Global Generalized Bianchi Identities for Invariant Variational Problems on Gauge-natural Bundles

Abstract

We derive both {\em local} and {\em global} generalized {\em Bianchi identities} for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the {\em a priori} introduction of a connection. The proof is based on a {\em global} decomposition of the {\em variational Lie derivative} of the generalized Euler--Lagrange morphism and the representation of the corresponding generalized Jacobi morphism on gauge-natural bundles. In particular, we show that {\em within} a gauge-natural invariant Lagrangian variational principle, the gauge-natural lift of infinitesimal principal automorphism {\em is not} intrinsically arbitrary. As a consequence the existence of {\em canonical} global superpotentials for gauge-natural Noether conserved currents is proved without resorting to additional structures.

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arXiv:math-ph/0311003v5 2 Jan 2005
Global Generalized Bianchi Identities for Invariant
Variational Problems on Gauge-natural Bundles∗
M. Palese and E. Winterroth†
Department of Mathematics, University of Torino
Via C. Alberto 10, 10123 Torino, Italy
e–mails: palese@dm.unito.it, ekkehart@dm.unito.it
Abstract
We derive both local and global generalized Bianchi identities for clas-
sical Lagrangian field theories on gauge-natural bundles. We show that
globally defined generalized Bianchi identities can be found without the
a priori introduction of a connection.
decomposition of the variational Lie derivative of the generalized Euler–
Lagrange morphism and the representation of the corresponding general-
ized Jacobi morphism on gauge-natural bundles. In particular, we show
that within a gauge-natural invariant Lagrangian variational principle,
the gauge-natural lift of infinitesimal principal automorphism is not in-
trinsically arbitrary. As a consequence the existence of canonical global
superpotentials for gauge-natural Noether conserved currents is proved
without resorting to additional structures.
The proof is based on a global
2000 MSC: 58A20,58A32,58E30,58E40,58J10,58J70.
Key words: jets, gauge-natural bundles, variational principles, generalized
Bianchi identities, Jacobi morphisms, invariance and symmetry proper-
ties.
1Introduction
Local generalized Bianchi identities for geometric field theories were introduced
[3, 5, 6, 19, 35] to get (after an integration by part procedure) a consistent
equation between local divergences within the first variation formula. It was also
stressed that in the general theory of relativity these identities coincide with the
contracted Bianchi identities for the curvature tensor of the pseudo-Riemannian
metric. We recall that in the classical Lagrangian formulation of field theories
the description of symmetries amounts to define suitable (vector) densities which
generate the conserved currents; in all relevant physical theories this densities
are found to be the divergence of skew–symmetric (tensor) densities, which are
called superpotentials for the conserved currents. It is also well known that the
∗Dedicated to Hartwig in occasion of his first birthday.
†Both of them supported by GNFM of INdAM and University of Torino.
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importance of superpotentials relies on the fact that they can be integrated
to provide conserved quantities associated with the conserved currents via the
Stokes theorem (see e.g. [11] and references quoted therein).
Subsequently, many attempts to “covariantize” such a derivation of Bianchi
identities and superpotentials have been made (see e.g. [4, 7, 9, 11, 12, 25, 26]
and the wide literature quoted therein) by resorting to background metrics
or (fibered) connections used to perform covariant integration by parts to get
covariant (variations of) currents and superpotentials. In particular, in [10]
such a covariant derivation was implicitly assumed to hold true for any choice
of gauge-natural prolongations of principal connections equal to prolongations
of principal connections with respect to a linear symmetric connection on the
basis manifold in the sense of [27, 28, 33].
In the present paper, we derive both local and global generalized Bianchi iden-
tities for classical field theories by resorting to the gauge-natural invariance of
the Lagrangian and via the application of the Noether Theorems [39]. In partic-
ular we show that invariant generalized Bianchi identities can be found without
the a priori introduction of a connection. The proof is based on a global de-
composition of the variational Lie derivative of the generalized Euler–Lagrange
morphism involving the definition - and its representation - of a new morphism,
the generalized gauge-natural Jacobi morphisms. It is in fact known that the
second variation of a Lagrangian can be formulated in terms of Lie derivative
of the corresponding Euler–Lagrange morphism [13, 14, 17, 20, 40]. As a conse-
quence the existence of canonical, i.e. completely determined by the variational
problem and its invariance properties, global superpotentials for gauge-natural
Noether conserved currents is proved without resorting to additional structures.
Our general framework is the calculus of variations on finite order jet of
fibered bundles. Fibered bundles will be assumed to be gauge-natural bundles
(i.e. jet prolongations of fiber bundles associated to some gauge-natural prolon-
gation of a principal bundle P [8, 22, 23, 27, 28, 32]) and variations of sections
are (vertical) vector fields given by Lie derivatives of sections with respect to
gauge-natural lifts of infinitesimal principal automorphisms (see e.g. [8, 24, 32]).
In this general geometric framework we shall in particular consider finite
order variational sequences on gauge-natural bundles. The variational sequence
on finite order jet prolongations of fibered manifolds was introduced by Krupka
as the quotient of the de Rham sequence of differential forms (defined on the
prolongation of the fibered manifold) with respect to a natural exact contact
subsequence, chosen in such a way that the generalized Euler-Lagrange and
Helmholtz-Sonin mappings can be recognized as some of its quotient mappings
[36, 37]. The representation of the quotient sheaves of the variational sequence
as sheaves of sections of tensor bundles given in [46] and previous results on
variational Lie derivatives and Noether Theorems [10, 15] will be used. Further-
more, we relate the generalized Bianchi morphism to the second variation of the
Lagrangian. A very fundamental abstract result due to Kol´ aˇ r concerning global
decomposition formulae of vertical morphisms, involved with the integration by
parts procedure [21, 29, 30], will be a key tool. In order to apply this results,
we stress linearity properties of the Lie derivative operator acting on sections
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of the gauge-natural bundle, which rely on properties of the gauge-natural lift
of infinitesimal principal automorphisms. The gauge-natural lift enables one to
define the generalized gauge-natural Jacobi morphism (i.e. a generalized Jacobi
morphism where the variation vector fields – instead of general deformations
– are Lie derivatives of sections of the gauge-natural bundle with respect to
gauge-natural lifts of infinitesimal automorphisms of the underlying principal
bundle), the kernel of which plays a very fundamental role.
The paper is structured as follows. In Section 2 we state the geometric
framework by defining the variational sequence on gauge natural-bundles and
by representing the Lie derivative of fibered morphisms on its quotient sheaves;
Section 3 is dedicated to the definition and the representation of the generalized
gauge-natural Jacobi morphism associated with a generalized gauge-natural La-
grangian. We stress some linearity properties of this morphism as a consequence
of the properties of the gauge-natural lift of infinitesimal right-invariant auto-
morphisms of the underlying structure bundle. In Section 4, by resorting to the
Second Noether Theorem, we relate the generalized Bianchi identities with the
kernel of the gauge-natural Jacobi morphism. We prove that the generalized
Bianchi identities hold true globally if and only if the vertical part of jet prolon-
gations of gauge-natural lifts of infinitesimal principal bundle automorphisms is
in the kernel of the second variation, i.e. of the generalized gauge-natural Jacobi
morphism.
Here, manifolds and maps between manifolds are C∞. All morphisms of
fibered manifolds (and hence bundles) will be morphisms over the identity of
the base manifold, unless otherwise specified.
2Variational sequences on gauge-natural bun-
dles
2.1Jets of fibered manifolds
In this Section we recall some basic facts about jet spaces. We introduce jet
spaces of a fibered manifold and the sheaves of forms on the s–th order jet space.
Moreover, we recall the notion of horizontal and vertical differential [32, 38, 42].
Our framework is a fibered manifold π : Y → X, with dimX = n and
dimY = n + m.
For s ≥ q ≥ 0 integers we are concerned with the s–jet space JsY of s–jet
prolongations of (local) sections of π; in particular, we set J0Y ≡ Y . We recall
the natural fiberings πs
these, the affine fiberings πs
s−1. We denote with V Y the vector subbundle of
the tangent bundle TY of vectors on Y which are vertical with respect to the
fibering π.
Charts on Y adapted to π are denoted by (xσ,yi). Greek indices σ,µ,...
run from 1 to n and they label basis coordinates, while Latin indices i,j,... run
from 1 to m and label fibre coordinates, unless otherwise specified. We denote
by (∂σ,∂i) and (dσ,di) the local basis of vector fields and 1–forms on Y induced
q: JsY → JqY , s ≥ q, πs: JsY → X, and, among
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by an adapted chart, respectively. We denote multi–indices of dimension n by
boldface Greek letters such as α = (α1,...,αn), with 0 ≤ αµ, µ = 1,...,n; by
an abuse of notation, we denote with σ the multi–index such that αµ = 0, if
µ ?= σ, αµ= 1, if µ = σ. We also set |α|:=α1+ ··· + αnand α!:=α1!...αn!.
The charts induced on JsY are denoted by (xσ,yi
particular, we set yi
the above coordinates are denoted by (∂α
In the theory of variational sequences a fundamental role is played by the
contact maps on jet spaces (see [36, 37, 38, 46]). Namely, for s ≥ 1, we consider
the natural complementary fibered morphisms over JsY → Js−1Y
α), with 0 ≤ |α| ≤ s; in
0≡ yi. The local vector fields and forms of JsY induced by
i) and (di
α), respectively.
D : JsY ×
XTX → TJs−1Y ,ϑ : JsY
×
Js−1YTJs−1Y → V Js−1Y ,
with coordinate expressions, for 0 ≤ |α| ≤ s − 1, given by
D= dλ⊗Dλ= dλ⊗(∂λ+ yj
α+λ∂α
j),ϑ= ϑj
α⊗∂α
j= (dj
α− yj
α+λdλ)⊗∂α
j.
The morphisms above induce the following natural splitting (and its dual):
JsY
×
Js−1YT∗Js−1Y =
?
JsY
×
Js−1YT∗X
?
⊕ C∗
s−1[Y ],(1)
where C∗
s−1[Y ]:=imϑ∗
sand ϑ∗
s: JsY
×
Js−1YV∗Js−1Y → JsY
s−1[Y ] ≃ JsY
×
Js−1YT∗Js−1Y .
We have the isomorphism C∗
×
Js−1YV∗Js−1Y . The role of the
splitting above will be fundamental in the present paper.
If f : JsY → I R is a function, then we set Dσf :=Dσf, Dα+σf :=DσDαf,
where Dσ is the standard formal derivative. Given a vector field Ξ : JsY →
TJsY , the splitting (1) yields Ξ◦πs+1
s
= ΞH+ΞV where, if Ξ = Ξγ∂γ+Ξi
then we have ΞH= ΞγDγand ΞV = (Ξi
ΞV the horizontal and the vertical part of Ξ, respectively.
The splitting (1) induces also a decomposition of the exterior differential on
Y , (πs
and vertical differential. The action of dHand dV on functions and 1–forms on
JsY uniquely characterizes dH and dV (see, e.g., [42, 46] for more details). A
projectable vector field on Y is defined to be a pair (Ξ,ξ), where Ξ : Y → TY
and ξ : X → TX are vector fields and Ξ is a fibered morphism over ξ. If
there is no danger of confusion, we will denote simply by Ξ a projectable vector
field (Ξ,ξ). A projectable vector field (Ξ,ξ), with coordinate expression Ξ =
ξσ∂σ+ ξi∂i, ξ = ξσ∂σ, can be conveniently prolonged to a projectable vector
field (jsΞ,ξ), whose coordinate expression turns out to be
α∂α
i,
α− yi
α+γΞγ)∂α
i. We shall call ΞH and
s−1)∗◦ d = dH+ dV, where dH and dV are defined to be the horizontal
jsΞ = ξσ∂σ+ (Dαξi−
?
β+γ=α
α!
β!γ!Dβξµyi
γ+µ)∂α
i,
where β ?= 0 and 0 ≤ |α| ≤ s (see e.g. [36, 38, 42, 46]); in particular, we
have the following expressions (jsΞ)H = ξσDσ, (jsΞ)V = Dα(ΞV)i∂α
i, with
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(ΞV)i= ξi− yi
From now on, by an abuse of notation, we will write simply jsΞHand jsΞV. In
particular, jsΞV : Js+1Y ×
σξσ, for the horizontal and the vertical part of jsΞ, respectively.
JsYJsY → Js+1Y ×
JsYJsV Y .
We are interested in the case in which physical fields are assumed to be sec-
tions of a fibered bundle and the variations of sections are generated by suitable
vector fields. More precisely, fibered bundles will be assumed to be gauge-
natural bundles and variations of sections are (vertical) vector fields given by
Lie derivatives of sections with respect to gauge-natural lifts of infinitesimal
principal automorphisms. Such geometric structures have been widely recog-
nized to suitably describe so-called gauge-natural field theories, i.e. physical
theories in which right-invariant infinitesimal automorphisms of the structure
bundle P uniquely define the transformation laws of the fields themselves (see
e.g. [8, 32]). In the following, we shall develop a suitable geometrical setting
which enables us to define and investigate the fundamental concept of conserved
quantity in gauge-natural Lagrangian field theories.
2.2Gauge-natural prolongations
First we shall recall some basic definitions and properties concerning gauge-
natural prolongations of (structure) principal bundles (for an extensive exposi-
tion see e.g. [32] and references therein; in the interesting paper [23] fundamental
reduction theorems for general linear connections on vector bundles are provided
in the gauge-natural framework).
Let P → X be a principal bundle with structure group G. Let r ≤ k be
integers and W(r,k)P := JrP ×
XLk(X), where Lk(X) is the bundle of k–frames
in X [8, 22, 32], W(r,k)G:=JrG⊙GLk(n) the semidirect product with respect
to the action of GLk(n) on JrG given by the jet composition and GLk(n) is the
group of k–frames in I Rn. Here we denote by JrG the space of (r,n)-velocities
on G.
Elements of W(r,k)P are given by (jx
section, t : I Rn→ X locally invertible at zero, with t(0) = x, x ∈ X. Elements
of W(r,k)G are (j0
at zero, with α(0) = 0.
rγ,j0
kt), with γ : X → P a local
rg,j0
kα), where g : I Rn→ G, α : I Rn→ I Rnlocally invertible
Remark 1 The bundle W(r,k)P is a principal bundle over X with structure
group W(r,k)G. The right action of W(r,k)G on the fibers of W(r,k)P is defined
by the composition of jets (see, e.g., [22, 32]).
Definition 1 The principal bundle W(r,k)P (resp. the Lie group W(r,k)G) is
said to be the gauge-natural prolongation of order (r,k) of P (resp. of G).
Remark 2 Let (Φ,φ) be a principal automorphism of P [32]. It can be pro-
longed in a natural way to a principal automorphism of W(r,k)P, defined by:
W(r,k)(Φ,φ) : (jx
rγ,j0
kt) ?→ (jφ(x)
r
(Φ ◦ γ ◦ φ−1),j0
k(φ ◦ t)).
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