The Problem of Invariance for Covariant hamiltonians

The Problem of Invariance for Covariant Hamiltonians (*).

J. MUNÄ OZMASQUEÂ(**) - M. EUGENIAROSADOMARIÂA(***)

ABSTRACT- The problem of invariance for the covariant Hamiltonians attached to first-order Lagrangian densities on the bundle of connections of a principal bundle and to first- and second-order Lagrangian densities on linear frame bundles, is solved.

1. Position of the problem.

In classical Mechanics, the Hamiltonian function attached to the La- grangian densityLˆL(t;qⁱ;q_ⁱ)dtonJ¹(R;Q)'RTQis given by,

Hˆq_ⁱ@L

@q_ⁱ L:

IfULdenotes the PoincareÂ-Cartan form attached toL, then U_Lˆp_idqⁱ Hdt

…1†

ˆs(L)k Hdt;

where s(L):RTQ!RTQ is the Legendre transformation, p_iˆ@L=@q_ⁱ, andkis the canonical 1-form onTQ.

As it was early observed in [9], the Hamiltonian function is not an in- variant concept when an arbitrary fibred manifoldt:E!Ris considered, generalizing the notion of «absolute time» (e.g., see [4, 15, 13] for a general (*) Supported by Ministerio of Ciencia y TecnologõÂa of Spain, under grant

#MTM2005-00173.

(**) Indirizzo dell'A.: Instituto de FõÂsica Aplicada, CSIC C/ Serrano 144, 28006-Madrid, Spain

E-mail: jaime@iec.csic.es

(***) Department of Applied Mathematics «G. Sansone», University of Florence, Via S. Marta 3, I-50139 Florence, Italy

E-mail: rosado@dma.unifi.it

development from this point of view), instead of the trivial direct product bundle RQ!R. In this case, an Ehresmann connection is needed in order to lift the vector field @=@t from R to E. Then, the Hamiltonian function is defined by contractingU_Lwith the horizontal lift of@=@t.

In the field theory - where no distinguished vector field exists on the ground manifold - the need of an Ehresmann connection is even greater, in order to attach a covariant Hamiltonian to each Lagrangian density; for example, see [13, 4.1], [14], and the definitions below.

Letp:M!N be a fibred manifold over a connected manifoldN, with nˆdimN, dimMˆm‡n, oriented by a volume formv. Throughout the paper, Latin (resp. Greek) indices run from 1 ton(resp.m).

We consider only coordinate systems (xⁱ) onN adapted tov, that is vˆdx¹^ ^dxⁿ; viˆdx¹^ ^dxcⁱ^ ^dxⁿ:

…2†

Letgbe an Ehresmann (or non-linear) connection on p, i.e.,gis a differ- ential 1-form on M with values in the vertical bundle V(p), such that g(X)ˆX, 8X2V(p) (cf. [13, 2.2]). According to [14], the covariant Ha- miltonian L^g associated to a Lagrangian density on J¹M, LˆLv, L2C¹(J¹M), with respect togis the Lagrangian density defined by

L^gˆ(p¹⁰)g w¹

^vL L;

…3†

where,p¹⁰:J¹M!Mdenotes the natural projection,w¹is theV(p)-valued 1-form onJ¹Mgoverning the contact structure on the jet bundle, which is given on a fibred coordinate system (xⁱ;y^a), byw¹ˆ(dy^a y^a_idxⁱ)@=@y^a (also see Section 3 below), andvLis the Legendre form attached toL, i.e., theV(p)-valuedp¹-horizontal (n 1)-form onJ¹Mgiven by

vLˆ( 1)^{i 1}@L

@y^a_ividy^a;

where (xⁱ;y^a;y^a_i) is the coordinate system induced from (xⁱ;y^a) on the 1-jet bundle (see Section 2.1 below for the details). Locally, we have

L^gˆ

g^a_i ‡y^a_i@L

@y^a_i L v:

…4†

From (3) we obtain the decomposition of the PoincareÂ-Cartan form ana- logous to (1), i.e.,

U_Lˆw¹^v_L‡L …5†

ˆ p¹⁰

g^vL L^g:

An automorphism of the fibred manifold p:M!N is a diffeomorphism F:M!M for which a diffeomorphism exists on the ground manifold f:N!N such that pFˆfp. The set of such automorphisms is de- noted by Aut (p) and its «Lie algebra» is the space Aut (p)X(M) of p- projectable vector fields on M, or equivalently, a vector field belongs to Aut (p) if and only if each transformationFt of its flow, belongs to Aut (p).

Given a subgroupG Aut (p), a Lagrangian densityLis said to beG- invariant if

F⁽¹⁾

LˆL; 8F2 G;

…6†

where F⁽¹⁾:J¹M!J¹M denotes the 1-jet prolongation of F. In- finitesimally, the equation (6) can be reformulated as follows:

L_X⁽¹⁾Lˆ0; 8X2Lie (G):

…7†

When a groupGof transformations ofMis given, a natural question arises:

(Q) Does a class of connectionsgexist such thatL^gisG-invariant for everyG-invariant Lagrangian densityL?

A natural answer to this question is to assume that the connectiong, itself, isG-invariant. In fact, for everyX2Aut (p) we can define the Lie derivative ofgwith respect toXas follows (see [6, § XI]):L_Xg(Y)ˆ[X;g(Y)] g([X;Y]), for everyY 2X(M), and the infinitesimalG-invariance ofgmeansL_Xgˆ0, 8X2Lie (G). Then, the following result holds:

PROPOSITION1.1. If g is infinitesimally G-invariant, then g solves the question (Q), i.e., L_X(1)Lˆ0, 8X2Lie (G), implies L_X(1)L^gˆ0, 8X2Lie (G).

PROOF. If Xˆuⁱ@=@xⁱ‡v^a@=@y^a, uⁱ2C¹(N), v^a2C¹(M), is the local expression ofX2Lie (G), then from the general formulas (11), (12) in Section 2.2 below, we obtain in particular,

X⁽¹⁾ˆuⁱ @

@xⁱ‡v^a @

@y^a‡v^a_i @

@y^a_i; v^a_i ˆ@v^a

@xⁱ‡y^b_i@v^a

@y^b y^a_j@u^j

@xⁱ: Furthermore, if gˆ(dy^a‡g^a_idxⁱ)@=@y^a, g^a_i 2C¹(M) is the local ex- pression of the Ehresmann connectiong(see the formula (13) in Section 3 below) we obtainL_XgˆC^a_i(X)dxⁱ@=@y^a, where

C_i^a(X)ˆX(g^a_i)‡v^a_i (g^b_j ‡y^b_j)@v^a_i

@y^b_j : …8†

We remark on the fact thatLXg(Y)ˆ0 ifYis ap-vertical vector field on M. HenceLXgisp-horizontal, i.e., it defines a section of the vector bundle pTNV(p).

Then, the formula (4) yields, L_X⁽¹⁾L^g ˆ X(g^a_i)‡v^a_i@L

@y^a_i

…9†

‡(g^a_i ‡y^a_i) X⁽¹⁾ @L

@y^a_i

‡@L

@y^a_i

@u^h

@x^h

v:

AsLisG-invariant,X⁽¹⁾Lˆ L@u^h=@x^h. Hence @

@y^a_i X⁽¹⁾

Lˆ @L

@y^a_i

@u^h

@x^h:

Moreover, by computing the bracket [@=@y^a_i;X⁽¹⁾], from the previous for- mula we obtain

X⁽¹⁾ @L

@y^a_i

‡@L

@y^a_i

@u^h

@x^hˆ@uⁱ

@x^h

@L

@y^a_h

@v^b

@y^a

@L

@y^b_i;

and substituting the right hand side of this equation for the left hand side into the equation (9) and taking the formula (8) into account, we finally

obtainL_X⁽¹⁾L^gˆC_i^a(X)(@L=@y^a_i)v p

Unfortunately, the infinitesimal G-invariance of g is a too restrictive condition to assure the existence of Ehresmann connections solving the question (Q). In fact, in general there is no solution to the system of equationsC^a_i(X)0,8X2Lie (G), whereC_i^a(X) is given by the formula (8), as proved in the next proposition. First, some preliminaries are required.

Everyp-projectable vector fieldX2X(M) vanishing at a pointy2M induces a linear map A_X;y:V_y(p)!V_y(p) by setting A_X;y(v)ˆ[Y;X]_y, 8v2V_y(p), whereY is anyp-vertical vector field onM such thatvˆY_y.

PROPOSITION1.2. If for every y2M, there exists a p-projectable vector field X2Lie (G)suchthat Xyˆ0, A_X;yˆ0, j¹_xX⁰ˆ0, where xˆp(y)and X⁰is the projection of X onto N, but j¹_yX6ˆ0, then there is no infinitesi- mallyG-invariant Ehresmann connection on p:M!N.

PROOF. With the same notations as in the proof of Proposition 1.1, the assumption is equivalent to the following: uⁱ(x)ˆ0, (@uⁱ=@x^j)(x)ˆ0, v^a(y)ˆ0, (@v^a=@y^b)(y)ˆ0, for all indicesa,b,i;j, but (@v^a=@xⁱ)(y)6ˆ0 for

some indicesa,i. Then, from the equation (8) for these indices, we obtain C_i^a(X)(y)ˆ(@v^a=@xⁱ)(y), thus leading us to a contradiction. p Fortunately, in the cases we shall deal with below, the vertical bundle V(p) decomposes asV(p)pTNW for a certain vector bundleW, so thatLXgcan be viewed as a section of the bundlep(TNTN)W, and the invariance ofL^g is assured by the assumption of the vanishing of the antisymmetric part of the tensor field L_Xg for every X. In these cases, there are plenty of Ehresmann connections answering the question (Q) above. Specifically, we deal with the following three cases: 1) First-order Lagrangian densities on the bundle of connections of a principal bundle; 2) First-order Lagrangian densities on the bundle of linear frames of a smooth manifold; and 3) The class of second-order Lagrangian densities on the bundle of linear frames of a smooth manifold, which admit a second- order Hamiltonian formalism, i.e., second-order Lagrangian densities whose corresponding covariant Hamiltonian is also of second order. In the first and third cases, the question (Q) imposes an algebraic condition on the antisymmetric part of the horizontal part of the Ehresmann connection under consideration, but in the second case the solution to (Q) leads one to a true system of partial differential equations. This asymmetry should deserve a subsequent analysis.

Finally, a summary of the contents of the present work is as follows: in Section 2, the notations and preliminary results on jet bundles, contact transformations and Lagrangian densities, which are used below, are introduced. In Section 3, the notion af an Ehresmann connection of ar- bitrary order on a fibred manifold p:M!N, is reviewed, several geo- metric properties of these non-linear connections are proved, and the action of the group Aut (p) on the space of Ehresmann connections is defined. Section 4 is devoted to state the basics of diffeomorphism in- variance on the bundle of linear framesp:FN!Nof a smooth manifold N. Namely, we first reduce the problem of determining DiffN-invariant Lagrangian densities onJ^r(FN) to that of DiffN-invariant functions and then, we explain why DiffN-invariance can be reduced to consider only the infinitesimal notion of invariance; i.e.,X(N)-invariance. Finally, bases (previously obtained) forX(N)-invariant functions for the casesrˆ1;2, are introduced. In Section 5 the invariance problem for first-order cov- ariant Hamiltonians, is considered in two particular cases: First, for the bundle of connections on an arbitrary principal bundle under the gauge group of the bundle; second, for the bundles of linear frames under the group of diffeomorphism on the ground manifold. In both cases, a geo-

metric characterization of Ehresmann connections providing invariant covariant Hamiltonians for every invariant Lagrangian, is given. In Section 6 a class of second-order Lagrangian densities admitting a sec- ond-order covariant Hamiltonian, is introduced. In Section 7, the in- variance problem for second-order Lagrangian densities on linear frame bundles belonging to the class defined in Section 6, is solved for second- order Ehresmann connections induced by a 2-jet field, the explicit form of which is given in Theorem 7.4.

2. Preliminaries and notations.

2.1 ±Jet bundle notations.

For every fibred manifold p:M!N there is a homomorphism Aut (p)!DiffN,F7!f, whose kernel is the subgroup Aut^v(p) of fibred (or vertical) diffeomorphisms inducing the identity over N. A system of coordinates (xⁱ;y^a) on an open subsetUMis said to be a fibred coordinate system for the submersionpif (xⁱ) is a coordinate system forNonp(U). The vertical bundle of p is the sub-bundle V(p)ˆ fX2TM:pXˆ0g. Let Mxˆp ¹(x) be the fibre overx. We haveTy(Mx)ˆVy(M),8y2Mx; i.e., the vertical tangent vectors aty are the vectors tangent to the fibre passing throughy. IfF:M!M⁰is a fibred map, thenF:TM!TM⁰mapsV(p) into V(p⁰). Let p^k:J^kM!N be the k-jet bundle of local sections of p:M!N, with natural projections p^kl:J^kM !J^lM, p^kl(j^k_xs)ˆj^l_xs, for kl, j^k_xs denoting thek-jet at xof a section s of pdefined on a neigh- bourhood of x2N. If Iˆ(i₁;. . .;i_n)2Nⁿ is a multi-index of order jIj ˆi₁‡. . .‡i_nthen we set@^jIjf=@x^Iˆ@^jIjf=(@x¹)ⁱ¹. . .(@xⁿ)ⁱⁿ, for every f 2C¹(Rⁿ). Multi-indices are added and subtracted componentwise and IJ means ihjh for 1hn. We set I!ˆi1! in! and

J

I ˆJ!=(J I)!I! whenever IJ. We denote by (i) the multi-index whose components are (i)_hˆdⁱ_h, and we set (ij)ˆ(i)‡(j), (ijk)ˆ

ˆ(i)‡(j)‡(k), etc. Hence the symbol (i1. . .ia) thus defined depends symmetrically on the indices i1;. . .;ia. The identity d^(hi)_(jk) ˆd^h_jdⁱ_k‡

‡d^h_kdⁱ_j d^h_id_k^jd^h_k for Kronecker's deltas of multi-indices, will be used.

A fibred coordinate system (xⁱ;y^a) for p on an open subset UM, induces a coordinate system (xⁱ;y^a_I), 0 jIj r, on (p^r0) ¹(U)ˆJ^rU; i.e., y^a_I(j^r_xs)ˆ(@^jIj(y^as)=@x^I)(x), withy^a₀ˆy^a. For first-order derivatives we simply writey^a_iinstead ofy^a_(i). Every fibred mapF:M!M⁰whose induced

map on the base manifoldsf:N!N⁰is a diffeomorphism, induces a map F^(r):J^rM!J^rM⁰by settingF^(r)(j^r_xs)ˆj^r_f(x)(Fsf ¹).

Below, we make repeatedly use of the affine-bundle structure of the projectionp^{r;r 1}:J^rM!J^{r 1}Mmodelled over the vector bundle

(p^{r 1})…pS^rTNV(p)† ˆS^rTN_J^{r 1}_MV(p);

which is expressed on a fibred coordinate system (xⁱ;y^a) as follows. We set (dx)^Iˆ(dx¹). . .ⁱ¹ (dx¹). . .(dxⁿ). . .ⁱⁿ (dxⁿ) for every multi-index Iˆ(i₁;. . .;i_n), where the symbolstands for the symmetric product. If

t_rˆX

jIjˆr

1

I!l_I(dx)^I_x0(@=@y^a)_y0; x₀ˆp(y₀)

is an element in S^rT_x0NVy0(p), then the r-jet j^r_x0s⁰ˆtr‡j^r_x0s is de- termined by the equations

j^r_x0¹s⁰ˆj^r_x0¹s;

@^r(y^as⁰)

@x^I (x0)ˆlI‡@^r(y^as)

@x^I (x0); jIj ˆr:

2.2 ±Contact forms and contact transformations.

Letp:M!N be a fibred manifold. A differential 1-formwonJ^rMis said to be a contact form if (j^rs)wˆ0 for every local sectionsofp. The set of contact forms is a sheaf ofC_J¹rM-modules, denoted byC^r_M, which is locally spanned by the forms

w^a_I ˆdy^a_I y^a_I‡(i)dxⁱ; jIj r 1:

…10†

For every X2X(N),DX:C¹(J^rM)!C¹(J^r‡1M) denotes the total deri- vative with respect toX, which is defined by D_Xf(j^r‡1_x s)ˆX_x(fj^rs) for everyf 2C¹(J^r‡1M). The operatorD_Xis the only derivation onJ¹Mthat satisfies: 1)D_Xprojects ontoX, and 2)w^a_I(D_X)ˆ0, for allI;a. In particular, we denote the total derivative with respect to the coordinatexⁱby

D_iˆ @

@xⁱ‡X¹

jIjˆ0

y^a_I‡(i) @

@y^a_I:

We setD^IˆD₁. . .ⁱ¹ D₁ D_n. . .ⁱⁿ D_n. Recall that [D_i;D_j]ˆ0.

The horizontalization of a-possibly vector-valued-covariant tensor field

tof degreeqonJ^rMis thep^r‡1-horizontal covariant tensor field hor(t) of degreeqonJ^r‡1Mdefined by hor(t)(j^r‡1_x s)ˆ(p^r‡1)((j^rs)t)(x). The total differential of a p^r-horizontal form v_k on J^rM is the p^r‡1-horizontal on J^r‡1M defined as follows: (Dv_k)_j^r‡1_{x s}ˆ(p^r‡1)(d((j^rs)v_k))_x. Hence, we haveD fdxⁱ¹^ ^dx^ik

ˆ(D_if)dxⁱ^dxⁱ¹^ ^dx^ik,8f 2C¹(J^rM).

IfFt:M!Mis the flow of ap-projectable vector fieldX2X(M), then (Ft)^r:J^rM!J^rMis the flow of a vector field onJ^rM, denoted byX^(r)and called the infinitesimal contact transformation of orderrassociated toX. The map X7!X^(r) is an injection of Lie algebras. If Xˆuⁱ@=@xⁱ‡v^a@=@y^a, whereuⁱ2C¹(N) andv^a2C¹(M), then (e.g., see [16, 19, 21]) we have

X^(r)ˆuⁱ @

@xⁱ‡X^r

jIjˆ0

v^a_I @

@y^a_I; v^a₀ˆv^a; …11†

v^a_IˆD^Iv^a X

J<I

I J

@^{jI Jj}uⁱ

@x^{I J} y^a_J‡(i); 1 jIj r:

…12†

2.3 ±Lagrangian densities.

A Lagrangian density of order r onp:M!N is a p^r-horizontal dif- ferential n-formLonp^r:J^rM!N. If N is orientable and oriented by a volume formv, then every Lagrangian density can be written in a unique way asLˆL(p^r)v, whereL2C¹(J^rM). Usually, we shall simply write LˆLv. In this case, we confine ourselves to consider coordinate systems (xⁱ) onNadapted tov, as shown in (2).

3. Ehresmann connections.

An Ehresmann (or non-linear) connection of orderronp:M!Nis a differential 1-form g^r on J^{r 1}M with values in the vertical sub-bundle V(p^{r 1}) such thatg^r(X)ˆX for everyX2V(p^{r 1}) (e.g., see [14, 20, 21]).

Once a connectiong^ris given, we have a decomposition of vector bundles T(J^{r 1}M)ˆV(p^{r 1})kerg^r, where kerg^r is called the horizontal sub- bundle determined byg^r. In the coordinate system onJ^{r 1}Minduced from a fibred coordinate system (x^j;y^a) forp, an Ehresmann connection can be written as

g^rˆX^{r 1}

jIjˆ0

(dy^a_I‡g^a_Ijdx^j) @

@y^a_I; g^a_Ij2C¹(J^{r 1}M):

…13†

There is a bijective correspondence between ther-th order connectionsg^r on p:M!N and the linear sections ~g^r:(p^{r 1})TN!T(J^{r 1}M) of the homomorphism (p^{r 1}):T(J^{r 1}M)!(p^{r 1})TN given by the following formula:Xˆg^r(X)‡~g^r(j^r_x ¹s;(p^{r 1})X),8X2T_j^r_x¹_s(J^{r 1}M).

The sections of the affine bundle p^{r;r 1}:J^rM!J^{r 1}M are usually calledr-th order jet fields (e.g., see [21, § 4.6, § 5.4]). Ifs:J^{r 1}M!J^rMis such a field andw^rˆP

jIjr 1w^a_I@=@y^a_Iis the structure form onJ^rM(see the formula (10) and [16]), theng^r_sˆsw^ris a connection of orderr. As an affine bundle admits global sections (e.g., see [11, I, Theorem 5.7]), this proves that every fibred manifold admits connections of any order. Fur- thermore, the map s7!g^r_sˆsw^r is an injection from jet fields into con- nections. In fact, with the same notations as in (13), we have

g^a_Ijˆ y^a_I‡(j); 0 jIj r 2;

g^a_Ijˆ y^a_I‡(j)s; jIj ˆr 1:

…14†

For rˆ1 the correspondences7!g¹_sˆsw¹ is bijective. Hence w¹ is the

«universal» connection form. Every r-th order jet fields:J^{r 1}M!J^rM can be viewed as a first-order jet fieldWs:J^{r 1}M!J¹(J^{r 1}M) along the projection p^{r 1}:J^{r 1}M!N, where W:J^rM!J¹(J^{r 1}M) is the natural embedding. Ther-th order connectionsw^rcoincides with the first-order connection induced byWs; i.e.,sw^r_Mˆ(Ws)w¹_Jr 1M.

PROPOSITION3.1. Every r-order connection g^r on p:M!N induces an affine map of affine bundles over J^{r 1}M,

hg^r:J^rM!Hom (p^{r 1})TN;V(p^{r 1}) given by hg^r(j^r_xs)ˆ (j^{r 1}s)g^r

x, whose associated linear mapping is the inclusion S^rTN_J^{r 1}_MV(p)TN_J^{r 1}_MV(p^{r 1}). Moreover, ifg^r_sˆsw^r for a jet field s:J^{r 1}M!J^rM, then imhg^r S^rTN_J^{r 1}_MV(p) and we have h_g^r(j^r_xs)‡s(j^r_x ¹s)ˆj^r_xs.

PROOF. From the formula (13) and the definition ofh_g^r we obtain

hg^r(j^r_xs)ˆX^{r 1}

jIjˆ0

@^jIj‡1(y^as)

@x^I‡(j) (x)‡g^a_Ij(j^r_x ¹s)

(dx^j)_x @

@y^a_I

j^rx¹s

;

thus proving that h_g^r is an affine map whose associated linear map is the injection of S^rTN_J^{r 1}_MV(p) into TN_J^{r 1}_MV(J^{r 1}M). Moreover, if

g^rˆsw^r, then from the formulas (14) we obtain h_g^r ˆ X

jIjˆr 1

y^a_I‡(j) y^a_I‡(j)sp^{r;r 1}

dx^j @

@y^a_I;

and the result follows. p

The action of the group Aut (p) on the space ofr-th order connections is defined by the formula Fg^rˆ(F^{r 1})g^r(F^{r 1})¹, 8F2Aut (p). As F^{r 1}:J^{r 1}M!J^{r 1}Mis a morphism of fibred manifolds over N, (F^{r 1}) transforms the vertical sub-bundleV(p^{r 1}) into itself; hence the previous definition makes sense.

4. Diffeomorphism invariance on linear frame bundles.

Let p:FN!N be the bundle of linear frames of N. A Lagrangian densityL defined onJ^r(FN) is said to be DiffN-invariant- or even, in- variant under diffeomorphisms-(resp. X(N)-invariant) if the following equation holds: (f~^r)LˆL, 8f2DiffN (resp. LX~^(r)Lˆ0, 8X2X(N)), wheref~(resp.X~ 2X(FN)) is the natural lift off(resp.X) toFN([11, VI,

§ 1, § 2]), andX~^(r) denotes ther-th jet prolongation ofX, as introduced in~ Section 2.2. We can write

Lˆ Lu¹^ ^uⁿ; …15†

where uˆ(u¹;. . .;uⁿ) is the canonical 1-form on FN ([11, III, § 2, p. 118]), and L 2C¹(J^r(FN)) is called the «canonical Lagrangian»

associated to L. A density L is DiffN-invariant (resp. X(N)-invariant) if and only ifL ~f^rˆ L, 8f2DiffN (resp. X~^(r)L ˆ0,8X2X(N)), asu is DiffN-invariant and hence, X(N)-invariant. The problem of de- termining invariant Lagrangian densities is thus reduced to that of determining invariant Lagrangian functions. Moreover, DiffN-in- variance implies X(N)-invariance and both notions are equivalent ex- cept when N is orientable and admits an orientation-reversing dif- feomorphism onto itself (see [18, § 2.1]). Because of this, we consider only X(N)-invariance.

Each coordinate system (xⁱ) on an open domainUN induces a co- ordinate system (xⁱ;xⁱ_j) onFUˆp ¹(U) by setting

uˆ (@=@x¹)_x;. . .;(@=@xⁿ)_x

(xⁱ_j(u)

; xˆp(u);

…16†

and also, according to Section 2.1, a coordinate system xⁱ;xⁱ_j;xⁱ_j;(k1...kq) , 1qr,k1. . .kq, onJ^r(FU) by

xⁱ_j;(k1...kq)(j^r_xs)ˆ @^q(xⁱ_js)=@x^k1. . .@x^kq (x):

From the local expression of the canonical 1-form uⁱˆyⁱ_jdx^j; (yⁱ_j)ˆ(xⁱ_j) ¹; …17†

we deduce that the relation betweenL andLin the Lagrangian density LˆLvˆ Lu¹^ ^uⁿisLˆ Ldet(y^a_b), or equivalently,L ˆLdet(x^a_b).

EveryX(N)-invariant Lagrangian onJ¹(FN) can be written as a dif- ferentiable function of the following¹₂n²(n 1) Lagrangians:

L^c_abˆ x^d_ax^e_b;d x^d_bx^e_a;d

y^c_e; a<b;

…18†

and every X(N)-invariant second-order Lagrangian on J²(FN) can be written as a differentiable function of the

1

2n²(n 1)‡1

6n²(n 1)(n‡1)ˆ1

6n²(n 1)(2n‡5)

Lagrangians:L^c_ab,a<b;L^c_ab;d,a<b,ad, whereL^c_ab;dis given as follows:

L^c_ab;dˆx^r_dDr L^c_ab …19†

ˆx^r_d x^u_a;rx^e_b;u x^u_b;rx^e_a;u

y^c_e‡x^r_d x^u_ax^e_b;(ru) x^u_bx^e_a;(ru) y^c_e x^r_dx^p_q;r x^u_ax^e_b;u x^u_bx^e_a;u

y^c_py^q_e:

The geometric meaning of such functions is the following. Ifs:U!FNis the section induced by a linear frame (X1;. . .;Xn) on an open neighbour- hood of a pointx2N, we denote byr^s the only linear connection onFU that parallelizes each vector field X_i; i.e., (r^s)_XjX_iˆ0, for i;jˆ1;. . .;n.

Then, we have

L^c_ab(j¹_xs)ˆ v^c…Torr^s…Xa;X_b††(x);

L^c_ab;d(j²_xs)ˆ v^c…r^sTorr^s†…Xd;Xa;Xb†(x);

…20†

where (v¹;. . .;vⁿ) denotes the dual coframe. Moreover, the inequalityad is due to the constraints imposed by Bianchi's first identity for the linear connection r^s; namely, L^h_abL^c_hd‡ L^h_bdL^c_ha‡L^h_daL^c_hbˆ L^c_bd;a‡L^c_da;b‡L^c_ab;d. For the details of these facts, see [8, 18].

5. First-order covariant Hamiltonians.

5.1 ±Covariant Hamiltonians on the bundle of connections.

First of all, we introduce some basic notations. For further details we refer the reader to [3, 6]. Letp:P!Nbe a principalG-bundle. Connections onPare identified to the splittings of the sequence of vector bundles overN (cfr. [1, 5, 7, 12]), 0!adP!T_GP ^p! TN!0, where adP!N is the adjoint bundle, i.e., the bundle associated withP and the adjoint re- presentation of G on its Lie algebra g, and T_GPˆ(TP)=G. Hence adPV(p)=G. The sections of the quotient bundleT_GPare identified to theG-invariant vector fields onP, and the sections of adPare identified to the gauge algebra (e.g., see [10]), that is, G(N;T_GP)ˆAutP, G(N;adP)ˆgauP. We think of gauP as being the «Lie algebra» of the gauge group GauP(cfr. [2, 5, 7, 10]).

Consequently, there exists a bundlep:C(P)!Nwhose global sections can be identified with the connections on P (e.g., see [1, 5, 7, 12]). This bundle is affine and modelled over TNadP. We denote by sG:N!C(P) the section of the bundle of connections induced tautologi- cally byG.

Let UM be an open subset such that p ¹(U)UG. For every B2gthere is a flow of gauge transformationsW^B_t:UG!UGgiven byW^B_t(x;g)ˆ(x;exp (tB)g). The infinitesimal generator ofW^B_t is denoted byB, which is an infinitesimal gauge transformation on p ¹(U).

LetB~ ˆB(modG) be the orbit ofB in adP. If (Ba) is a basis ofg, then (B~a) is a basis ofG(U;adP)ˆgaup ¹(U). Accordingly, every gauge field X2gauPcan be written as

Xˆg^aB~a; g^a2C¹(U):

…21†

Coordinates onC(P) are introduced as follows: Let (U;xⁱ) be a coordinate domain in N such that p ¹(U)UG. We can define functions A^a_j:p ¹(U)!R, by setting

@=@x^j_h(Gx)

x ˆ @=@x^j

x A^a_j(G_x)(B~_a)_x; …22†

for every connection Gx at a pointx2U, where h(Gx) denotes the hor- izontal lift with respect toG_x. The functions (x^j;A^a_j) constitute a coordinate system onp ¹(U). According to the general notations introduced in Sec- tion 2.1, we denote by (x^j;A^a_j;A^a_j;k) the induced coordinate system on J¹C(P).

Each gauge transformation F2GauP acts on connections by pulling connection forms back (see [11, II, Proposition 6.2-(b)]):G⁰ˆFG, where v_G⁰ ˆ(F ¹)vG. Hence a unique diffeomorphism F_C:C(P)!C(P) exists such thatF_Cs_G ˆs_FG, for every connectionG onP.

IfF_tis the flow of a gauge fieldX2gauP, then (F_t)_Cis a flow onC(P) and the corresponding infinitesimal generator is denoted byXC; its local expression is (see [3, 6]),

X_Cˆ c^b_agg^aA^g_i @g^b

@xⁱ

@

@A^b_i; …23†

wherec^a_bgare the structural constants,‰Bb;BgŠ ˆc^a_bgBa, andXis given by the formula (21).

According to the general equation (7), a LagrangianL2C¹(J¹C(P)) is gauge invariant ifX_C⁽¹⁾Lˆ0,8X2gauP. SinceX_C⁽¹⁾isp¹-vertical, we have L_X(1)

C (Lv)ˆX_C⁽¹⁾(L)v. Hence the gauge invariance of the densityLvis re- duced to that of the Lagrangian function.

The classification of gauge invariant Lagrangian densities is given by the geometric formulation of Utiyama's theorem ([2, 7, 23]), which estab- lishes that a LagrangianLis gauge invariant if and only if there exists a smooth function L: ^²TNadP!R, which in turn must also be in- variant under the natural action of the adjoint representation on

^²TNadP, such thatLˆLV, whereV:J¹C(P)! ^²TNadPis the curvature mapping, i.e.,V(j¹_xsG)ˆ(VG)_x.

LetR^a_jk,j<k, be the coordinate system induced by (x^j) and (B_a) on the curvature bundle; i.e.,h₂ˆP

j<kR^a_jk(h₂)(dx^j)_x^(dx^k)_x(B~_a)_x, for every 2- covector h₂2 ^²T_xN(adP)_x. Then, the equations of the curvature mapping are (e.g., see [6]),

R^a_jkVˆA^a_j;k A^a_k;j c^a_bgA^b_jA^g_k: …24†

From the general formula (13) we obtain the local expression of a first- order Ehresmann connection g¹ on C(P) in the coordinates (xⁱ;A^a_i) in- troduced in the formula (22); precisely,

g¹ ˆ dA^a_j ‡g^a_jkdx^k @

@A^a_j; g^a_jk2C¹(C(P)):

…25†

Ifp:M!Nis an affine bundle modelled over a vector bundleq:W!N, then we have an isomorphismpW!V(p) of vector bundles overM, that maps the pair (y;w)2Mq ¹(p(y)) onto the tangent vectorx(y;w)2Vy(p) given byx(y;w)(f)ˆlim

t!0

1t(f(y‡tw) f(y)),8f 2C¹(M).

Asp:C(P)!N is an affine bundle modelled overTNadP, a first- order Ehresmann connectiong¹onC(P) can be viewed as a 1-form onC(P) with values in the vector bundlep(TNadP). By taking its horizontal part, we conclude that horg¹ defines a section of the vector bundle (p¹)(TNTNadP), or even a morphism of fibred manifolds overN,

horg¹:J¹C(P)!TNTNadP:

LEMMA 5.1. Withthe same notations as above, the section corre- sponding to dx^jB~a in the isomorphism p(TNadP)!V(p), is

@=@A^a_j.

PROOF. According to the formula (22), for everyG_x2C(P), we have A^b_i G_x‡t(dx^jB~_a)_x

ˆA^b_i(G_x) td^j_id^b_a: Hencex Gx;(dx^jB~a)_x

ˆ @=@A^a_j

Gx. p

THEOREM5.2. Letp:P!N be a principal G-bundle. Ifg¹ is a first- order Ehresmann connection on the bundle of connections p:C(P)!N of P satisfying the following equation:

alt (horg¹)ˆV;

…26†

V being the curvature mapping andalt:²TNadP! ^²TNadP the antisymmetrization operator, then the covariant Hamiltonian L^g1vof every gauge invariant Lagrangian L on J¹C(P)is gauge invariant.

For every principal bundle P, first-order Ehresmann connections globally defined on C(P)satisfying the equation(26), always exist.

PROOF. According to [6, Theorem 1-(2)], ifg¹ satisfies the following equation:

D^a_jkˆg^a_jk g^a_kjˆ c^a_bgA^b_jA^g_k; j<k;

…27†

then L^g1 is gauge invariant for every gauge invariant Lagrangian L on J¹C(P). Moreover, by applying Lemma 5.1 to the formula (25) we obtain

g¹ ˆ dA^a_j ‡g^a_jkdx^k

dx^jB~a;

and by taking the horizontal part, horg¹ˆ (A^a_j;k‡g^a_jk)dx^kdx^jB~a. Hence alt (horg¹)ˆP

j<k(A^a_j;k A^a_k;j‡D^a_j;k)dx^j^dx^kB~_a, and the for- mula (26) in the statement follows from the equations of the curvature mapping in the formula (24) wheneverg¹satisfies the equation (27) above.

Finally, as remarked in Section 3, there is a one-to-one correspondence between first-order Ehresmann connections and first-order jet fields given by g¹_sˆsw¹. The sections s for which the corresponding Ehresmann connection g¹_s satisfies the equation (26), are the smooth sections of an affine sub-bundle of the bundlep¹⁰:J¹C(P)!C(P), which is modelled over the vector sub-bundle p(S²TNadP)p(²TNadP), thus

proving the last part of the statement. p

If the centre ofgis trivial, then the equation (26) in Theorem 5.2 is also necessary for the covariant Hamiltonian of every gauge invariant La- grangian on J¹C(P) to be gauge invariant; see [6, Theorem 1-(2)]. More- over, since XC isp-vertical,LXCg¹ defines a section of the vector bundle p(²TNadP), as remarked in the proof of Proposition 1.1, and the antisymmetric part of this section vanishes for all gauge fieldX, if and only if g¹ solves the question (Q); see [6, Proposition 6] for the details of the proof.

Although Theorem 5.2 guarantees the existence of Ehresmann con- nections solving the question (Q) for the bundle of connections of a principal bundle, we remark on the fact that there is no gauge invariant Ehresmann connection on C(P). In fact, once a point G_x2C(P) and a coordinate system (xⁱ) centred at xhave been fixed, from the formulas (21), (23) we conclude that the vector field X_C for g^a(x)ˆ0, (@g^a=@xⁱ)(x)ˆ0,8a;i, but (@²g¹=@(x¹)²)(x)6ˆ0, satisfies the conditions of Proposition 1.2.

5.2 ±Covariant Hamiltonians on linear frame bundles.

Let (U;xⁱ) be a coordinate domain inN. According to (13), on the induced coordinate system (xⁱ;xⁱ_j) forp:FN!N (see (16) above), an Ehresmann connection g¹ on FN can be written as g¹ˆ(dxⁱ_j‡gⁱ_jkdx^k)@=@xⁱ_j, with gⁱ_jk2C¹(FN).

LetL^g1be the covariant Hamiltonian associated to a Lagrangian density LonJ¹(FN) with respect tog¹. As the covariant HamiltonianL^g1 is a La- grangian density, it also admits a canonical LagrangianL^g1 2C¹(J¹(FN)) defined, according to the formula (15), byL^g1 ˆ L^g1u¹^ ^uⁿ, where

L^g1ˆgⁱ_jk‡xⁱ_j;k @L

@xⁱ_j;k L; gⁱ_jk2C¹(FN):

…28†

LEMMA 5.3. The covariant Hamiltonian L^g1 of everyX(N)-invariant Lagrangian LisX(N)-invariant if and only if all the functions(L^c_ab)^g1, a<b, areX(N)-invariant.

PROOF. If the covariant HamiltonianL^g1 of everyX(N)-invariant La- grangianLisX(N)-invariant, then all the functions (L^c_ab)^g1,a<b, are, in particular,X(N)-invariant. Next, we prove the converse. IfXˆuⁱ@=@xⁱ, uⁱ2C¹(N), then

X~ ˆuⁱ @

@xⁱ‡x^l_j@uⁱ

@x^l

@

@xⁱ_j; …29†

and from the formulas (11), (12) above forrˆ1, we obtain X~⁽¹⁾ˆ uⁱ @

@xⁱ‡vⁱ_j @

@xⁱ_j‡vⁱ_jk @

@xⁱ_j;k; vⁱ_j ˆ @uⁱ

@x^lx^l_j; vⁱ_jkˆ @uⁱ

@x^rx^r_j;k @u^r

@x^kxⁱ_j;r‡ @²uⁱ

@x^k@x^rx^r_j: 8>

>>

>>

>>

<

>>

>>

>>

>: …30†

From the formulas (28), (30), we obtain, after some computations, X~⁽¹⁾(L^g1)ˆC_jkⁱ(X)~ @L

@xⁱ_j;k; …31†

where

Cⁱ_jk(X)~ ˆXg~ ⁱ_jk‡vⁱ_jk g^r_st‡x^r_s;t@vⁱ_jk

@x^r_s;t: …32†

As mentioned above, every invariant Lagrangian can be written as a smooth function of the Lagrangians L^c_ab defined in the formula (18).

Accordingly, we can write L ˆFP¹, for a certain function F2C¹(R^{12n2(n 1)}), where P¹:J¹(FN)!R^{12n2(n 1)} is the map whose com- ponents are the functionsL^c_ab, i.e.,t^c_abP¹ˆ L^c_ab, where (t^c_ab),a<b, are the standard coordinates onR^{12n2(n 1)}. Hence, substituting

@L

@xⁱ_j;kˆ @F

@t^c_abP¹ @L^c_ab

@xⁱ_j;k into the equation (31), we obtain

X~⁽¹⁾(L^g1)ˆ @F

@t^c_abP¹

X~⁽¹⁾ (L^c_ab)^g1

;

and the result follows. p

The notion of torsion of an Ehresmann connection g¹ onFN, can be introduced by imitating that of a linear connection; e.g., see [11, III, § 2].

LetB(v) be the standard horizontal vector field associated tov2Rⁿ(see [11, III, § 2]) with respect tog¹; precisely,B(v) is the onlyg¹-horizontal vector field onFNsuch thatp(B(v)_u)ˆu(v) for everyu2F_xN, where, here, the linear frameuis considered to be a linear isomorphismu:Rⁿ!TxN. Then, we define the torsion ofg¹as being the mapT^g1:FN! ^²(Rⁿ)Rⁿgiven by T^g1(u)(v;w)ˆdu(B(v);B(w))(u),8u2FN, 8v;w2Rⁿ, and u is the ca- nonical 1-form onFNintroduced in Section 4 above. Let (v_i) be the standard basis ofRⁿwith dual basis (vⁱ), and let (T^g1)ⁱ_jk(u),j<k, be the components of T^g1(u) in such basis, i.e.,

T^g1(u)ˆX

j<k

(T^g1)ⁱ_jk(u)v^j^v^kvi:

From the local expression B(v_a)ˆx^k_a(@=@x^k gⁱ_jk@=@xⁱ_j) of the standard horizontal vector field associated tov_aand that ofuin the formula (17), we obtain

(T^g1)^c_abˆ x^k_agⁱ_bk x^k_bgⁱ_ak y^c_i: …33†

THEOREM5.4. Letg¹be an Ehresmann connection onp:J¹(FN)!N.

The covariant Hamiltonian L^g1 associated withevery X(N)-invariant Lagrangian density L on J¹(FN)is X(N)-invariant, if and only if the torsion ofg¹is constant.

PROOF. LetL^g1 2C¹(J¹(FN)) be the canonical Lagrangian associated to the covariant Hamiltonian L^g1. We must prove that X~⁽¹⁾(L^g1)ˆ0, for everyX2X(N) and everyX(N)-invariant LagrangianL, if and only if the torsion ofg¹is constant. As follows from Lemma 5.3, we only need to impose X~⁽¹⁾((L^c_ab)^g1)ˆ0,8X2X(N), and all indicesa<b,c. Taking the following identity:

@L^c_ab

@xⁱ_j;k ˆ x^k_ad_b^j x^k_bd_a^j y^c_i

into account, from the formula (33) we obtain (L^c_ab)^g1ˆ gⁱ_jk‡xⁱ_j;k@L^c_ab

@xⁱ_j;k L^c_abˆgⁱ_jk@L^c_ab

@xⁱ_j;k ˆ(T^g1)^c_ab: …34†

HenceX~⁽¹⁾(L^c_ab)^g1 ˆX(T~ ^g1)^c_ab, which allows us to conclude the proof. p