Symmetries and Gauge Symmetries in Multisymplectic First and Second
SYMMETRIES AND GAUGE SYMMETRIES IN MULTISYMPLECTIC FIRST AND SECOND-ORDER LAGRANGIAN FIELD THEORIES: ELECTROMAGNETIC AND GRAVITATIONAL FIELDS 1JORDI GASET *, 2NARCISO ROMAN´ -ROY † 1Department of Physics, Universitat Autonoma` de Barcelona, Bellaterra, Spain. 2Department of Mathematics, Universitat Politecnica` de Catalunya, Barcelona, Spain. July 20, 2021 Abstract Symmetries and, in particular, Cartan (Noether) symmetries and conserved quantities (conser- vation laws) are studied for the multisymplectic formulation of first and second order Lagrangian classical field theories. Noether-type theorems are stated in this geometric framework. The concept of gauge symmetry and its geometrical meaning are also discussed in this formulation. The results are applied to study Noether and gauge symmetries for the multisymplectic description of the elec- tromagnetic and the gravitational theory; in particular, the Einstein–Hilbert and the Einstein–Palatini approaches. Key words: 1st and 2nd-order Lagrangian field theories, Higher-order jet bundles, Multisymplectic forms, symmetries, gauge symmetries, conservation laws, Noether theorem, Hilbert-Einstein action, Einstein-Palatini approach. MSC 2020 codes: Primary: 53D42, 70S05, 83C05; Secondary: 35Q75, 35Q76, 53Z05, 70H50, 83C99. arXiv:2107.08846v1 [math-ph] 19 Jul 2021 Contents 1 Introduction 2 2 Lagrangian field theories in jet bundles 3 2.1 Higher-order jet bundles. Multivector fields in jet bundles...................... 3 2.2 First andsecond-orderLagrangianfieldtheories . ..................... 4 3 Symmetries, conservation laws and gauge symmetries 6 3.1 Symmetries and conserved quantities for Lagrangian fieldtheories ................. 6 3.2 Gauge symmetries, gauge vector fields and gauge equivalence ................... 9 4 Electromagnetic field 12 5 Gravitational field (General Relativity) 13 5.1 TheHilbert-Einsteinaction . ............... 13 5.2 The Einstein-Palatini action (metric-affine model) .
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