Massive higher spin fields in curved spacetime and necessity of non-minimal couplings

Masafumi Fukuma,Junji Yamamoto,Katsuta Sakai,Hikaru Kawai

doi:10.1093/ptep/ptw080

Masafumi Fukuma, Junji Yamamoto + Show 2 more

Open Access

https://doi.org/10.1093/ptep/ptw080

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Abstract

Free massive higher spin fields in weak background gravitational fields are discussed. Contrary to the spin one case, higher spin fields should have nontrivial non-minimal couplings to the curvature. A precise analysis is given for the spin 2 case, and it is shown that two conditions should be satisfied among five non-minimal coupling constants, which we derive both in the Hamiltonian and Lagrangian formalisms. It is checked that the linearized limit of the massive gravity theory indeed has the non-minimal couplings that satisfy the conditions. We also discuss the form of the non-minimal couplings for the spin 3 case.

Highlights

Attempts to construct massive higher spin field theories first appeared with the papers written byFierz and Pauli, who formulated a free field theory of massive spin 2 particles in the Minkowski space [1,2]
In this paper we have obtained the Lagrangian that describes a free massive spin 2 or spin 3 particle propagating in the general gravitational background to the first order in the curvature.The Lagrangians contain non-minimal couplings
For the spin 2 case, the coefficients have three free parameters, and, in particular, the coupling constant associated with the Riemann tensor is arbitrary

Summary

Introduction

Fierz and Pauli, who formulated a free field theory of massive spin 2 particles in the Minkowski space [1,2]. In the spin 1 case, the divergence of the EOM always results in a first-order differential equation corresponding to the transverse condition, irrespective of how non-minimal couplings are introduced. The Lagrangian takes the form of higher spins, there is no choice of non-minimal couplings so as to cancel the right-hand side of (2.22) Another problem will emerge when formally substituting (2.22) into (2.18), since it results in fourth-order differential equations with respect to time. If the Hamiltonian has only linear terms in h00 after the elimination of h0i , the variables h00 and h0i will disappear from the system, leaving two constraints This means that the system has ten (= 6 + 6 − 2) DOF, which agree with those of a massive spin 2 field. Describes a massive spin 2 field with correct DOF in an arbitrary curved background

Analysis based on the Lagrangian

Discussion