Squared Weyl and Dirac fields with the Sommers-Sen connection associated with ... distribution

Generalization of the Sommers–Sen spinor connection for spinor fields, associated with the distribution V 3 4 is made and on its basis the equations for Weyl and Dirac null vector fields on complexificated V 3 4 are obtained. We interpret the obtained results by examining the interaction of spinor fields with inertial forces.


Introduction
As it is known, the four-dimensional description of relativistic elds for a number of problems, especially for the comparison of theoretical provisions and experimental results, must be substituted by a 3+1 description.For the spinor elds the original method of their description in the 3+1 form was proposed in 1] and developed in 2].The obtained in 2] Sen{Witten equation is widely used in the gravitational eld energy problem investigations 3{5].But this does not allow us to study all the variety of physical e ects in the interactions of spinor elds with inertial forces, because this method is based on the foliation of curved space-time by space-like hypersurfaces and thus on nonrotatory frames of reference.In our work 6] we introduced a covariant derivative of spinor elds associated with the space-like distribution V 3  4 , which generalizes the Sommers{Sen covariant derivative and obtained on this basis a 3+1 equation for Weyl and Rarita{Schwinger elds in arbitrary frames of reference, not only in nonrotatory ones.In this work we obtain for an arbitrary frame of reference the equations for a complex 3-vector, which are correspondent to Weyl spinors and bispinors.These squared equations of Weyl and Dirac elds are, in fact, a 3+1 splitting of the Penrose{Rindler 7] tensor form of spinor di erential equations.

V.Pelykh
In section 2 we brie y review the technique for obtaining 3+1 spinor equations with a spinor connection on nonintegrable manifolds.In section 3 we give a spinor representation of tensor elds on the V 3  4 distribution.

Generalization of the Sommers-Sen derivative on distributions
Let us consider the oriented manifold V 4 of class C (X; X 1 ) ?! D X X 1 ; X;X 1 2 ?(S); which for arbitrary vectors X;X 1 and functions g; f 2 C 1 (V 4 ) satis es the conditions D X (X 1 + X 2 ) = D X X 1 + D X X 2 ; (1) D X (fX 1 ) = X(f)X 1 + fD X X 1 ; (2) D fX+gX 1 X 2 = fD X X 2 + gD X 1 X 2 : (3) is called a covariant derivative or anholonomic connection on the distribution V m 4 .The homomorphism ?(B 0 ) : ?(TV 3 ) ?! ?(V m 4 ) of cross-sections is correspondent to projections B 0 of bundle TV 4 on subbundle V m 4 : 4 be a one-dimensional time-like distribution over V 4 ; its unitary crosssection u is identi ed with the eld of 4-velocity of some frame of reference, and the integral curve of cross-section u | with its time lines.The normal rigging V 3 4 of the distribution V 1  4 is a geometrical image of the physical space for the appropriate frame of reference and is nonintegrable in general.The vectors, which belong to V 1  4 ; are called time vectors, and those belonging to V 3 4 | spatial vectors.Further we require that the second Stiefel{Whitney class w 2 of manifold V 4 equal zero.Then V 4 permits the SL(2; C)spinor structure.Let us denote by S r;s (V 4 ) the C 1 module of spinor elds of (r; s) valence on V 4 : Let us introduce the C 1 module of S r;s (V 3 4 ) spinor elds of (r; s) valence on V 4 associated with the distribution V 3  4 in the following way: its elements are spinor elds of the form 'T A:::L M:::Q and The de ned in such a way C 1 module of S r;s (V 3 4 ) is a module of SU(2) spinor elds.Let us call the module S r;s (V 3 4 ) with the basis limited to the hypersurface a module of SU(2) spinor elds on the anholonomic hypersurface .In a particular case, when this hypersurface is ordinary and space{like, the module of SU (2) spinor elds on it coincides with the module of Sommers{Sen spinor elds.
We introduce the antisymmetric tensor A of anholonomicity of V 3 4 ; A 2 V 3 4 .Let T : ?(V 3 4 ) ?(V 3 4 ) ?! ?(P);T = ?(C0 ) X 1 ; X 2 ]; X 1 ; X 2 2 ?(P):Then T = 4A u.In the coordinate basis on some open domain in V 4 tensor A has the components 8] A = 1  2 h h u u ] : Let us introduce a spatial covariant derivative for spinor elds associated with the distribution V 3  4 as the mapping ?(V 3 4 ) S 1;0 (V 3 4 ) ?! S 1;0 (V 3 4 ); (X AB ; C ) ?! D AB C ; X AB 2 S 1;0 (V 3 4 ) S 1;0 (V 3 4 ) ; (4) determined by the condition where r B _ A is a spinor representation of an operator of the covariant derivative on V 4 ; in agreement with the metrical connection.The action D AB on spinors of a higher valence extends in accordance with the Leibnitz rule and the action on vector elds satis es the condition (1){(3).Reducing the SL(2; C) operator of the covariant derivative to the SU(2) operator, we obtain: The rst term denoted by p 2 2 " AB (u r) is a time derivative, the second one is represented in terms of space derivative D AB in the rigging.Finally, we obtain the action of r AB on spinor C in the form: We obtain the generalized 3+1 form of the Weyl equation r A _ A A = 0 carrying out the SL(2; C) !SU(2) reduction and using ( 6).Then we have Therefore, the Weyl spinor A 2 S 1;0 (V 3 4) is determined by the geometric properties of V 4 and by both the geometric and equally physical properties of V 3  4 .These properties are determinated by the acceleration spinor F AB 2 S 2;0 (V 3 4 ), the angular velocity spinor A ABCD 2 S 4;0 (V 3 4 ) and the rate{of{strain spinor ABCD 2 S 4;0 (V 3 4 ).These spinors are uniquely expressed by the Schouten rst order curvature tensors of V 3 4 and vector u 2 V 1 4 .

Spinor representation of tensor fields on the V 3 4 distributions
Let spinor eld T 2 S 2r;2s (V 3  4 ).If T is symmetric in all pairs of indices, then T 2 V 3 4 .The projector from TV 4 into V 1 4 is u u, the projector from TV 4 into V 3   4   is h = g + u u.
It is easy to characterise the SU(2) representation of a space projected tensor The overline denotes the components of space projected tensors, the symbol denotes the components of time projected tensors, A _ A are the Pauli spin matrices and the unit matrix which are referred to as a space{time tetrad.The AB and l AB matrices are given by the formulas and, respectively, For the matrices AB we obtain the necessary in the following consideration identities which substitute for the normalization and orthogonalization identities of the Pauli matrices: We obtain the matrices m CA in the form m CA = p 2(n 0 !m CA + n 1 m CA + n 2 m CA + n 3 m CA ); l C B = l CA " BC ; where The rst sum in (8) equals zero, therefore, further we consider only the second sum in which we distinguish the terms with the products y 1 z 2 .In this case the products of the necessary matrices give: Let X = ?A B and Y = ?A B be two complex null spatial vector elds.Then, the squared Dirac equation, which describes the evolutionof X and Y elds and is written in terms of the tensors determined on the distribution and in the rigging, is the system of two equations: < dX;u >= ?X ?

Discussion
The proposed in 6] and in this paper method for the investigation of an interaction between spinor elds and inertial forces requires the use of a nonintegrable subbundle.Unlike 1,2], the spinor derivatives are determined here by the intrinsic geometry of distribution.This is de ned by the physical sense of the problem and not by the application of the tetrad formalism which is not necessary here.In a particular case of the integrable V 3  4 distribution, both the tetrad and monad methods determine the spinors in terms of the intrinsic geometry of foliation.
We ascertain the appearance of additional di erences between the evolutions of the Weyl, Dirac and Maxwell elds, since the interaction of these elds with the inertial eld is described by the term which includes not only the angular velocity vector of the frame of reference, but also its angular velocity.

4 :
Let us also obtain the necessary for further investigations tensor representation of the spinor form y C(A z B)