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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) . ....................... 15 6 Conclusions and outlook 18 *e-mail: [email protected]. (ORCID: 0000-0001-8796-3149). †e-mail: [email protected]. (ORCID: 0000-0003-3663-9861). J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.2 1 Introduction It is well known that symmetries have enormous relevance in physical theories and, in general, in the treatment and resolution of differential equations modelling them. This is due to the fact that the presence of symmetries leads to the existence of conservation laws or conserved quantities which, in addition to helping the integration of these equations, highlight fundamental properties of physical systems. In this sense, the work of E. Noether at the beginning of the 20th century provides fundamental results on this topic [27]. From a geometric perspective, symmetries of mechanical systems and classical field theories are usually stated by demanding the invariance of the underlying geometric structures and/or the dynamical elements which characterize these systems (for a review in the case of mechanics see, for instance, [35] and the references cited therein). In the case of the multisymplectic description of field theories, these are the multisymplectic forms which are defined in the jet bundles and the multimomentum bundles where the theory is developed. These forms are constructed from the Lagrangian which describes the system, using the canonical elements which these bundles are endowed with [1, 12, 15, 22, 25, 30, 31, 39]. The study of symmetries and their associated conservation laws in this framework has been carried out in many papers (see, for instance, [11, 14, 18, 19, 38] and the references therein). One of the main characteristics of certain kinds of physical theories is the so-called gauge invariance, which is a consequence of the existence of a particular type of symmetry called gauge symmetry. This is a property which is associated with physical systems described by singular Lagrangians. Gauge symme- tries have their own geometric characterization which is related to the fact that the multisymplectic forms constructed from the (singular) Lagrangians are degenerated (and then they are called premultisymplectic forms). In this review paper, our aim is to present an accurate geometric description of symmetries in classi- cal field theories of first and second-order type. First, we introduce the standard classical symmetries of Noether type and state the Noether theorem which gives the way of obtaining the corresponding associ- ated conservation laws. Second, we discuss in detail the geometrical meaning and the characteristics of gauge symmetries; clarifying several geometric aspects that are not usually analysed in most treaties on this subject. Finally, we apply the results to the case of the two fundamental classical theories: Electro- magnetism (Maxwell theory) and Gravitation (General Relativity); in this last case, considering the two more basic models; i.e., the Einstein–Hilbert and the Einstein–Palatini approaches (the first and third are first-order field theories, but the second is second-order). The organization of the paper is as follows: Section 2 is devoted to summarize the multisymplectic Lagrangian formulation of first and second-order classical field theories. In Section 3 we study symme- tries, conservation laws and gauge symmetries in this multisymplectic setting. The analysis of symme- tries of Electromagnetism is analyzed in Section 4 and, finally, in Section 5 we study Noether and gauge symmetries of both models in General Relativity. All the manifolds are real, second countable and C∞. The maps and the structures are C∞. Sum over repeated indices is understood. Along this paper, we use the notation of multi-indices: a multi-index I is an element of Zm where every component is positive, the ith position of the multi-index is denoted m I(i), and |I| = I(i) is the length of the multi-index. The equality |I| = k means that the expression i=1 X Zm j is taken for every multi-index of length k. Furthermore, the element 1i ∈ is defined as 1i(j) = δi . Finally, n(ij) is a combinatorial factor with n(ij) = 1 for i = j, and n(ij) = 2 for i 6= j. J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.3 2 Lagrangian field theories in jet bundles 2.1 Higher-order jet bundles. Multivector fields in jet bundles (See [13, 26, 39] for details). π Let E −→ M be a fiber bundle over an orientable m-dimensional manifold M, with dim E = m+n. The kth-order jet bundle of the projection π is the manifold of the k-jets (equivalence classes) of local k k k sections of π, φ ∈ Γ(π), and is denoted J π. Points in J π are denoted by jxφ, with x ∈ M and φ ∈ Γ(π) being a representative of the equivalence class. J kπ is endowed with the following natural projections: if 1 6 r 6 k, k k r k k k k πr : J π −→ J π π : J π −→ E π¯ : J π −→ M k r k k jxφ 7−→ jxφ jxφ 7−→ φ(x) jxφ 7−→ x s k k k k k k k where πr ◦ πs = πr , π0 = π , πk = IdJkπ, and π¯ = π ◦ π . We denote ω the volume form in M and all its pull-backs to every J rπ. If (xi,yα), 1 6 i 6 m, 1 6 α 6 n, are local coordinates in E adapted to the bundle structure, such that ω =dx1 ∧ . ∧ dxm ≡ dmx; then local coordinates in J kπ are denoted i α (x ,yI ), with 0 6 |I| 6 k. If φ ∈ Γ(π), the kth prolongation of φ to J kπ is denoted jkφ ∈ Γ(¯πk). Then, a section ψ ∈ Γ(¯πk) is holonomic if jk(πk ◦ ψ)= ψ; that is, ψ is the kth prolongation of the section φ = πk ◦ ψ ∈ Γ(π). In an analogous way, let Φ: E → E be a diffeomorphism and and ΦM : M → M the diffeomor- phism induced on the basis; then the canonical lift of Φ to J kπ is the map jkΦ: J kπ −→ J kπ defined by k k k −1 k k (j Φ)(jx φ) := j (Φ ◦ φ ◦ ΦM )(ΦM (x)) ; for jx φ ∈ J π . Then, if Y ∈ X(E), the canonical lift of Y to J kπ is the vector field j1Y ∈ X(J kπ) whose associated local one-parameter groups of diffeomorphisms are the canonical lifts of the local one-parameter groups ∂ ∂ of diffeomorphisms of Y . In coordinates, if Y = f i + gα ; for instance, for k = 1 we have that ∂xi ∂yα ∂ ∂ ∂gα ∂f j ∂f j ∂gα ∂ j1Y = f i + gα + − yα + yβ + yβ . ∂xi ∂yα ∂xi j ∂xi i ∂yβ i ∂yβ ∂yα i Another special kind of vector fields are the coordinate total derivatives [34, 39]: k ∂ α ∂ Di = i + yI+1i α , ∂x ∂yI |XI|=0 ∞ k and for f ∈ C (J π), we have that Dif = L(Di)f. An m-multivector field in J kπ is a skew-symmetric contravariant tensor field of order m in J kπ. The set of m-multivector fields in J kπ is denoted Xm(J kπ). A multivector field X ∈ Xm(J kπ) is locally k k k k decomposable if, for every jxφ ∈ J π, there is an open neighbourhood U ⊂ J π, with jx φ ∈ U, and X1,...,Xm ∈ X(U) such that X|U = X1 ∧ . ∧ Xm. Locally decomposable m-multivector fields are locally associated with m-dimensional distributions D ⊂ TJ kπ, and multivector fields associated with the same distribution make an equivalence class {X} in the set Xm(J kπ). Then, X is integrable if its associated distribution is integrable. In particular, X is holonomic if it is integrable and its integral sections are holonomic sections of π¯k. X Xm k k k k A multivector field ∈ (J π) is π¯ -transverse if, at every point jxφ ∈ J π, we have that k∗ m m k ( (X)(¯π β)) k 6= 0; for every β ∈ Ω (M) such that β k k 6= 0.
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