Unconstrained Lagrangian Formulation for Bosonic Continuous Spin Theory in Flat Spacetime of Arbitrary Dimension

TL;DR

This paper introduces two new unconstrained Lagrangian formulations for bosonic continuous spin theory in any flat spacetime dimension, simplifying previous models by removing constraints and delta functions, and connecting to higher spin theories.

Contribution

The authors develop two unconstrained, conventional Lagrangians for bosonic continuous spin fields in arbitrary dimensions, extending and simplifying prior formulations.

Findings

Lagrangians do not include delta functions or higher derivatives.

Reproduces higher spin theory Lagrangian in the zero CS parameter limit.

Confirms modified Wigner equations through gauge fixing and equations of motion.

Abstract

We have discovered two unconstrained forms of free Lagrangian for continuous spin(CS) theory in arbitrary flat spacetime dimension for bosonic case. These Lagrangians, unlike that by Schuster and Toro, do not include delta functions and are conventional. The first form consists of five kinds of totally symmetric helicity fields and one kind of gauge parameter. By introducing auxiliary creation and annihilation operators, each is combined into a state vector in Fock space, including all ranks one by one. The Lagrangian imposes no constraints, such as trace conditions, on these fields or the gauge parameter field. Additionally, the Lagrangian does not contain higher-order derivative terms. In the limit as CS parameter $\mu$ approaches zero, it naturally reproduces a Lagrangian for helicity fields in higher spin(HS) theory, known as unconstrained quartet formulation. Permitting third-order derivatives, we also obtain the second unconstrained form of Lagrangian that can be written in terms of three kinds of fields, including $\mu$, similar to the formulation by Francia and Sagnotti. Partial gauge fixing and partial use of equations of motion(EOM) on this Lagrangian yield a Fronsdal-like Lagrangian with a single double-traceless field, including $\mu$. By imposing further gauge fixing on the field in the EOM with respect to divergence and trace, we confirm the reproduction of the modified Wigner equations already known in literature.