First-order Lagrangian Theory Research Articles

A first-order Lagrangian theory of fields with arbitrary spin

The bundles suitable for a description of higher-spin fields can be built in terms of a 2-spinor bundle as the basic “building block”. This allows a clear, direct view of geometric constructions aimed at a theory of such fields on a curved spacetime. In particular, one recovers the Bargmann–Wigner equations and the [Formula: see text]-dimensional representation of the angular-momentum algebra needed for the Joos–Weinberg equations. Looking for a first-order Lagrangian field theory we argue, through considerations related to the 2-spinor description of the Dirac map, that the needed bundle must be a fibered direct sum of a symmetric “main sector” — carrying an irreducible representation of the angular-momentum algebra — and an induced sequence of “ghost sectors”. Then one indeed gets a Lagrangian field theory that, at least formally, can be expressed in a way similar to the Dirac theory. In flat spacetime, one gets plane-wave solutions that are characterized by their values in the main sector. Besides symmetric spinors, the above procedures can be adapted to anti-symmetric spinors and to Hermitian spinors (the latter describing integer-spin fields). Through natural decompositions, the case of a spin-2 field describing a possible deformation of the spacetime metric can be treated in terms of the previous results.

Symplectic structures related with higher order variational problems

In this paper, we derive the symplectic framework for field theories defined by higher order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher order system of partial differential equations to a constrained first-order one, the symplectic structures naturally arising in the dynamics of a first-order Lagrangian theory, and the importance of the Poincaré–Cartan form for variational problems, are all well-established facts. However, their adequate combination corresponding to higher order theories is missing in the literature. Here we obtain a consistent and truly finite-dimensional canonical formalism, as well as a higher order version of the Poincaré–Cartan form. In our exposition, the rigorous global proofs of the main results are always accompanied by their local coordinate descriptions, indispensable to work out practical examples.

The derivation of Einstein equations from invariance principles

In a first-order Lagrangian theory of gravitation, the author proves rigorously that general coordinate transformations are Noetherian symmetries if and only if the Lagrangian is a linear combination of a cosmological term and of a Lagrangian giving Einstein equations. Only the pure gravity case is considered.