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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 954 (2020) 115006

www.elsevier.com/locate/nuclphysb

Canonical quantization of massive symmetric rank-two

tensor in string theory

Hayun Park, Taejin Lee

∗

Department of Physics, Kangwon National University, Chuncheon 24341, Republic of Korea

Received 6 November 2019; received in revised form 13 February 2020; accepted 28 March 2020

Available online 31 March 2020

Editor: Stephan Stieberger

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 sym-

metric

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 ex-

plicit

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 lo-

cal

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 La-

grangian

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 sym-

metric

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.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Corresponding author.

E-mail address: taejin@kangwon.ac.kr (T. Lee).

https://doi.org/10.1016/j.nuclphysb.2020.115006

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

2 H. Park, T. Lee / Nuclear Physics B 954 (2020) 115006

1. Introduction

Ever since Dirac [1] attempted to quantize Einstein’s gravity in the framework of the canoni-

cal Hamiltonian formulation, the problem of quantization of the spin-two particle has remained

unsolved. When we perform a canonical quantization of Einstein’s gravity, a number of difﬁcul-

ties may be encountered [2–5]: Among others, Einstein’s

gravity is highly non-linear, and it is

difﬁcult to apply the usual perturbation theory. If we expand Einstein’s gravity in terms of per-

turbative metrics near the ﬂat Minkowski space-time, we obtain an inﬁnite number of interaction

terms. Second, because Einstein’s gravity does not possess dimensionless coupling, the pertur-

bative quantum

theory of Einstein’s gravity cannot be renormalized in four dimensions. String

theory is anticipated to provide reasonable measures to address such difﬁculties associated with

the application of perturbation theory to Einstein’s gravity. In this respect, it is worth noting that

the graviton scattering amplitudes of Einstein’s gravity [6–11]arise from

the closed string scat-

tering amplitude in the low-energy approximation where α



→ 0. By replacing Einstein’s gravity

with closed string theory in the high-energy region, it is expected that the problem of ultraviolet

divergence in quantum gravity could be resolved.

In the present work, 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. (For a

review of massive gravity, see

Refs. [13,14].) In open string theory [15], the massive spin-two

ﬁeld is generated as a component of the massive symmetric rank-two tensor multiplet, which

additionally contains a vector ﬁeld and a scalar ﬁeld. These additional ﬁelds may be considered

as Stueckelberg ﬁelds [16]to ensure local gauge symmetry,

as will be shown later. The massless

limit of the massive spin-two ﬁeld corresponds to the high energy limit in open string theory

where α



→∞. Thus, canonical quantization of the massive spin-two ﬁeld is also intrinsically

important when exploring the high energy limit of open string theory [17,18].

Introducing Stueckelberg ﬁelds to massive spin-two ﬁeld is not new: If we introduce two

Stueckelberg ﬁelds to the Fierz-Pauli model, describing a massive spin-two particle, we may

obtain a well-behaved massive spin-two propagator which reduces smoothly to the propagator

of massless graviton in the zero-mass limit [13,19]. However, there seems to be

no unique way

of ﬁxing couplings between Stueckelberg ﬁelds to massive spin-two ﬁeld. On the other hand, in

string theory the massive spin-two ﬁeld arises as a component of the massive symmetric rank-two

tensor multiplet, together with two Stueckelberg ﬁelds of which couplings are uniquely ﬁxed by

the BRST symmetry of

string theory. Here, we will discuss this speciﬁc form of massive spin-two

ﬁeld theory, which differs from those previously studied in literature.

The purpose of this study is twofold: The ﬁrst is to perform a canonical quantization of the

massive symmetric rank-two tensor theory established by Siegel and Zwiebach (SZ) using the

BRST-invariant open string ﬁeld theory. The second is to clarify its relation to the linear theory

of massive gravity studied by

Fierz and Pauli (FP) and to propose an alternative, yet equivalent

model, that may not share the undesired features of the FP model in the massless limit. 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 ob-

tained from the massive symmetric rank-two tensor theory, by choosing the transverse-traceless

(TT) gauge [20–24]. The problematic features of the FP model, such as the discontinuity of the

massless limit [25,26], which is termed

as the van Dam-Veltman-Zakharov (vDVZ) discontinu-

ity, may be attributed to inadequate choice of gauge ﬁxing for the massive symmetric rank-two

tensor theory of Siegel and Zwiebach with small mass. By an explicit construction, we show

H. Park, T. Lee / Nuclear Physics B 954 (2020) 115006 3

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.

2. Preliminary: canonical quantization of the Proca model

As a preliminary study, it may be instructive to ﬁrst discuss the Proca model [27] that describes

a massive vector particle because it is simple and shares many similarities with the massive

symmetric rank-two tensor. A massive spin-one ﬁeld is described by the Proca model whose

Lagrangian is given as follows:

=−

μν

−

(1)

where F

μν

= ∂

− ∂

. Throughout this paper, we will use η

μν

= diag(−, +, ..., +) as the

d-dimensional space-time metric. Performing the Hamiltonian analysis of the Proca action, we

ﬁnd two second class constraints [28,29], namely ϕ

and ϕ

= 

= 0, (2a)

= ∂



− m

= 0, (2b)



(x), ϕ



)



= m

δ(x − x



) = 0, (2c)

where 

, μ = 0, 1, ..., d − 1are canonical momenta that are conjugate to A

. This pair of

second class constraints compels us to use the Dirac brackets instead of the Poisson brackets

in the Hamiltonian formulation of the model. Usually, canonical quantization with second class

constraints often turns out to be cumbersome and complicated [1]. This problem may be re-

solved by introducing

an auxiliary scalar ﬁeld called Stueckelberg ﬁeld [16]. With the help of a

Stueckelberg ﬁeld, φ, we can recast the model into a more convenient form as follows:

=−

μν

−



+ ∂



. (3)

This Lagrangian Eq. (3)is now invariant under a local gauge transformation

δA

= ∂

, δφ =−. (4)

By imposing a gauge ﬁxing condition φ = 0, the gauge-invariant Lagrangian Eq. (3) reduces

to the Proca Lagrangian Eq. (1). Thus, the Proca Lagrangian Eq. (1) can be understood as a

gauge-ﬁxed version of the gauge-invariant Lagrangian Eq. (3).

The advantage of using the gauge-invariant Lagrangian is that we do not need to employ the

Dirac brackets to perform the canonical analysis: The Hamiltonian system of the gauge-invariant

model possesses only ﬁrst class constraints, which generate a local gauge transformation. Be-

cause we can use the usual Poisson brackets to

carry out the Hamiltonian formulation, quantiza-

tion of the system is simpler and easier than that with the Dirac brackets. Deﬁning the canonical

momenta as



∂L

∂

,π

∂L

∂

, (5)

we ﬁnd



= F

,π

= m



φ + A



(6)

and a primary constraint

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