Nonpropagating ghost in covariant $f(Q)$ gravity

A general affine connection $\varGamma^{\alpha}_{\,\mu\nu}$ can be decomposed into three parts:

$\displaystyle\varGamma^{\alpha}_{\,\mu\nu}=\left\{{}^{\,\alpha}_{\mu\nu}\right\}+K_{\,\mu\nu}^{\alpha}+L_{\,\mu\nu}^{\alpha}\,.$

(1)

Here, $\left\{{}^{\,\alpha}_{\mu\nu}\right\}$ is Levi-Civita connection

$\displaystyle\left\{{}^{\,\alpha}_{\mu\nu}\right\}=\frac{1}{2}g^{\alpha\lambda}\left(\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\mu\lambda}-\partial_{\lambda}g_{\mu\nu}\right)\,,$

(2)

$K_{\,\mu\nu}^{\alpha}$ is contortion

$\displaystyle K^{\alpha}_{\,\mu\nu}=\frac{1}{2}g^{\alpha\lambda}\left(T_{\mu\lambda\nu}+T_{\nu\lambda\mu}+T_{\lambda\mu\nu}\right)\,,$

(3)

and $L_{\,\mu\nu}^{\alpha}$ is disformation

$\displaystyle L^{\alpha}_{\,\mu\nu}=\frac{1}{2}g^{\alpha\lambda}\left(Q_{\lambda\mu\nu}-Q_{\mu\lambda\nu}-Q_{\nu\lambda\mu}\right)\,.$

(4)

Torsion tensor $T^{\lambda}_{\,\mu\nu}$ and nonmetricity tensor $Q_{\alpha\mu\nu}$ are defined as

	$\displaystyle T^{\alpha}_{\ \mu\nu}$	$\displaystyle\equiv\varGamma^{\alpha}_{\ \mu\nu}-\varGamma^{\alpha}_{\ \nu\mu}\,,$		(5)
	$\displaystyle\begin{split}Q_{\alpha\mu\nu}&\equiv\hat{\nabla}_{\alpha}g_{\mu\nu}\\ &=\partial_{\alpha}g_{\mu\nu}-g_{\nu\sigma}\varGamma^{\sigma}_{\ \mu\alpha}-g_{\sigma\mu}\varGamma^{\sigma}_{\ \nu\alpha}\,.\end{split}$			(6)

Using two types of trace of nonmetricity tensor

$\displaystyle Q_{\alpha}\equiv Q_{\alpha}{}_{\mu}{}^{\mu}\,,\quad\tilde{Q}_{\alpha}\equiv Q^{\mu}_{\ \mu\alpha}\,,$

(7)

we define the nonmetricity conjugate

$\displaystyle P^{\alpha\mu\nu}=-\frac{1}{4}Q^{\alpha\mu\nu}+\frac{1}{2}Q^{(\mu\nu)\alpha}+\frac{1}{4}\left(Q^{\alpha}-\tilde{Q}^{\alpha}\right)g^{\mu\nu}-\frac{1}{4}g^{\alpha(\mu}Q^{\nu)}\,,$

(8)

and nonmetricity scalar

$\displaystyle\begin{split}Q&=Q_{\alpha\mu\nu}P^{\alpha\mu\nu}\\ &=-\frac{1}{4}Q^{\alpha\mu\nu}Q_{\alpha\mu\nu}+\frac{1}{2}Q^{\nu\mu\alpha}Q_{\alpha\mu\nu}+\frac{1}{4}Q^{\alpha}Q_{\alpha}-\frac{1}{2}\tilde{Q}^{\alpha}Q_{\alpha}\,.\end{split}$

(9)

The action of STEGR is

$\displaystyle S=\int\mathrm{d}^{4}x\left(\sqrt{-g}Q_{\alpha\mu\nu}P^{\alpha\mu\nu}+\lambda_{\alpha}{}^{\beta\mu\nu}R^{\alpha}{}_{\beta\mu\nu}+\lambda_{a}{}^{\mu\nu}T^{\alpha}_{\ \mu\nu}\right)\,.$

(10)

Variation of the action (10) with respect to Lagrange multipliers $\lambda_{\alpha}{}^{\beta\mu\nu}$ and $\lambda_{a}{}^{\mu\nu}$ generates the torsion-free condition

$\displaystyle T^{\alpha}_{\,\mu\nu}\stackrel{{\scriptstyle!}}{{=}}0$

(11)

and vanishing curvature

$\displaystyle{R}_{\ \beta\mu\nu}^{\alpha}(\varGamma)\equiv\partial_{\mu}\varGamma^{\alpha}_{\ \nu\beta}-\partial_{\nu}\varGamma_{\ \mu\beta}^{\alpha}+\varGamma_{\ \mu\lambda}^{\alpha}\varGamma_{\ \nu\beta}^{\lambda}-\varGamma_{\ \nu\lambda}^{\alpha}\varGamma_{\ \mu\beta}^{\lambda}\stackrel{{\scriptstyle!}}{{=}}0\,.$

(12)

The former demands the general affine connection to be symmetric to its lowered two indices, as the latter implies that the connection must have the form $\varGamma^{\lambda}_{\,\mu\nu}=(A^{-1})^{\lambda}_{\ \alpha}\partial_{\mu}A^{\alpha}_{\ \nu}$ Zhao (2022).

Combining two conditions on the affine connection as in Eqs. (11) and (12), we find that the matrix $A^{\alpha}{}_{\beta}=\partial_{\beta}\xi^{\alpha}$, and the affine connection consequently takes the following form:

$\displaystyle\varGamma^{\lambda}_{\,\mu\nu}=\frac{\partial x^{\lambda}}{\partial\xi^{\alpha}}\frac{\partial^{2}\xi^{\alpha}}{\partial x^{\mu}\partial x^{\nu}}\,.$

(13)

Here, four scalar fields $\xi^{\alpha}(x)$ are the arbitrary functions of coordinates. Note that the connection is symmetric with respect to two lower indices because two partial derivatives of $\xi^{\alpha}$ commute each other. Moreover, one finds $A^{\alpha}_{\mu}$ corresponds to the tetrad $e^{a}_{\mu}$ for Weitzenböck connection in the teleparallel gravity.

A special choice $\xi^{\alpha}(x)=x^{\alpha}$ makes the connection vanish $\varGamma^{\lambda}_{\ \mu\nu}=0$. This choice is called the coincident gauge. Under the coincident gauge, the covariant derivative is replaced by the partial derivative $\nabla_{\alpha}\mapsto\partial_{\alpha}$, and the nonmetricity tensor $Q_{\alpha\mu\nu}$, deformation tensor $L^{\alpha}_{\,\mu\nu}$, nonmetricity scalar $Q$, and the traces $Q_{\alpha}$, $\tilde{Q}_{\alpha}$ reduce to

	$\displaystyle Q_{\alpha\mu\nu}\mapsto\mathring{Q}_{\alpha\mu\nu}$	$\displaystyle=\partial_{\alpha}g_{\mu\nu}\,,$		(14)
	$\displaystyle L^{\alpha}_{\,\mu\nu}\mapsto\mathring{L}_{\,\mu\nu}^{\alpha}$	$\displaystyle=-\left\{{}^{\,\alpha}_{\mu\nu}\right\}\,,$		(15)
	$\displaystyle\begin{split}Q\mapsto\mathring{Q}&=g^{\mu\nu}\left(\mathring{L}_{\ \sigma\mu}^{\alpha}\mathring{L}_{\ \nu\alpha}^{\sigma}-\mathring{L}_{\ \sigma\alpha}^{\alpha}\mathring{L}_{\ \mu\nu}^{\sigma}\right)\\ &=g^{\mu\nu}\left(\left\{{}^{\,\alpha}_{\sigma\mu}\right\}\left\{{}^{\,\sigma}_{\nu\alpha}\right\}-\left\{{}^{\,\alpha}_{\sigma\alpha}\right\}\left\{{}^{\,\sigma}_{\mu\nu}\right\}\right)\,,\end{split}$			(16)
	$\displaystyle\begin{split}Q^{\alpha}-\tilde{Q}^{\alpha}\mapsto\mathring{Q}^{\alpha}-\mathring{\tilde{Q}}^{\alpha}&=g^{\mu\nu}\mathring{L}^{\,\alpha}_{\,\mu\nu}-g^{\mu\alpha}\mathring{L}^{\,\nu}_{\,\mu\nu}\\ &=-g^{\mu\nu}\left\{{}^{\,\alpha}_{\mu\nu}\right\}+g^{\mu\alpha}\left\{{}^{\,\nu}_{\mu\nu}\right\}\,.\end{split}$			(17)

Writing the Einstein-Hilbert action as follows

$\displaystyle\begin{split}\mathcal{S}_{{\mathrm{EH}}}&=\int d^{4}x\,\sqrt{-g}\,g^{\mu\nu}\mathcal{R}_{\mu\nu}\left(\left\{{}^{\,\alpha}_{\mu\nu}\right\}\right)\\ &=\int d^{4}x\,\sqrt{-g}g^{\mu\nu}\left(\left\{{}^{\,\alpha}_{\sigma\mu}\right\}\left\{{}^{\,\sigma}_{\nu\alpha}\right\}-\left\{{}^{\,\alpha}_{\sigma\alpha}\right\}\left\{{}^{\,\sigma}_{\mu\nu}\right\}\right)\\ &\quad+\int d^{4}x\,\partial_{\alpha}\left[\sqrt{-g}\,(g^{\mu\nu}\left\{{}^{\,\alpha}_{\mu\nu}\right\}-g^{\mu\alpha}\left\{{}^{\,\nu}_{\mu\nu}\right\})\right]\,\end{split}$

(18)

we find that the Ricci scalar can be expressed by the nonmetricity scalar and divergence of two independent traces of the nonmetricity tensor, where the divergence term is reduced to the boundary term after the integration. This result indicates the equivalence between GR and STEGR up to the boundary term. Moreover, for general connection, we find the relation between $\mathcal{R}$:

$\displaystyle\mathcal{R}=Q-{\nabla}_{\alpha}\left(Q^{\alpha}-\tilde{Q}^{\alpha}\right)$

(19)

Under the general coordinate transformation $\{X^{\mu}\}\rightarrow\{x^{\mu}\}$, the connection is transformed as

$\displaystyle\varGamma^{\lambda}_{\ \mu\nu}(x)=\frac{\partial x^{\lambda}}{\partial X^{\alpha}}\frac{\partial X^{\rho}}{\partial x^{\mu}}\frac{\partial X^{\sigma}}{\partial x^{\nu}}\varGamma^{\alpha}_{\ \rho\sigma}(X)+\frac{\partial x^{\lambda}}{\partial X^{\alpha}}\frac{\partial^{2}X^{\alpha}}{\partial x^{\mu}\partial x^{\nu}}~{}.$

(20)

Provided that the connection vanishes in the coordinate system $\{X^{\mu}\}$, the first term vanishes in Eq. (20), and the second term reproduces Eq. (13). Thus, we can identify a set of scalar fields $\{\xi^{\mu}\}$ as such a special coordinate system $\{X^{\mu}\}$. Note that the above argument partially includes the definition of the local Lorentz frame, although the space-time is not necessarily flat in the current setup.

Equation (13) also suggests that the index of the scalar fields is contracted within the connection, and it does not show up as a free index. Therefore, one can expect that the index $\alpha$ in $\xi^{\alpha}$ is related to the inertial symmetry rather than the space-time index. To distinguish it from the space-time index, hereafter, we denote the scalar field as $\xi^{a}$:

$\displaystyle\varGamma^{\lambda}_{\,\mu\nu}=\frac{\partial x^{\lambda}}{\partial\xi^{a}}\frac{\partial^{2}\xi^{a}}{\partial x^{\mu}\partial x^{\nu}}\,.$

(21)

We can find that Eq. (21) is invariant under the following transformation for $\xi^{a}$:

$\displaystyle\xi^{a}(x)\rightarrow\bar{\xi}^{a}(x)=\mathcal{M}^{a}_{\ b}\xi^{b}(x)+\zeta^{a}\,.$

(22)

Here, $\mathcal{M}^{a}_{\ b}$ is a $4\times 4$ nondegenerate constant matrix, and $\zeta^{a}$ is a constant vector. In the coincident gauge $\xi^{a}=x^{a}$, the above transformation leads to the linear transformation between two coordinate systems,

$\displaystyle\bar{x}^{a}=\mathcal{M}^{a}_{\ b}x^{b}+\zeta^{a}\,.$

(23)

Equation (23) represents the affine transformation, and the connection (21) is purely inertial, which always can transform to zero under Eq. (23). By imposing on the flat space-time, the Minkowski metric is invariant under the transformation, we find that matrix $\mathcal{M}^{a}_{\ b}$ corresponds to the Lorentz transformation, and Eq. (23) is reduced to the Poincaré transformation in the coincident gauge.

Thus, any two different frames under the coincident gauge are related by transformation (23), naturally to introduce the metric $f_{ab}(x)$ in the coincident gauge. Under the coordinate transformation $\{\xi^{a}\}\rightarrow\{x^{\mu}\}$, the space-time metric is transformed as $f_{ab}(x)\rightarrow g_{\mu\nu}(x)$ written in terms of the Jacobi matrix:

$\displaystyle g_{\mu\nu}=\frac{\partial\xi^{a}}{\partial x^{\mu}}\frac{\partial\xi^{b}}{\partial x^{\nu}}f_{ab}\,.$

(24)

We present an explicit example in Appendix A to better demonstrate how $f_{ab}$ transforms under Eq. (24). It is apparent that $\xi^{a}$ plays a role in restoring the general covariance while fixing $\xi^{a}$ breaks the diffeomorphism explicitly. A similar structure can be found in the relation between the tetrad and metric if we read $\partial_{\mu}\xi^{a}=e^{a}_{\mu}$ and $f_{ab}=\eta_{ab}$. Moreover, Eq. (24) shows up in the mass term of the de Rham-Gabadadze-Tolley (dRGT) massive gravity de Rham et al. (2011, 2012); Hassan and Rosen (2011); Hinterbichler (2012); Molaee and Shirzad (2018) , and $\xi^{a}=x^{a}$ is called the unitary gauge therein. The above arguments suggest that we can treat four scalar fields $\xi^{a}$ as the Stüeckelberg fields associated with the diffeomorphism as discussed in the dRGT massive gravity. Hereafter we will handle the generic $\xi^{a}$ to investigate the $f(Q)$ gravity in a covariant way.

Let us consider the $f(Q)$ gravity by replacing $Q$ with a function of $Q$:

$\displaystyle S_{f(Q)}=\int d^{4}x\sqrt{-g}f(Q)\,.$

(25)

By introducing an auxiliary scalar field $\phi$, one can consider the following action

$\displaystyle S_{f(Q)}=\int d^{4}x\sqrt{-g}\left\{f^{\prime}(\phi)Q-\left[\phi f^{\prime}(\phi)-f(\phi)\right]\right\}\;.$

(26)

By varying the above action with respect to $\phi$, one gets $f^{\prime\prime}(\phi)(Q-\phi)=0$. Provided that $f^{\prime\prime}(\phi)\neq 0$, this equation implies $\phi=Q$ and restores the original action. Moreover, we can redefine the scalar field $\varphi\equiv f^{\prime}(\phi)$ and $V(\varphi)=\phi f^{\prime}(\phi)-f(\phi)$. The Lagrangian density of our interest then takes the following form

$\displaystyle S_{f(Q)}=\int d^{4}x\sqrt{-g}\left[\varphi Q-V(\varphi)\right]~{}.$

(27)

Using Eq. (19), we obtain

$\displaystyle S_{f(Q)}=\int d^{4}x\sqrt{-g}\left[\varphi\mathcal{R}-V(\varphi)+\varphi\nabla_{\mu}\left(Q^{\mu}-\tilde{Q}^{\mu}\right)\right]~{}.$

(28)

The third term includes the covariant derivative with respect to the Levi-Civita connection $\nabla_{\mu}$, and we can utilize the ordinary formula for the divergence,

$\displaystyle\int d^{4}x\sqrt{-g}~{}\nabla_{\mu}A^{\mu}=\int d^{4}x~{}\partial_{\mu}\left(\sqrt{-g}A^{\mu}\right)\,,$

(29)

and ignore the surface integration. The integration by parts reduces the original action (25) to the following form:

$\displaystyle S_{f(Q)}=\int d^{4}x\sqrt{-g}\left[\varphi\mathcal{R}-V(\varphi)-\partial_{\mu}\varphi\cdot\left(Q^{\mu}-\tilde{Q}^{\mu}\right)\right]~{}.$

(30)

Moreover, by the transformation, the nonmetricity tensor $Q_{\alpha\beta\gamma}$ and its traces $Q^{\mu}$ and $\tilde{Q}^{\mu}$ are transformed as below:

$\displaystyle\begin{split}Q_{\alpha\beta\gamma}\rightarrow Q^{\prime}_{\alpha\beta\gamma}&=\hat{\nabla}_{\alpha}g^{\prime}_{\beta\gamma}\\ &=\hat{\nabla}_{\alpha}\left(e^{-\Phi}g_{\beta\gamma}\right)\\ &=e^{-\Phi}Q_{\alpha\beta\gamma}-g_{\beta\gamma}e^{-\Phi}\partial_{\alpha}\Phi\;,\end{split}$

(33)

$\displaystyle\begin{split}Q^{\mu}\rightarrow Q^{\prime\mu}&=g^{\prime\mu\alpha}g^{\prime\beta\gamma}Q^{\prime}_{\alpha\beta\gamma}\\ &=e^{2\Phi}g^{\mu\alpha}g^{\beta\gamma}\left(e^{-\Phi}Q_{\alpha\beta\gamma}-g_{\beta\gamma}e^{-\Phi}\partial_{\alpha}\Phi\right)\\ &=e^{\Phi}Q^{\mu}-4e^{\Phi}g^{\mu\alpha}\partial_{\alpha}\Phi\;,\end{split}$

(34)

$\displaystyle\begin{split}\tilde{Q}^{\mu}\rightarrow\tilde{Q}^{\prime\mu}&=g^{\prime\mu\alpha}g^{\prime\beta\gamma}Q^{\prime}_{\beta\alpha\gamma}\\ &=e^{2\Phi}g^{\mu\alpha}g^{\beta\gamma}\left(e^{-\Phi}Q_{\beta\alpha\gamma}-g_{\alpha\gamma}e^{-\Phi}\partial_{\beta}\Phi\right)\\ &=e^{\Phi}\tilde{Q}^{\mu}-e^{\Phi}g^{\mu\alpha}\partial_{\alpha}\Phi\;.\end{split}$

(35)

In Eq. (33), the general affine connection in the covariant derivative $\hat{\nabla}$ are not transformed. From the above, we can rewrite the original nonmetricity tensor $Q$ by new ones $Q^{\prime}$ and scalar field $\Phi$ after the conformal transformation,

$\displaystyle\begin{split}Q^{\mu}&=e^{-\Phi}Q^{\prime\mu}+4g^{\mu\alpha}\partial_{\alpha}\Phi\\ &=e^{-\Phi}Q^{\prime\mu}+4e^{-\Phi}g^{\prime\mu\alpha}\partial_{\alpha}\Phi\;,\end{split}$

(36)

$\displaystyle\begin{split}\tilde{Q}^{\mu}&=e^{-\Phi}\tilde{Q}^{\prime\mu}+g^{\mu\alpha}\partial_{\alpha}\Phi\\ &=e^{-\Phi}\tilde{Q}^{\prime\mu}+e^{-\Phi}g^{\prime\mu\alpha}\partial_{\alpha}\Phi\;.\end{split}$

(37)

Thus, the third term in Eq. (30) is given as

$\displaystyle\begin{split}\partial_{\mu}\varphi\cdot\left(Q^{\mu}-\tilde{Q}^{\mu}\right)&=-e^{-\Phi}\partial_{\mu}\Phi\cdot e^{-\Phi}\left(Q^{\prime\mu}+4g^{\prime\mu\alpha}\partial_{\alpha}\Phi-\tilde{Q}^{\prime\mu}-g^{\prime\mu\alpha}\partial_{\alpha}\Phi\right)\\ &=-e^{-2\Phi}\left[\partial_{\mu}\Phi\cdot\left(Q^{\prime\mu}-\tilde{Q}^{\prime\mu}\right)+3\partial^{\alpha}\Phi\partial_{\alpha}\Phi\right]\;.\end{split}$

(38)

By substituting Eqs. (32) and (38) into Eq. (30), the action of $f(Q)$ gravity is written by the Einstein gravity with the minimally coupled scalar field:

$\displaystyle\begin{split}S&=\int d^{4}x\sqrt{-g^{\prime}}e^{2\Phi}\left\{\mathrm{e}^{-2\Phi}\left[\mathcal{R}^{\prime}-3g^{\prime\mu\nu}\nabla^{\prime}_{\mu}\partial_{\nu}\Phi-\frac{3}{2}g^{\prime\mu\nu}\left(\partial_{\mu}\Phi\right)\left(\partial_{\nu}\Phi\right)\right]\right.\\ &\qquad\qquad\qquad\qquad\qquad\left.-V(\Phi)+e^{-2\Phi}\left[\partial_{\mu}\Phi\cdot\left(Q^{\prime\mu}-\tilde{Q}^{\prime\mu}\right)+3\partial^{\alpha}\Phi\partial_{\alpha}\Phi\right]\right\}\\ &=\int d^{4}x\sqrt{-g^{\prime}}\,\left[\mathcal{R}^{\prime}+\frac{3}{2}\partial^{\alpha}\Phi\partial_{\alpha}\Phi-U(\Phi)+\partial_{\mu}\Phi\cdot\left(Q^{\prime\mu}-\tilde{Q}^{\prime\mu}\right)\right]\;.\end{split}$

(39)

Here, we defined $U(\Phi)\equiv e^{2\Phi}V(\Phi)$ and ignored the divergence term.

In Eq. (39), the sign of the kinetic term of scalar field $\Phi$ is positive, indicating that it is ghost mode. The conformal transformation of the first and second terms in Eq. (30) reproduces the minimally coupled scalar field $\Phi$, which is the well-known result in $f(R)$ gravity. However, since the third term, the nonmetricity tensor, in Eq. (30) gives rise to a new kinetic term with the opposite sign, the sign of the kinetic term of the scalar field is finally reversed.

As a classical theory, the ghost fields have negative kinetic energy and generate the instability of the system. In a quantum context, the energy is bounded below by the vacuum, even if there is a ghost. The ghosts, however, generate negative norm states, which give the negative probability; therefore, the existence of the ghosts conflicts with the Copenhagen interpretation of the quantum theory, and the theory is not physically acceptable. In the case of the gauge theory, if we quantize the system by using the BRS symmetry Becchi et al. (1976), the ghosts appear in the combinations of zero norm states in the physical states, which satisfy the constraints coming from the BRS symmetry or gauge symmetry, and therefore the negative norm states are eliminated Kugo and Ojima (1978, 1979). The situation in $f(Q)$ gravity could be similar. Although there appear to be ghosts even in $f(Q)$ gravity, the propagation of the ghost fields is excluded as the physical degrees of freedom by the constraints, as we will see in the next section.

The nonmetricity scalar $Q$ includes the metric and its first derivative as in Eqs. (6) and (9), while the connection includes the first and second derivatives of Stüeckelberg fields as in Eq. (21). Thus, the Lagrangian density of $f(Q)$ gravity (27) is symbolically given as

$\displaystyle\sqrt{-g}\mathcal{L}\,\left(\partial_{\nu}\xi^{a},\partial_{\mu}\partial_{\nu}\xi^{a},g_{\mu\nu},\partial_{\beta}g_{\mu\nu}\right)\,.$

(40)

To handle the higher-order derivatives in the Lagrangian density, we can introduce new variables into the $f(Q)$ theory in a way that does not alter the theory Motohashi et al. (2016). By introducing four space-time vectors $A^{a}_{\;\mu}$ as new variables, we can rewrite the Lagrangian density (40) as ²²2 Taking the variation of (41) with respect to $\omega_{\;a}^{\nu}$, we can get $\partial_{\nu}\xi^{a}=A_{\;\nu}^{a}$. By inserting this relation back into the equation of motion, it is easy to check that the Lagrangian (41) is equivalent to (40).

$\displaystyle\sqrt{-g}\mathcal{L}^{eq}=\sqrt{-g}\mathcal{L}\,(A^{a}_{\;\nu},\partial_{\mu}A^{a}_{\;\nu},g_{\mu\nu},\partial_{\beta}g_{\mu\nu})+\omega_{\;a}^{\nu}(\partial_{\nu}\xi^{a}-A_{\;\nu}^{a})\,.$

(41)

We note that by rewriting $\partial_{\nu}\xi^{a}=A_{\;\nu}^{a}$, the affine connection Eq. (21) goes back the original form suggested by Eq. (12).

Using Eq. (41), we can reformulation the action Eq. (30) into following form ³³3We present the detail calculation in Appendix B,

	$\displaystyle S_{f(Q)}^{eq}$	$\displaystyle=\int d^{4}x\,\left[\sqrt{-g}\,\left(\varphi\mathcal{R}-V\right)-\sqrt{-g}\partial_{\alpha}\varphi\cdot\left(Q^{\alpha}-\tilde{Q}^{\alpha}\right)+\omega_{\;a}^{\nu}(\partial_{\nu}\xi^{a}-A_{\;\nu}^{a})\right]$
		$\displaystyle=S_{f(R)}-\int d^{4}x\,\left\{\sqrt{-g}\,\partial_{\alpha}\varphi\left[g^{\mu\nu}(A^{-1})^{\alpha}_{\ a}-g^{\mu\alpha}(A^{-1})^{\nu}_{\ a}\right]\cdot\nabla_{\mu}A^{a}_{\ \nu}-\omega^{\nu}_{\ a}\left(\partial_{\nu}\xi^{a}-A^{a}_{\ \nu}\right)\right\}$
		$\displaystyle=S_{f(R)}-\int d^{4}x\,\left[\sqrt{-g}\,C_{a}^{\alpha\mu\nu}\partial_{\alpha}\varphi\nabla_{\mu}A^{a}_{\ \nu}-\omega^{\nu}_{\ a}\left(\partial_{\nu}\xi^{a}-A^{a}_{\ \nu}\right)\right]\,.$		(42)

To simplify the expression in Eq. (42), we defined a tensor $C_{a}^{\alpha\mu\nu}$ as

$\displaystyle C_{a}^{\alpha\mu\nu}\equiv g^{\mu\nu}B^{\alpha}_{\ a}-g^{\mu\alpha}B^{\nu}_{\ a}\quad\text{where}\quad B^{\alpha}_{\ a}=(A^{-1})^{\alpha}_{\ a}\,.$

(43)

Where the term $S_{f(R)}$ in action (42) is the scalar-tensor description of $f(R)$ gravity,

$\displaystyle S_{f(R)}=\int d^{4}x\,\sqrt{-g}\left[\varphi\mathcal{R}-V(\varphi)\right]\,.$

(44)

We note that the nonmetricity tensor in Eq. (30) induces the coupling between the scalar field $\varphi$ and vector field $A^{a}_{\ \mu}$ in Eq. (42). This term corresponds to the origin of the ghost scalar mode after the conformal transformation of metric, as we observed in the previous section.

In this subsection, we investigate the origin of the ghost mode under the Lagrangian formulation by performing the $3+1$ decomposition of the action (42) in terms of ADM variables. We decompose space-time into 3-dimensional spacelike hypersurfaces $\Sigma_{t}$ and their normal vector $n^{\mu}$, which satisfies the normalization condition $n^{\mu}n_{\mu}=-1$

$\displaystyle n_{\mu}$

$\displaystyle=\left(-N,0,0,0\right)\,,\ n^{\mu}=\left(1/N,-N^{i}/N\right)\,.$

(45)

$N$ is the lapse function and $N^{i}$ is the shift vector lying on the $\Sigma_{t}$. We also introduce the three-dimensional induced metric $h_{\mu\nu}$ defined by

$\displaystyle\begin{split}h_{\mu\nu}&=g_{\mu\nu}+n_{\mu}n_{\nu}=\left(\begin{array}[]{cc}N_{i}N^{i}&N_{i}\\ N_{i}&h_{ij}\end{array}\right)\,,\\ h^{\mu\nu}&=g^{\mu\nu}+n^{\mu}n^{\nu}=\left(\begin{array}[]{cc}0&0\\ 0&h^{ij}\end{array}\right)\,.\end{split}$

(46)

Inversely, we can express the metric in terms of $N$, $N^{i}$, and $h_{ij}$,

$\displaystyle\begin{split}g_{\mu\nu}&=\left(\begin{array}[]{cc}-N^{2}+N_{i}N^{i}&N_{i}\\ N_{i}&h_{ij}\end{array}\right)\,,\\ g^{\mu\nu}&=\left(\begin{array}[]{cc}-N^{-2}&N^{-2}N^{i}\\ N^{-2}N^{i}&h^{ij}-\frac{N^{i}N^{j}}{N^{2}}\end{array}\right)\,.\end{split}$

(47)

Moreover, we can decompose the space-time vector $A_{\mu}^{a}$ into tangent $\bar{A}_{\mu}^{a}$ and normal part $A_{*}^{a}$ separately:

$\displaystyle A_{\mu}^{a}=\bar{A}_{\mu}^{a}-A_{*}^{a}n_{\mu}\quad\text{with}\quad A_{*}^{a}=A_{\alpha}^{a}n^{\alpha}\,.$

(48)

For the tangent part, we have $A_{0}^{a}=\bar{A}_{0}^{a}+A_{*}^{a}N$ and $A_{i}^{a}=\bar{A}_{i}^{a}$. Provided by these two relations, the normal component of $A_{\mu}^{a}$ can be expressed by

$\displaystyle\begin{split}A_{*}^{a}=&A_{0}^{a}\cdot n^{0}+A_{i}^{a}n^{i}\\ &=\frac{1}{N}(\bar{A}_{0}^{a}+A_{*}^{a}N-N^{i}\bar{A}_{i}^{a})\,,\end{split}$

(49)

from which we get the relation $\bar{A}_{0}^{a}=N^{i}\bar{A}_{i}^{a}$. After straightforward calculations presented in Appendix C, one can obtain the different components of the covariant derivative of four vectors $A_{\mu}^{a}$:

$\displaystyle\begin{split}\nabla_{0}A_{0}^{a}&=N\dot{A}_{*}^{a}-\mathcal{K}_{ij}\left(A_{*}^{a}N^{i}N^{j}+N\bar{A}^{ia}N^{j}+N\bar{A}^{ja}N^{i}\right)\\ &-N^{i}A^{a}_{\ast}\mathcal{D}_{i}N-N^{i}\bar{A}^{a}_{j}\mathcal{D}_{i}N^{j}-N\bar{A}^{ia}\mathcal{D}_{i}N+N^{i}\dot{\bar{A}}_{i}^{a}\,,\\ \nabla_{i}A_{0}^{a}&=-\mathcal{K}_{ij}\left(A_{*}^{a}N^{j}+N\bar{A}^{ja}\right)+N\mathcal{D}_{i}A_{*}^{a}+N^{j}\mathcal{D}_{i}\bar{A}_{j}^{a}\,,\\ \nabla_{0}A_{i}^{a}&=\dot{\bar{A}}_{i}^{a}-\mathcal{K}_{ij}\left(A_{*}^{a}N^{j}+N\bar{A}^{ja}\right)-A^{a}_{\ast}\mathcal{D}_{i}N-\bar{A}^{a}_{j}\mathcal{D}_{i}N^{j}\,,\\ \nabla_{i}A_{j}^{a}&=\mathcal{D}_{i}\bar{A}_{j}^{a}-A_{*}^{a}\mathcal{K}_{ij}\,.\end{split}$

(50)

Subsequently, by using the same approach, the $3+1$ decomposition of $B_{a}^{\alpha}$ is given by

$\displaystyle B_{a}^{0}=-B_{a}^{*}\frac{1}{N}~{},~{}B_{a}^{i}=\bar{B}_{a}^{i}+B_{a}^{*}\frac{N^{i}}{N}~{},~{}$

(51)

the calculation details of Eq. (51) and the $3+1$ decomposition of the interaction term $C_{a}^{\alpha\mu\nu}$ have also shown explicitly in Appendix C. It is worth mentioning that since the $A_{a}^{\mu}$ and $B_{a}^{\mu}$ are not independent space-time vectors, they are related by the $B_{a}^{\mu}A_{\nu}^{a}=\delta^{\mu}_{\;\nu}$, which lead to the following important relations:

$\displaystyle A_{*}^{a}B_{a}^{*}=A_{\alpha}^{a}n^{\alpha}B_{a}^{\beta}n_{\beta}=\delta_{\;\alpha}^{\beta}n^{\alpha}n_{\beta}=-1~{},~{}$

(52)

and

$\displaystyle B_{a}^{\mu}A_{\mu}^{a}=B_{a}^{0}A_{0}^{a}+B_{a}^{i}A_{i}^{a}=\bar{B}_{a}^{i}\bar{A}_{i}^{a}-B_{a}^{*}A_{*}^{a}=4~{},~{}$

(53)

from which, one concludes that the tangent parts of those vectors also enjoy the relation $\bar{B}_{a}^{i}\bar{A}_{j}^{a}=\delta^{i}_{j}$. In particular, we have

$\displaystyle\bar{A}_{j}^{a}B_{a}^{*}=A_{j}^{a}(B_{a}^{\alpha}n_{\alpha})=-\delta_{j}^{0}N=0~{}.~{}$

(54)

It shows the tangent part of $A_{a}^{\mu}$ and the normal part of $B_{a}^{\mu}$ are orthogonal to each other and vice versa, i.e., $A_{*}^{a}\bar{B}_{a}^{i}=0$. Putting together all our results and utilizing relation (52)-(54), we finally obtain the full $3+1$ decomposition of the action (42),

$\displaystyle\begin{split}\mathcal{L}^{eq}_{f(Q)}&=\,\mathcal{L}_{f(R)}-\mathcal{L}_{\text{int}}+\omega_{a}^{\nu}(\partial_{\nu}\xi^{a}-A_{\nu}^{a})\\ &=\,N\sqrt{h}\varphi\left({}^{(3)}\mathcal{R}+\mathcal{K}^{ij}\mathcal{K}_{ij}-\mathcal{K}^{2}-\frac{V(\varphi)}{\varphi}\right)\\ &\quad+\sqrt{h}\dot{\varphi}\left(-\frac{1}{N}\bar{B}^{i}_{a}\dot{\bar{A}}^{a}_{i}+B^{\ast}_{a}\mathcal{D}_{i}\bar{A}^{ia}+\frac{N^{i}}{N}\bar{B}^{j}_{a}\mathcal{D}_{i}\bar{A}^{a}_{j}+\frac{1}{N}\mathcal{D}_{i}N^{i}\right)+\sqrt{h}\dot{A}^{a}_{\ast}\bar{B}^{i}_{a}\mathcal{D}_{i}\varphi\\ &\quad+\sqrt{h}\mathcal{D}_{i}\varphi\mathcal{D}^{i}N+\sqrt{h}\frac{N^{i}}{N}\left(\bar{B}^{j}_{a}\dot{\bar{A}}^{a}_{j}-\mathcal{D}_{j}N^{j}-N^{j}\bar{B}^{k}_{a}\mathcal{D}_{j}\bar{A}^{a}_{k}\right)\mathcal{D}_{i}\varphi\\ &\quad-\sqrt{h}N\left(B^{\ast}_{a}\mathcal{D}^{i}A^{a}_{\ast}+\bar{B}^{i}_{a}\mathcal{D}^{j}\bar{A}^{a}_{j}-\bar{B}_{ja}\mathcal{D}^{i}\bar{A}^{ja}\right)\mathcal{D}_{i}\varphi\\ &\quad-\sqrt{h}N^{i}\left(B^{\ast}_{a}\mathcal{D}_{j}\bar{A}^{ja}\mathcal{D}_{i}\varphi+\bar{B}^{j}_{a}\mathcal{D}_{i}A^{a}_{\ast}\mathcal{D}_{j}\varphi\right)\\ &\quad+\omega_{a}^{0}\left(\dot{\xi}^{a}-NA_{*}^{a}-N^{i}\bar{A}^{a}_{i}\right)+\omega_{a}^{i}(D_{i}\xi^{a}-\bar{A}_{i}^{a})\,,\end{split}$

(55)

where $\mathcal{K}_{ij}$ is extrinsic curvature of the hypersurface,

$\displaystyle\mathcal{K}_{ij}=\frac{1}{2N}\left(\dot{h}_{ij}-D_{i}N_{j}-D_{j}N_{i}\right)\,.$

(56)

In the Lagrangian formulation, the field equations obtained from (55) involve only first-order time derivatives of $\varphi$ and ${A}_{*}^{a}$. However, wave functions require second-order derivatives. In other words, the action (55) strongly indicates these scalar fields cannot propagate as physical DOF. Furthermore, we can glimpse the prospect of the Hamiltonian formulation. The canonical momentum variable with respect to the scalar field $\varphi$ is given as

$\displaystyle p=\sqrt{h}\left[-\frac{1}{N}\bar{B}^{i}_{a}\dot{\bar{A}}^{a}_{i}+B^{\ast}_{a}\mathcal{D}_{i}\bar{A}^{ia}+\frac{N^{i}}{N}\bar{B}^{j}_{a}\mathcal{D}_{i}\bar{A}^{a}_{j}+\frac{1}{N}\mathcal{D}_{i}N^{i}\right]\,.$

(57)

Moreover, we consider the constraint from the Lagrange multiplier $\omega^{\nu}_{\ a}$. We obtain $\nabla_{\mu}A^{a}_{\ \nu}=\nabla_{\nu}A^{a}_{\ \mu}$ after we use the constraint $A^{a}_{\ \mu}=\partial_{\mu}\xi^{a}$ Langlois and Noui (2016). Since the dummy indices $\mu,\nu$ are symmetric, we find that $\nabla_{i}A_{0}^{a}$ equals to $\nabla_{0}A_{i}^{a}$, which induces a relation $\dot{\bar{A}}_{i}^{a}=\mathcal{D}_{i}A_{0}^{a}$. If we use $\dot{\bar{A}}^{a}_{i}=\mathcal{D}_{i}A^{a}_{0}$ and $\mathcal{D}_{i}\bar{A}^{a}_{j}=\mathcal{D}_{j}\bar{A}^{a}_{i}$, Eq. (57) is reduced to the following form:

$\displaystyle p\approx\sqrt{h}(B_{a}^{\ast}D_{i}\bar{A}^{ia}-\bar{B}_{a}^{i}D_{i}A_{*}^{a})~{}\,,$

(58)

which generates constraints between phase space variables on the hypersurfaces $\Sigma_{t}$. In other words, we can eliminate the phase space variables $\varphi$ and its corresponding conjugate momentum $p$ by using Eq. (58) and other constraints. It indicates that the ghost scalar mode discussed in Sec. III is nonphysical; thus, we can conclude that the covariant $f(Q)$ gravity theory is free from the ghosts.