Canonical quantization of massive symmetric rank-two tensor in string theory

Hayun Park,Taejin Lee

doi:10.1016/j.nuclphysb.2020.115006

Abstract

The canonical quantization of a massive symmetric rank-two tensor in string theory, which contains two Stueckelberg fields, was studied. As a preliminary study, we performed a canonical quantization of the Proca model to describe a massive vector particle that shares common properties with the massive symmetric rank-two tensor model. By performing a canonical analysis of the Lagrangian, which describes the symmetric rank-two tensor, obtained by Siegel and Zwiebach (SZ) from string field theory, we deduced that the Lagrangian possesses only first class constraints that generate local gauge transformation. By explicit calculations, we show that the massive symmetric rank-two tensor theory is gauge invariant only in the critical dimension of open bosonic string theory, i.e., d=26. This emphasizes that the origin of local symmetry is the nilpotency of the Becchi-Rouet-Stora-Tyutin (BRST) operator, which is valid only in the critical dimension. For a particular gauge imposed on the Stueckelberg fields, the gauge-invariant Lagrangian of the SZ model reduces to the Fierz-Pauli Lagrangian of a massive spin-two particle. Thus, the Fierz-Pauli Lagrangian is a gauge-fixed version of the gauge-invariant Lagrangian for a massive symmetric rank-two tensor. By noting that the Fierz-Pauli Lagrangian is not suitable for studying massive spin-two particles with small masses, we propose the transverse-traceless (TT) gauge to quantize the SZ model as an alternative gauge condition. In the TT gauge, the two Stueckelberg fields can be decoupled from the symmetric rank-two tensor and integrated trivially. The massive spin-two particle can be described by the SZ model in the TT gauge, where the propagator of the massive spin-two particle has a well-defined massless limit.

Highlights

Ever since Dirac [1] attempted to quantize Einstein’s gravity in the framework of the canonical Hamiltonian formulation, the problem of quantization of the spin-two particle has remained unsolved
We note some of the problems of the Fierz and Pauli (FP) model in the massless limit from the point of view of canonical quantization and propose an alternative theory for the massive spin-two field, which can be obtained from the massive symmetric rank-two tensor theory, by choosing the transverse-traceless (TT) gauge [20,21,22,23,24]
We have studied the massive symmetric rank-two tensor in open string theory in the framework of canonical quantization that contains two Stueckelberg fields

Summary

INTRODUCTION

Ever since Dirac [1] attempted to quantize Einstein’s gravity in the framework of the canonical Hamiltonian formulation, the problem of quantization of the spin-two particle has remained unsolved. We approach the canonical quantization of the spin-two field from a different angle: we discuss the massive symmetric rank-two tensor, which is closely related to the linear theory of massive gravity [12], within the framework of open string theory. We note some of the problems of the FP model in the massless limit from the point of view of canonical quantization and propose an alternative theory for the massive spin-two field, which can be obtained from the massive symmetric rank-two tensor theory, by choosing the transverse-traceless (TT) gauge [20,21,22,23,24]. We show that in the TT gauge, the propagator of the massive spin-two field smoothly reduces to that of a massless graviton in the massless limit without encountering any discontinuities

PRELIMINARY

MASSIVE SYMMETRIC RANK-TWO TENSOR IN STRING THEORY

CANONICAL QUANTIZATION OF THE FIERZ-PAULI LAGRANGIAN

CANONICAL QUANTIZATION OF MASSIVE SYMMETRIC RANK-TWO TENSOR

MASSIVE SYMMETRIC RANK-TWO TENSOR IN THE TRANSVERSE-TRACELESS GAUGE

DISCUSSIONS AND CONCLUSIONS