Abstract
We propose a framework for Lorentz-invariant Lagrangian field theories where Ostrogradsky's scalar ghosts could be absent. A key ingredient is the generalized Kronecker delta. The general Lagrangians are reformulated in the language of differential forms. The absence of higher order equations of motion for the scalar modes stems from the basic fact that every exact form is closed. The well-established Lagrangian theories for spin-0, spin-1, p-form, spin-2 fields have natural formulations in this framework. We also propose novel building blocks for Lagrangian field theories. Some of them are novel nonlinear derivative terms for spin-2 fields. It is nontrivial that Ostrogradsky's scalar ghosts are absent in these fully nonlinear theories.
Highlights
The problem of ghost-like degrees of freedom are encountered in the construction of theories with large numbers of spacetime indices, either from high spin fields or high order derivatives
High spin fields are dangerous because some tensor indices become derivative indices for the longitudinal modes, while high derivative Lagrangians are dangerous due to Ostrogradsky’s instability
We propose that the Lagrangians should be some differential forms
Summary
The problem of ghost-like degrees of freedom are encountered in the construction of theories with large numbers of spacetime indices, either from high spin fields or high order derivatives. It is tempting to develop a unifying framework for local, ghost-free, Lorentz-invariant, Lagrangian field theories, with antisymmetrization being a key ingredient. It is clearly not an easy task to keep track of all the degrees of freedom in full generality and to make sure they are all free of ghost-like instability. Ωk could be the curvature two-forms and some exact forms constructed from matter fields The use of these elaborate building blocks can help us to understand why higher derivative terms are forbidden from appearing in equations of motion: the absence of ghost-like degrees of freedom becomes a consequence of a basic property of exterior derivative d 2 = 0,. Whose geometric interpretation by Stokes’s theorem is the boundary of a boundary vanishes
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