Tensor Lagrangian Research Articles

Background Independence and Gauge Invariance in General Relativity Part 1—The Classical Theory

The implications of the principles of general and manifest covariance, together with those of the objectivity principle, are considered for the purpose of establishing a DeDonder–Weyl-type Hamiltonian variational formulation for classical general relativity. Based on the analysis of the Einstein–Hilbert variational principle, it is shown that only synchronous variational principles permit the construction of fully 4−tensor Lagrangian and Hamiltonian theories of this type. In addition, the possible validity of an extended Hamiltonian formulation in which Lagrangian variables include also the Ricci tensor is investigated and shown to occur provided the classical cosmological constant is non-vanishing.

Restricted Galileons

We study Galileon theories that emerge in ghost-free massive gravity. In particular, we focus on a sub-class of these theories where the Galileons can be completely decoupled from the tensor Lagrangian. These Galileons differ from generic ones -- they have interrelated coefficients of the cubic and quartic terms, and most importantly, a non-standard coupling to external stress-tensors, governed by the same coefficient. We show that this theory has no static stable spherically symmetric solutions that would interpolate from the Vainshtein region to flat space; these two regions cannot be smoothly matched for the sign of the coefficient for which fluctuations are stable. Instead, for this sign choice, a solution in the Vainshtein domain is matched onto a cosmological background. Small fluctuations above this solution are stable, and sub-luminal. We discuss observational constraints on this theory, within the quantum effective Lagrangian approach, and argue that having a graviton mass of the order of the present-day Hubble parameter, is consistent with the data. Last but not least, we also present a general class of cosmological solutions in this theory, some of which exhibit the de-mixing phenomenon, previously found for the self-accelerated solution.

Tensor Lagrangians, Lagrangians Equivalent to the Hamilton-Jacobi Equation and Relativistic Dynamics

We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and magnetic fields of Maxwell's equations with sources. We introduce and analyse Lagrangians which are equivalent to the Hamilton-Jacobi equation and recast them to relativistic equations.

General tensor Lagrangians from the gravitational Higgs mechanism

The gravitational Higgs mechanism proposed by 't Hooft in arXiv:0708.3184 involves the spacetime metric gμν as well as the induced metric where ϕa (a = 0, …, 3), as we call it, break all four diffeomorphisms spontaneously via the vacuum expectation values ⟨ϕa⟩ ∝ xa. In this framework, we construct and analyze the most general action density in terms of various invariants involving the curvature tensors, connexion coefficients, and the contractions and the determinants of the two metric fields. We show that this action admits a consistent expansion about the flat background such that the resulting Lagrangian possesses several novel features not found in the linearized Einstein–Hilbert Lagrangian with Fierz–Pauli mass term (LEHL-FP): (i) its kinetic part generalizes that of LELHL-FP by weighing the corresponding structures with certain coefficients generated by invariants, (ii) the entire Lagrangian is ghost– and tachyon–free for mass terms not necessarily in the Fierz–Pauli form, and, (iii) a consistent mass term is generated with no apparent need to higher derivative couplings.

Reconstruction of the scalar–tensor Lagrangian from aΛCDM background and Noether symmetry

We consider scalar–tensor theories and reconstruct their potentialU(Φ) and couplingF(Φ) by demandinga background ΛCDM cosmology. In particular we impose a background cosmic historyH(z) provided by the usualflat ΛCDM parameterizationthrough the radiation (weff = 1/3), matter (weff = 0)and de Sitter (weff = −1) eras. The cosmological dynamical system which is constrained to obey theΛCDM cosmic history presents five critical points in each era, one of whichcorresponding to the standard General Relativity (GR). In the cases thatdiffer from GR, the reconstructed coupling and potential are of the formF(Φ)∼Φ2 andU(Φ)∼F(Φ)m, wherem is a constant relatingthe two functions F(Φ) and U(Φ): this assumption is necessary to find out the fixed points of the dynamical system. Thisclass of scalar tensor theories is also theoretically motivated by a completely independentapproach: imposing maximal Noether symmetry on the scalar–tensor Lagrangian. Thisapproach independently (i) provides the form of the coupling and the potential asF(Φ)∼Φ2 and U(Φ)∼F(Φ)m, (ii) provides a conserved charge related to the potential and the coupling, and (iii) allowsthe derivation of exact solutions by first integrals of motion.

On different lagrangian formalisms for vector resonances within chiral perturbation theory

We study the relation of vector Proca field formalism and antisymmetric tensor-field formalism for spin-one resonances in the context of the large NC inspired chiral resonance Lagrangian systematically up to the order O(p6) and give a transparent prescription for the transition from vector to antisymmetric tensor Lagrangian and vice versa. We also discuss the possibility to describe the spin-one resonances using an alternative “mixed” first order formalism, which includes both types of fields simultaneously, and compare this one with the former two. We also briefly comment on the compatibility of the above lagrangian formalisms with the high-energy constraints for a concrete VVP correlator.

On ghost-free tensor Lagrangians and linearized gravitation

The most general linear Lagrangian for massless and massive tensor fields without ghosts and tachyons is determined. Transversality is used to eliminate massless ghosts. The linear Einstein and Fierz-Pauli Lagrangians are unique for massless and massive tensor fields containing spin two. Hence, for gravitation, as for electromagnetism, gauge invariance and locality follow from positivity of the free-field energy. Symmetric and antisymmetric tensor theories with continuous zero-mass limit are found, but contain only spin one and spin zero.

Tensor lagrangians and generalized conservation laws for free fields

The existence of tensor Lagrangians for noninteracting massless fields of arbitrary spin is demonstrated. In particular, for spin one, the Euler-Lagrange equations for a vector Lagrangian are equivalent to Maxwell’s equations. Conserved quantities are derived in the usual fashion from these generalized Lagrangians by means of Noether’s theorem. A tensor Lagrangian and corresponding conserved quantities are also exhibited for the Dirac field with nonzero mass. A very natural quantization assumption which applies to all massless fields is made, and the physical interpretation of the generalized conserved quantities in terms of number operators is given.