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The relation between the Jacobi morphism and the Hessian in gauge-natural field theories

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Abstract

We generalize a classic result, due to Goldschmidt and Sternberg, relating the Jacobi morphism and the Hessian for first-order field theories to higher-order gauge-natural field theories. In particular, we define a generalized gauge-natural Jacobi morphism where the variation vector fields are Lie derivatives of sections of the gauge-natural bundle with respect to gauge-natural lifts of infinitesimal principal automorphisms, and we relate it to the Hessian. The Hessian is also very simply related to the generalized Bergmann-Bianchi morphism, whose kernel provides necessary and sufficient conditions for the existence of global canonical superpotentials. We find that the Hamilton equations for the Hamiltonian connection associated with a suitably defined covariant strongly conserved current are satisfied identically and can be interpreted as generalized Bergmann-Bianchi identities and thus characterized in terms of the Hessian vanishing.

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Author information

Correspondence to M. Palese.

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 377–389, August, 2007.

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Palese, M., Winterroth, E. The relation between the Jacobi morphism and the Hessian in gauge-natural field theories. Theor Math Phys 152, 1191–1200 (2007). https://doi.org/10.1007/s11232-007-0102-4

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Keywords

  • jet
  • gauge-natural bundle
  • second variational derivative
  • generalized Jacobi morphism