[Paper Review] Unitarity of Singh-Hagen model in $D$ dimensions

E. L. Mendonça, R. Schimidt Bittencourt|arXiv (Cornell University)|Feb 13, 2019

Algebraic structures and combinatorial models7 references4 citations

TL;DR

This paper provides a complete set of spin-projection operators for totally symmetric rank-3 fields in $D$ dimensions, derives the propagator of the Singh-Hagen model, and confirms its unitarity and positivity in both massive and massless cases. The analysis verifies that the theory describes only a single spin-3 particle with no ghost states, establishing its physical consistency across dimensions through Hamiltonian constraint analysis and spectral decomposition.

ABSTRACT

The particle content of the Singh-Hagen model ($SH$) in $D$ dimensions is revisited. We suggest a complete set of spin-projection operators acting on totally symmetric rank-3 fields. We give a general expression for the propagator and determine the coefficients of the $SH$ model confirming previous results of the literature. Adding totally symmetric source terms we provide an unitarity analysis in $D$ dimensions.

Motivation & Objective

• To construct a complete basis of spin-projection operators for totally symmetric rank-3 tensor fields in $D$ dimensions.
• To re-express the Singh-Hagen model's Lagrangian using these projectors and derive its propagator in general $D$.
• To verify the unitarity of the model by analyzing the sign of the imaginary part of the residue of the transition amplitude with source terms.
• To confirm the positivity of the massless Hamiltonian and the correct counting of degrees of freedom via first-class constraints.
• To demonstrate that the theory describes only a physical spin-3 particle, free of unphysical modes or ghost states.

Proposed method

• The authors define a set of six orthonormal, idempotent spin-projection operators $P^{(s)}_{ij}$ acting on rank-3 symmetric tensors, generalizing known constructions to arbitrary $D$ dimensions.
• They use the transverse projector $\theta_{\mu\nu} = \eta_{\mu\nu} - \partial_\mu \partial_\nu / \Box$ and longitudinal projector $\omega_{\mu\u u} = \partial_\mu \partial_\nu / \Box$ to build the projectors via symmetrized combinations.
• The Lagrangian is decomposed into bilinear forms using these projectors, allowing the identification of contributions from spin-3, spin-2, spin-1, and spin-0 sectors.
• Unitarity is tested by coupling the theory to external sources and computing the residue of the transition amplitude, confirming a positive imaginary part for physical states.
• For the massless case, the canonical Hamiltonian and constraints are derived in $D$ dimensions, and the constraints are shown to be first-class, ensuring correct physical degrees of freedom.
• The reduced Hamiltonian is constructed using constraints as strong equalities, proving the theory describes only a spin-3 particle with a positive-definite energy spectrum.

Experimental results

Research questions

• RQ1 What is the complete set of spin-projection operators for totally symmetric rank-3 fields in $D$ dimensions, and how do they decompose the field into irreducible Lorentz representations?
• RQ2 Does the Singh-Hagen model in $D$ dimensions describe only a physical spin-3 particle, or are there unphysical degrees of freedom such as ghosts?
• RQ3 Is the transition amplitude of the model unitary, as indicated by the sign of the imaginary part of the residue in the presence of sources?
• RQ4 Is the massless version of the model unitary and positive-definite, as confirmed by the Hamiltonian structure and constraint analysis?
• RQ5 How do the coefficients in the Lagrangian affect the particle content, and can they be removed via field redefinitions?

Key findings

• A complete, orthonormal basis of six spin-projection operators for rank-3 symmetric tensors is constructed in $D$ dimensions, generalizing previous results in $D=3+1$.
• The propagator of the Singh-Hagen model is derived using the general bilinear Lagrangian decomposition, confirming the presence of only a single spin-3 particle in the spectrum.
• The coefficients $a$ and $t$ in the model’s Lagrangian are found to be removable via field redefinitions, confirming the model’s physical content is independent of these parameters.
• Unitarity is confirmed by showing the imaginary part of the residue of the transition amplitude is positive, indicating a physical propagating mode.
• In the massless case, the canonical Hamiltonian is derived and shown to be positive-definite after reduction via first-class constraints, confirming the absence of ghost states.
• The reduced Hamiltonian is expressed solely in terms of spin-projection operators, proving the theory describes exactly one spin-3 particle with the correct number of physical degrees of freedom.