The composition algebra based methodology (CAM) (Wolk in Pap Phys 9:090002, 2017, Phys Scr 94:025301, 2019, Adv Appl Clifford Algebras 27(4):3225, 2017, J Appl Math Phys 6:1537, 2018, Phys Scr 94:105301, 2019) has previously been shown to generate the pre-Higgs Standard Model Lagrangian. In this paper the symmetry of general covariance is incorporated into CAM. The Riemann curvature tensor thereby arises, from which gravity-field Lagrangians are constructed. A Higgs-like scalar field coupled to the spacetime metric tensor also manifests.

The Riemann Curvature Tensor and Higgs Scalar Field within CAM Theory

We present canonical procedures for the manipulation of whole classes of Lagrangians that share the same transformation law and functional dependence but are otherwise arbitrary in functional form, and for the derivation therefrom of generalized conserved quantities. The techniques are demonstrated on the class of scalar density LagrangiansL=L
G+L
EM, whereL
G is a function of the metric and its first and second derivatives andL
EM is a function of the metric and a vector potential and its first derivative, which generate the Einstein-Maxwell equations (without cosmological constant). These procedures should be of interest to those studying alternate formulations of general relativity, those deriving new field theories, and others working with general of modified Lagrangians.

Variational Techniques on Classes of Lagrangians

Weyl’s theorem is extended making use of the theory of concomitants to obtain a Lagrangian density for the massless bosonic fields without dimensional constants. It turns out to be quadratic in the gravitational field and encompasses all the theories that usually appear in the literature. It is shown that the gauge invariance of the Lagrangian follows from the invariance of the field equations.

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Nissani N

Papers

Quadratic Lagrangian for general relativity theory.