[Paper Review] Gauge-natural field theories and Noether Theorems: canonical covariant conserved currents

Marcella Palese, Ekkehart Winterroth|ArXiv.org|Dec 7, 2005

Geometric Analysis and Curvature Flows27 references3 citations

TL;DR

This paper establishes a canonical, covariant derivation of conserved currents in gauge-natural field theories by proving the naturality property $ L^{js}_{\bar{\Xi}_H} \omega(\lambda, K) = 0 $ using iterated Lie derivatives and Kol\'a\v{r}'s invariant decomposition. The key contribution is the existence of a global, connection-independent superpotential for conserved currents, ensuring covariant Bergmann–Bianchi identities and a generalized energy–momentum tensor density.

ABSTRACT

Recently we found that canonical gauge-natural superpotentials are obtained as global sections of the {\em reduced} $(n-2)$-degree and $(2s-1)$-order quotient sheaf on the fibered manifold $\bY_{\zet} imes_{\bX} \mathfrak{K}$, where $\mathfrak{K}$ is an appropriate subbundle of the vector bundle of (prolongations of) infinitesimal right-invariant automorphisms $\barΞ$. In this paper, we provide an alternative proof of the fact that the naturality property $\cL_{j_{s}\barΞ_{H}}ω(λ, \mathfrak{K})=0$ holds true for the {\em new} Lagrangian $ω(λ, \mathfrak{K})$ obtained contracting the Euler--Lagrange form of the original Lagrangian with $\barΞ_{V}\in \mathfrak{K}$. We use as fundamental tools an invariant decomposition formula of vertical morphisms due to Kolář and the theory of iterated Lie derivatives of sections of fibered bundles. As a consequence, we recover the existence of a canonical generalized energy--momentum conserved tensor density associated with $ω(λ, \mathfrak{K})$.

Motivation & Objective

• To resolve globality issues in conserved current constructions for gauge-natural field theories.
• To establish a connection-independent, canonical derivation of superpotentials for Noether currents.
• To prove the naturality of the modified Lagrangian $ \omega(\lambda, K) $ under horizontal Lie derivatives.
• To recover a generalized energy–momentum tensor density via the First Noether Theorem.

Proposed method

• Uses the variational Lie derivative and quotient morphisms in finite-order variational sequences.
• Applies Kol\'a\v{r}'s invariant decomposition formula for vertical morphisms.
• Employs iterated Lie derivatives of sections of fibered bundles to analyze symmetry properties.
• Defines the modified Lagrangian $ \omega(\lambda, K) $ as the contraction of the Euler–Lagrange form with $ \bar{\Xi}_V \in K $.
• Derives the conservation law $ DH(-j^s\mathcal{L}_{\bar{\Xi}_V} \rfloor p^{DV} \omega(\lambda,K)) = 0 $ via horizontal splitting.
• Utilizes the exactness of the variational sequence to ensure global existence of the superpotential.

Experimental results

Research questions

• RQ1 How can conserved currents in gauge-natural field theories be derived in a globally and covariantly consistent manner?
• RQ2 What is the role of the Second Noether Theorem in generating generalized symmetries and conserved quantities?
• RQ3 Can a canonical superpotential for conserved currents be constructed without fixing a connection a priori?
• RQ4 Under what conditions does the modified Lagrangian $ \omega(\lambda, K) $ satisfy the naturality property $ L^{js}_{\bar{\Xi}_H} \omega(\lambda, K) = 0 $?
• RQ5 How do generalized Bergmann–Bianchi identities emerge from the structure of the variational sequence and iterated Lie derivatives?

Key findings

• The naturality property $ L^{js}_{\bar{\Xi}_H} \omega(\lambda, K) = 0 $ holds true for the modified Lagrangian $ \omega(\lambda, K) $, ensuring its invariance under horizontal symmetries.
• The existence of a canonical, global sheaf morphism $ \nu(\lambda, K) \in \mathcal{V}^{n-2}_{2s-1}(Y^\zeta \times_X K) $ is established, providing a connection-independent superpotential.
• The conservation law $ DH(-j^s\mathcal{L}_{\bar{\Xi}_V} \rfloor p^{DV} \omega(\lambda,K)) = 0 $ is derived, confirming the covariant conservation of the current.
• The generalized energy–momentum tensor density $ \epsilon(\lambda, K) - \tilde{\epsilon}(\lambda, K) $ is shown to be globally conserved via a 'strong' conservation law.
• The theory ensures that $ \omega(\lambda, K) $ is gauge-natural invariant, extending the invariance of the original Lagrangian $ \lambda $.
• The construction avoids fixing a principal connection a priori, enabling truly covariant field theories with global superpotentials.