Massive gravity from a first-quantized perspective

The theory of General Relativity (GR) has been the subject of modification attempts since the early years following Einstein’s formulation. Despite the considerable success of GR in the description of fundamental aspects of our reality, several phenomena persist unexplained, prompting the study of new physics. An intriguing modification of gravitational theories suggests that gravity is propagated by a massive spin $2$ particle: a massive graviton. The question of whether the graviton possesses mass remains an unresolved inquiry to date. This prospect, far from being dismissible on the basis of theoretical and experimental arguments deRham:2016nuf , would entail profound consequences, which may account for the enduring history of theoretical explorations dedicated to this possibility. The first theory of massive gravity dates back to 1939 when Fierz and Pauli proposed a relativistic action for a massive spin $2$ particle on flat spacetime which is now known as Fierz-Pauli (FP) theory Fierz:1939zz ; Fierz:1939ix . While appearing deceptively simple in its formulation, the theory already reveals the roots of several issues that have consistently plagued every effort to improve massive gravity beyond the linear level. Specifically, two primary obstacles arise: the vDVZ discontinuity vanDam:1970vg ; Zakharov:1970cc , namely the different predictions between linear GR and the FP theory in the massless limit, and the presence of a Boulware-Deser (BD) ghost, an extra degree of freedom which inevitably appears at the non-linear level Boulware:1972yco . To this date, only a promising non-linear theory of massive gravity free from the DB ghost instability has been formulated, which is known as dRGT theory deRham:2010ik ; deRham:2010kj ; deRham:2011rn . The interested reader may refer to excellent reviews for a comprehensive exploration of this subject Hinterbichler:2011tt ; deRham:2014zqa ; Schmidt-May:2015vnx .
This paper represents the first attempt to contribute to the ongoing investigation from an alternative perspective by employing a worldline approach Schubert:2001he . The primary goal is to achieve a first-quantized description of massive gravity. To this end, we exploit the first-quantized models known as $O({\mathcal{N}})$ spinning particles. These are mechanical models with ${\mathcal{N}}$ local supersymmetry on the worldline that have been shown to describe, on Minkowski, a spin $s=\frac{{\mathcal{N}}}{2}$ particle in first quantization and constitute an alternative to conventional second-quantized field theories in the study of quantum field theory (QFT) Berezin:1976eg ; Brink:1976uf ; Gershun:1979fb ; Howe:1988ft ; Howe:1989vn ; Siegel:1988ru ; Bastianelli:2008nm ; Corradini:2010ia ; Bastianelli:2014lia ; Corradini:2015tik . In recent years, considerable effort has been invested in exploring the consistent coupling of the spinning particles to more general backgrounds beyond flat spacetime. It was shown in Howe:1988ft that for ${\mathcal{N}}>2$, worldline supersymmetry transformation rules leave the spinning particle action invariant only if the target spacetime is flat. Subsequent progress was made in Kuzenko:1995mg and in Bastianelli:2008nm , where it was established how to couple the spinning particle to (A)dS spaces and arbitrary conformally flat spaces respectively. A significant advance has been recently achieved realizing that the technique of BRST quantization – originally developed in the context of the path integral quantization of Lagrangian gauge theories Becchi:1975nq – provides a way to investigate different backgrounds rather efficiently. An appealing aspect of BRST quantization relies on the automatic generation of the complete spectrum of spacetime fields essential for the Batalin-Vilkoviski (BV) quantization of the corresponding spacetime quantum field theory Batalin:1981jr ; Batalin:1983ggl . The BRST quantization of the spinning particle models led to a first-quantized description of Yang-Mills theory from the ${\mathcal{N}}=2$ spinning particle Dai:2008bh and Einstein gravity from the ${\mathcal{N}}=4$ spinning particle Bonezzi:2018box . The latter analysis showed quantum consistency in coupling the model to a more general set of backgrounds, namely to Einstein spaces. Furthermore, this formalism has been extended to include the NS-sector of supergravity in Bonezzi:2020jjq .
In this work, we intend to apply this procedure to reproduce linearized massive gravity (LMG) on both flat and curved spacetime. As a warm-up, we address the case of the massive ${\mathcal{N}}=2$ spinning particle and its connection to the Proca theory. This preliminary investigation serves a dual purpose: introducing the relevant methodologies and exploring the potential impact of the mass as an obstruction to the BRST algebra. One viable approach to incorporate a mass term is the Scherk-Schwarz mechanism Scherk:1979zr , which entails a dimensional reduction of the massless model. Recently, another method has been developed consisting of the introduction of auxiliary oscillators describing the internal degrees of freedom of the spinning particle Carosi:2021wbi . In this work, both methods are employed to confer mass to the graviton on a flat background, allowing for an in-depth exploration of the main features of the massive ${\mathcal{N}}=4$ supersymmetric worldline theory. Notably, these two approaches appear to yield different outcomes, with the first method providing the correct BV spectrum and field equations of linearized massive gravity. Conversely, the second approach encounters difficulties already at the initial stage. Subsequently, we attempt to couple the model to a generic curved background, and our findings suggest that the nilpotency of the BRST charge requires the background to be Einstein with cosmological constant set to zero.

The paper is organized as follows. In section 2 we outline the main features of the $O({\mathcal{N}})$ massless spinning particle models, providing the basis for the upcoming analysis. Section 3 provides an overview of the BRST quantization procedure, with emphasis on the first-quantized massless graviton both on flat and curved spacetime. In section 4 we discuss how to confer a mass to a spinning particle model. We begin by taking the ${\mathcal{N}}=2$ spinning particle as an illustrative example to elucidate the role of the mass in the BRST algebra, showing that the model correctly reproduces the Proca theory on a general background. We then proceed to give a mass to the graviton and check whether the theory correctly reproduces that of Fierz-Pauli. Finally, in section 5 we attempt to couple the model to a generic curved background, establishing the consistency of the BRST quantization exclusively on a subset of viable spacetimes. Conclusions and possible outlooks are presented in section 6.

The so-called $O({\mathcal{N}})$ spinning particles represent prominent examples where gauge symmetries allow for a manifestly Lorentz covariant formulation. This, in turn, gives rise to a constrained Hamiltonian system with first-class constraints new-book . These kinds of constraints, denoted as $C_{\alpha}$ (with $\alpha$ spanning the number of constraints), satisfy upon quantization a graded algebra of the form¹¹1With the following notation for the graded commutator: $$[\cdot,\cdot\}=\begin{cases*}\{\cdot,\cdot\}\quad\text{anticommutator if both variables are fermionic,}\\ [\cdot,\cdot]\quad\;\;\text{commutator otherwise.}\end{cases*}$$

$$[C_{\alpha},C_{\beta}\}=f_{\alpha\beta}{}^{\gamma}C_{\gamma}$$

(1)

with some structure functions $f_{\alpha\beta}{}^{\gamma}$. The phase space action of such models depends on the particle spacetime coordinates $x^{\mu}$ together with ${\cal N}$ real fermionic superpartners $\Xi^{\mu}_{I}$ introduced to describe the spin degrees of freedom. In the case of a flat $d$-dimensional target spacetime, the action is given by²²2The Minkowski metric $\eta_{\mu\nu}\sim(-,+,\cdots,+)$ is used to raise and lower spacetime indices. Indices named $\mu,\nu,\dots$ refer to spacetime indices $(\mu,\nu=0,1,\dots,d-1)$, while those named $I,J,\dots$ stand for internal $SO({\mathcal{N}})$ indices $(I,J=1,\dots,{\mathcal{N}})$.

$$S=\int d\tau\left[p_{\mu}\dot{x}^{\mu}+\frac{i}{2}\Xi_{\mu}\cdot\dot{\Xi}^{\mu}-e\,H-i\chi^{I}\,q_{I}-a^{IJ}\,J_{IJ}\right]\ ,$$

(2)

where a dot indicates a contraction of the internal indices. The theory described by (2) should be regarded as a one-dimensional field theory living on the worldline, which provides a first-quantized description of relativistic particles of spin $s$. A few remarks regarding its field content are in order.

The canonical coordinates $(x^{\mu},p_{\mu},\Xi^{\mu}_{I})\,$ upon quantization are subject to the canonical commutation relations (CCR)

$$[x^{\mu},p_{\nu}]=i\,\delta^{\mu}_{\nu}\ ,\quad\{\Xi^{\mu}_{I},\Xi^{\nu}_{J}\}=\delta_{IJ}\,\eta^{\mu\nu}\ .$$

(3)

The worldline supergravity multiplet in one dimension $(e,\chi^{I},a_{IJ})$ contains the einbein $e$ which gauges worldline translations, the ${\mathcal{N}}$ gravitinos $\chi$ which gauge the worldline supersymmetry, and the gauge field $a$ for the symmetry which rotates by a phase the worldline fermions and gravitinos, the $R$-symmetry. In (2) they act as Lagrange multipliers for the suitable first-class constraints, which are the Hamiltonian $H$, the supercharges $q_{I}$, and the $R$-symmetry algebra generators $J_{IJ}$, choosing a specific order for the latter to avoid ambiguities

$$H:=\frac{1}{2}\,p^{\mu}p_{\mu}\ ,\quad q_{I}:=\Xi_{I}^{\mu}\,p_{\mu}\ ,\quad J_{IJ}:=i\,\Xi_{[I}^{\mu}\,\Xi_{J]_{\mu}}\ .$$

(4)

Hamiltonian and supercharges together form the following one-dimensional algebra:

$$\{q_{I},q_{J}\}=2\,\delta_{IJ}\,H\ ,\quad[q_{I},H]=0\ .$$

(5)

These constraints have to be introduced to ensure the mass-shell condition and to eliminate negative norm states, ensuring the consistency of the model with unitarity at the quantum level Corradini:2015tik . On the other hand, regarding the aforementioned $R$-symmetry algebra one finds:

$\displaystyle\begin{split}[J_{IJ},q_{K}]&=i\left(\delta_{JK}q_{I}-\delta_{IK}q_{J}\right)\ ,\\ [J_{IJ},J_{KL}]&=i\left(\delta_{JK}J_{IL}-\delta_{IK}J_{JL}-\delta_{JL}J_{IK}+\delta_{IL}J_{JK}\right)\ .\end{split}$

(6)

The gauging of the $R$-symmetry group is optional and can be used to constrain the model to deliver pure spin $s$ states and have the minimal amount of degrees of freedom: to count them, one can construct the path integral on the one-dimensional torus of the free spinning particles, which has been achieved for all ${\mathcal{N}}$ in Bastianelli:2007pv . Equations (5) together with (6) form the so-called $O({\mathcal{N}})$-extended worldline supersymmetry algebra: it displays ${\mathcal{N}}$ supercharges $q_{I}$ which close on the Hamiltonian $H$ and which transform in the vector representation of $SO({\mathcal{N}})$, whose Lie algebra is described by the second line of (6). In the following, we will refer only to (5) as “${\mathcal{N}}$ SUSY algebra”, since it will be appropriate to distinguish the two algebras in light of the future BSRT procedure. It is rather convenient to work with a complex redefinition of the original fermionic variables $\Xi_{I}\to(\xi_{i},\bar{\xi}^{i})$, namely

$$\xi^{\mu}_{i}:=\tfrac{1}{\sqrt{2}}(\Xi^{\mu}_{i}+i\,\Xi^{\mu}_{i+2})\ ,\quad\bar{\xi}^{\mu i}:=\tfrac{1}{\sqrt{2}}(\Xi^{\mu}_{i}-i\,\Xi^{\mu}_{i+2})\ ,\quad\text{with}\;i=1,\dots,\tfrac{{\mathcal{N}}}{2}$$

(7)

for the case of even ${\mathcal{N}}$, i.e. for particles with integer spin. The respective CCR (3) become

$$\{\bar{\xi}^{i}_{\mu},\xi_{j}^{\nu}\}=\delta^{i}_{j}\,\delta^{\nu}_{\mu}\ ,\quad\{\bar{\xi}^{i}_{\mu},\bar{\xi}^{j}_{\nu}\}=0=\{\xi_{i}^{\mu},\xi_{j}^{\nu}\}\ .$$

(8)

Accordingly, both the supercharges and the $R$-symmetry generators split under the complex redefinition: we report the explicit expressions for the particle models of interest, namely the spin one and the spin two. By choosing a specific Fock vacuum, an arbitrary state $\ket{\Psi}$ in the Hilbert space is isomorphic to the wavefunction $\Psi(x,\xi)$, on which the conjugated momenta act as derivatives

$$p_{\mu}=-i\partial_{\mu}\ ,\quad\bar{\xi}_{\mu}=\frac{\partial}{\partial\xi^{\mu}}\ .$$

(9)

Thus, for the case ${\mathcal{N}}=2$, the spin one model, one has the splitting $q_{I}\to(q,\bar{q})$ and $J_{IJ}\to\mathrm{J}$ as follows

$$\begin{array}[]{l}q:=-i\,\xi^{\mu}\partial_{\mu}\\[5.69054pt] \bar{q}:=-i\,\partial^{\mu}\frac{\partial}{\partial\xi^{\mu}}\end{array}\ ,\qquad\mathrm{J}:=\xi^{\mu}\frac{\partial}{\partial\xi^{\mu}}-\frac{d}{2}\ ,$$

(10)

where the shift $-\tfrac{d}{2}$ in the definition of the $R$-symmetry generator is a quantum effect stemming from an antisymmetric ordering for the Grassmann variables Bastianelli:2005vk . For the ${\mathcal{N}}=4$ case, the spin two, one has $q_{I}\to(q_{i},\bar{q}^{i})$ and the $SO(4)$-symmetry generators are realized maintaining manifest covariance only under a $u(2)$-subalgebra $J_{IJ}\to(\mathrm{J}_{i}{}^{j},\mathrm{Tr}^{ij},\mathrm{G}_{ij})$, explicitly realized as³³3In the following, whether a dot $\cdot$ indicates contractions on internal or spacetime indices should be clear within the context.

$$\begin{array}[]{l}q_{i}:=-i\,\xi_{i}^{\mu}\partial_{\mu}\\[5.69054pt] \bar{q}^{i}:=-i\,\partial^{\mu}\frac{\partial}{\partial\xi_{i}^{\mu}}\end{array}\ ,\qquad\begin{array}[]{l}\mathrm{J}_{i}{}^{j}:=\xi_{i}\cdot\frac{\partial}{\partial\xi_{j}}-\tfrac{d}{2}\,\delta_{i}^{j}\\[5.69054pt] \mathrm{Tr}^{ij}:=\frac{\partial^{2}}{\partial\xi_{i}\cdot\partial\xi_{j}}\\[5.69054pt] \mathrm{G}_{ij}:=\xi_{i}\cdot\xi_{j}\end{array}\ .$$

(11)

Note that the only surviving components of the $\mathrm{Tr}$ and $\mathrm{G}$ operators are those with $i\neq j$.

There are at least two methods for dealing with first-class constraints and constructing canonical quantization of gauge systems: the Dirac method and the BRST quantization. In the former approach, the procedure entails turning the constraints into operators $\hat{C}_{\alpha}$ and imposing that an element $\ket{\Psi}$ of the Hilbert space is physical if it gets annihilated by the constraint operators, i.e.

$$\hat{C}_{\alpha}\ket{\Psi}=0\ .$$

(12)

This procedure has been extensively employed in Bastianelli:2008nm ; Bastianelli:2014lia to quantize both massless and massive spinning particle models. It is a relatively straightforward and powerful method, however, its application fails when the constraints algebra ceases to be first-class. These considerations eventually led to the BRST analysis of Dai:2008bh ; Bonezzi:2018box ; Bonezzi:2020jjq , which is reviewed in the following to delineate the main steps of the quantization procedure, with the intention of applying them to the massive case.

Take the ${\mathcal{N}}=4$ spinning particle with a four-dimensional target space as an instructive example. The internal indices take values $i=1,2$ and the ${\mathcal{N}}=4$ worldline SUSY algebra is explicitly realized as

$\displaystyle\{q_{i},\bar{q}^{j}\}=2\,\delta_{i}^{j}\,H\ ,\quad[q_{i},H]=[\bar{q}^{i},H]=\{q_{i},q_{j}\}=\{\bar{q}^{i},\bar{q}^{j}\}=0\ .$

(13)

The initial step of the BRST procedure entails an enlargement of the Hilbert space to realize ghost-antighost pairs of operators $(g^{\alpha},P_{\alpha})$ associated with each constraint $C_{\alpha}$, with opposite Grassmann parity of the latter – that is, anticommuting ghosts for bosonic constraints and commuting ghosts for fermionic constraints – and canonical graded commutation relation

$$[P_{\alpha},g^{\beta}\}=\delta^{\beta}_{\alpha}\ .$$

(14)

Hence, we assign the fermionic pair $(c,b)$ to the Hamiltonian, and the bosonic superghost pairs $(\bar{\gamma}^{i},\beta_{i})$ and $(\gamma_{i},\bar{\beta}^{i})$ to the supercharges $q_{i}$ and $\bar{q}^{i}\,$ respectively, obeying

$$\{b,c\}=1\ ,\quad[\beta_{i},\bar{\gamma}^{j}]=\delta_{i}^{j}\ ,\quad[\bar{\beta}^{j},\gamma_{i}]=\delta_{i}^{j}\ ,$$

(15)

with ghost number assignments

	$\displaystyle{\mathrm{gh}(c,\gamma_{i},\bar{\gamma}^{i})=+1}\ ,$		(16)
	$\displaystyle{\mathrm{gh}(b,\beta_{i},\bar{\beta}^{i})=-1}\ .$		(17)

The second step consists of constructing the BRST operator ${\mathbfcal{Q}}$. It is realized as follows

$${\mathbfcal{Q}}:=g^{\alpha}C_{\alpha}-\frac{1}{2}(-1)^{\epsilon(g_{\beta})}g^{\beta}g^{\alpha}f_{\alpha\beta}{}^{\gamma}P_{\gamma}\ ,$$

(18)

where $\epsilon(g_{\beta})$ is the Grassmann parity⁴⁴4We employ the following convention: the parity of $g^{\alpha}$ is $0$ if $g^{\alpha}$ is Grassmann even and $1$ if $g^{\alpha}$ is Grassmann odd. of the ghost $g^{\beta}$. Equation (18) is exact if the structure functions $f_{\alpha\beta}{}^{\gamma}$ are constant, and can be derived by demanding the following properties for the BRST operator:

• •

  It has to be anticommuting and of ghost number $+1$.

• •

  It has to act on the operators, corresponding to the original phase space variables prior to quantization, as the gauge transformations with the ghost variables replacing the gauge parameters. This, together with the latter requirement, is enough to constraint the structure to be ${\mathbfcal{Q}}=g^{\alpha}C_{\alpha}+\dots$

• •

  Finally, the BRST charge has to be nilpotent. This determines the second structure of (18) as can be checked by direct computation of $\{{\mathbfcal{Q}},{\mathbfcal{Q}}\}$.

Note that the BRST charge is nilpotent by construction as long as the associated algebra is first-class. In more general cases, higher-order terms may appear and need to be determined by the nilpotency condition. Given its significance for future discussions, we shall show it for the case of the ${\mathcal{N}}=4$ spinning particle. The BRST operator associated with the first-class algebra (13) is

$${\mathbfcal{Q}}=c\,H+\gamma_{i}\,\bar{q}^{i}+\bar{\gamma}^{i}\,q_{i}-2\bar{\gamma}^{i}\gamma_{i}\,b\ .$$

(19)

To verify its nilpotency, it is first convenient to define $\bm{\nabla}:=\gamma\cdot\bar{q}+\bar{\gamma}\cdot q$, such that for any operator of the form (19) one always finds the following potential obstruction-terms to its nilpotency

$${\mathbfcal{Q}}^{2}=-2\,\bar{\gamma}\cdot\gamma\,H+\bm{\nabla}^{2}-2c\,\cancel{\left[H,\bm{\nabla}\right]}\ ,$$

(20)

which are vanishing in the present case since

$$\bm{\nabla}^{2}=\gamma_{i}\bar{\gamma}^{j}\{\bar{q}^{i},q_{j}\}\implies-2\,\bar{\gamma}\cdot\gamma\,H+\bm{\nabla}^{2}=0\ .$$

(21)

Notice the critical role played by the presence of the first-class algebra (13) in both the cancellation of (20) and of (21). The same does not hold anymore in more general cases. In the next section, we will illustrate how to address this issue.

The careful reader might wonder about the absence of a set of ghosts associated with the $so(4)$ constraints. The key idea of the BRST procedure, as first developed in Dai:2008bh providing a first-quantized description of the gluon, is to treat the $R$-symmetry constraints and the SUSY ones on different footings. The formers are imposed as constraints on the BRST Hilbert space, thus defining precisely the general dependence of the “string field” $\Psi$ on the spacetime fields content.⁵⁵5Regarding terminology, we deliberately confuse “string field” and “BRST wavefunction” since $\Psi$ plays the same role in the worldline theory as it would in string field theory: its expansion as a linear combination of first-quantized states display coefficients which correspond to ordinary particle fields Gomis:1994he ; Thorn:1988hm . It is within this restricted Hilbert space that the cohomology of the BRST charge has to be studied. The procedure remains consistent as long as the $R$-symmetry constraints (11) commute with the BRST charge: in order to achieve that, it is necessary to extend $(\mathrm{J},\mathrm{Tr},\mathrm{G})\to({\mathcal{J}},{{\cal T}\!r},{\mathcal{G}})$ as follows

$$\begin{split}{\mathcal{J}}_{i}{}^{j}&:=\xi_{i}\cdot\bar{\xi}^{j}+\gamma_{i}\bar{\beta}^{j}-\bar{\gamma}^{j}\beta_{i}-2\,\delta_{i}^{j}\ ,\\[5.69054pt] {{\cal T}\!r}^{ij}&:=\bar{\xi}^{i}\cdot\bar{\xi}^{j}+\bar{\gamma}^{i}\bar{\beta}^{j}-\bar{\gamma}^{j}\bar{\beta}^{i}\ ,\\[5.69054pt] {\mathcal{G}}_{ij}&:=\xi_{i}\cdot\xi_{j}+\gamma_{i}\beta_{j}-\gamma_{j}\beta_{i}\ .\end{split}$$

(22)

The relevant $so(4)$ generators to be imposed as constraints on the BRST Hilbert space are the two number operators ${\mathcal{J}}_{i}:={\mathcal{J}}_{i}{}^{i}$ ($i$ not summed), namely the diagonal entries of ${\mathcal{J}}_{i}{}^{j}$, the Young antisymmetrizer ${\mathcal{Y}}:={\mathcal{J}}_{1}^{2}$, and finally the trace ${{\cal T}\!r}$, which implement the maximal reduction of the model. We collectively denote these constraints as $\mathcal{T}_{\alpha}:=({\mathcal{J}},{\mathcal{Y}},{{\cal T}\!r})$. The BRST system is defined as follows

$$\begin{split}&{\mathbfcal{Q}}\,\Psi=0\;,\quad\delta\Psi={\mathbfcal{Q}}\,\Lambda\;,\\[5.69054pt] &{\cal T}_{\alpha}\,\Psi=0\;,\quad{\cal T}_{\alpha}\,\Lambda=0\;,\end{split}$$

(23)

and its consistency is ensured by $[{\mathbfcal{Q}},\mathcal{T}_{\alpha}]=0$. This is indeed equivalent to saying that we are studying the cohomology of ${\mathbfcal{Q}}$ on the restricted Hilbert space of fixed $R$-charge defined by $\mathcal{H}_{\mathrm{red}}:=\ker\mathcal{T}_{\alpha}$. The system above will serve as the starting point for BRST quantization in both the massless and massive cases.

In this section, we present two methods to confer a mass to a spinning particle model. The first one, the Scherk-Schwarz mechanism, is analogous to the Kaluza-Klein compactification. In the second method, the auxiliary oscillators approach, the model is treated as a truncation of the RNS open superstring Green:2012pqa . We employ both methods to give mass to the graviton on a flat background and subsequently discuss the results. However, before delving into the graviton case, we begin by examining the ${\mathcal{N}}=2$ scenario, in order to introduce the techniques and investigate the role of the mass as a potential obstruction to nilpotency of the BRST charge.