The content of thi article where publihed in General Relativity and Gravitation, Vol. 24, p. 501, 1992 PHYSICAL GEOMETRY AND FIELD QUANTIZATION Gutavo González-Martín Departamento de Fíica, Univeridad Simón Bolívar, Apartado 89000, Caraca 1080-A, Venezuela. Web page URL http:\\prof.ub.ve\ggonzalm\ A geometrical model for the proce of field quantization i propoed, within the context of the unified theory of connection and frame introduced in previou publication. A phyical geometry produced by matter determine locally a canonical bracket operation for generalized Jacobi vector field and provide a geometrical model for the exitence of fermionic and boonic operator field end their rule of quantization.
1. Introduction. FIELD QUANTIZATION. We may raie the following quetion: Can we obtain quantum field theory from thi geometry without recurring to claical or quantum mechanic? Thi quetion i in line with the ideal apiration indicated by Schrödinger [1 ]. The reult of previou work, which provide a poible geometric unification of gravitation, electromagnetim and Dirac equation, give u upport in the hope of obtaining more intereting reult from further analyi of the propoed geometrical theory, in particular in connection with quantum phenomena. It may be hown that the ection admit an interpretation imilar to generalized wavefunction. The procedure of quantum mechanic to determine phyical reult may arie canonically from the geometric propertie of the theory. In quantum field theory the field are conidered operator and form an algebra. It appear poible that thi algebra arie from geometrical propertie of ection in the principal bundle, which are propertie related to group element. Here we dicu thi quetion, which i intereting ince the group i aociated to a Clifford algebra with an anticommuting algebraic tructure. The potulate of Newton claical mechanic [2 ] are baed on the concept of point particle and the free motion along traight line of Euclidean geometry. A claical field i uually een a a mechanical ytem where claical mechanic i applied. With the advent of General Relativity and Gauge Theorie, there hould be a recognition that the geometry of the phyical world i not a imple a that provided by claical Greek geometry. The idea of a phyical geometry determined by the ditribution of ma and energy [3,4 ] i attractive a a criterion for etting the fundamental law of nature. It wa natural when quantum mechanic wa born [5,6 ] to bae it development on claical mechanic and Euclidean geometry. Potulate dicloed the difficultie with the imultaneou meaurement of poition and velocity of a point particle. Another approach, with hindight, may be the realization that a point i not an appropriate geometrical element on which to uperpoe phyical potulate. The concept of field i cloe to modern geometrical idea and point in the direction of etablihing the phyical potulate on a general geometry, away from prejudice introduced by claical geometry and claical mechanic. In conequence, we dicu a generalization of field theory, directly to geometry, bypaing the intermediate tep of mechanic. Why hould we introduce thee intermediate concept which hitory of phyic proved need reviion in relativitic and quantum mechanic? Mechanic may be een, rather than a a fundamental theory, a a implification when the evolution of matter can be approximated a the motion of a point particle.
3 2. Linearization of Field. In the theory of connection and frame developed in previou chapter, where the fundamental equation are D Ω = k J = 4πα J, ( 2.1) 1 J = ε e~ αβγµ κ edx dx dx 3! µ α β γ, ( 2.2) µ α µ 2κ e+ κ u$ e = 0, ( 2.3) µ $ µ α the matter field are repreented by the coframe e which i a ection in a principal bundle E. The interaction i repreented by a connection with curvature form Ω. Both the frame and the connection are determined by the equation, in term of an orthonormal frame u and an orthonormal et κ that generate the geometric algebra [7 ] aociated to pacetime. If we introduce the bundle of connection W, a connection may be taken a a ection of thi affine bundle. When we conider an equation relating the connection and the frame, we are dealing with differentiable manifold of ection of the fiber bundle E, W and nonlinear differential operator which define differentiable map between thee manifold of ection. The technique neceary for attacking thi problem i known a global nonlinear analyi [8 ]. Generally, it i accepted that the proce of quantization require the exitence of a claical mechanical theory which i quantized by ome fundamental rule. Intead, we think of the phyically dependent geometry giving rie directly to field and in an approximate way to both claical and quantum theorie. We make the conjecture that the proce of field quantization i the technique of replacing the nonlinear problem, jut indicated, by a linear problem obtained by variation of the nonlinear map, reducing Banach differentiable manifold of ection to Banach linear pace and the nonlinear differentiable map between manifold of ection, to linear map between Banach pace. From a geometrical point of view, thi mean working at the tangent pace of thee ection manifold at ome particular point (ection). Certain manifold, convenient to deal with bundle ection, are the jet bundle of order k, indicated by J k E, which are, eentially, manifold of ection equal at a point modulu derivative [9] of higher order k+1. Of interet i the manifold of olution γ, which i the ubmanifold of all the ection that atify a given nonlinear differential equation. Section obtained from a olution by the action of the tructure group are equivalent olution. The quotient of γ by thi equivalence relation are the phyical olution.
4 If we have a variational problem, it critical ection (olution) may be characterized geometrically a follow: A ection i critical if and only if the Euler-Lagrange form i zero on the 1-jet prolongation [10] j of the ection, ( D f ) Λ + η = 0. ( 2.4) j The et of all the critical ection form the differentiable manifold γ of olution of the variational problem. On thi manifold there are vector field whoe flow generate the pace of olution. Thee field are ection of the tangent pace Tγ. Intead of tudying γ it i poible to tudy the linear pace Tγ. We may introduce in Tγ a vector ψ repreenting a olution of the aociated varied equation. A quantum operator field may be related to a Jacobi field on the bundle in conideration. Really a olution to the equation correpond to a mapping between two uch manifold of olution, one γ E, correponding to the bundle E, repreenting a matter frame olution and the other γ W correponding to the bundle W, repreenting the interaction connection olution. A ingle manifold of olution may be obtained by combining γ E and γ W, within a product manifold related to E and W, but thi approach lead to unneceary mathematical complication for our purpoe. Here we hall conider both manifold eparately. 3. Frame Solution. For the manifold of frame olution γ E we hall conider it tangent Tγ. We define a Jacobi vector field a a ection V, of *T v E, induced by the ection from the vertical ub-bundle of TE, uch that the correponding 1-jet prolongation atifie ( ) L jv DΛ + fη = 0. ( 3.1) Here indicate the Lie derivative. The pace of all the Jacobi vector field form the tangent pace of γ at a given ection, denoted by Tγ. The lat equation i a linearization of the nonlinear field eq. (2.4) which i eq.(5.2.1) in another form. A Jacobi field may be een a the tangent vector to a differentiable curve of olution t of the nonlinear equation at a given olution. When we have uch a curve, we may conider it to be the integral curve of an extended Jacobi vector field V which take the value V, at each point of the curve. The fiber of the bundle E i the group SL(2,Q). The Lie algebra A of thi group i enveloped by a geometric Clifford algebra with the natural product of matrice of the orthonormal ubet of the Clifford algebra. The element of A may be expreed a fourth degree polynomial in term of the orthonormal ubet of the Clifford algebra.
5 When variation are taken, a uual, in term of the fiber coordinate, the algebraic propertie of the fiber are not brought to full ue in the theory. In order to extract the additional information contained in the fiber which, apart from being a manifold, i related to algebra A, we note that in many cae we have to work with the fiber repreentative by the bundle homeomorphim. We are dealing then with element of G. The vertical pace of *T v E are homeomorphic to the vertical pace of TG. Thi mean that the fiber of *T v E i A, taken a a vector pace. The aociation of the algebraic tructure of A to the vertical pace of *T v E depend on the image of the ection in G given by the principal bundle homeomorphim. When an oberver frame i choen, the local homeomorphim i fixed and the algebraic tructure of A may be aigned to the vertical pace. A variation of the oberver h produce an effect equivalent to a variation of the ection, and hould be taken into conideration. The total phyical variation i due to a variation of the ection and/or a variation of the oberver in the compoition mapping ho, defined on a local chart U ho U : U ( G G). ( 3.2) A variation of ho i repreented by a variation vector, a generalized Jacobi vector, valued in the Clifford algebra. We are dealing with a double algebraic tructure, one related to the algebra A, of the fiber of TE and the other to the Lie algebra of vector field F on the jet bundle JE. Thi allow u to aign a canonical algebraic tructure to the Jacobi field, which hould be defined in term of thee natural tructure in each of the related pace. For intance, if we ue both algebraic tructure and calculate the following Lie bracket of vector, 0 1 [ xκ µ yκ ν],, ( 3.3) where µ, v are coordinate on the jet bundle and x, y the component, the reult i not a vector becaue of the anticommuting propertie of the orthonormal et. Neverthele, if we calculate the anticommutator, the reult i a vector. In particular, it i alo known [11 ] that the Clifford algebra, taken a a graded vector pace, i iomorphic to the exterior algebra of the aociated tangent pace, which in our cae i *TM. There are two canonical product, called the exterior and the interior product, that may be defined between any two monomial in the algebra uing the Clifford product. For any element κ α of the orthonormal et the product of a monomial a of degree p by κ α give a mapping p p+ 1 p κ α : A A + A 1. ( 3.4)
6 The firt component of the map i the exterior product κ α a and the econd component i the interior product κ α a. Then we may write καa= κα a+ κα a. ( 3.5) Due to the aociative property of the Clifford product we may extend thi decompoition to product of monomial. We may define the grade of the product αβ, indicated by gr(αβ), a the number of interior product in the Clifford product. The grade i equal to the number of common element in the monomial. For example, (κ 0 κ 1 ) (κ 2 κ 0 κ 3 ) i of grade one. Thi decompoition may be applied to the product of any two element of A. It i clear that the maximum grade i the number of orthonormal element and that the product of grade zero i the exterior product. The ue of the exterior product intead of the Clifford product turn the algebra A into a Gramann algebra iomorphic to the exterior algebra of differential form on the tangent pace. Thi fact lead u to look for gradation of the Lie algebra tructure [12 ]. The tructure group of the theory wa choen a the imple group of inner automorphim of the Clifford algebra. In thi ene, the group, it Lie algebra and their repective product arie from the Clifford algebra. In fact the Lie bracket i equal to the commutator of Clifford product. To avoid an unneceary factor of 2 we may define a Lie product a the anti ymmetrized product (1/2 of the commutator). We may conider that the element of A alo atify the potulate to form a ring, in the ame manner a the complex number may be conidered a an algebra over R or a a field. In thi manner, the element of A play the role of generalized number, the Clifford number. The candidate for the ring product are naturally the geometric Clifford product, the exterior Gramann product, the interior product and the Lie product. A mentioned, all thee product arie from the Clifford product. In fact, for any two monomial α, β A, all the other product α β are either zero or equal to the Clifford product αβ. The geometric Clifford product i more general and fundamental, and the other may be obtained a retriction of the Clifford product. Beide, the Lie product i not aociative and may need generalization of the algebraic tructure. On the other hand, the fiber bundle of frame, E, i a principal bundle and it vertical tangent bundle TE ha for fiber a Lie algebra tructure inherited from the group, it would be more natural that the choen product be cloed in the algebra o that the reult be alo valued in the Lie algebra. For the moment, we take the more general geometric Clifford product a the product of the ring tructure of A and we hall pecialize to other product when needed. It i convenient, then, to work with the univeral enveloping aociative algebra of the Lie algebra of vector field F. In thi manner we have an aociative product defined and we may repreent the bracket a commutator of the element. The vertical vector on JE may be taken a a module over the ring A. Uing the canonical algebraic tructure we may define a bracket operation on the element of the product A F. For any monomial α,β,γ
7 A and V,W,Z F, define {, } [, ] αv βw = α β V W = γz, ( 3.6) turning the module into an algebra A. Thi bracket atifie { αv, βw} ( ) αβ { βw, αv} = 1, ( 3.7) γα αβ ( 1 ){ αv, { βw, γz} } ( 1 ) βw, { γz, αv} { } βγ ( ) { γz, { αv, βw} } + + 1 = 0, ( 3.8) where αβ i the gradation of the product αβ or the the bracket, equal to the number of permutation given by αβ = α β gr ( αβ), ( 3.9) in term of the degree α of the monomial in A and the grade of the product αβ. For any two element ψ, Φ of the algebra A which are polynomial in A, the bracket i defined a the um of the bracket of the monomial component, { } ( ) ( ) { φψ, } = φ n, ψ m nm,. ( 3.10) Another way of looking at thi bracket operation i to conider that the action of a vector ψ on a calar field f on M give a ection a of the bundle AM with A a the fiber, ( ) Ψ f a α = ψ Ea α f = a. ( 3.11) With thi undertanding, the bracket of vector monomial in A may be defined by it action on function, a follow: ( ) ( ( ) ( ) ( ) ( )) ( n) ( m) ( n) ( m { } ) ΦΨ m n Φ Ψ Φ Ψ ( ) Ψ Φ, f = f + 1 f. ( 3.12) Thi definition agree with the previou one, eq. (3.6). In order to ee thi, we decompoe the vector in term of a bae and apply the firt definition. The previou formula are valid in general if we retrict the Clifford product to the exterior or the interior product taking the proper number of permutation for each cae. For example, in the cae of Gramann product, gr(αβ) i alway zero. If we retrict to the Lie product, the bracket i alway ymmetric in eq. (3.6) and in the graded Jacobi identity, eq. (3.8).
8 4. Connection Solution. For the manifold of connection olution γ W, we may proceed imilarly, taking in conideration that the bundle of connection W ha a different geometrical tructure than the bundle of frame E. Conider a principal bundle p:e M with tructural group G and correponding Lie algebra A. Denote by T G E the bundle of G-invariant vector field on E. Denote by AdE the adjoint bundle of E, which i the fiber bundle aociated with E by the adjoint repreentation of G. It i the ubbundle of T G E defined by G-invariant vector field that are tangent to the fiber (vertical). It i a bundle of Lie algebra. A connection on the principal bundle may be defined by an plitting of the hort exact equence of vector bundle [13 ], π 0 AdE TG E TM 0, ( 4.1) where the plitting ω:tm TG E = H V ( 4.2) i a homomorphim defining the horizontal ubpace of the principal bundle. Thee horizontal ubpace define a connection form ω. A connection on E may be identified with a ection of the bundle of connection π:w M defined a follow. For a point m M, let W m be the et of homomorphim ω: TM TE uch that π o ω = I. ( 4.3) Define W = U M W m letting π be the natural projection of W onto M. It may be een that each point w m, of the vertical pace W m of the fiber bundle W correpond to a vector pace complement of AdE in T G E. It i known [7] that the pace of linear complement of a vector ubpace in a vector pace ha a natural affine tructure. Therefore the fiber of the bundle W i an affine pace with linear part L(T G E / AdE, AdE) L(TM, AdE). We define Jacobi vector field V, a ection of T v W and aociate quantum operator to the prolongation of the extended Jacobi vector. A difference arie, in thi cae, becaue the algebraic tructure of the fiber of W, which i not a principal bundle, differ from that of the fiber of E. A connection in E i defined by giving a ection in W. Thi define horizontal vector ubpace in TE. The fiber of the manifold of connection W i the pace of complement of the vertical pace in T G E e. The fiber of W i an affine pace and we ay that W i an affine bundle. Intuitively, we may conider an affine pace a a plane P of n dimenion without a
9 defined origin. The affine tructure of the pace allow that any point o P be defined a origin, turning the pace into a vector pace V, Θ 0 : P V. ( 4.4) The algebraic tructure of the tangent pace to the fiber of W i then iomorphic to TP p and conequently iomorphic to V with the uual operation of addition of vector. Correpondingly the vertical vector on JW form a vector pace over a commutative ring and the bracket defined in the previou ection reduce to the Lie bracket, ince no ign permutation arie under commutation. 5. Bracket a Derivation. In both cae we have that the bracket i defined uing the natural geometric product related to the algebraic tructure of the fiber of the correponding bundle E and W. It i clear that the bracket i a derivation, { } { } { } ΞΦΨ, = ΞΦΨ, + ΦΞΨ,, ( 5.1) and we have, therefore, a generalized Lie derivative with repect to the prolongation of Jacobi field given by { jv jw} D jv jw =,, ( 5.2) where the bracket i the anticommutator or the commutator depending on the number of permutation for frame and i alway the commutator for connection. Phyically thi mean that matter field hould be fermionic and that interaction field hould be boonic. The quantum operator in quantum field theory may be identified with the prolongation of the extended Jacobi vector field. Thee bracket, eq. ( 5.2) are equivalent to the potulate of quantum theory that give the change in ome quantum operator field Φ produced by the tranformation generated by ome other operator Ψ. In the preent context, thi equation i not a eparate potulate, but rather it i jut the reult of taking the D derivative with repect to a direction tangent to a curve in the manifold of olution γ and it i due to the geometry of the bundle. The D derivative may be generalized to tenorial form valued over the ring A. Thee derivative repreent the variation of ection along ome direction in Tγ, which correpond to a generator of ome tranformation on the jet bundle along a vertical direction. The complete geometrical formulation of thi phyical variational problem may be carried by contructing the extenion of the vertical vector pace T v E e to a module over the ring A. In a manner imilar to the complexification of a real vector pace we conider the dual (T v E e ) and define mapping
10 v ( e ) V: T E A. ( 5.3) Thee mapping are element of a right A-module which we deignate by A T v E e. If we form the union of thee pace over the manifold E we get a fiber bundle A T v E e over E. Further, we may contruct the bundle AT v JE e over JE and T v E over M. The tandard geometric verion of variation may be generalized to thee phyical variation by ubtituting appropriately thee bundle for the bundle T v E, T v J and T v E repectively, uing the D derivative and keeping track of the non commutative product. [14] 6. Geometric Quantum Field Theory. The operator bracket lead to the quantization relation of quantum field theory uing Schwinger action principle. The generator of a variation F may be written in the jet bundle formalim by the appropriate term in eq. (2.9), appendix C, F jv Π. ( 6.1) It may be een that the element entering in the expreion are tenorial form on the jet bundle that inherit the algebraic propertie of the fiber and therefore have the propertie of operator. The generating function F determine the variation a indicated in eq. ( 5.2). We hall write the generator F in the equivalent tandard expreion [15 ] ued in quantum theory obtaining ( ) { Ψ } ( ) µ δψ x = F =, ψ x, Π δψdσ µ, ( 6.2) σ for the field operator Ψ variation. In thi expreion we recognize that the element to be quantum operator, a defined above, which obey the defined bracket operation. If we expre the bracket in thi relation a in eq. (5.1), we get µ { Ψ Π } ( ) µ Π ( ){ Ψ( ) ( )} µ ( ) ( ) ( ) δψ x = x, y δψ y dσ ± y x, δψ y dσ µ. ( 6.3) σ It i clear that the reultant commutator for the connection operator and anticommutator or commutator for the frame operator are none other than the quantization relation in quantum field theory for boon field and fermion field. The lat equation implie that { Ψ () x () y } σ,δψ = 0, ( 6.4)
11 { Ψ( x ), Π µ ( y )} δ µ ( x, y ) =, ( 6.5) where the bracket i interpreted a the commutator or anticommutator depending on the gradation of the bracket according to the type of field. From thi formulation, the geometrical reaon for the exitence of both boonic and fermionic field alo become clear. Another advantage of thi geometric formulation i that the operator nature of the field may be explicitly given in term of the tangent vector operator on the pace of olution and the Clifford algebra matrice. A tangent vector may be conidered a an infiniteimal action on function on a manifold. In particular for γ, the manifold of ection that are olution, the quantum operator may be conidered to act on a olution ection, called the ubtratum or background ection, producing olution perturbation, called quantum excitation. The ubtratum or background ection may take the place of the vacuum in conventional quantum field theory. If the manifold of ection i taken a a Hilbert manifold it tangent pace are Hilbert linear pace. Thi mean that the quantum operator, which act on tangent pace, would act on Hilbert pace a uually aumed in quantum theory. If the manifold in quetion admit local harmonic ection we may introduce fundamental harmonic excitation. The energy of thee field excitation ytem may be defined in the uual manner leading to the concept of a quantum particle. The contruction of creation, annihilation and particle number operator of field excitation may alo be carried out tarting from thi geometrical approach. Neverthele, if thi geometric model i taken eriouly, field (econd) quantization reduce to a technique for calculating perturbation to exact olution of theoretical geometrical equation. There may be other technique for calculating thee perturbation. In fact, it i known that quantum reult may be obtained without quantization of the electromagnetic field. See, for example, an alternate technique to QED [16 ] where the elf-field i taken a fundamental. In General Relativity, the elf-field reaction term do not appear a eparate feature in the non-linear equation but they appear in the linearized equation obtained uing a perturbative technique. Once the geometrical non phenomenological tructure of the ource term J i known, the exact equation of motion for the field decribing matter, including particle, would be the conervation law for J. Thi relation, being an integrability condition on the field equation, include all elf-reaction term of the matter on itelf. There hould be no worrie about infinitie produced by elf reaction term. A phyical ytem would be repreented by matter field and interaction field which are olution to the et of imultaneou equation. When a perturbation i performed on the equation, for example to obtain linearity of the equation, the plitting of the equation into equation of different order bring in the concept of field produced by the ource, force produced by the field and therefore elf-reaction term. We hould not look at elf reaction a a fundamental feature but, rather, a an indication of the need to ue non-linear equation.
12 7. Summary. It wa hown [17] that if we take into conideration the geometrical tructure of the fiber bundle E or W, related to the algebraic tructure of their fiber, a proce of variation of the equation of the theory lead to an interpretation of the extended Jacobi field a quantum operator. It i poible to define a bracket operation which become the commutator for the Jacobi field aociated to the connection and become the anticommutator or commutator, according to the gradation, for thoe aociated to the frame. Thi bracket operation lead the quantization relation of quantum field theory for boonic interaction field and fermionic matter field. 8. Jet Bundle. APPENDIX. Conider a C º bundle E. Let S(E) denote the pace of all ection of E and C k (E) the ubpace of C k ection. Let e E be given uch that π(e) = m and U be an open neighborhood of m with local coordinate x µ and uch that there exit a trivialization ϕ: EU U Em (8.1) and let W be an open neighborhood of e m, in the fiber E m with local coordinate y i. Indicate partial derivative of order α by α α = α1... αn x1 x, (8.2) n where α = {α 1,...α n,} are n-tuple of nonnegative integer and α =α 1 + +α n. We define an equivalence relation on the ection at a point, by tating that (x, S 1 (x)) i equivalent to (x, S 2 (x )) if x = x, (8.3) α i α i ( y ( 1) ) = y ( 2) m ( ) m (8.4) and indicate the quotient of S(E) by thi equivalence relation by J k (E) m. For a ection let j k (m) be the cla of ection with equal derivative up to order k, at m. Uniquely we can take the J k (E) m a fiber of a C vector bundle J k E over M o that j k C (J k E) for all ection C (E), which we call the k th order jet bundle of bundle E [18, 19]. The linear map
13 k ( ) ( ) j : C E C J E k (8.5) i called the k-jet prolongation map of a ection. Two equivalent ection of J k E have the ame partial derivative at the pecified point. The J 1 E ha for vertical pace at a point, all clae of ection that have the ame value and the ame firt partial derivative at the pecified point. There i a natural projection p from J k E to E defined by ending a point j k m, J k E over m M to the point of E that ha for fiber at m the order zero equivalence cla of j k m. We have ( ) p j =. (8.6) k m m The vertical pace J k E e i the et of all clae of ection, defined near m=π(e), uch that (m)=e, relative to the equivalence relation 1 2 if and only if ( k 1 ) m ( k 2 ) m. The element of J k E e defined by the ection i called the k-jet of at m, indicated by j k m The function (x µ,y i,z i µ ) defined on p -1 [ϕ -1 (U W)] by the rule ( ) ( ) µ µ x j1 = x m, (8.7) m ( ) i ( ) ( ) i y j1 = y m, (8.8) z m i y 1 m = ( ), (8.9) x ( j ) m ( ) i µ µ where y i = y i (x µ ) are the equation of the ection relative to the coordinate x µ, y i and the trivialization ϕ that we have choen, form a ytem of local coordinate on J 1 E, called canonical coordinate on J 1 E. It can be een that thi formalim provide a natural etting for the dicuion of differential equation. One can think of a differential equation ytem a defined by a et of relation of the form µ i i i i ( µ µν µν ξ ) F x, y, z, z... z... = 0, (8.10) A where the z coordinate are linked to the partial derivative of the independent variable y i in term of the dependent variable x µ. Geometrically, the function F A repreent urface in the jet bundle J k E determining a ubpace S J k E. A olution of the ytem may be thought of a a ection in J k E atifying j k S. (8.11) It i poible to introduce a fundamental form aociated to the jet bundle that may be ued to geometrically characterize variational problem. For thi purpoe we hall follow the work of García [20, 21, 22]. Let u tranport the fiber of TE v to J 1 E contructing the
14 induced bundle p*te v a follow. For a given point j(m) J 1 E we obtain a point of E, p( j ( m) ) ( m) 1 = E. (8.12) Let p*te v be the et {j 1 (m), v (m) } where v (m) TE v indicate a vector at (m) atifying ( ) ( ) m p( jm ( ) 1 ) = qv ( ) = m, (8.13) where q : TE v E. The bundle p*te v ha the projection v 1 q : p TE J E q ( jm ( ) v ( )) = jm ( ) 1 m 1, (8.14),. (8.15) Now we can define a differentiable 1-form on J 1 E, called tructure form Θ, valued on p*te v. If U i an open neighborhood of e in J 1 E with canonical coordinate (x µ,y i,z i µ) then p* / y i i a bae for the fiber of p*te v and Θ ha the following expreion: Θ:TJ 1 E p T v E, (8.16) a 1 Θ( V) = θ ( V) p V TJ E a, (8.17) y where the θ i are the ordinary 1-form a a a α θ = dy z dx. (8.18) α It follow then that the 1-jet prolongation j l of a ection on E i the unique ection on J 1 E uch that po j 1 =, (8.19) Θ j = 0. (8.20) It i convenient to introduce the following definition. An infiniteimal contact tranformation i generated by a vector field V, a ection of TJ k E, by mean of the Lie derivative, if it atifie L Θ = h Θ, (8.21) V where h i a homomorphim of the fiber of p*te v and i the uual bilinear product. It may be proved that there are prolongation of vector in analogy with prolongation of ection. Let V be a ection in TE. Then there exit a unique p-projectable infiniteimal contact tranformation jv, a ection of TJ 1 E, uch that p jv = V. We hall call thi jv the 1-jet prolongation of V. The map V jv i an injection of the Lie algebra of vector
15 field on E into the algebra of vector field on J 1 E. 9. Critical Section And Jacobi Vector. If we have an orientation η on the bae manifold M which we hall call the volume element of M, we can inject it by (π o p)* in the algebra of differential form on J 1 E. We can peak of the 4-form Lη in J 1 E where L i the lagrangian. We can define a functional on the et of differentiable ection of E by () A = Lη, (9.1) j where j i the 1-jet prolongation of. The functional A i defined on the et S of thoe ection uch that the integral exit. We hall call the differential of the functional A at a given ection S, denoted by δa the linear functional on the pace {j 1 V} of the infiniteimal contact tranformation with compact upport defined by ( ) ( η) δa jv = L L. (9.2) j jv We hall ay that a ection i critical when ( jv) δa = 0. (9.3) We may define an (m-1)-form Λ, called the Legendre tranformation form, by m 1 1 v Λ: TJ E p TE, (9.4) with the following local expreion: Λ= Λ where i i pdy, (9.5) J L = ε dx dx dx α β m Λ i m! z i µαβ µ L L 1. (9.6) We can alo define the Poincaré-Cartan form Π, aociated to the given variational problem a the m-form Π: m 1 TJ E R, (9.7) Π = Θ Λ Lη, (9.8)
16 where the exterior product i taken with repect to the bilinear product defined by the notion of duality. The expreion for the Lie derivative in eq. (9.2) may be calculated. For every infiniteimal canonical tranformation jv we have j V ( Lη ) = Θ( jv ) ( DΛ + fη) d( jv Π ) + Θ Λ L, (9.9) where Λ i a p*te v* valued (m-1)-form on J 1 E and f i a unique ection of p*te v*. The ymbol ¾ indicate the interior product of form. We hall call the reultant m-form DΛ+fη, the Euler-Lagrange form aociated to the given variational problem. Noticing that Θ j = 0 and neglecting the exact form in eq. (9.9) we get from eq. (9.2) that a ection S i critical if and only if the Euler-Lagrange form i zero on the 1-jet extenion of the ection (Euler equation) ( D f ) Λ + η = 0. (9.10) j Given a ection of E, let Ξ be the pace of all ection V of *T v E. By identifying M with (M) by the ection, every V Ξ, define a vertical vector field of E defined on (M). Then there i a vector field V of J 1 E uniquely defined on j 1 (M) uch that pv = V, (9.11) ( L ) V ( ) Θ = 0. (9.12) j M From the definition of jet prolongation we may how that V i the value taken on j 1 (M) by the 1-jet prolongation j 1 V of ome local extenion V of V. If i critical we can define the heian of L at a the bilinear ymmetric functional 2 δ A : Ξ Ξ R, (9.13) δ 2 A ( V V ) = L L ( Lη) = Θ( jv ) L ( DΛ fη) jv jv jv + j j,. (9.14) The kernel of (δ 2 L) i the ubpace of Ξ defined by thoe vector field V Ξ, uch that ( DΛ + fη) 0 L. (9.15) jv = Let u define the Jacobi vector field on the critical ection a thoe vector field that atify the lat equation for the econd variation (or linear varied field equation).
17 Reference 1 E. Schrödinger, Space-time Structure, 1 t ed. (Univerity Pre, Cambridge), p. 1 (1963). 2 I. Newton, in Sir IaacNewton Mathematical Principle of Natural Phyloophy and hi Sytem of the World, edited by F. Cajori (Univ. of California Pre, Berkeley and Lo Angele) (1934). 3 A. Eintein, M. Gromann, Zeit. Math. Phy. 62, 225 (1913). 4 A. Eintein The Meaning of Relativity, 5 th ed. (Princeton Univ. Pre, Princeton), p. 55, 98, 133, 166 (1956). 5 E. Schrödinger, Ann. Phyik 79, 489 (1925). 6 W. Heienberg, Z. Phyik, 33, 879 (1925). 7 I. Porteou, Topological Geometry, (Van Notrand Reinhold, London), ch 13 (1969). 8 R: S. Palai, Foundation of Global Non linear Analyi, (W. A. Benjamin, New York) (1968). 9 See ap. C 10 See ap. C 11 M. Atiyah, Topology Seminar Note, (Harvard Univerity, Cambridge) p. 23 (1962). 12 R. Hermann, preprint HUTP-77/AO12 (1977). 13 P: L. Garcia, Rep. on Math. Phy. 13, 337 (1978). 14 See ap. C 15 J. Schwinger, Phy. Rev. 82, 914 (1951). 16 A. O. Barut, Phy. Rev. 133B, 839 (1964). 17 G. González-Martín, Gen. Rel. Grav. 24, 501 (1992). 18 R: S. Palai, Foundation of Global Non Linear Analyi, (W. A. Benjamin, New York) (1968). 19 R. Hermann, The Geometry of Non Linear Differential Equation, Bäcklund Tranformation and Soliton (Math. Sci. Pre, Brookline) (1976). 20 P. L. García, Symp. Math. 14, 219 (1974). 21 P. L. García, J. Diff. Geom. 12, 209 (1977). 22 P. L. García, Rep. on Math. Phy. 13, 337 (1978).