Quantum Lagrangian of the Hořava theory and its nonlocalities

Jorge Bellorín1, Claudio Bórquez2 and Byron Droguett3

Department of Physics, Universidad de Antofagasta, 1240000 Antofagasta, Chile.

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

> We perform the BFV quantization of the $2+1$ projectable and the $3+1$
> nonprojectable versions of the Hořava theory. This is a Hamiltonian
> formalism, and noncanonical gauges can be used with it. In the projectable
> case, we show that the integration on canonical momenta reproduces the
> quantum Lagrangian known from the proof of renormalization of Barvinsky et
> al. This quantum Lagrangian is nonlocal, its nonlocality originally arose as
> a consequence of getting regular propagators. The matching of the BFV
> quantization with the quantum Lagrangian reinforces the program of
> quantization of the Hořava theory. We introduce a local gauge-fixing
> condition, hence a local Hamiltonian, that leads to the nonlocality of the
> Lagrangian after the integration. For the case of the nonprojectable theory,
> this procedure allows us to obtain the complete (nonlocal) quantum
> Lagrangian that takes into account the second-class contraints. We compare
> with the integration in general relativity, making clear the relationship
> between the underlying anisotropic symmetry of the Hořava theory and the
> nonlocality of its quantum Lagrangian.

##  1 Introduction

Several studies have been devoted to the consistent quantization of the Hořava
theory [1]. Some of the analyses performed under the framework of quantum
field theory can be found in Refs. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14]. Other approaches of quantization, as causal dynamical triangulations and
loop quantum gravity has been done, for example in Refs. [15, 16, 17, 18, 19].
A fundamental advance is the renormalizability proof of the projectable
version presented in Ref. [2]. The difference between the projectable and the
nonprojectable versions of the Hořava theory is that in the former the lapse
function is restricted to be a function only on time, a condition that can be
imposed consistently in the Hořava theory, whereas in the latter it can be a
general function of time and space. An interesting feature of the proof of
renormalizability is the introduction of nonlocal gauge-fixing conditions,
which leads to a nonlocal quantum Lagrangian. The nonlocal gauges were
motivated by the goal of obtaining regular propagators for all quantum modes,
such that the renormalizability can be achieved in a similar way to the case
of Lorentz-violating gauge theories [22, 23, 24]. The condition of regularity
implies that the propagators have no divergences in space valid for each time
and viceversa. For the case of the Hořava theory, the propagators acquire
anisotropic higher order in momentum space.

Due to the emphasis on the symmetry, quantization of gauge field theories are
usually performed in the Lagrangian formalism, rather than in the Hamiltonian
formalism. The standard procedure for fixing the gauge is the Faddeev-Popov
method [20], together with its associated Becchi-Rouet-Stora-Tyutin (BRST)
symmetry [21]. Nevertheless, the quantization of the Hořava theory using the
Hamiltonian formalism deserves to be considered. In particular, the
quantization of the nonprojectable case is a delicate issue since it is a
theory with second-class constraints. The analogous of the Hamiltonian
constraint of general relativity acquires a second-class behavior in the
nonprojectable Hořava theory, which can be related to the reduction of the
gauge symmetry. The Hamiltonian formalism provides a natural framework for the
quantization of theories with second-class constraints. Indeed, the
contribution to the measure of these constraints is defined in the phase space
[25]. Analyses on the Hamiltonian formulation and the dynamics of the degrees
of freedom of the Hořava theory can be found in Refs. [26, 27, 28, 29, 30,
31].

The nonlocal gauge-fixing conditions introduced in the projectable case are
noncanonical gauges, in the sense that they involve a Lagrange multiplier. If
one wants to use this kind of gauges in the Hamiltonian formalism, then an
extension of the phase space is required. Motivated by this, two of us
presented the Batalin-Fradkin-Vilkovisky (BFV) quantization of the $2+1$
nonprojectable Hořava theory in Ref. [32]. The BFV formalism provides a quite
general framework for quantization of systems with constraints, with the
particularity that first-class constraints are not imposed explicitly and
their Lagrange multipliers are promoted to be part of the canonical variables.
The BFV formalism was first presented in Ref. [33] as a way to introduce
noncanonical gauge-fixing conditions in the Hamiltonian formalism. This
extension allows us to introduce relativistic gauges in the phase space, which
is a way to establish the unitarity of relativistic gauge theories. The
formalism was extended in Ref. [34] to the case of systems with fermionic
constraints, together with establishing an essential connection with the BRST
symmetry. In Ref. [35], the BFV quantization was extended to the case of
systems with (bosonic and fermionic) second-class constraints, and for the
general case when the Hamiltonian and the BRST charge have expansions of
certain order on the ghosts fields. In Ref. [36] the formalism was applied to
general relativity. In the general BFV scheme of quantization, the gauge
symmetry is fixed by the choice of a fermionic function, and the resulting
gauge-fixed path integral enjoys a BRST symmetry operating on the canonical
variables, as we commented.

Since the quantization of the Hořava theory has been focused in the Lagrangian
and the Hamiltonian BFV formalisms, a natural question is to ask whether both
approaches are equivalent. Indeed, in Ref. [2] it is indicated that the
introduction of an extra variable, that can be interpreted as the conjugate
momentum of the shift vector, eliminates the nonlocality of the final
Lagrangian. This suggests that we perform a complete Hamiltonian analysis of
the quantum theory. In this paper we undertake this problem, with interest in
connecting with the gauge-fixing conditions known from the renormalization of
the projectable case [2]. We study the BFV quantization of the $2+1$
projectable and $3+1$ nonprojectable versions of the Hořava theory. We
emphasize that this provides a consistent quantization in the sense that it is
based on a canonical phase space that admits the gauges of interest. This is
particularly critical for the nonprojectable case. After we present the BFV
quantization, we get the quantum Lagrangian for each case by means of
integration on the canonical momenta, including the ghosts. To achieve this,
we specialize the form of the gauge-fixing condition, by specifying its
functional dependence on the momenta (it depends on a particular momentum).
This leads us to introduce operators that, after the integration, yield the
nonlocalities on the quantum Lagrangian. In the case of the projectable theory
we arrive at the same quantum Lagrangian defined in Ref. [2]. In the
nonprojectable case we obtain the quantum Lagrangian of the theory where the
second-class constraints have been taken into account. In this way the quantum
Lagrangian of the nonprojectable case is consistent. We compare the same
method of integration in general relativity, whose BFV quantization was
presented in Ref. [36], noticing that the relativity leads to a local quantum
Lagrangian.

This paper is organized as follows. In section 2 we present the BFV
quantization of the projectable theory and the integration to obtain the
quantum Lagrangian. We first develop the formalism for a Hořava theory defined
in a general spatial dimension $d$, and eventually we specialize to the $2+1$
case. In section 3 we present the same approach for the $3+1$ nonprojectable
case, presenting the BFV quantization and the quantum Lagrangian. In section 4
we compare with general relativity. Finally, we present some conclusions.

##  2 Projectable Hořava theory

###  2.1 Classical theory

The Hořava theory [1], both in the projectable and nonprojectable cases, is
based on a given foliation that has an absolute physical meaning. The aim is
to get an anisotropic scaling at the ultraviolet that favor the
renormalizability of the theory, where a parameter $z$ measures the degree of
anisotropy. To hold this anisotropic scaling, the dimensions of the space and
time are defined to be

$[t]=-z,\qquad[x^{i}]=-1\,.$ (2.1)

The order $z$ is fixed by the criterium of power-counting renormalizability,
which yields $z=d$, where $d$ is the spatial dimension of the foliation. The
Arnowitt-Deser-Misner variables $N$, $N^{i}$ and $g_{ij}$ are used to describe
the gravitational dynamics on the foliation. The allowed coordinate
transformations on the foliation,

$\delta t=f(t),\qquad\delta x^{i}=\zeta^{i}(t,\vec{x})\,,$ (2.2)

lead to the gauge symmetry of the foliation-preserving diffeomorphisms,

$\displaystyle\delta N$ $\displaystyle=$
$\displaystyle\zeta^{k}\partial_{k}N+f\dot{N}+\dot{f}N\,,$ (2.3)
$\displaystyle\delta N_{i}$ $\displaystyle=$
$\displaystyle\zeta^{k}\partial_{k}N_{i}+N_{k}\partial_{i}\zeta^{k}+\dot{\zeta}^{j}g_{ij}+f\dot{N}_{i}+\dot{f}N_{i}\,,$
(2.4) $\displaystyle\delta g_{ij}$ $\displaystyle=$
$\displaystyle\zeta^{k}\partial_{k}g_{ij}+2g_{k(i}\partial_{j)}\zeta^{k}+f\dot{g}_{ij}$
(2.5)

(strictly, the spatial diffeomorphisms are the gauge transformations). The
condition that defines the projectable version is that the lapse function is
restricted to be a function only of time, $N=N(t)$, a condition that is
preserved by the transformation (2.3). In this section we summarize the
canonical formulation of the projectable case, dealing with an arbitrary
number $d$ of spatial dimensions. The Hamiltonian analysis of the projectable
case, taking the infrared effective action, was done in Ref. [26]. Further
analyses, with different boundary conditions, can be found in Ref. [28]. The
quantization of the same model under the scheme of loop quantum gravity has
been studied in Ref. [19].

The Lagrangian of the projectable theory is given by

$\mathcal{L}=\sqrt{g}N\Big{(}K^{ij}K_{ij}-\lambda
K^{2}-\mathcal{V}[g_{ij}]\Big{)}\,,$ (2.6)

where the extrinsic curvature is defined by

$K_{ij}=\frac{1}{2N}\Big{(}\dot{g}_{ij}-2\nabla_{(i}N_{j)}\Big{)}$ (2.7)

and $\mathcal{V}[g_{ij}]$, called the potential, is built from invariants of
the spatial curvature and their derivatives, up to the order $2z$.

In the Hamiltonian formulation the canonical pair is $(g_{ij}\,,\pi^{ij})$,
whereas $N(t)$ and $N^{i}(t,\vec{x})$ enter as Lagrange multipliers. Since
$N(t)$ is a function only of time, there is an associated global constraint,
given in terms of a spatial integral. This constraint is

$\int
d^{d}x\mathcal{H}=0\,,\quad\mathcal{H}\equiv\frac{1}{\sqrt{g}}\Big{(}\pi^{ij}\pi_{ij}+\frac{\lambda}{1-d\lambda}\pi^{2}\Big{)}+\sqrt{g}\mathcal{V}\,.$
(2.8)

Throughout this paper we assume that $\lambda$ does not take the critical
value $\lambda=1/d$. This global constraint does not eliminate a complete
functional degree of freedom. The local constraint of the theory is the
momentum constraint,

$\mathcal{H}_{i}=-2\nabla^{k}\pi_{ki}\,.$ (2.9)

The primary Hamiltonian is

${H}_{0}=\int d^{d}x\mathcal{H}_{0}=N\int d^{d}x\mathcal{H}\,.$ (2.10)

Since $N$ is a function of time in the projectable theory, we take advantage
of the symmetry of reparameterizing the time, Eqs. (2.2) and (2.3), to set
$N=1$. With this setting the primary Hamiltonian density is equivalent to
$\mathcal{H}$. Due to their importance in the BFV quantization, and since the
Hamiltonian is equivalent to $\mathcal{H}$, we show the following two brackets
between constraints,

$\displaystyle\left\\{\int d^{d}x\epsilon^{k}\mathcal{H}_{k}\,,\int
d^{d}y\eta^{l}\mathcal{H}_{l}\right\\}=\int
d^{d}x\mathcal{H}_{l}\mathcal{L}_{\vec{\epsilon}}\eta^{l},$ (2.11)
$\displaystyle\left\\{\int d^{d}x\epsilon^{k}\mathcal{H}_{k}\,,\rho\int
d^{d}y\mathcal{H}\right\\}=0\,.$ (2.12)

In the above $\rho$ is a test function only of time whereas $\epsilon^{k}$ and
$\eta^{k}$ are test functions of time and space.

###  2.2 BFV quantization

The initial consideration in the BFV formalism is that the constrained system
under quantization must be involutive. This means that, given a Hamiltonian
$\mathcal{H}_{0}$ and a set of functions $G_{a}$, the following relations are
satisfied

$\displaystyle\left\\{G_{a},G_{b}\right\\}=U^{c}_{ab}G_{c},$ (2.13)
$\displaystyle\left\\{\mathcal{H}_{0},G_{a}\right\\}=V^{b}_{a}G_{b}.$ (2.14)

To avoid writing huge expressions, we use a simplification on the notation of
brackets: we insert densities instead of spatial integrals, such as
$\\{A\,,B\\}\rightarrow\\{\int d^{d}xA\,,\int d^{d}yB\\}$. The first-class
constraints are part of the definition of the $G_{a}$ functions. The other
part is given by the canonical momenta conjugated to the Lagrange multipliers
of the first-class constraints, since these multipliers are promoted to
canonical variables in the BFV extension of the phase space. The extended
phase space is completed with the canonical pair of fermionic ghosts
$(\eta^{a},\mathcal{P}_{a})$, where each pair is incorporated for each
function $G_{a}$.

To apply this formalism to the projectable Hořava theory, we identify the
momentum constraint $\mathcal{H}_{i}$ as the only first-class constraint,
being the shift vector $N^{i}$ its Lagrange multiplier. By denote by $\pi_{i}$
the canonical momentum conjugated to $N^{i}$. Thus, the functions are
$G_{a}=(\mathcal{H}_{i}\,,\pi_{i})$. Since $\pi_{i}$ commutes with itself and
with $\mathcal{H}_{i}$, the algebra (2.13) reduces to the algebra of
$\mathcal{H}_{i}$,

$\\{\mathcal{H}_{i}\,,\mathcal{H}_{j}\\}=U_{ij}^{k}\mathcal{H}_{k}\,.$ (2.15)

This corresponds to the algebra of spatial diffeomorphisms, as shown in
(2.11), and we take the definition of $U_{ij}^{k}$ from it. $U_{ab}^{c}=0$ for
$a,b,c>i$. The primary Hamiltonian is identified in (2.10), hence the bracket
(2.14) corresponds to (2.12), such that $V_{a}^{b}=0$. By incorporating the
ghost fields, the full BFV phase space of the projectable Hořava theory is
given by the canonical pairs $(g_{ij},\pi^{ij})$, $(N_{i},\pi^{i})$ and
$(\eta_{a},\mathcal{P}^{a})$. The ghosts can be split in the two sets,
$(\eta_{1}^{i},\mathcal{P}^{1}_{i})$, $(\eta_{2}^{i},\mathcal{P}^{2}_{i})$.
The gauge-fixing condition is incorporated in the path integral by means of a
fermionic function $\Psi$, which is a given functional on the extended phase
space. Thus, the BFV path integral of the projectable Hořava theory is given
by

$Z=\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}\eta^{a}\mathcal{D}\mathcal{P}_{a}\exp\left[i\int
dtd^{d}x\Big{(}\pi^{ij}\dot{g_{ij}}+\pi_{k}\dot{N}^{k}+\mathcal{P}_{a}\dot{\eta}^{a}-\mathcal{H}_{\Psi}\Big{)}\right]\,.$
(2.16)

In this formalism the ghosts eliminate the unphysical quantum degrees of
freedom that should be eliminated by the first-class constraints. Indeed, in
$d$ spatial dimensions the canonical pairs $(g_{ij},\pi^{ij})$,
$(N_{i},\pi^{i})$ amount for $d(d+3)$ degrees of freedom, and the ghosts
$(\eta_{1}^{i},\mathcal{P}^{1}_{i})$, $(\eta_{2}^{i},\mathcal{P}^{2}_{i})$ sum
$4d$ degrees. After subtracting, one gets $d(d-1)$ physical degrees of freedom
in the phase space of the quantum theory. In $d=2$ this yield $2$ degrees of
freedom, which represent the scalar mode of the $2+1$ projectable theory in
canonical variables. In $d=3$ the degrees of freedom are six, which are the
two tensorial modes plus the extra scalar mode. Since the Hořava theory has
anisotropic scaling, it is important to write down the dimensions of the
several fields. This is

$\begin{array}[]{lll}&[g_{ij}]=0\,,&[\pi^{ij}]=d=z\,,\\\\[4.30554pt]
&[N^{k}]=z-1\,,&[\pi_{k}]=1+d-z=1\,,\\\\[4.30554pt]
&[\eta_{1}^{i}]=[\mathcal{P}^{2}_{i}]=(d-z)/2=0\,,&[\eta_{2}^{i}]=[\mathcal{P}^{1}_{i}]=(d+z)/2=z\,.\end{array}$
(2.17)

In the general BFV formalism, the gauge-fixed quantum Hamiltonian is defined
by

$\mathcal{H}_{\Psi}=\mathcal{H}_{1}+\left\\{\Psi,\Omega\right\\}\,.$ (2.18)

The Poisson bracket is extended to include fermionic variables,

$\\{A,B\\}=\dfrac{\delta^{\mbox{\tiny R}}A}{\delta
q^{r}}\dfrac{\delta^{\mbox{\tiny L}}B}{\delta
p^{r}}-(-1)^{n_{A}n_{B}}\dfrac{\delta^{\mbox{\tiny R}}B}{\delta
q^{r}}\dfrac{\delta^{\mbox{\tiny L}}A}{\delta p^{r}}\,,$ (2.19)

where R and L denote right and left derivatives and $n_{A}$ is 0 or 1
depending on whether $A$ is a boson or a fermion. $\Omega$ is the generator of
the BRST symmetry. According to the extension of the BFV formalism presented
in Ref. [35], $\Omega$ and $\mathcal{H}_{1}$ are defined in terms of
expansions in the ghost fields,

$\displaystyle\Omega$
$\displaystyle=G_{a}\eta^{a}+\sum_{k=1}^{s}\mathcal{P}_{b_{k}}\ldots\mathcal{P}_{b_{1}}\Omega^{b_{1}\ldots
b_{k}}\,,$ (2.20) $\displaystyle\mathcal{H}_{1}$
$\displaystyle=\mathcal{H}_{0}+\sum_{k=1}^{s}\mathcal{P}_{b_{k}}\ldots\mathcal{P}_{b_{1}}\mathcal{H}_{1}^{b_{1}\ldots
b_{k}},$ (2.21)

where $s$ represents the rank of theory. The coefficient functions of the
first order in $\mathcal{P}_{a}$ are given by

$\Omega^{a}=-\frac{1}{2}U^{a}_{bc}\eta^{b}\eta^{c}\,,\qquad\mathcal{H}_{1}^{a}=V^{a}_{b}\eta^{b}\,.$
(2.22)

The rest of coefficients, up to the order $s$ of the theory, are obtained by
recurrence relations, starting from the first-order ones [35]. An essential
condition of the BFV formalism is that $\Omega$ and $\mathcal{H}_{1}$ must
satisfy

$\\{\Omega\,,\Omega\\}=0\,,\qquad\\{\mathcal{H}_{1}\,,\Omega\\}=0\,.$ (2.23)

The first one is a nontrivial condition since $\Omega$ is a fermionic
variables. These conditions support the BRST symmetry of the quantum theory.

The projectable Hořava theory is of first order, that is, $\Omega$ ends at the
first order in the ghosts, whereas $\mathcal{H}_{1}$ is of zeroth order,

$\displaystyle\Omega=G_{a}\eta^{a}-\frac{1}{2}U^{c}_{ab}\eta^{a}\eta^{b}\mathcal{P}_{c}=\mathcal{H}_{k}\eta_{1}^{k}+\pi_{k}\eta_{2}^{k}-\frac{1}{2}U^{k}_{ij}\eta_{1}^{i}\eta_{1}^{j}\mathcal{P}_{k}^{1}\,,$
(2.24) $\displaystyle\mathcal{H}_{1}=\mathcal{H}_{0}=\mathcal{H}\,.$ (2.25)

The conditions (2.23) are satisfied as follows. We have the bracket of
$\Omega$ with itself,

$\\{\Omega\,,\Omega\\}=\\{\mathcal{H}_{i}\eta^{i}_{1}\,,\mathcal{H}_{j}\eta^{j}_{1}\\}-\\{\mathcal{H}_{i}\eta^{i}_{1}\,,U^{k}_{mn}\eta_{1}^{m}\eta_{1}^{n}\mathcal{P}_{k}^{1}\\}+\frac{1}{4}\\{U^{k}_{ij}\eta_{1}^{i}\eta_{1}^{j}\mathcal{P}_{k}^{1}\,,U^{l}_{mn}\eta_{1}^{m}\eta_{1}^{n}\mathcal{P}_{l}^{1}\\}\,.$
(2.26)

The first two brackets are equal,

$\\{\mathcal{H}_{i}\eta^{i}_{1}\,,\mathcal{H}_{j}\eta^{j}_{1}\\}=\\{\mathcal{H}_{i}\eta^{i}_{1}\,,U^{k}_{mn}\eta_{1}^{m}\eta_{1}^{n}\mathcal{P}_{k}^{1}\\}=U^{i}_{jk}\eta_{1}^{j}\eta_{1}^{k}\mathcal{H}_{i}\,,$
(2.27)

hence cancel themselves. The last bracket is proportional to the structure
$\eta_{1}^{j}\eta_{1}^{m}\eta_{1}^{n}$ $U^{k}_{ij}U^{i}_{mn}$, which is zero
by the Jacobi identity. Therefore $\\{\Omega\,,\Omega\\}=0$. Next,

$\\{\mathcal{H}_{1}\,,\Omega\\}=\\{\mathcal{H}\,,\Omega\\}=\\{\mathcal{H}\,,\eta_{1}^{i}\mathcal{H}_{i}\\}=0\,,$
(2.28)

where the last equality follows from (2.12)111Recalling that we mean spatial
integrals inside the brackets.. Therefore, we obtain the BFV gauge-fixed
Hamiltonian of the projectable Hořava theory,

$\mathcal{H}_{\Psi}=\mathcal{H}+\left\\{\Psi\,,\mathcal{H}_{k}\eta_{1}^{k}+\pi_{k}\eta_{2}^{k}-\frac{1}{2}U^{k}_{ij}\eta_{1}^{i}\eta_{1}^{j}\mathcal{P}_{k}^{1}\right\\}\,.$
(2.29)

According to the original BFV formulation, $\Psi$ can adopt a form suitable
for relativistic gauges. It turns out that this form is also suitable for the
anisotropic symmetry of the Hořava theory. First, we deal with gauge-fixing
conditions of the general structure

$\Phi^{i}=-\dot{N}^{i}+\chi^{i}=0\,,$ (2.30)

where the phase-space functional $\chi^{i}$ is the part of the gauge-fixing
condition that can be chosen. Thus, the specific BFV fermionic gauge-fixing
function is

$\Psi=\mathcal{P}_{i}^{1}N^{i}+\mathcal{P}_{i}^{2}\chi^{i}\,.$ (2.31)

With this choice the gauge-fixed Hamiltonian becomes

$\mathcal{H}_{\Psi}=\mathcal{H}+\mathcal{H}_{k}N^{k}+\mathcal{P}_{k}^{1}\eta_{2}^{k}-\mathcal{P}^{1}_{i}\left(N^{j}\partial_{j}\eta_{1}^{i}+N^{i}\partial_{j}\eta_{1}^{j}\right)+\\{\mathcal{P}_{i}^{2}\chi^{i}\,,\Omega\\}\,.$
(2.32)

Throughout this paper we assume that the gauge-fixing condition $\chi^{l}$
does not depend on the ghosts fields, then the Hamiltonian takes the form

$\mathcal{H}_{\Psi}=\mathcal{H}+\mathcal{H}_{k}N^{k}+\mathcal{P}_{k}^{1}\eta_{2}^{k}-\mathcal{P}^{1}_{i}\left(N^{j}\partial_{j}\eta_{1}^{i}+N^{i}\partial_{j}\eta_{1}^{j}\right)+\pi_{k}\chi^{k}+\mathcal{P}_{i}^{2}\\{\chi^{i}\,,\mathcal{H}_{k}\\}\eta_{1}^{k}+\mathcal{P}_{i}^{2}\dfrac{\delta\chi^{i}}{\delta
N^{l}}\eta_{2}^{l}\,.$ (2.33)

Therefore, the BFV path integral for the projectable Hořava theory in the
gauge (2.30) – (2.31) becomes

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}\eta_{1}^{i}\mathcal{D}\mathcal{P}^{1}_{i}\mathcal{D}\eta_{2}^{i}\mathcal{D}\mathcal{P}^{2}_{i}\\\
&\exp\left[i\int
dtd^{d}x\Big{(}\pi^{ij}\dot{g_{ij}}+\pi_{k}\dot{N}^{k}+\mathcal{P}^{1}_{i}\dot{\eta}_{1}^{i}+\mathcal{P}^{2}_{i}\dot{\eta}_{2}^{i}-\mathcal{H}-\mathcal{H}_{k}N^{k}\right.\\\
&\left.-\mathcal{P}_{k}^{1}\eta_{2}^{k}+\mathcal{P}^{1}_{i}\left(N^{j}\partial_{j}\eta_{1}^{i}+N^{i}\partial_{j}\eta_{1}^{j}\right)-\pi_{k}\chi^{k}-\mathcal{P}_{i}^{2}\\{\chi^{i}\,,\mathcal{H}_{k}\\}\eta_{1}^{k}-\mathcal{P}_{i}^{2}\dfrac{\delta\chi^{i}}{\delta
N^{l}}\eta_{2}^{l}\Big{)}\right]\,.\end{split}$ (2.34)

The generator of the BRST symmetry $\Omega$ acts on the canonical fields by
means of the canonical transformation

$\tilde{\varphi}=\varphi+\left\\{\varphi,\Omega\right\\}\epsilon\,,$ (2.35)

where $\epsilon$ is the fermionic parameter of the transformation. The
transformation of the fields is

$\begin{array}[]{lll}&\delta_{\Omega}g_{ij}=2g_{k(i}\nabla_{j)}\eta^{k}_{1}\epsilon\,,&\delta_{\Omega}\pi^{ij}=-2\pi^{k(i}\nabla_{k}\eta^{j)}_{1}\epsilon+\nabla_{k}(\pi^{ij}\eta^{k}_{1})\epsilon\,,\\\\[4.30554pt]
&\delta_{\Omega}N^{k}=\eta^{k}_{2}\epsilon\,,&\delta_{\Omega}\pi_{k}=0\,,\\\\[4.30554pt]
&\delta_{\Omega}\eta^{i}_{1}=-\frac{1}{2}U^{i}_{jk}\eta^{j}_{1}\eta^{k}_{1}\epsilon\,,&\delta_{\Omega}\mathcal{P}^{1}_{i}=\mathcal{H}_{i}\epsilon-U^{k}_{ij}\eta^{j}_{1}\mathcal{P}^{1}_{k}\epsilon\,,\\\\[4.30554pt]
&\delta_{\Omega}\eta^{k}_{2}=0\,,&\delta_{\Omega}\mathcal{P}^{2}_{k}=\pi_{k}\epsilon\,.\end{array}$

###  2.3 Quantum Lagrangian

We continue with working on an arbitrary spatial dimensionality $d$,
eventually we specialize to the $d=2$ case. For the BFV quantization we have
defined the structure of the gauge-fixing condition (2.30), which has the part
$\chi^{i}$ unspecified. To arrive at the quantum Lagrangian, we impose
conditions on the functional form of $\chi^{i}$ that allow us to perform the
integration on the several canonical momenta. These conditions allow us to
make a connection with the same gauge fixing used in the proof of
renormalizability of the projectable theory.

We start with the integration on the momentum $\pi_{i}$. The term
$-\pi_{k}\chi^{k}$ in the action of (2.34) suggests to demand that $\chi^{i}$
has a linear dependence on $\pi_{i}$, leading to a quadratic term in $\pi_{i}$
in the Hamiltonian, otherwise a higher order dependence on this variable could
lead to a violation of unitarity, which is contradiction to the spirit of the
Hořava theory and its anisotropic symmetry. Therefore, we assume the structure
of the gauge-fixing condition

$\chi^{k}=\mathfrak{D}^{ki}\pi_{i}+\Gamma^{k}[g_{ij},N^{k}]\,,$ (2.36)

where $\Gamma^{k}$ is a functional that may depend only on $g_{ij}$ and
$N^{k}$. The restriction that $\Gamma^{k}$ does not depend on the momentum
$\pi^{ij}$ allows us to perform the integration straightforwardly. According
to the anisotropic dimensional assignments (2.17), the gauge-fixing condition
must satisfy $[\chi^{k}]=2z-1$, hence the dimension of the operator
$\mathfrak{D}^{ij}$ must be

$[\mathfrak{D}^{ij}]=3z-d-2=2z-2\,.$ (2.37)

Below we give explicitly the operator $\mathfrak{D}^{ij}$ and the gauge-fixing
form $\Gamma^{i}$ in the perturbative framework. Nevertheless, many operations
can be carried out without recurring to perturbations and for general
$\Gamma^{i}$. Hence we stay for a while on nonperturbative variables, using
only the fact that $\mathfrak{D}^{ij}$ is a flat operator (does not depend on
any field variable).

By setting the form (2.36) for the gauge-fixing condition, the last three
terms of the action of Eq. (2.34) become

$-\pi_{k}\mathfrak{D}^{ki}\pi_{i}-\pi_{k}\Gamma^{k}-\mathcal{P}_{i}^{2}\\{\Gamma^{i}\,,\mathcal{H}_{k}\\}\eta_{1}^{k}-\mathcal{P}_{i}^{2}\dfrac{\delta\Gamma^{i}}{\delta
N^{l}}\eta_{2}^{l}\,.$ (2.38)

We may complete the square involving $\pi_{i}$ and then integrate on the
shifted variable, obtaining the path integral

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\eta^{a}\mathcal{D}\mathcal{P}_{a}\\\
&\exp\left[i\int
dtd^{d}x\Big{(}\pi^{ij}\dot{g_{ij}}+\mathcal{P}_{a}\dot{\eta}^{a}+\frac{1}{4}(\dot{N}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{N}^{l}-\Gamma^{l})-\mathcal{H}-\mathcal{H}_{k}N^{k}\right.\\\
&\left.-\mathcal{P}_{k}^{1}\eta_{2}^{k}+\mathcal{P}^{1}_{i}\left(N^{j}\partial_{j}\eta_{1}^{i}+N^{i}\partial_{j}\eta_{1}^{j}\right)-\mathcal{P}_{i}^{2}\\{\Gamma^{i}\,,\mathcal{H}_{k}\\}\eta_{1}^{k}-\mathcal{P}_{i}^{2}\dfrac{\delta\Gamma^{i}}{\delta
N^{l}}\eta_{2}^{l}\Big{)}\right]\,.\end{split}$ (2.39)

Since $\mathfrak{D}^{ij}$ is a local operator, its inverse
$\mathfrak{D}_{kl}^{-1}$, which has arisen by the integration, is a nonlocal
operator.

Now we move to the ghost sector. The following change of notation is useful
for the final the quantum Lagrangian:

$\begin{split}\eta_{1}^{i}\rightarrow
C^{i}\,,\qquad&\mathcal{P}^{1}_{i}\rightarrow\bar{\mathcal{P}}_{i}\,,\\\
\eta_{2}^{i}\rightarrow\mathcal{P}^{i}\,,\qquad&\mathcal{P}^{2}_{i}\rightarrow\bar{C}_{i}\,.\end{split}$
(2.40)

We may perform the integration of the Grassmann variables $\mathcal{P}^{i}$
and $\bar{\mathcal{P}}_{i}$, which arise in the action (2.39) in the terms

$-\bar{\mathcal{P}}_{k}\mathcal{P}^{k}+\bar{\mathcal{P}}_{k}\left(\dot{C}^{k}-N^{j}\partial_{j}C^{k}-N^{k}\partial_{j}C^{j}\right)+\mathcal{P}^{k}\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
N^{k}}\right)\;.$ (2.41)

The bilinear $-\bar{\mathcal{P}}_{k}\mathcal{P}^{k}$ can be completed, such
that the Gaussian integration on these Grassmann variables can be performed
(without consequences on the measure). After these steps of integration, the
path integral becomes

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\bar{C}_{i}\mathcal{D}C^{i}\\\
&\exp\left[i\int
dtd^{d}x\Big{(}\pi^{ij}\dot{g_{ij}}+\frac{1}{4}(\dot{N}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{N}^{l}-\Gamma^{l})-\mathcal{H}-\mathcal{H}_{k}N^{k}\right.\\\
&\left.+\left(\dot{C}^{k}-N^{j}\partial_{j}C^{k}-N^{k}\partial_{j}C^{j}\right)\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
N^{k}}\right)-\bar{C}_{i}\\{\Gamma^{i}\,,\mathcal{H}_{k}\\}C^{k}\Big{)}\right]\,.\end{split}$
(2.42)

Now we focus the integration on $\pi^{ij}$. A significant part of the
computations can be continued on nonperturbative grounds. Since this is
interesting on its own, in appendix A we show this nonperturbative integration
for the case of the projectable theory. In what follows we adopt a
perturbative approach. We consider perturbations around the analogous of the
Minkowski spacetime, given by $g_{ij}=\delta_{ij}$, $\pi^{ij}=0$, $N^{i}=0$,
$\bar{C}_{i}=C^{i}=0$.

We comment that for the $d=2$-dimensional case we take the operator
$\mathfrak{D}^{ij}$ as

$\mathfrak{D}^{ij}=\delta_{ij}\Delta+\kappa\partial_{i}\partial_{j}\,,\quad\mathfrak{D}^{-1}_{ij}=\frac{\delta_{ij}}{\Delta}-\frac{\kappa}{1+\kappa}\frac{\partial_{i}\partial_{j}}{\Delta^{2}}\,,$
(2.43)

where $\kappa$ is an arbitrary constant. The inverse $\mathfrak{D}^{-1}_{ij}$
is a nonlocal operator of dimension $-2$ in $d=2$. The operator
$\mathfrak{D}_{ij}^{-1}$ (2.43) was introduced in the gauge-fixing condition
used in Ref. [2], with the aim of introducing the nonlocality that finally
leads to regular propagators. This version of the operator $\mathfrak{D}^{ij}$
for the $d=2$ case arises in several steps of the integration for arbitrary
dimension $d$, with a fixed value of $\kappa$. For this reason we denote these
special cases as

$\displaystyle\mathfrak{D}^{ij}_{1}=\delta_{ij}\Delta+\partial_{i}\partial_{j}\,,$
(2.44)
$\displaystyle\mathfrak{D}_{2}^{ij}=\delta_{ij}\Delta+\frac{1+\lambda}{1-\lambda}\partial_{i}\partial_{j}\,,$
(2.45)
$\displaystyle\mathfrak{D}^{ij}_{3}=\delta_{ij}\Delta+(1-2\lambda)\partial_{i}\partial_{j}\,.$
(2.46)

The inverse of $\mathfrak{D}_{2}^{ij}$ is also required,

$\mathfrak{D}_{2ij}^{-1}=\frac{\delta_{ij}}{\Delta}-\frac{1+\lambda}{2}\frac{\partial_{i}\partial_{j}}{\Delta^{2}}\,.$
(2.47)

Note that the operator $\mathfrak{D}_{2}^{ij}$ cannot be extended to the
relativistic limit $\lambda=1$.

We denote the perturbative variables

$g_{ij}-\delta_{ij}=h_{ij}\,,\qquad\pi^{ij}=p^{ij}\,,\qquad N^{i}=n^{i}\,,$
(2.48)

and the ghosts $\bar{C}_{i}$, $C^{i}$ are considered perturbative variables of
first order. The quantum action given in (2.42), expanded up to quadratic
order, results

$\begin{split}S=&\int
dtd^{d}x\left\\{p^{ij}\dot{h}_{ij}+\frac{1}{4}(\dot{n}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{n}^{l}-\Gamma^{l})-\mathcal{H}-n^{k}\mathcal{H}_{k}\right.\\\
&\left.+\dot{C}^{k}\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
n^{k}}\right)-\bar{C}_{i}\\{\Gamma^{i},\mathcal{H}_{k}\\}C^{k}\right\\}\,,\end{split}$
(2.49)

where

$\displaystyle\mathcal{H}_{j}$ $\displaystyle=$
$\displaystyle-2\partial_{k}p^{kj}-2\partial_{k}(h_{ij}p^{ki})+p^{kl}\partial_{j}h_{kl}\,,$
(2.50) $\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle
p^{ij}p^{ij}+\frac{\lambda}{1-d\lambda}p^{2}+\sqrt{g}\mathcal{V}\,.$ (2.51)

We perform the transverse-longitudinal decomposition

$h_{ij}=h_{ij}^{TT}+\frac{1}{d-1}\left(\delta_{ij}-\frac{\partial_{i}\partial_{j}}{\Delta}\right)h^{T}+\partial_{(i}h_{j)}\,,$
(2.52)

and similarly for $p^{ij}$. In $d=2$ dimensions the ${TT}$ mode must be absent
from this decomposition. Thus, the action (2.49) becomes

$\begin{split}S=&\int
dtd^{d}x\left[p_{TT}^{ij}\dot{h}_{ij}^{TT}+\frac{1}{(d-1)}p^{T}\dot{h}^{T}-\frac{1}{2}p^{i}\mathfrak{D}^{ij}_{1}\dot{h}_{j}+\frac{1}{4}(\dot{n}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{n}^{l}-\Gamma^{l})\right.\\\
&-p^{ij}_{TT}p^{ij}_{TT}-\frac{1-\lambda}{(d-1)(1-d\lambda)}(p^{T})^{2}+\frac{1}{2}p^{i}\left(\delta_{ij}\Delta+\left(1+\frac{2\lambda}{1-d\lambda}\right)\partial_{i}\partial_{j}\right)p^{j}\\\
&\left.-\frac{2\lambda}{1-d\lambda}p^{T}\partial_{i}p^{i}+n^{i}\mathfrak{D}^{ij}_{1}p^{j}-\sqrt{g}\mathcal{V}+\dot{C}^{k}\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
n^{k}}\right)-\bar{C}_{i}\\{\Gamma^{i},\mathcal{H}_{k}\\}C^{k}\right]\,.\end{split}$
(2.53)

Note that the $(p^{T})^{2}$ term disappears in the relativistic limit
$\lambda=1$, hence we assume that $\lambda$ does not take this value. By
integrating $p^{ij}_{TT}$ and $p^{T}$, the action takes the form222Note that,
since we are assuming that $\Gamma^{k}$ does not depend on $p^{ij}$, the last
term in (2.53) does not depend on this momentum.

$\begin{split}S=&\int\,dt\,d^{d}x\left\\{\frac{1}{4}\dot{h}_{ij}^{TT}\dot{h}_{ij}^{TT}+\frac{1-d\lambda}{4(1-\lambda)(d-1)}(\dot{h}^{T})^{2}+\frac{1}{4}(\dot{n}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{n}^{l}-\Gamma^{l})\right.\\\
&+\frac{\lambda}{1-\lambda}p^{i}\partial_{i}\dot{h}^{T}+p^{i}\mathfrak{D}^{ij}_{1}\left(-\frac{1}{2}\dot{h}_{j}+n^{j}\right)+\frac{1}{2}p^{i}\mathfrak{D}_{2}^{ij}p^{j}-\sqrt{g}\mathcal{V}\\\
&\left.+\dot{C}^{k}\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
n^{k}}\right)-\bar{C}_{l}\\{\Gamma^{l}\,,\mathcal{H}_{k}\\}C^{k}\right\\}\,.\end{split}$
(2.54)

The last integration is on $p^{i}$. The square involving this variable can be
completed,

$\frac{1}{2}\mathfrak{D}_{2}^{ij}\left(p^{i}+\mathfrak{D}^{-1}_{2ik}B^{k}\right)\left(p^{j}+\mathfrak{D}^{-1}_{2jl}B^{l}\right)-\frac{1}{2}B^{i}\mathfrak{D}_{2ij}^{-1}B^{j},$
(2.55)

where

$\displaystyle B^{k}=$
$\displaystyle\frac{\lambda}{1-\lambda}\partial_{k}\dot{h}^{T}-\mathfrak{D}^{kl}_{1}\left(\frac{\dot{h}_{l}}{2}-n^{l}\right)\,,$
(2.56) $\displaystyle-\frac{1}{2}{B}^{k}\mathfrak{D}^{-1}_{2kl}{B}^{l}=$
$\displaystyle-\frac{1}{8}\dot{h}^{l}\mathfrak{D}^{kl}_{3}\dot{h}_{k}+\frac{1}{2}\dot{h}_{l}\mathfrak{D}^{kl}_{3}n^{k}-\frac{1}{2}{n}^{l}\mathfrak{D}^{kl}_{3}{n}^{k}$
$\displaystyle+\frac{\lambda^{2}}{4(1-\lambda)}(\dot{h}^{T})^{2}+\frac{1}{2}\lambda\dot{h}_{k}\partial_{k}\dot{h}^{T}-\lambda
n^{k}\partial_{k}\dot{h}^{T}\,.$ (2.57)

After the Gaussian integration, the action (2.54) becomes

$\begin{split}S=&\int
dtd^{d}x\left\\{\frac{1}{4}\dot{h}_{ij}^{TT}\dot{h}_{ij}^{TT}+\frac{(1-d\lambda)}{4(1-\lambda)(d-1)}(\dot{h}^{T})^{2}+\frac{1}{4}(\dot{n}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{n}^{l}-\Gamma^{l})\right.\\\
&\left.-\sqrt{g}\mathcal{V}-\frac{1}{2}{B}^{k}\mathfrak{D}^{-1}_{2kl}{B}^{l}+\dot{C}^{k}\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
n^{k}}\right)-\bar{C}_{i}\\{\Gamma^{i}\,,\mathcal{H}_{k}\\}C^{k}\right\\}\,.\end{split}$
(2.58)

So far, the potential $\mathcal{V}$ and the factor $\Gamma^{i}$ of the gauge-
fixing condition have been left unspecified, hence all the above formulas for
projectable Hořava theory are valid in any spatial dimension $d$, except for
the fact that in the $d=2$ case the $h_{ij}^{TT}$ mode must be dropped from
all expressions. Now, to continue on obtaining the quantum Lagrangian, we
specialize to the $d=2$ case, specifying the potential and the gauge-fixing
condition completely. The potential of the $d=2$ projectable Hořava theory, up
to second order in perturbations, becomes

$\sqrt{g}\mathcal{V}=\mu\sqrt{g}R^{2}=\mu(\Delta h^{T})^{2}\,.$ (2.59)

The operator $\mathfrak{D}^{ij}$ is defined in (2.43). For the factor
$\Gamma^{i}$ we take the form introduced in Ref. [2], which was obtained by
considering the anisotropic scaling of the variables of the Hořava theory,

$\Gamma^{k}=2c_{1}\Delta\partial_{l}h_{kl}+2c_{2}\Delta\partial_{k}h+c_{3}\partial_{k}\partial_{i}\partial_{j}h_{ij}\,,$
(2.60)

where $c_{1},c_{2},c_{3}$ are constants. In the transverse–longitudinal
decomposition it takes the form

$\Gamma^{k}=c_{1}\Delta^{2}h_{k}+\gamma\Delta\partial_{k}\partial_{l}h_{l}+2c_{2}\Delta\partial_{k}h^{T}\,,$
(2.61)

where $\gamma=c_{1}+2c_{2}+c_{3}$. Now we may write explicitly several
elements of the action (2.58) for the $d=2$ case. We have the terms that
involve the time derivative of the shift vector,

$\begin{split}&\frac{1}{4}(\dot{n}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{n}^{l}-\Gamma^{l})=\frac{1}{4}\dot{n}^{l}\mathfrak{D}^{-1}_{kl}\dot{n}^{k}-\frac{1}{2}\dot{n}^{k}\Big{(}c_{1}\left(\delta_{kl}\Delta-\rho\kappa\partial_{k}\partial_{l}\right)+\rho\gamma\partial_{k}\partial_{l}\Big{)}h_{l}\\\
&-\rho
c_{2}\dot{n}^{k}\partial_{k}h^{T}+\frac{1}{4}h_{k}\left[c_{1}^{2}\delta_{lk}\Delta^{3}+\left(2c_{1}\gamma+\gamma^{2}-\rho\kappa\left(\gamma+c_{1}\right)^{2}\right)\Delta^{2}\partial_{k}\partial_{l}\right]h_{l}\\\
&+\rho c_{2}(\gamma+c_{1})h_{k}\Delta^{2}\partial_{k}h^{T}-\rho
c_{2}^{2}h^{T}\Delta^{2}h^{T}\,,\end{split}$ (2.62)

where $\rho=(1+\kappa)^{-1}$. In the ghost sector we have the bracket

$\int
d^{2}x\bar{C}_{i}\\{\Gamma^{i}\,,\mathcal{H}_{k}\\}C^{k}=-2\int\,d^{2}xC^{k}(c_{1}\delta_{kl}\Delta^{2}+\gamma\Delta\partial_{k}\partial_{l})\bar{C}_{l}\,.$
(2.63)

The action in $2+1$ dimensions takes the form

$\begin{split}S=&\int\,dt\,d^{2}x\left\\{\frac{1-\lambda}{4}(\dot{h}^{T})^{2}-\left(\mu+\rho
c_{2}^{2}\right)h^{T}\Delta^{2}h^{T}+\frac{1}{4}\dot{n}^{l}\mathfrak{D}^{-1}_{kl}\dot{n}^{k}+\frac{\lambda}{2}\dot{h}_{k}\partial_{k}\dot{h}^{T}\right.\\\
&-\frac{1}{8}\dot{h}_{l}\mathfrak{D}^{kl}_{3}\dot{h}_{k}+\rho
c_{2}(\gamma+c_{1})h_{k}\Delta^{2}\partial_{k}h^{T}-\frac{1}{2}n^{l}\mathfrak{D}^{kl}_{3}n^{k}+\left(\lambda-\rho
c_{2}\right)\dot{n}^{k}\partial_{k}h^{T}\\\
&-\frac{1}{2}\dot{n}^{k}\left[\left(1+c_{1}\right)\delta_{lk}\Delta+\left(1-2\lambda+\rho(\gamma-\kappa
c_{1})\right)\partial_{k}\partial_{l}\right]h_{l}\\\
&+\frac{1}{4}h_{k}\left[c_{1}^{2}\delta_{kl}\Delta^{3}+\left(2c_{1}\gamma+\gamma^{2}-\rho\kappa\left(\gamma+c_{1}\right)^{2}\right)\Delta^{2}\partial_{k}\partial_{l}\right]h_{l}\\\
&\left.+\dot{C}^{k}\dot{\bar{C}}_{k}+2C^{k}(c_{1}\delta_{kl}\Delta^{2}+\gamma\Delta\partial_{k}\partial_{l})\bar{C}_{l}\right\\}.\end{split}$
(2.64)

We notice the presence of odd derivatives in time or space in (2.64), which
are also the terms that mix $n^{i}$ and the components of $h_{ij}$333
Actually, the odd derivatives are only in time. We see this explicitly in the
$3+1$ nonprojectable case, where we present the analogous formulas, Eqs.
(3.31) and (3.34), written with the decomposition on the vectors fields.. We
see that these odd terms cancel if we set444This identification is equivalent
to the one used in Ref. [2] for the Lagrangian theory.

$c_{1}=-1\,,\qquad c_{2}=\lambda(1+\kappa)\,,\qquad c_{3}=-2\kappa\,.$ (2.65)

By adjusting these constants, the final quantum path integral of the
projectable $2+1$ Hořava theory, written in Lagrangian variables and at second
order in perturbations, is

$\begin{split}Z=&\int\mathcal{D}h^{T}\mathcal{D}h_{i}^{T}\mathcal{D}h^{L}\mathcal{D}n^{iT}\mathcal{D}n^{L}\mathcal{D}\bar{C}_{i}\mathcal{D}C^{i}\\\
&\exp\left[i\int
dtd^{2}x\left[h^{T}\Big{(}-\frac{1}{4}(1-\lambda)\partial_{t}^{2}-\left(\mu+(1+\kappa)\lambda^{2}\right)\Delta^{2}\Big{)}h^{T}\right.\right.\\\
&+C^{k}\left[-\partial_{t}^{2}-2\delta_{kl}\Delta^{2}+2\left(2(\lambda-1)(1+\kappa)+1\right)\Delta\partial_{k}\partial_{l}\right]\bar{C}_{l}\\\
&-\frac{1}{4}n^{kT}\Big{(}\Delta^{-1}\partial_{t}^{2}+2\Delta\Big{)}n^{kT}+n^{L}\left(\frac{1}{4(1+\kappa)}\partial_{t}^{2}+(1-\lambda)\Delta^{2}\right)n^{L}\\\
&+h^{T}\Big{(}\frac{1}{2}\lambda\partial_{t}^{2}-2\lambda(\lambda-1)(1+\kappa)\Delta^{2}\Big{)}h^{L}+\frac{1}{8}{h}_{k}^{T}\Big{(}\Delta\partial_{t}^{2}+2\Delta^{3}\Big{)}h^{k}{}^{T}\\\
&\left.-h^{L}\Big{(}\frac{1}{4}(1-\lambda)\partial_{t}^{2}+(\lambda-1)^{2}(1+\kappa)\Delta^{2}\Big{)}h^{L}\right]\,,\end{split}$
(2.66)

where we have also decomposed the vectors,

$\begin{split}&n^{i}=n^{i}{}^{T}+\partial^{i}n^{L}\,,\qquad\partial_{i}n^{i}{}^{T}=0\,,\\\
&h_{i}=h_{i}{}^{T}+\partial_{i}\Delta^{-1}h^{L}\,,\qquad\partial_{i}h_{i}{}^{T}=0\,.\end{split}$
(2.67)

The quantum Lagrangian of Eq. (2.66) coincides with the one presented in Ref.
[2]. Those authors used a Faddeev-Popov procedure for fixing the gauge, hence
they get the usual parameter $\sigma$ associated to the averaging on the
gauge-fixing condition. To match exactly both Lagrangians, we must set
$\sigma=1/4$. At the end, the nonlocality only affects the time-derivative of
the shift vector (and all propagators are regular [2]).

Finally, we make a comment on the cubic order in perturbations in the ghost
sector. We take the ghost sector of the action given in (2.42). Its expansion
up to cubic order, imposing the gauge (2.60), is

$\begin{split}S_{gh}=&\int
dtd^{2}x\left\\{\dot{C}^{k}\dot{\bar{C}}_{k}+\dot{\bar{C}}_{k}\partial_{i}C^{k}n^{i}-C^{k}\partial_{i}\dot{\bar{C}}_{k}n^{i}-\dot{\bar{C}}_{k}C^{i}\partial_{i}n^{k}+C^{i}\left[-\delta_{ij}\Delta^{2}\right.\right.\\\
&\left.+(2\lambda-1+2\kappa(\lambda-1))\Delta\partial_{i}\partial_{j}\right]\bar{C}_{j}-\Delta\partial_{k}\bar{C}_{l}(\partial_{k}C^{j}h_{lj}+\partial_{l}C^{j}h_{kj}+C^{j}\partial_{j}h^{kl})\\\
&\left.+\lambda(1+\kappa)\Delta\partial_{i}\bar{C}_{i}(2\partial_{l}C^{k}h_{lk}+\partial_{k}hC^{k})-\kappa\partial_{k}\partial_{l}\partial_{j}\bar{C}_{j}(2\partial_{k}C^{i}h_{li}+\partial_{i}h_{kl}C^{i})\right\\}.\end{split}$
(2.68)

This is equal to the cubic order in the ghost sector of Ref. [2], except for
an additional term we find, which is
$-C^{k}\partial_{i}\dot{\bar{C}}_{k}n^{i}$.

##  3 $3+1$ Nonprojectable theory

###  3.1 Classical theory

In the nonprojectable theory the lapse function $N$ is allowed to depend on
time and space, hence it represents a complete functional degree of freedom.
In this case a large class of terms that depend on the vector
$a_{i}=\partial_{i}\ln N$ arise in the Lagrangian [37]. We focus the
nonprojectable theory in $3+1$ dimensions. The Lagrangian has the general form
shown in (2.6), but eliminating the restriction of projectability on $N$. The
criterium of power-countig renormalizability requires us to include a term of
order $z=3$ in $3+1$ dimensions. The total Lagrangian, containing the
$z=1,2,3$ orders has many terms. In this analysis we take for the potential
only the $z=3$ terms that contribute to the propagators, which are the
dominant terms in the propagators in the ultraviolet regime. They are [9]

$\mathcal{V}=-\alpha_{3}\nabla^{2}R\nabla_{i}a^{i}-\alpha_{4}\nabla^{2}a_{i}\nabla^{2}a^{i}-\beta_{3}\nabla_{i}R_{jk}\nabla^{i}R^{jk}-\beta_{4}\nabla_{i}R\nabla^{i}R\,,$
(3.1)

where $\alpha_{3},\alpha_{4},\beta_{3},\beta_{4}$ are coupling constants.

In the nonprojectable theory the lapse function $N$ and its conjugate momentum
$P_{N}$ are part of the canonical variables. There is no time derivative of
$N$ in the Lagrangian, hence

$\theta_{1}\equiv P_{N}=0$ (3.2)

is a constraint of the theory. The classical Hamiltonian, obtained by a
Legendre transformation, is

$H_{0}=\int
d^{3}x\sqrt{g}N\left(\frac{\pi^{ij}\pi_{ij}}{g}+\frac{\lambda}{1-3\lambda}\frac{\pi^{2}}{g}+\mathcal{V}\right)\,.$
(3.3)

The rest of constraints are the momentum constraint,
$\mathcal{H}_{i}=-2\nabla^{k}\pi_{ki}$, and the constraint

$\begin{split}\theta_{2}\equiv&\frac{N}{\sqrt{g}}\left(\pi^{ij}\pi_{ij}+\frac{\lambda}{1-3\lambda}\pi^{2}\right)+\sqrt{g}N\mathcal{V}-\alpha_{3}\sqrt{g}\nabla^{2}(N\nabla^{2}R)\\\
&+2\alpha_{4}\sqrt{g}\left(\nabla^{i}\nabla^{2}(N\nabla^{2}a_{i})\right)=0\,.\end{split}$
(3.4)

In the definition of the phase space, the main qualitative difference between
the projectable and nonprojectable cases is the activation of the lapse
function as a degree of freedom and the arising of the constraint $\theta_{2}$
(3.4) in the side of the nonprojectable theory. The last two terms of the
constraint $\theta_{2}$ are total derivatives of sixth order, hence the
integral of $\theta_{2}$ is equal to the primary Hamiltonian (3.3),

$H_{0}=\int d^{3}x\,\theta_{2}\,.$ (3.5)

Actually, when the $z=1$ terms are included in the potential, there is a
boundary contribution remaining from the integral of $\theta_{2}$. Moreover, a
term proportional to the so called Arnowitt-Deser-Misner energy is required
for the differentiability of one of the $z=1$ terms. Therefore, the general
statement is that the primary Hamiltonian of the $3+1$ nonprojectable Hořava
theory can be written as the integral of $\theta_{2}$ plus boundary terms.
Since in this analysis we focus on the $z=3$ terms, we can discard these
boundary terms.

###  3.2 BFV quantization

Since the nonprojectable theory has second-class constraints, the definitions
of the BFV quantization must be adapted, according to Ref. [35]. The
involution is defined in terms of Dirac brackets,

$\left\\{G_{a},G_{b}\right\\}_{D}=U^{c}_{ab}G_{c},\;\;\;\left\\{\mathcal{H}_{0},G_{a}\right\\}_{D}=V^{b}_{a}G_{b}\,,$
(3.6)

where Dirac brackets are defined by

$\left\\{F,R\right\\}_{D}=\left\\{F,R\right\\}-\left\\{F,\theta_{A}\right\\}\mathbb{M}^{-1}_{AB}\left\\{\theta_{B},R\right\\},\;\;\;\mathbb{M}_{AB}=\left\\{\theta_{A},\theta_{B}\right\\}.$
(3.7)

The implementation of the BFV quantization of the $3+1$ case is parallel to
the $2+1$ case shown in Ref. [32]. Here we present the summary. The matrix of
Poisson brackets of the second-class constraints has a triangular form,

$\mathbb{M}=\begin{pmatrix}0&\\{\theta_{1},\theta_{2}\\}\\\
-\\{\theta_{1},\theta_{2}\\}&\\{\theta_{2},\theta_{2}\\}\end{pmatrix}\,.$
(3.8)

Since the primary Hamiltonian $\mathcal{H}_{0}$ is equivalent to the second-
class constraint $\theta_{2}$, its Dirac bracket is zero with any quantity,
hence $V_{a}^{b}=0$. The Dirac bracket of the momentum constraint
$\mathcal{H}_{i}$ with itself is equivalent to its Poisson bracket,

$\left\\{\mathcal{H}_{i}(x),\mathcal{H}_{j}(y)\right\\}_{D}=\left\\{\mathcal{H}_{i}(x),\mathcal{H}_{j}(y)\right\\}\,,$
(3.9)

This leads to the algebra of spatial diffeomorphisms, as in the projectable
case, hence the coefficients $U_{ij}^{k}$ are the same, and $U_{ab}^{c}=0$ for
$a,b,c>i$.

We perform the BFV extension of the phase space in a similar way to the
projectable case. The Lagrange multipliers form a new canonical pair
$(N^{i},\pi_{i})$. The ghosts are the canonical pairs
$(\eta^{a},\mathcal{P}_{a})$. Thus, the full phase space is given by the pairs
$(g_{ij},\pi^{ij})$, $(N,P_{N})$, $(N_{i},\pi^{i})$ and
$(\eta_{a},\mathcal{P}^{a})$. The BFV path integral of the nonprojectable
Hořava theory is given by

$Z=\int\mathcal{D}V\delta(\theta_{1})\delta(\theta_{2})e^{iS},$ (3.10)

where the measure and the action are given by

$\displaystyle\mathcal{D}V$
$\displaystyle=\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N\mathcal{D}P_{N}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}\eta^{a}\mathcal{D}\mathcal{P}_{a}\times\sqrt{\det\mathbb{M}}\,,$
(3.11) $\displaystyle S$ $\displaystyle=\int
dtd^{3}x\left(\pi^{ij}\dot{g}_{ij}+P_{N}\dot{N}+\pi_{k}\dot{N}^{k}+\mathcal{P}_{a}\dot{\eta}^{a}-\mathcal{H}_{\Psi}\right).$
(3.12)

Unlike the projectable case, in this case the second-class constraints must be
imposed explicitly. By comparing the quantum degrees of freedom with the
projectable case, here we see that the canonical pair $(N,P_{N})$ has been
added to the phase space, but at the same time the imposition of the two
second-class constraints $\theta_{1},\theta_{2}$ compensates the pair
$(N,P_{N})$. Hence the balance is the same of the projectable case: there are
$d(d-1)$ physical degrees of freedom in the quantum phase space. In the $3+1$
foliation, they are the two tensorial modes and the extra scalar mode.

The BRST charge $\Omega$ and $\mathcal{H}_{1}$ take the same definition of
Eqs. (2.20) and (2.21). The quantum gauge-fixed Hamiltonian is

$\mathcal{H}_{\Psi}=\mathcal{H}_{1}+\left\\{\Psi,\Omega\right\\}_{D}\,.$
(3.13)

The nonprojectable Hořava theory is a theory of rank one, then

$\displaystyle\Omega$
$\displaystyle=G_{a}\eta^{a}-\frac{1}{2}U^{c}_{ab}\eta^{a}\eta^{b}\mathcal{P}_{c}=\mathcal{H}_{k}\eta^{k}_{1}+\pi_{k}\eta^{k}_{2}-\frac{1}{2}U^{k}_{ij}\eta^{i}_{1}\eta^{j}_{1}\mathcal{P}^{1}_{k}\,,$
(3.14) $\displaystyle\mathcal{H}_{1}$ $\displaystyle=\mathcal{H}_{0}\,.$
(3.15)

In the case of the second-class constraints the consistency conditions for the
BFV quantization are

$\\{\Omega\,,\Omega\\}_{D}=0\,,\qquad\\{\mathcal{H}_{1}\,,\Omega\\}_{D}=0\,.$
(3.16)

The first condition holds following the same steps of the projectable case,
but operating with Dirac brackets in this case. The second condition holds
because $\mathcal{H}_{1}=\mathcal{H}_{0}$, and this is equivalent to a second-
class constraint, hence its Dirac bracket is always zero. The gauge-fixed
quantum Hamiltonian takes the form

$\mathcal{H}_{\Psi}=\mathcal{H}_{0}+\left\\{\Psi,\mathcal{H}_{k}\eta_{1}^{k}+\pi_{k}\eta^{k}_{2}-\frac{1}{2}U^{k}_{ij}\eta^{i}_{1}\eta^{j}_{1}\mathcal{P}_{k}^{1}\right\\}_{D}.$
(3.17)

As we did in the projectable case, we can adopt the form of the gauge-fixing
condition used in the general BFV formalism, originally introduced for
relativistic theories. Thus, the gauge-fixing condition $\Phi^{i}=0$ and the
associated fermionic function $\Psi$ take the forms given in (2.30) and
(2.31), respectively. The Hamiltonian takes the form

$\begin{split}{\mathcal{H}}_{\Psi}=&\mathcal{H}_{0}+\mathcal{H}_{k}N^{k}+\mathcal{P}_{k}^{1}\eta_{2}^{k}-\mathcal{P}^{1}_{i}\left(N^{j}\partial_{j}\eta_{1}^{i}+N^{i}\partial_{j}\eta_{1}^{j}\right)+\pi_{k}\chi^{k}\\\
&+\mathcal{P}_{i}^{2}\\{\chi^{i}\,,\mathcal{H}_{k}\\}\eta_{1}^{k}+\mathcal{P}_{i}^{2}\dfrac{\delta\chi^{i}}{\delta
N^{l}}\eta_{2}^{l}\,.\end{split}$ (3.18)

Due to the form (3.8), the measure of the second-class constraints simplifies
to $\sqrt{\det\mathbb{M}}=\det\\{\theta_{1}\,,\theta_{2}\\}$. Thus, this
measure can be incorporated to the Lagrangian by means of the ghosts fields
$\bar{\varepsilon},\varepsilon$,

$\det\left\\{\theta_{1},\theta_{2}\right\\}=\int\mathcal{D}\bar{\varepsilon}\mathcal{D}\varepsilon\exp\left(i\int
dtd^{3}x\;\bar{\varepsilon}\left\\{\theta_{1},\theta_{2}\right\\}\varepsilon\right).$
(3.19)

Taking the definition of $\theta_{2}$ given in (3.4), the bracket
$\\{\theta_{1}\,,\theta_{2}\\}$ results

$\begin{split}\left\\{\theta_{1}(x),\theta_{2}(y)\right\\}&=\frac{\theta_{2}}{N}\delta_{xy}+2\sqrt{g}\left\\{-\alpha_{3}\nabla^{i}\left(\nabla_{i}\delta_{xy}\nabla^{2}R-\delta_{xy}a_{i}\nabla^{2}R\right)\right.\\\
&\left.+\alpha_{4}\left[\nabla_{i}\nabla^{2}(\delta_{xy}\nabla^{2}a^{i})-N\nabla^{2}a^{i}\nabla^{2}\nabla_{i}\left(\frac{\delta_{xy}}{N}\right)\right.\right.\\\
&\left.\left.+\nabla_{i}\nabla^{2}\left(N\nabla^{2}\nabla^{i}\left(\frac{\delta_{xy}}{N}\right)\right)-\frac{\delta_{xy}}{N}\nabla_{i}\nabla^{2}(N\nabla^{2}a^{i})\right]\right\\}\,,\end{split}$
(3.20)

where $\delta_{xy}\equiv\delta(x^{i}-y^{i})$. Once we have obtained this
bracket, we may integrate the variable $P_{N}$ without further consequences,
since it vanishes due to the constraint $\theta_{1}=0$. The constraint
$\theta_{2}$ can be incorporated to the Lagrangian by means of a Lagrange
multiplier, which we denote by $\xi$. Thus, the BFV path integral of the
nonprojectable Hořava theory in $3+1$ dimensions takes the form

$\begin{split}Z_{\Psi}=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}\eta_{1}^{i}\mathcal{D}\mathcal{P}^{1}_{i}\mathcal{D}\eta_{2}^{i}\mathcal{D}\mathcal{P}^{2}_{i}\mathcal{D}\xi\mathcal{D}\bar{\varepsilon}\mathcal{D}\varepsilon\\\
&\exp\left[i\int
dtd^{3}x\Big{(}\pi^{ij}\dot{g_{ij}}+\pi_{k}\dot{N}^{k}+\mathcal{P}^{1}_{i}\dot{\eta}_{1}^{i}+\mathcal{P}^{2}_{i}\dot{\eta}_{2}^{i}-\mathcal{H}_{0}-\mathcal{H}_{k}N^{k}-\mathcal{P}_{k}^{1}\eta_{2}^{k}\right.\\\
&+\mathcal{P}^{1}_{i}\left(N^{j}\partial_{j}\eta_{1}^{i}+N^{i}\partial_{j}\eta_{1}^{j}\right)-\pi_{k}\chi^{k}-\mathcal{P}_{i}^{2}\\{\chi^{i}\,,\mathcal{H}_{k}\\}\eta_{1}^{k}-\mathcal{P}_{i}^{2}\dfrac{\delta\chi^{i}}{\delta
N^{l}}\eta_{2}^{l}\\\
&\left.+\xi\theta_{2}+\bar{\varepsilon}\left\\{\theta_{1}\,,\theta_{2}\right\\}\varepsilon\Big{)}\right]\,.\end{split}$
(3.21)

###  3.3 Quantum Lagrangian

By adapting the discussion done in section 2 about the structure of the gauge-
fixing condition to the nonprojectable case, we set

$\chi^{i}=\mathfrak{D}^{ij}\pi_{j}+\Gamma^{i}[g_{ij},N,N^{i}]\,.$ (3.22)

The Gaussian integration on $\pi_{i}$ leads to the path integral

$\begin{split}Z_{\Psi}=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N\mathcal{D}N^{k}\mathcal{D}\eta_{1}^{i}\mathcal{D}\mathcal{P}^{1}_{i}\mathcal{D}\eta_{2}^{i}\mathcal{D}\mathcal{P}^{2}_{i}\mathcal{D}\xi\mathcal{D}\bar{\varepsilon}\mathcal{D}\varepsilon\\\
&\exp\left[i\int
dtd^{3}x\Big{(}\pi^{ij}\dot{g_{ij}}+\mathcal{P}^{1}_{i}\dot{\eta}_{1}^{i}+\mathcal{P}^{2}_{i}\dot{\eta}_{2}^{i}+\frac{1}{4}(\dot{N}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{N}^{l}-\Gamma^{l})\right.\\\
&-\mathcal{H}_{0}-\mathcal{H}_{k}N^{k}-\mathcal{P}_{k}^{1}\eta_{2}^{k}+\mathcal{P}^{1}_{i}\left(N^{j}\partial_{j}\eta_{1}^{i}+N^{i}\partial_{j}\eta_{1}^{j}\right)\\\
&\left.-\mathcal{P}_{i}^{2}\\{\Gamma^{i}\,,\mathcal{H}_{k}\\}\eta_{1}^{k}-\mathcal{P}_{i}^{2}\dfrac{\delta\Gamma^{i}}{\delta
N^{l}}\eta_{2}^{l}+\xi\theta_{2}+\bar{\varepsilon}\left\\{\theta_{1},\theta_{2}\right\\}\varepsilon\Big{)}\right]\,.\end{split}$
(3.23)

The next integration we perform is over the BFV ghosts that are canonical
momenta. We perform the same change of notation (2.40). For the terms of the
action that depend on $\mathcal{P}^{a}$ and $\bar{\mathcal{P}}_{a}$ it is
possible to carry out the integration after completing the bilinear in these
variables, as in (2.41). The action of the ghost sector results

$\begin{split}S_{\text{ghost}}&=\int
dtd^{3}x\left\\{\left(\dot{C}^{k}-N^{j}\partial_{j}C^{k}-N^{k}\partial_{j}C^{j}\right)\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\dfrac{\delta\Gamma^{i}}{\delta
N^{k}}\right)-\bar{C}_{i}\left\\{\Gamma^{i}\,,\mathcal{H}_{k}\right\\}_{D}C^{k}\right\\}.\end{split}$
(3.24)

Now we adopt the perturbative variables defined in (2.48), adding $N-1=n$. For
the $d=3$ nonprojectable theory we take [2]

$\mathfrak{D}^{ij}=\delta_{ij}\Delta^{2}+\kappa\Delta\partial_{i}\partial_{j}\,,\quad\mathfrak{D}^{-1}_{ij}=\frac{\delta_{ij}}{\Delta^{2}}-\frac{\kappa}{1+\kappa}\frac{\partial_{i}\partial_{j}}{\Delta^{3}}\,.$
(3.25)

The momentum constraint $\mathcal{H}_{j}$ is given in (2.50), and the
Hamiltonian density $\mathcal{H}_{0}$ takes the form

$\begin{split}\mathcal{H}_{0}=&p^{ij}p^{ij}+\frac{\lambda}{1-3\lambda}p^{2}-\alpha_{3}n\Delta^{2}\left(\partial_{i}\partial_{j}h_{ij}-\Delta
h\right)+\alpha_{4}n\Delta^{3}n\\\
&+\left(\frac{\beta_{3}}{2}+\beta_{4}\right)h_{kj}\partial_{j}\partial_{k}\partial_{n}\partial_{l}\Delta
h_{nl}+\left(\frac{\beta_{3}}{4}+\beta_{4}\right)\left(h\Delta^{3}h-2h\partial_{j}\partial_{k}\Delta^{2}h_{jk}\right)\\\
&+\frac{\beta_{3}}{4}\left(h_{kj}\Delta^{3}h_{kj}-2h_{kj}\partial_{l}\partial_{k}\Delta^{2}h_{lj}\right).\end{split}$
(3.26)

Therefore, the path integral becomes

$\begin{split}Z=&\int\mathcal{D}h_{ij}\mathcal{D}p^{ij}\mathcal{D}n\mathcal{D}n^{k}\mathcal{D}\bar{C}_{i}\mathcal{D}C_{i}\mathcal{D}\bar{\varepsilon}\mathcal{D}\varepsilon\mathcal{D}\xi\\\
&\exp\left[i\int
dtd^{3}x\left(p^{ij}\dot{h}_{ij}-\mathcal{H}_{0}-n^{k}\mathcal{H}_{k}+\frac{1}{4}(\dot{n}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{n}^{l}-\Gamma^{l})\right.\right.\\\
&\left.\left.+\xi\theta_{2}+\bar{\varepsilon}\left\\{\theta_{1},\theta_{2}\right\\}\varepsilon+\dot{C}^{k}\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
n^{k}}\right)-\bar{C}_{i}\left\\{\Gamma^{i},\mathcal{H}_{k}\right\\}_{D}C^{k}\right)\right]\,.\end{split}$
(3.27)

We make the decomposition (2.52) on the fields. The second class constraint
$\theta_{2}$ and the measure of the second-class constraints, given by the
bracket (3.20), contribute to the perturbative action with the following
terms, respectively,

$\xi\theta_{2}=\xi\left(\alpha_{3}\Delta^{3}h^{T}+2\alpha_{4}\Delta^{3}n\right)\,,\quad\bar{\varepsilon}\left\\{\theta_{1},\theta_{2}\right\\}\varepsilon=-2\alpha_{4}\bar{\varepsilon}\Delta^{3}\varepsilon\,,$
(3.28)

where the Lagrange multiplier $\xi$ is regarded as a perturbative variable.
After these steps, the Gaussian integration on $p^{ij}_{TT}$ and $p^{T}$ can
be done by completing squares (assuming again $\lambda\neq 1$). This yields
the action

$\begin{split}S&=\int
dtd^{3}x\left\\{\frac{1}{4}\dot{h}_{ij}^{TT}\dot{h}_{ij}^{TT}+\frac{(1-3\lambda)}{8(1-\lambda)}(\dot{h}^{T})^{2}+\frac{1}{4}\left(\dot{n}^{i}-\Gamma^{i}\right)\mathfrak{D}^{-1}_{ij}\left(\dot{n}^{j}-\Gamma^{j}\right)\right.\\\
&\left.+\frac{\lambda}{1-\lambda}p^{i}\partial_{i}\dot{h}^{T}+p^{l}\mathfrak{D}_{1}^{kl}\left(-\frac{1}{2}\dot{h}_{k}+n^{k}\right)+\frac{1}{2}p^{i}\mathfrak{D}_{2}^{ij}p^{j}-\alpha_{3}n\Delta^{3}h^{T}-\alpha_{4}n\Delta^{3}n\right.\\\
&\left.-\frac{\beta_{3}}{4}h^{TT}_{ij}\Delta^{3}h^{TT}_{ij}-\left(\frac{3\beta_{3}}{8}+\beta_{4}\right)h^{T}\Delta^{3}h^{T}+\alpha_{3}\xi\Delta^{3}h^{T}+2\alpha_{4}\xi\Delta^{3}n\right.\\\
&\left.+\dot{C}^{k}\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
n^{k}}\right)-2\bar{C}_{l}\frac{\delta\Gamma^{l}}{\delta
h_{ij}}\partial_{j}C^{i}-2\alpha_{4}\bar{\varepsilon}\Delta^{3}\varepsilon\right\\}\,,\end{split}$
(3.29)

where the operators $\mathfrak{D}_{1}^{ij}$ and $\mathfrak{D}_{2}^{ij}$ are
the same of the $d=2$ case defined in (2.44) and (2.45). The last integration
is on $p^{i}$. We integrate in a similar way to how it was done in the
projectable case, obtaining

$\begin{split}S&=\int
dtd^{3}x\left\\{\frac{1}{4}\dot{h}_{ij}^{TT}\dot{h}_{ij}^{TT}+\frac{(1-3\lambda)}{8(1-\lambda)}(\dot{h}^{T})^{2}+\frac{1}{4}\left(\dot{n}^{i}-\Gamma^{i}\right)\mathfrak{D}^{-1}_{ij}\left(\dot{n}^{j}-\Gamma^{j}\right)\right.\\\
&\left.-\frac{\beta_{3}}{4}h^{TT}_{ij}\Delta^{3}h^{TT}_{ij}-\left(\frac{3\beta_{3}}{8}+\beta_{4}\right)h^{T}\Delta^{3}h^{T}-\alpha_{3}n\Delta^{3}h^{T}-\alpha_{4}n\Delta^{3}n+\alpha_{3}\xi\Delta^{3}h^{T}\right.\\\
&\left.+2\alpha_{4}\xi\Delta^{3}n-\frac{1}{2}B^{j}\mathfrak{D}_{2jl}^{-1}B^{l}+\dot{C}^{k}\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
n^{k}}\right)-2\bar{C}_{l}\frac{\delta\Gamma^{l}}{\delta
h_{ij}}\partial_{j}C^{i}-2\alpha_{4}\bar{\varepsilon}\Delta^{3}\varepsilon\right\\}\,,\end{split}$
(3.30)

where $B^{i}$ is defined in Eq. (2.56). Finally, we make the decomposition on
the vector variables shown in (2.67). In particular,

$\begin{split}&-\frac{1}{2}B^{i}\mathfrak{D}_{2ij}^{-1}B^{j}=-\frac{\lambda^{2}}{4(1-\lambda)}h^{T}\partial^{2}_{t}h^{T}+\frac{\lambda}{2}h^{T}\partial_{t}^{2}h^{L}-\frac{1}{2}\dot{n}^{iT}\Delta
h_{i}^{T}-\lambda\dot{n}^{L}\Delta h^{T}\\\ &+(1-\lambda)\dot{n}^{L}\Delta
h^{L}+\frac{1}{8}h_{i}^{T}\partial^{2}_{t}\Delta
h_{i}^{T}-\frac{1}{2}n^{iT}\Delta
n^{iT}-\frac{(1-\lambda)}{4}h^{L}\partial^{2}_{t}h^{L}+(1-\lambda)n^{L}\Delta^{2}n^{L}\,.\end{split}$
(3.31)

Now we define the factor $\Gamma^{i}$ of the gauge-fixing condition, adopting
the analysis of Ref. [2]. Those authors found that the appropriate gauge
fixing condition in $d=3$, preserving the anisotropy of the Hořava theory, is
given by

$\Gamma^{i}=2c_{1}\Delta^{2}\partial_{j}h_{ij}+2c_{2}\Delta^{2}\partial_{i}h+c_{3}\Delta\partial_{i}\partial_{j}\partial_{k}h_{jk}\,.$
(3.32)

The notation on the constants $c_{1,2,3}$ has been put intentionally equal to
the projectable case (2.60). In terms of the transverse-longitudinal
decomposition (2.52), this is

$\Gamma^{i}=c_{1}\Delta^{3}h_{i}+2c_{2}\Delta^{2}\partial_{i}h^{T}+\gamma\Delta^{2}\partial_{i}\partial_{j}h_{j}.$
(3.33)

We have the expansion of the term,

$\begin{split}&\frac{1}{4}\left(\dot{n}^{i}-\Gamma^{i}\right)\mathfrak{D}^{-1}_{ij}\left(\dot{n}^{j}-\Gamma^{j}\right)=\frac{1}{4}\dot{n}^{iT}\frac{1}{\Delta^{2}}\dot{n}^{iT}-\frac{1}{4}\rho\dot{n}^{L}\frac{1}{\Delta}\dot{n}^{L}-\frac{1}{2}c_{1}\dot{n}^{iT}\Delta
h_{i}^{T}+\rho c_{2}\dot{n}^{L}\Delta h^{T}\\\ &+\rho\nu\dot{n}^{L}\Delta
h^{L}+\frac{1}{4}c_{1}^{2}h_{i}^{T}\Delta^{4}h_{i}^{T}-\rho
c_{2}^{2}h^{T}\Delta^{3}h^{T}-\rho c_{2}\nu
h^{T}\Delta^{3}h^{L}-\frac{1}{4}\rho\nu^{2}h^{L}\Delta^{3}h^{L},\end{split}$
(3.34)

with $\nu=2c_{1}+2c_{2}+c_{3}$. As in the projectable case, the terms with a
odd time derivative in (3.31) and (3.34) can be canceled by an appropriate
setting of the constants $c_{1,2,3}$, which coincides with (2.65) since the
notation on these constants is the same. With this choice, the final path
integral in the Lagrangian formalism of the $3+1$ nonprojectable Hořava
theory, with the $z=3$ potential, results

$\begin{split}Z=&\int\mathcal{D}h_{ij}^{TT}\mathcal{D}h^{T}\mathcal{D}h_{i}^{T}\mathcal{D}h^{L}\mathcal{D}n\mathcal{D}n^{iT}\mathcal{D}n^{L}\mathcal{D}\bar{C}_{i}^{T}\mathcal{D}C^{iT}\mathcal{D}\bar{C}^{L}\mathcal{D}C^{L}\mathcal{D}\xi\mathcal{D}\bar{\varepsilon}\mathcal{D}\varepsilon\\\
&\exp\left[i\int
dtd^{3}x\left\\{\bar{C}_{k}^{T}\left(\partial_{t}^{2}+2\Delta^{3}\right)C^{kT}-\bar{C}^{L}\left[\partial_{t}^{2}\Delta+4(1-\lambda)(1+\kappa)\Delta^{4}\right]C^{L}\right.\right.\\\
&\left.-\frac{1}{4}h_{ij}^{TT}\left(\partial_{t}^{2}+\beta_{3}\Delta^{3}\right)h^{TT}_{ij}-h^{T}\left[\frac{1-2\lambda}{8}\partial_{t}^{2}+\left(\frac{3\beta_{3}}{8}+\beta_{4}+\lambda^{2}(1+\kappa)\right)\Delta^{3}\right]h^{T}\right.\\\
&\left.+\frac{1}{8}h_{i}^{T}\left(\partial^{2}_{t}\Delta+2\Delta^{4}\right)h_{i}^{T}-\frac{1-\lambda}{4}h^{L}\left[\partial^{2}_{t}+4(1-\lambda)(1+\kappa)\Delta^{3}\right]h^{L}\right.\\\
&\left.-\frac{1}{4}n^{iT}\left(\frac{\partial_{t}^{2}}{\Delta^{2}}+2\Delta\right)n^{iT}+\frac{1}{4}\rho
n^{L}\left(\frac{\partial_{t}^{2}}{\Delta}+4(1-\lambda)(1+\kappa)\Delta^{2}\right)n^{L}\right.\\\
&\left.+\frac{\lambda}{2}h^{T}\left(\partial_{t}^{2}+4(1-\lambda)(1+\kappa)\Delta^{3}\right)h^{L}-\alpha_{3}n\Delta^{3}h^{T}-\alpha_{4}n\Delta^{3}n\right.\\\
&\left.\left.+\alpha_{3}\xi\Delta^{3}h^{T}+2\alpha_{4}\xi\Delta^{3}n-2\alpha_{4}\bar{\varepsilon}\Delta^{3}\varepsilon\right\\}\right]\,.\end{split}$
(3.35)

In the set of propagators derived from this action, shown in appendix B,
almost all of them are regular. The nonregular ones arise when the variables
associated to the second-class constraints, $\xi$ and
$\bar{\varepsilon},\varepsilon$, are involved. This confirms that the nonlocal
Lagrangian (3.35) leads to regular propagators for the original field
variables, including the ghosts associated to the gauge fixing [2], but the
presence of nonregular propagators persists, associated to the fact that the
theory has second-class constraints, unlike the projectable case.

##  4 Comparison with General Relativity

As it is well known, the classical canonical action of general relativity
written in ADM variables is

$S=\int
dtd^{3}x\Big{(}\dot{g}_{ij}\pi^{ij}-N\mathcal{H}-N^{k}\mathcal{H}_{k}\Big{)}\,.$
(4.1)

The constraints are given by

$\displaystyle\mathcal{H}_{i}$ $\displaystyle=$
$\displaystyle-2\nabla^{k}\pi_{ki}\,,$ (4.2) $\displaystyle\mathcal{H}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{g}}\left(\pi^{ij}\pi_{ij}-\frac{1}{2}\pi^{2}\right)-\sqrt{g}R\,,$
(4.3)

and both constraints are of first class. $N$ and $N^{i}$ play the role of
Lagrange multipliers. We denote them collectively by
$\mathcal{H}_{a}=(\mathcal{H},\mathcal{H}_{i})$, and $N^{a}=(N,N^{i})$.

For the BFV quantization [36] we introduce the canonical pair
$(N^{a},\pi_{a})$, hence we have the functions
$G_{A}=(\mathcal{H}_{a},\pi_{a})$. For each of these functions we define the
pair of fermionic ghosts $(\eta_{A}\,,\mathcal{P}^{A})$, which can be split as
$(\eta_{1}^{a},\mathcal{P}^{1}_{a})$, $(\eta_{2}^{a},\mathcal{P}^{2}_{a})$.
The involution relations $\\{G_{A}\,,G_{B}\\}=U_{AB}^{C}G_{C}$ lead to the
algebra of spacetime diffeomorphisms. There is an essential qualitative
difference with the Hořava theory, since in general relativity the
coefficients $U_{ab}^{c}$ depend on the canonical fields. This fact has
important consequences in the BFV quantization [33, 36]. The gauge-fixed BFV
path integral takes the form

$\begin{split}Z=\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{a}\mathcal{D}\pi_{a}\mathcal{D}\eta^{A}\mathcal{D}\mathcal{P}_{A}\exp\left[i\int
dtd^{3}x\Big{(}\dot{g}_{ij}\pi^{ij}+\pi_{a}\dot{N}^{a}+\mathcal{P}_{A}\dot{\eta}^{A}-\mathcal{H}_{\Psi}\Big{)}\right]\,.\end{split}$
(4.4)

In the $3+1$-dimensional spacetime the two canonical pairs
$(g_{ij},\pi^{ij})$, $(N^{a},\pi^{a})$ sum $20$ degrees of freedom. The ghosts
$(\eta_{1}^{a},\mathcal{P}^{1}_{a})$, $(\eta_{2}^{a},\mathcal{P}^{2}_{a})$ sum
$16$ degrees. The substraction yields the usual four physical degrees of
freedom in the phase space of quantum general relativity. The BRST charge
takes the form

$\Omega=G_{A}\eta^{A}=\mathcal{H}_{a}\eta_{1}^{a}+\pi_{a}\eta_{2}^{a}-\frac{1}{2}U_{ab}^{c}\eta_{1}^{a}\eta_{1}^{b}\mathcal{P}^{1}_{c}\,.$
(4.5)

The gauge-fixed quantum Hamiltonian is defined by Eq. (2.18), with
$\mathcal{H}_{1}=0$. The appropriate form of the gauge-fixing fermionic
function is given in (2.31), which, considering the four spacetime directions,
takes the form $\Psi=\mathcal{P}_{a}^{1}N^{a}+\mathcal{P}_{a}^{2}\chi^{a}$.
Thus, the gauge-fixed Hamiltonian results

$\begin{split}\mathcal{H}_{\Psi}=&N^{a}\mathcal{H}_{a}+\mathcal{P}_{a}^{1}\eta_{2}^{a}+\pi_{a}\chi^{a}+\mathcal{P}_{a}^{2}\\{\chi^{a}\,,\mathcal{H}_{b}\\}\eta_{1}^{b}+\mathcal{P}_{a}^{2}\\{\chi^{a}\,,\pi_{b}\\}\eta_{2}^{b}\\\
&-\frac{1}{2}\mathcal{P}_{a}^{2}\\{\chi^{a},U_{cd}^{e}\\}\eta_{1}^{c}\eta_{1}^{d}\mathcal{P}_{e}^{1}+N^{a}\eta_{1}^{b}U_{ba}^{c}\mathcal{P}_{c}^{1}\,.\end{split}$
(4.6)

We proceed to the construction of the quantum Lagrangian. For the integration
on $\pi_{a}$ we adopt the same strategy we used in the Hořava theory,
considering in this case the four directions of spacetime diffeomorphisms. We
take a gauge-fixing condition in the form

$\chi^{a}=\mathfrak{D}^{ab}\pi_{b}+\Gamma^{a}[g_{ij},N,N^{i}]\,.$ (4.7)

The isotropic scaling in general relativity is

$[\pi_{a}]=3,\qquad[\chi^{a}]=1,\qquad[\mathfrak{D}^{ab}]=-2\,.$ (4.8)

Therefore, $\mathfrak{D}^{ab}$ is nonlocal whereas its inverse
$\mathfrak{D}_{ab}^{-1}$ is a local operator. After the integration on
$\pi_{a}$, the quantum action takes the form

$\begin{split}S=&\int\,dt\,d^{3}x\Big{(}\dot{g}_{ij}\pi^{ij}+\frac{1}{2}(\dot{N}^{a}-\Gamma^{a})\mathfrak{D}^{-1}_{ab}(\dot{N}^{b}-\Gamma^{b})+\mathcal{P}_{A}\dot{\eta}^{A}-N^{a}\mathcal{H}_{a}-\mathcal{P}_{a}^{1}\eta_{2}^{a}\\\
&-N^{a}\eta_{1}^{b}U_{ba}^{c}\mathcal{P}_{c}^{1}-\mathcal{P}_{a}^{2}\\{\Gamma^{a}\,,\mathcal{H}_{b}\\}\eta_{1}^{b}-\mathcal{P}_{a}^{2}\dfrac{\delta\Gamma^{a}}{\delta
N^{b}}\eta_{2}^{b}+\frac{1}{2}\mathcal{P}_{a}^{2}\\{\Gamma^{a},U_{bc}^{d}\\}\eta_{1}^{b}\eta_{1}^{c}\mathcal{P}_{d}^{1}\Big{)}\,.\end{split}$
(4.9)

The ghost sector is given by

$-\bar{\mathcal{P}}_{a}\mathcal{P}^{a}+\bar{\mathcal{P}}_{a}\left(\dot{C}^{a}-\frac{1}{2}\bar{C}_{d}\\{\Gamma^{d}\,,U_{bc}^{a}\\}C^{b}C^{c}+N^{d}C^{b}U_{bd}^{a}\right)+\mathcal{P}^{a}\left(\dot{\bar{C}}_{a}+\bar{C}_{e}\dfrac{\delta\Gamma^{e}}{\delta
N^{a}}\right)\,.$ (4.10)

By integrating on the corresponding Grassmann variables we get the action

$\begin{split}S=&\int\,dt\,d^{3}x\Big{(}\dot{g}_{ij}\pi^{ij}+\frac{1}{2}(\dot{N}^{a}-\Gamma^{a})\mathfrak{D}^{-1}_{ab}(\dot{N}^{b}-\Gamma^{b})-N^{a}\mathcal{H}_{a}\\\
&+\left(\dot{C}^{a}-\frac{1}{2}\bar{C}_{d}\\{\Gamma^{d},U_{bc}^{a}\\}C^{b}C^{c}+N^{d}C^{b}U_{bd}^{a}\right)\left(\dot{\bar{C}}_{a}+\bar{C}_{e}\dfrac{\delta\Gamma^{e}}{\delta
N^{a}}\right)-\bar{C}_{a}\\{\Gamma^{a},\mathcal{H}_{b}\\}C^{b}\Big{)}\,.\end{split}$
(4.11)

We now perform perturbations, obtaining the second-order action

$\begin{split}S=&\int\,dt\,d^{3}x\left[-\left(p^{TT}_{ij}-\frac{1}{2}\dot{h}_{ij}^{TT}\right)^{2}+\frac{1}{4}\dot{h}^{TT}_{ij}\dot{h}^{TT}_{ij}+\frac{1}{2}p^{T}\dot{h}^{T}+\frac{1}{2}p^{k}\Delta
p^{k}\right.\\\
&+p^{k}(-\partial_{k}p^{T}+(\delta_{kj}\Delta+\partial_{k}\partial_{j})(n^{j}-\frac{1}{2}\dot{h}_{j}))+\frac{1}{4}h^{TT}_{ij}\partial^{2}h^{TT}_{ij}-\frac{1}{8}h^{T}\partial^{2}h^{T}-n\partial^{2}h^{T}\\\
&\left.+\frac{1}{2}(\dot{N}^{a}-\Gamma^{a})\mathfrak{D}^{-1}_{ab}(\dot{N}^{b}-\Gamma^{b})-\bar{C}_{a}\\{\Gamma^{a},\mathcal{H}_{b}\\}C^{b}+\dot{C}^{a}\left(\dot{\bar{C}}_{a}+\bar{C}_{e}\dfrac{\delta\Gamma^{e}}{\delta
N^{a}}\right)\right]\,.\end{split}$ (4.12)

Here we face another qualitative difference with respect to the Hořava
gravity. The Lagrangian in (4.12) has no $(p^{T})^{2}$ term, unlike the
Lagrangian in Eq. (2.53). This is a consequence of the relativistic structure
behind the Hamiltonian of general relativity, which implies the frozen of the
scalar mode. Hence, we change the order of integration in this case, by
performing first the integration on the longitudinal component of the momentum
$p^{i}$. This brings the terms to the Lagrangian

$\begin{split}&+\frac{1}{2}(p^{T})^{2}+p^{T}(-2\partial_{k}n^{k}+\partial_{k}\dot{h}_{k})-\frac{1}{2}n^{k}(\delta_{kl}\Delta+3\partial_{k}\partial_{l})n^{l}\\\
&-\frac{1}{8}\dot{h}_{k}(\delta_{kl}\Delta+3\partial_{k}\partial_{l})\dot{h}_{l}+\frac{1}{2}n^{k}(\delta_{kl}\Delta+3\partial_{k}\partial_{l})\dot{h}_{l}\,.\end{split}$
(4.13)

Now we perform the integration on $p^{T}$ and $p^{ij}_{TT}$, obtaining

$\begin{split}S=&\int
dtd^{3}x\Big{(}\frac{1}{4}\dot{h}_{ij}^{TT}\dot{h}_{ij}^{TT}-\frac{1}{8}\dot{h}_{k}^{T}\Delta\dot{h}_{k}^{T}+\frac{1}{4}h^{TT}_{ij}\Delta
h^{TT}_{ij}-\frac{1}{8}h^{T}\Delta h^{T}-n\Delta h^{T}-\frac{1}{2}n^{kT}\Delta
n^{kT}\\\
&+\frac{1}{2}n^{kT}\Delta\dot{h}^{T}_{k}-\frac{1}{8}\Big{(}(\dot{h}^{T})^{2}+\dot{h}^{T}(-8\Delta
n^{L}+4\dot{h}^{L})\Big{)}-\bar{C}_{a}\\{\Gamma^{a},\mathcal{H}_{i}\\}C^{i}\\\
&+\dot{C}^{a}(\dot{\bar{C}}_{a}+\bar{C}_{e}\dfrac{\delta\Gamma^{e}}{\delta
N^{a}})+\frac{1}{2}(\dot{n}^{a}-\Gamma^{a})\mathfrak{D}_{ab}^{-1}(\dot{n}^{b}-\Gamma^{b})\Big{)}\,.\end{split}$
(4.14)

Therefore, the resulting quantum Lagrangian is completely local as far as the
remaining part $\Gamma^{a}$ of the gauge-fixing condition is local.

## Conclusions

We have seen that the BFV quantization is suitable for the Hořava theory, both
in its projectable and nonprojectable versions, and varying the dimension of
the foliation. This extends the analysis that two of us performed in Ref.
[32]. The BFV formalism provides a rich framework to study the quantum
dynamics of the Hořava gravity, in particular by incorporating the BRST
symmetry in terms of the canonical variables. In the past it has been used to
establish the unitarity of gauge theories, thanks to the ability of
introducing a bigger class of gauge-fixing conditions in the Hamiltonian
formalism [33, 36].

We have seen that the BFV version of the projectable (three-dimensional)
theory reproduces the quantum Lagrangian presented in Ref. [2], which was
obtained by fixing the gauge following the Faddeev-Popov procedure. Our
results reinforces the consistency of the quantization of the theory. We have
performed the integration on momenta after specifying the dependence that the
gauge-fixing condition has on them. Specifically, we have introduced a linear
dependence on the momentum conjugated to the shift vector. Guided by a
criterium of anisotropic scaling, we have incorporated an operator that
balances the momentum in the gauge-fixing condition. It turns out that, in
both versions of the Hořava theory, this operator introduces a nonlocality in
the Lagrangian after the integration. Thus, we have arrived at the same result
obtained in [2] of having a nonlocal quantum Lagrangian, in our case starting
from a self-consistent Hamiltonian formulation provided by the BFV formalism.
The original Hamiltonian theory is completely local. In Ref. [2] it was
pointed out that the final nonlocality of the quantum Lagrangian, restricted
to the kinetic term of the shift vector, can be eliminated by introducing the
conjugated momentum of the shift vector. We have corroborated this in an
inverse way, starting from the complete, self-consistent and local Hamiltonian
formulation and ending with the nonlocal Lagrangian. With the aim of having a
further comparison, we have performed the same procedure in general
relativity, taking into account the relativistic isotropy of its field
variables. In this case the operator introduced in the gauge fixing-condition
is nonlocal and the quantum Lagrangian resulting after the integration is
local (whenever the dependence of the gauge-fixing condition on the rest of
variables is local). Thus, we see an interesting relationship between the
anisotropy of the underlying symmetry and the nonlocality of the quantum
Lagrangian. The relationship has been established on very basic grounds, since
it comes from the integration of the Hamiltonian theory.

## Acknowledgments

C. B. is partially supported by the grant CONICYT PFCHA/DOCTORADO BECAS
CHILE/2019 – 21190960. C. B. is a graduate student in the “Doctorado en Física
Mención Física-Matemática” PhD program at the Universidad de Antofagasta. B.
D. is partially supported by the Universidad de Antofagasta grant PROYECTO
ANT1756, Chile.

##  Appendix A Nonperturbative integration on $\pi^{ij}$

All the dependence that the action given in (2.42) has on $\pi^{ij}$ is
contained in the three terms

$\pi^{ij}\dot{g}_{ij}-\mathcal{H}-N^{k}\mathcal{H}_{k}=-\frac{1}{\sqrt{g}}\mathcal{G}_{ijkl}\pi^{ij}\pi^{kl}+2K_{kl}\pi^{kl}-\sqrt{g}\mathcal{V}\,,$
(A.1)

where

$\displaystyle\mathcal{G}_{ijkl}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(g_{ik}g_{jl}+g_{il}g_{jk})-\frac{\lambda}{1-d\lambda}g_{ij}g_{kl}\,,$
(A.2) $\displaystyle{K}_{kl}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\dot{g}_{kl}-2\nabla_{(k}N_{l)}\right)\,.$
(A.3)

After completing the square involving $\pi^{ij}$ and making the Gaussian
integration, from the terms in (A.1) there results

$\sqrt{g}G^{ijkl}K_{ij}K_{kl}-\sqrt{g}\mathcal{V}=\mathcal{L}_{\mbox{\tiny
cl}}\,,$ (A.4)

where ${G}^{ijkl}=\frac{1}{2}(g^{ik}g^{jl}+g^{il}g^{jk})-\lambda
g^{ij}g^{kl}$, such that it is the inverse of $\mathcal{G}_{ijkl}$ [1], and
$\mathcal{L}_{\mbox{\tiny cl}}$ coincides with the classical Lagrangian (2.6).
A factor of $(\det\mathcal{G})^{-1/2}$ arises in the measure after the
integration. Therefore, the path integral takes the form

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}N^{k}\mathcal{D}\bar{C}_{i}\mathcal{D}C^{i}(\det\mathcal{G})^{-1/2}\exp\left[i\int
dtd^{d}x\Big{(}\mathcal{L}_{\mbox{\tiny
cl}}+\frac{1}{4}(\dot{N}^{k}-\Gamma^{k})\mathfrak{D}^{-1}_{kl}(\dot{N}^{l}-\Gamma^{l})\right.\\\
&\left.+\left(\dot{C}^{k}-N^{j}\partial_{j}C^{k}-N^{k}\partial_{j}C^{j}\right)\left(\dot{\bar{C}}_{k}+\bar{C}_{i}\frac{\delta\Gamma^{i}}{\delta
N^{k}}\right)-\bar{C}_{i}\\{\Gamma^{i}\,,\mathcal{H}_{k}\\}C^{k}\Big{)}\right]\,.\end{split}$
(A.5)

##  Appendix B Propagators of the nonprojectable theory

By making Fourier transforms in time and space in the Lagrangian (3.35), and
inverting the resulting operator between the squared fields, we obtain the
propagators of the nonprojectable theory,

$\displaystyle\langle h_{ij}^{TT}h_{ij}^{TT}\rangle=P_{1}\,,\qquad\langle
h^{T}h^{T}\rangle=\frac{2(1-\lambda)}{(1-3\lambda)}P_{2}\,,\qquad k^{2}\langle
h_{i}^{T}h_{i}^{T}\rangle=2P_{3}\,,$ $\displaystyle\langle
h^{L}h^{L}\rangle=\frac{1}{1-\lambda}P_{4}+\frac{2\lambda^{2}}{(1-\lambda)(1-3\lambda)}P_{2}\,,\qquad\langle
h^{T}h^{L}\rangle=\frac{4\lambda}{(1-3\lambda)}P_{2}\,,$ $\displaystyle\langle
nn\rangle=\frac{\alpha_{3}^{2}(1-\lambda)}{2\alpha_{4}(1-3\lambda)}P_{2}\,,\qquad\langle
nh^{T}\rangle=-\frac{2\alpha_{3}(1-\lambda)}{\alpha_{4}(1-3\lambda)}P_{2}\,,$
$\displaystyle\langle
nh^{L}\rangle=-\frac{2\alpha_{3}\lambda}{\alpha_{4}(1-3\lambda)}P_{2}\,,\qquad\langle
n^{iT}n^{iT}\rangle=k^{4}P_{3}\,,\qquad k^{2}\langle
n^{L}n^{L}\rangle=(1+\kappa)k^{4}P_{4}\,,$
$\displaystyle\langle\bar{C}_{k}^{T}C^{kT}\rangle=P_{3}\,,\qquad
k^{2}\langle\bar{C}^{L}C^{L}\rangle=P_{4}\,,\qquad\langle
n\xi\rangle=-2P_{5}\,,$
$\displaystyle\langle\xi\xi\rangle=-P_{5}\,,\qquad\langle\bar{\varepsilon}\varepsilon\rangle=-2P_{5}\,,$

where

$\displaystyle P_{1}=\frac{4}{\omega^{2}+\beta_{3}k^{6}}\,,$ $\displaystyle
P_{2}=4\left[\omega^{2}+\frac{(1-\lambda)}{\alpha_{4}(1-3\lambda)}\left(\alpha_{4}\left(3\beta_{3}+8\beta_{4}\right)-2\alpha_{3}^{2}\right)k^{6}\right]^{-1}\,,$
$\displaystyle P_{3}=\frac{4}{\omega^{2}+2k^{6}}\,,\qquad
P_{4}=\frac{4}{\omega^{2}+4(1-\lambda)(1+\kappa)k^{6}}\,,$ $\displaystyle
P_{5}=\frac{1}{\alpha_{4}k^{6}}\,.$

In (LABEL:propagtaors) we have written the propagators of $h_{i}^{T}$,
$n^{L}$, and $\bar{C}^{L},C^{L}$ multiplied by $k$ factors since these
variables must be compensated with spatial derivatives in the composition of
the original tensor variables. We adopt the definition of regular propagators
given in Ref. [2], which is based on the analysis of renormalizability of
Lorentz-violating gauge theories of Refs. [22, 23, 24]. $P_{1},P_{2},P_{3}$,
and $P_{4}$ are regular, but $P_{5}$ is not regular.555Actually, to hold the
regularity of $P_{1},P_{2},P_{3}$, and $P_{4}$ several conditions on the
constants must be imposed, since the coefficients of $\omega$ and $k^{6}$ must
be positive (after a Wick rotation) and unwanted ghosts should be avoided. The
number of independent constants is enough to have nonempty sets where these
conditions holds. Thus, the nonregularity is located on the variables
$\xi,\bar{\varepsilon},\varepsilon$ associated to the second-class
constraints.

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