. 127.! For a universal problem the uniqueness is obvious if we prove the existence of solution C(G). To do this we use tensor algebra pr L (F ) 5 % L qs (F ) 5 % `i50 T i (F ), where T 0 ~F !5k and T i ~F !5F ^ ••• ^ F for i.0. Let I(G) be the bilateral ideal generated by elements of form e (u)5u ^ u2G(u)•1 where uPF and 1 is the unity element of algebra L~F !. Every element from I(G) can be written as S i l i e (u i ) m i , where l i , m i PL~F ! and u i PF . Let C(G)5L~F !/I(G) and define j:F →C(G) as the composition of monomorphism i:F →L 1~F !,L~F ! and projection p:L~F !→C(G). In this case pair (C(G), j) is the solution of our problem. From the general properties of tensor algebras the homomorphism w:F →A can be extended to L~F !, i.e., the diagram

is commutative, where r is a monorphism of algebras. Because ( w (u)) 2 5G(u)•1, then r vanishes on ideal I(G) and in this case the necessary homomorphism t is defined. As a consequence of uniqueness of r, the homomorphism t is unique. Tensor d-algebra L~F ! can be considered as a Z/2 graded algebra. Really, let us introduce L ~0!~F !5( `i51 T 2i ~F ! and L ~1!~F !5( `i51 T 2i11 ~F !. Setting I ( a ) (G)5I(G)ùL ~a!~F ! @~a! 5~1!,~2!#, we have I(G)5I (0) (G) % I (1) (G). Define C ( a ) (G) as p~L ~a!~F !!, where p:L~F !→C(G) is the canonical projection. Then C(G)5C (0) (G) % C (1) (G) and in consequence we obtain that the Clifford d-algebra is Z/2 graded. It is obvious that Clifford d-algebra functorially depends on pair ~F , G!. If f :F →F 8 is a homomorphism of k-vector spaces, such that G 8 ( f (u))5G(u), where G and G 8 are, respectively, metrics on F and F 8, then f induces an homomorphism of d-algebras C ~ f ! :C ~ G ! →C ~ G 8 ! with identities C( w • f )5C( w )C( f ) and C(IdF ) 5 IdC(F ) . If Aa and B b are Z/2-graded d-algebras, then their graded tensorial product Aa ^ B b is defined as a d-algebra for k-vector d-space A a ^ B b with the graded product induced as (a ^ b)(c ^ d)5(21) ab ac ^ bd, where bPB b and cPAa~a,b50,1!. Now we reformulate for d-algebras the Chevalley theorem.19 Theorem 2: The Clifford d-algebra C(hF % v F ,g1q! is naturally isomorphic to C(g) ^ˆ C(q).

n:hF →C(g) and n 8 : v F →C(q) be canonical maps and map m:hF % v F →C(g) ^ˆ C(q) is defined as m(x, y)5n(x) ^ 111 ^ n 8 (y), xPhF , yP v F . We Proof:

Let

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Sergiu I. Vacaru: Spinors in locally anisotropic spaces

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have (m(x, y)) 2 5[(n(x)) 2 1(n 8 (y)) 2 ]•15[g(x)1q(y)]. Taking into account the universality property of Clifford d-algebras we conclude that m induces the homomorphism

z :C ~ hF

% vF

,g % q ! →C ~ hF ,g ! ^ˆ C ~ v F ,q ! .

We also can define a homomorphism

y :C ~ hF , g ! ^ˆ C ~ v F , q ! →C ~ hF

% vF

, g % q!

by using formula y (x ^ y)5 d (x) d 8 (y), where homomorphisms d and d8 are, respectively, induced by imbeddings of hF and v F into hF % v F :

d :C ~ hF , g ! →C ~ hF

% vF

,g % q ! ,

d 8 :C ~ v F , q ! →C ~ hF

% vF

,g % q ! .

Because xPC ( a ) (g) and yPC ( a ) (q), d (x) d 8 (y) 5 ( 2 1) a 8 d 8 (y) d (x). Superpositions of homomorphisms z and y lead to identities

y z 5IdC ~ hF , g ! ^ˆ C ~v F , q ! ,

~13!

z y 5IdC ~ hF , g ! ^ˆ C ~v F , q ! . Really, d-algebra C(hF

%vF

,g1q! is generated by elements of type m(x, y). Calculating

y z ~ m ~ x, y !! 5 y ~ n ~ x ! ^ 111 ^ n 8 ~ y !! 5 d ~ n ~ x !! d ~ n 8 ~ y !! 5m ~ x, 0! 1m ~ 0, y ! 5m ~ x, y ! , we prove the first identity in ~13!. On the other hand, d-algebra C(hF , g) ^ˆ C( v F , q! is generated by elements of type n(x) ^ 1 and ( z y )(1 ^ n 8 (y)) and 1 ^ n 8 (y). Because ( z y )(n(x) ^ 1)5 c ( d (n(x))5n(x) ^ 1 5 c ( d 8 (n(y))51 ^ n 8 (y), we prove the second identity in ~13!. Following from the above-mentioned properties of homomorphisms z and y we can assert that the natural isomorphism is explicitly constructed. h In consequence of theorem 2 we conclude that all operations with Clifford d-algebras can be reduced to calculations for C(hF , g! and C( v F , q! which are usual Clifford algebras of dimension 2 n and, respectively, 2 m .16,20 Of special interest is the case when k5R and F is isomorphic to vector space Rp1q,a1b provided with quadratic form 2x 21 2•••2x 2p 1•••1x 2p1q 2y 21 2•••2y 2a 1•••1y 2a1b . In this case, (x) (x) (y) the Clifford algebra, denoted as (C p,q , C a,b ), is generated by symbols e (x) 1 , e 2 ,...,e p1q , e 1 , (y) (y) 2 2 2 e 2 ,...,e a1b satisfying properties (e i ) 521 (1