BRST symmetry and unitarity of the Hořava theory

Jorge Bellorín1,a, Claudio Bórquez1,b and Byron Droguett2,c

1Department of Physics, Universidad de Antofagasta, 1240000 Antofagasta,
Chile.

2Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad Santo
Tomás, Sede Arica 1000000, Chile.

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

> We present an analysis on the BRST symmetry transformations of the Hořava
> theory under the BFV quantization, both in the nonprojectable and
> projectable cases. We obtain that the BRST transformations are intimately
> related to a particular spatial diffeomorphism along one of the ghost vector
> fields. We show explicitly the invariance of the quantum action and the
> nilpotence of the BRST transformations, using this diffeomorphism largely.
> The BRST symmetry is verified in the whole phase space, inside and outside
> the constrained surface. When restricted to the constrained surface, the
> BRST transformations are completely local. The consistency of the BRST
> symmetry is a fundamental feature of a quantum field theory, specially for
> renormalization. The BFV quantization is independent of the chosen gauge-
> fixing condition. This allows us to study the unitarity of the quantum
> Hořava theory, covering the gauge required for renormalization. We prove
> that, if the solution for the lapse function exists, then the unitarity of
> the theory is established.

##  1 Introduction

There have been important advances in the quantization of the Hořava theory
[1], which is a proposal of a theory of quantum gravity aimed to solve the
problem of the (perturbative) nonrenormalizability of general relativity. The
theory introduces a foliation of spacelike hypersurfaces along a timelike
direction, such that the foliation has an absolute physical meaning. The group
of diffeomorphisms that preserve the given foliation is the gauge group of the
theory. In this way, spatial derivatives are increased without increasing the
order in time derivatives. This is the core for the renormalizability. There
are two different versions of the theory that are compatible with this gauge
symmetry: the nonprojectable and the projectable cases. Their definitions are
given in the next section. The nonprojectable version has field equations
that, at the large-distance limit, are closer to the Einstein equations than
the ones of the projectable case. From this point of view, the nonprojectable
version acquires a central relevance in the context of the Hořava theory.

A significant advance in the quantization is the proof of the complete
renormalization of the projectable case presented in Ref. [2]; see also Ref.
[3]. The projectable version is a theory with only first-class constraints,
which implies that, in terms of the Lagrangian formulation, it can be
quantized as the class of gauge theories as Yang-Mills theory or perturbative
general relativity. A remarkable feature of the renormalization showed in [2]
is the introduction of a nonlocal gauge-fixing condition; hence a nonlocal
quantum Lagrangian. On the other hand, standard renormalizability requires a
local Lagrangian. The reconciliation between these two aspects is achieved in
the Hamiltonian formulation: the canonical Lagrangian with the required gauge-
fixing condition is indeed local. This was first observed in Ref. [2]. In Ref.
[4] we performed the complete quantization of the projectable case in the
Hamiltonian formalism using the Batalin-Fradkin-Vilkovisky (BFV) scheme of
quantization [5, 6, 7]. The starting point to motivate this analysis is that
the gauge-fixing condition required for renormalization is a noncanonical
condition. The BFV quantization is suitable for such a gauge condition since
it was precisely designed for relativistic gauges that are not canonical, like
the Lorentz gauge for Yang-Mills theory, when the problem of the unitarity of
those theories is focused. In [4], we obtained the local Hamiltonian, and the
integration on momenta led to the same quantum Lagrangian of Ref. [2] with the
nonlocal gauge-fixing condition. A further advance in the quantization of the
projectable theory is the proof that the $2+1$ theory is asymptotically free
[8], among other developments that can be found in the literature.

The quantization of the nonprojectable case presents a fundamental difference
with respect to the projectable case: the presence of second-class
constraints. Indeed, the proof of renormalization of the nonprojectable case
is still pending. We have studied the quantization of the nonprojectable
version using also the BFV formalism [4, 9, 10]. In this scheme the same
noncanonical gauge-fixing condition of the projectable case [2], together with
the second-class constraints can be incorporated. The main point is that the
second-class constraints affect the measure of the path integral [11]. The
analysis of Ref. [12] shows that the measure of the second-class constraints
changes the behavior of some propagators, pointing to regular propagators, as
required for renormalization [2]. We obtained a further advance using the BFV
formalism: in Ref. [10] we found that, although the quantum measure leads
still to nonregular propagators for some fields, the divergences of loops
formed with these fields cancel between them completely. The rest are loops
formed with regular propagators. These results are encouraging to continue on
the program of the quantization of the nonprojectable case under the BFV
formalism.

A fundamental aspect of the quantization of a field theory with gauge symmetry
is the Becchi-Rouet-Stora-Tyutin (BRST) symmetry, specially to get the final
renormalization. The BFV quantization is intimately linked to this class of
symmetry [6]. The Hamiltonian is presented as a gauge-fixed Hamiltonian, where
the gauge fixing is done by operating with the BRST charge. The main theorem
developed in the original papers of the BFV formalism ensures that the path
integral is independent of the chosen gauge-fixing function. The BRST charge
is formally defined using the first-class constraints and their gauge algebra
for a given theory. Its action on the fields is defined by a canonical
transformation: with Poisson brackets for the case of a theory with first-
class constraints only, and with Dirac brackets when second-class constraints
are present. Thus, in the case of the nonprojectable theory the whole
procedure applies using Dirac brackets.

Our first aim in this paper is to develop a detailed study of the BRST
symmetry in both versions of the Hořava theory. We obtain explicit expressions
for these transformations and study how the invariance of the action is
achieved. We show this explicitly, beyond the formal definitions of the
general BFV quantization. Since we pursue to make a general study, we develop
the BRST transformations on nonperturbative variables and in the whole phase
space, including the subset where the second-class constraints are not
satisfied (the exception is the transformation of the gauge-fixing terms,
which are defined in terms of perturbative variables). We find the interesting
result that most of the BRST transformations can be modeled in terms of a
spatial diffeomorphism along one of the ghost vector fields. We take this
result as the basis to operate the BRST symmetry on the quantum action.
Moreover, we show explicitly the nilpotence of the BRST transformations, where
the diffeomorphism along the ghost field plays a prominent role. For a
possible application to renormalization, it is important to elucidate the
local or nonlocal character of the BRST transformations. We discuss how this
is related to the second-class constraints.

Our second objective is to take advantage of the well-posed BFV quantization
to study the unitarity of the Hořava theory, which is a fundamental feature of
the $S$ matrix. We remark that the BFV formalism was developed precisely to
show the unitarity of relativistic theories (Yang-Mills theory, general
relativity) when relativistic gauges are used. The unitarity is achieved by
making use of the capacity of the formalism to change the gauge condition. In
this paper we show that, once the BFV quantization has been completed under an
appropriate gauge condition, and assuming that the solution of a second-class
constraint exists, the integration on all unphysical variables leads to a
canonical path integral with the correct degrees of freedom on the integration
and with measure 1. We give evidence for the existence of the solution of the
second-class constraint under certain limits.

##  2 BFV quantization of the nonprojectable case

Hořava’s gravitational theory is formulated using the Arnowitt-Deser-Misner
(ADM) variables of the given foliation, $N(t,\vec{x})$, $N^{i}(t,\vec{x})$,
and $g_{ij}(t,\vec{x})$. The foliation-preserving diffeomorphisms are defined
by $\delta t=f(t)$, $\delta x^{i}=\zeta^{i}(t,\vec{x})$. Their action of the
ADM variables is

$\displaystyle\delta N$ $\displaystyle=$
$\displaystyle\zeta^{k}\partial_{k}N+f\dot{N}+\dot{f}N\,,$ (2.1)
$\displaystyle\delta N_{i}$ $\displaystyle=$
$\displaystyle\zeta^{k}\partial_{k}N_{i}+N_{k}\partial_{i}\zeta^{k}+\dot{\zeta}^{j}g_{ij}+f\dot{N}_{i}+\dot{f}N_{i}\,,$
(2.2) $\displaystyle\delta g_{ij}$ $\displaystyle=$
$\displaystyle\zeta^{k}\partial_{k}g_{ij}+2g_{k(i}\partial_{j)}\zeta^{k}+f\dot{g}_{ij}\,.$
(2.3)

Among these transformations, only the spatial diffeomorphisms ($f=0$,
$\zeta^{i}(t,\vec{x})$ arbitrary) constitute a gauge symmetry in the strict
sense. The nonprojectable case is defined by the condition of the lapse
function $N(t,\vec{x})$ that can be a function of time and the space, whereas
in the projectable case it is restricted to be a function only of time (and,
consequently, it is an spurious degree of freedom). The nonprojectable case
exhibits a dynamics closer to the one of general relativity. In this section,
we study the nonprojectable case.

Given the foliation, the action of the nonprojectable theory is [1, 13]

$S=\int dtd^{2}x\sqrt{g}N\left(K_{ij}K^{ij}-\lambda
K^{2}-\mathcal{V}\right)\,.$ (2.4)

The kinetic terms of the Lagrangian are defined in terms of the extrinsic
curvature tensor,

$K_{ij}=\frac{1}{2N}\left(\dot{g}_{ij}-2\nabla_{(i}N_{j)}\right)\,,\qquad
K=g^{ij}K_{ij}\,.$ (2.5)

$\mathcal{V}$ is called the potential. It contains the higher-order spatial
derivatives, whose order is identified with the parameter $z$, such that the
higher-order terms are of $2z$ order in spatial derivatives. In this paper it
is sufficient to consider the $(2+1)$-dimensional theory, since the $3+1$ case
is conceptually equal. The $2+1$ theory requires a potential of $z=2$ order
for power-counting renormalizability. The complete potential is [14]

$\displaystyle\mathcal{V}$ $\displaystyle=$ $\displaystyle-\beta R-\alpha
a^{2}+\alpha_{1}R^{2}+\alpha_{2}a^{4}+\alpha_{3}Ra^{2}+\alpha_{4}a^{2}\nabla_{i}a^{i}$
(2.6)
$\displaystyle+\alpha_{5}R\nabla_{i}a^{i}+\alpha_{6}\nabla^{i}a^{j}\nabla_{i}a_{j}+\alpha_{7}(\nabla_{i}a^{i})^{2}\,,$

where

$a_{i}=\frac{\partial_{i}N}{N}\,,\quad a^{2}\equiv a_{k}a^{k}\,.$ (2.7)

$\nabla_{i}$ and $R$ are the covariant derivative and the Ricci scalar of the
spatial metric $g_{ij}$. The coupling constants of the theory are $\lambda$,
$\beta$, $\alpha$, $\alpha_{1}$,…,$\alpha_{7}$. We use shorthand for frequent
combinations of these coupling constants,

$\sigma=\frac{1-\lambda}{1-2\lambda}\,,\quad\bar{\sigma}=\frac{\lambda}{1-2\lambda}\,,\quad\alpha_{67}=\alpha_{6}+\alpha_{7}\,.$
(2.8)

In the Hamiltonian formulation [15, 16, 17] the canonically conjugate pairs
are $(g_{ij},\pi^{ij})$ and $(N,P_{N})$. After the Legendre transformation,
the primary classical Hamiltonian results in

$H_{0}=\int
d^{2}x\mathcal{H}_{0}\,,\quad\mathcal{H}_{0}\equiv\sqrt{g}N\left(\frac{\pi^{ij}\pi_{ij}}{g}+\bar{\sigma}\frac{\pi^{2}}{g}+\mathcal{V}\right)\,.$
(2.9)

The constraint that corresponds to the involutive functions under Dirac
brackets in the BFV quantization is

$\mathcal{H}_{i}\equiv-2g_{ij}\nabla_{k}\pi^{kj}=0\,.$ (2.10)

As it is well known from general relativity, this is the generator of the
spatial diffeomorphisms on the canonical pair $(g_{ij},\pi^{ij})$. The second-
class constraints are

$\displaystyle\theta_{1}$ $\displaystyle\equiv$
$\displaystyle\mathcal{H}_{0}+\sqrt{g}\Big{(}2\alpha\nabla_{i}(Na^{i})-4\alpha_{2}\nabla_{i}(Na^{2}a^{i})-2\alpha_{3}\nabla_{i}(NRa^{i})+2\alpha_{6}\nabla^{i}\nabla^{j}(N\nabla_{j}a_{i})$
$\displaystyle+\alpha_{4}\left(\nabla^{2}(Na^{2})-2\nabla_{i}(Na^{i}\nabla_{j}a^{j})\right)+\alpha_{5}\nabla^{2}(NR)+2\alpha_{7}\nabla^{2}(N\nabla_{i}a^{i})\Big{)}=0\,,$
$\displaystyle\theta_{2}$ $\displaystyle\equiv$ $\displaystyle P_{N}=0\,.$
(2.12)

An important feature of the nonprojectable theory is that the primary
Hamiltonian (2.9) can be written as the integral of one of the second-class
constraints [18],

$H_{0}=\int d^{2}x\,\theta_{1}\,.$ (2.13)

This holds because the terms in $\theta_{1}$ that are not contained in
$\mathcal{H}_{0}$ are total derivatives that vanish after spatial
integration.111In $d=3$ spatial dimensions, there are nonzero boundary terms
in (2.13). They ensure the differentiability of the functional, similar to the
Hamiltonian formulation of general relativity (see Refs. [16, 17]). The phase
space of the classical theory is spanned by $\\{g_{ij},\pi^{ij},N,P_{N}\\}$.
These are eight functional degrees of freedom. The theory has the constraints
$\mathcal{H}_{i}$, $\theta_{1}$, and $\theta_{2}$, which sum up four
conditions. The gauge symmetry of the spatial diffeomorphism eliminate two
more degrees, leaving a physical phase space of dimension two. This is the
physical phase space of a scalar mode.222In three spatial dimensions the field
variables $\\{g_{ij},\pi^{ij},N,P_{N}\\}$ sum $14$, the constraints
$\mathcal{H}_{i}$, $\theta_{1}$, and $\theta_{2}$ give five and the spatial
diffeomorphisms three. The result is a physical phase space of dimension six:
two tensorial modes and one extra scalar mode.

We present a summary of the BFV quantization of the nonprojectable Hořava
theory using a specific gauge-fixing condition. Further details about the
procedure can be found in Refs. [4, 9]. The phase space is extended by adding
the canonical pairs $(N^{i},\pi_{i})$, $(C^{i},\bar{\mathcal{P}}_{i})$, and
$(\bar{C}_{i},\mathcal{P}^{i})$ to it. The last two pairs are the BFV ghosts.
The definition of the BFV path integral is

$Z=\int\mathcal{D}Ve^{iS},$ (2.14)

where the measure and the action are given, respectively, by

$\displaystyle\mathcal{D}V=\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N\mathcal{D}P_{N}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}C^{i}\mathcal{D}\bar{\mathcal{P}}_{i}\mathcal{D}\bar{C}_{i}\mathcal{D}\mathcal{P}^{i}\times\delta(\theta_{1})\delta(\theta_{2})\sqrt{\det\\{\theta_{p},\theta_{q}\\}}\,,\;\;\;$
(2.15) $\displaystyle S=\int
dtd^{2}x\left(\pi^{ij}\dot{g_{ij}}+P_{N}\dot{N}+\pi_{i}\dot{N}^{i}+\bar{\mathcal{P}}_{i}\dot{C}^{i}+\mathcal{P}^{i}\dot{\bar{C}}_{i}-\mathcal{H}_{\Psi}\right)\,.$
(2.16)

The quantum gauge-fixed Hamiltonian density is defined by

$\mathcal{H}_{\Psi}=\mathcal{H}_{0}+\\{\Psi,\Omega\\}_{\text{D}}\,,$ (2.17)

where $\Omega$ is the generator of the BRST symmetry, $\Psi$ is a gauge-fixing
fermionic function and $\\{\,,\\}_{\text{D}}$ indicates Dirac brackets.
According to the general formalism, to obtain $\Omega$ the algebra of spatial
diffeomorphisms,

$\\{\mathcal{H}_{i}\,,\mathcal{H}_{j}\\}=U_{ij}^{k}\mathcal{H}_{k}\,,$ (2.18)

is required. By using the explicit form of $U^{k}_{ij}$, the BRST charge
becomes333In Refs. [4, 9] we presented the BFV quantization using a simplified
notation: integrals needed for the definition of some functionals like
$\Omega$ were omitted.

$\Omega=\int
d^{2}x\left(\mathcal{H}_{i}C^{i}+\pi_{i}\mathcal{P}^{i}-C^{i}\partial_{i}C^{j}\bar{\mathcal{P}}_{j}\right)\,.$
(2.19)

The delta $\delta(\theta_{2})$ in the measure forces $P_{N}$ to be a spurious
variable after the integration. Nevertheless, it is interesting to keep
$P_{N}$ active in the formalism, specially to study the BRST transformations
in the whole phase space. We promote the two deltas $\delta(\theta_{1})$ and
$\delta(\theta_{2})$ to the quantum canonical Lagrangian by means of the
integration on the Lagrange multipliers $\mathcal{A}$ and $\mathcal{B}$,
respectively. Moreover, the particular algebra of constraints yields the
important simplification

$\sqrt{\det\\{\theta_{p},\theta_{q}\\}}=\det\\{\theta_{1},\theta_{2}\\}\,.$
(2.20)

This determinant can also be incorporated to the canonical Lagrangian by means
of a pair of fermionic ghosts, which we denote by $\eta,\bar{\eta}$. Thus, the
measure of the second-class constraints becomes

$\delta(\theta_{1})\delta(\theta_{2})\sqrt{\det\\{\theta_{p},\theta_{q}\\}}=\int\mathcal{D}\mathcal{A}\mathcal{D}\mathcal{B}\mathcal{D}\bar{\eta}\mathcal{D}\eta\exp\left[i\int
dtd^{d}x\left(\mathcal{A}\theta_{1}+\mathcal{B}\theta_{2}+\bar{\eta}\\{\theta_{1},\theta_{2}\\}\eta\right)\right]\,.$
(2.21)

The main theorem of the BFV formalism guarantees that the path integral (2.14)
is independent of the fermionic function $\Psi$. A form of the gauge-fixing
condition, and hence of $\Psi$, that has been used in the quantization of the
Hořava theory [4, 9, 10] is taken from the relativistic theories

$\displaystyle\dot{N}^{i}-\chi^{i}=0\,,$ (2.22)
$\displaystyle\Psi=\bar{\mathcal{P}}_{i}N^{i}+\bar{C}_{i}\chi^{i}\,,$ (2.23)

where $\chi^{i}$ is a free functional part of the gauge condition that must be
specified to complete it. With the form of $\Omega$ given in (2.19) and $\Psi$
in (2.23), and assuming that $\chi^{i}$ does not depend on the BFV ghosts, we
obtain that the gauge-fixed Hamiltonian (2.17) takes the generic form

$\begin{split}\mathcal{H}_{\Psi}=\,&\mathcal{H}_{0}+\mathcal{H}_{i}N^{i}+\bar{\mathcal{P}}_{i}\mathcal{P}^{i}-\bar{\mathcal{P}}_{i}\left(N^{j}\partial_{j}C^{i}+N^{i}\partial_{j}C^{j}\right)+\pi_{i}\chi^{i}\\\
&+\bar{C}_{i}\\{\chi^{i}\,,\mathcal{H}_{j}\\}_{\mathrm{D}}C^{j}+\bar{C}_{i}\dfrac{\delta\chi^{i}}{\delta
N^{j}}\mathcal{P}^{j}\,.\end{split}$ (2.24)

We may take the $\chi^{i}$ factor that connects to the renormalizable
projectable theory [2, 4]. The perturbative variables are defined by

$g_{ij}=\delta_{ij}+h_{ij}\,,\quad\pi^{ij}=p^{ij}\,,\quad N=1+n\,,\quad
N^{i}=n^{i}\,.$ (2.25)

On the rest of quantum fields, we keep the original notation, considering them
of perturbative order. We denote the trace of $h_{ij}$ by $h=h_{kk}$. In $2+1$
dimensions, the required gauge-fixing condition is

$\chi^{i}=\mathfrak{D}^{ij}\pi_{j}-2\Delta\partial_{j}h_{ij}+2\rho\Delta\partial_{i}h-2\kappa\partial_{ijk}h_{jk}\,,$
(2.26)

where

$\mathfrak{D}^{ij}=\delta_{ij}\Delta+\kappa\partial_{ij}\,,\quad\Delta\equiv\partial_{kk}$
(2.27)

$\rho=\lambda(1+\kappa)$, $\tilde{\rho}=1-2\lambda+2\kappa(1-\lambda)$ for
future use, and $\kappa$ is an arbitrary constant. We use the shorthand
$\partial_{ij\cdots k}=\partial_{i}\partial_{j}\cdots\partial_{k}$. The
coefficients in $\chi^{i}$ has been chosen to simplify the resulting
propagators of the quantum fields. Under the gauge-fixing condition (2.26),
the path integral takes the form

$\begin{split}Z=&\int\mathcal{D}h_{ij}\mathcal{D}p^{ij}\mathcal{D}n\mathcal{D}P_{N}\mathcal{D}n^{i}\mathcal{D}\pi_{i}\mathcal{D}C^{i}\mathcal{D}\bar{\mathcal{P}}_{i}\mathcal{D}\bar{C}_{i}\mathcal{D}\mathcal{P}^{i}\mathcal{D}\mathcal{A}\mathcal{D}\mathcal{B}\mathcal{D}\bar{\eta}\mathcal{D}\eta\\\
&\exp\left[i\int
dtd^{2}x\Big{(}p^{ij}\dot{h}_{ij}+P_{N}\dot{n}+\pi_{i}\dot{n}^{i}+\bar{\mathcal{P}}_{i}\dot{C}^{i}+\mathcal{P}^{i}\dot{\bar{C}}_{i}-\mathcal{H}_{0}-\mathcal{H}_{i}n^{i}-\bar{\mathcal{P}}_{i}\mathcal{P}^{i}\right.\\\
&+\bar{\mathcal{P}}_{i}(n^{j}\partial_{j}C^{i}+n^{i}\partial_{j}C^{j})-\pi_{i}\mathfrak{D}^{ij}\pi_{j}+2\pi_{i}(\Delta\partial_{j}h_{ij}-\rho\Delta\partial_{i}h+\kappa\partial_{ijk}h_{jk})\\\
&\left.+2\bar{C}^{i}\left(\delta_{ij}\Delta^{2}+\tilde{\rho}\Delta\partial_{ij}\right)C_{j}+\mathcal{A}\theta_{1}+\mathcal{B}\theta_{2}+\bar{\eta}\\{\theta_{1}\,,\theta_{2}\\}\eta\Big{)}\right]\,.\end{split}$
(2.28)

In this action, the objects $\mathcal{H}_{0}$, $\mathcal{H}_{i}$,
$\theta_{1}$, and $\\{\theta_{1}\,,\theta_{2}\\}$ must be expanded at the
required order in perturbations.

##  3 BRST symmetry

For a theory with second-class constraints the BRST transformations are
defined by means of a canonical transformation with Dirac brackets,

$\delta_{\Omega}\varphi=\\{\varphi\,,\Omega\\}_{\text{D}}\epsilon\,,$ (3.1)

where $\varphi$ stands for a generic object that depends on the canonical
variables and $\epsilon$ is the fermionic parameter of the BRST
transformation. The BRST charge of the Hořava theory is given in Eq. (2.19).
On the canonical fields $g_{ij}$ and $\pi^{ij}$, the result of the BRST
transformation is a diffeomorphism along the vector $C^{i}\epsilon$,444In Ref.
[2], using perturbative variables under the Lagrangian formalism of the
projectable theory, the BRST transformation on $h_{ij}$ was identified as a
diffeomorphism along $C^{i}$.

$\displaystyle\delta_{\Omega}g_{ij}=\partial_{k}g_{ij}C^{k}\epsilon+2g_{k(i}\partial_{j)}C^{k}\epsilon\equiv\delta^{\text{diff}}_{C\epsilon}g_{ij}\,,$
(3.2)
$\displaystyle\delta_{\Omega}\pi^{ij}=\partial_{k}\pi^{ij}C^{k}\epsilon-2\pi^{k(i}\partial_{k}C^{j)}\epsilon+\pi^{ij}\partial_{k}C^{k}\epsilon\equiv\delta^{\text{diff}}_{C\epsilon}\pi^{ij}\,.$
(3.3)

Interestingly, the BRST transformations of other fields also involve the
diffeomorphism along $C^{i}\epsilon$. This is the case of $N$. Let us see how
this works in detail. As a consequence of dealing with Dirac brackets, there
arises the factor

$\\{\theta_{1}\,,\theta_{2}\\}=\frac{\delta\theta_{1}}{\delta N}\,.$ (3.4)

The direct computation from (3.1) yields

$\displaystyle\delta_{\Omega}N=-W\\{\theta_{1}\,,\mathcal{H}_{i}C^{i}\\}\epsilon\,,$
(3.5) $\displaystyle W\equiv\left(\frac{\delta\theta_{1}}{\delta
N}\right)^{-1}\,.$ (3.6)

We may write this transformation in a more illustrative form. $\theta_{1}$ is
a spatial density that depends exclusively on the canonical fields $g_{ij}$,
$\pi^{ij}$, and $N$. $\mathcal{H}_{i}$, defined in (2.10), is the generator of
the spatial diffeomorphisms on the canonical pair $(g_{ij},\pi^{ij})$. The
complete generator that also acts on the pair $(N,P_{N})$ is

$\mathcal{H}_{i}+\theta_{2}\partial_{i}N\,.$ (3.7)

Thus, in (3.5) we may recover the spatial diffeomorphism on $\theta_{1}$ if we
complete the generator (3.7), compensating with an extra term,

$\delta_{\Omega}N=-W\delta^{\text{diff}}_{C\epsilon}\theta_{1}+W\\{\theta_{1}\,,\theta_{2}\partial_{i}NC^{i}\\}\epsilon\,.$
(3.8)

Due to the functional dependence of $\theta_{1}$, the last bracket in this
equation is simplified:
$\\{\theta_{1}\,,\theta_{2}\partial_{i}NC^{i}\\}\epsilon=\\{\theta_{1}\,,\theta_{2}\\}\partial_{i}NC^{i}\epsilon$.
Hence, Eq. (3.8) becomes

$\delta_{\Omega}N=-W\delta^{\text{diff}}_{C\epsilon}\theta_{1}+\partial_{i}NC^{i}\epsilon=-W\delta^{\text{diff}}_{C\epsilon}\theta_{1}+\delta^{\text{diff}}_{C\epsilon}N\,.$
(3.9)

Another interesting case is the transformation of the ghost field $C^{i}$
itself. Its BRST transformation can be interpreted as a spatial diffeomorphism
on it along the same vector $C^{i}\epsilon$,

$\delta_{\Omega}C^{i}=\partial_{j}C^{i}C^{j}\epsilon\equiv\frac{1}{2}\delta^{\text{diff}}_{C\epsilon}C^{i}\,.$
(3.10)

This diffeomorphism of $C^{i}$ along itself is not zero due the Grassmann
nature of $C^{i}$ and $\epsilon$. The last field whose BRST transformation
involves a diffeomorphism is $\bar{\mathcal{P}}_{i}$,

$\delta_{\Omega}\bar{\mathcal{P}}_{i}=\left(\partial_{j}\bar{\mathcal{P}}_{i}C^{j}+\bar{\mathcal{P}}_{j}\partial_{i}C^{j}+\bar{\mathcal{P}}_{i}\partial_{j}C^{j}\right)\epsilon+\mathcal{H}_{i}\epsilon=\delta^{\text{diff}}_{C\epsilon}\bar{\mathcal{P}}_{i}+\mathcal{H}_{i}\epsilon\,.$
(3.11)

The rest of transformations on the other fields are computed straightforwardly
from (3.1). In summary, the BRST transformations of the quantized theory are
given by

$\begin{array}[]{ll}\delta_{\Omega}g_{ij}=\delta^{\text{diff}}_{C\epsilon}g_{ij}\,,&\delta_{\Omega}\pi^{ij}=\delta^{\text{diff}}_{C\epsilon}\pi^{ij}\,,\\\\[4.30554pt]
{\displaystyle\delta_{\Omega}N=\delta^{\text{diff}}_{C\epsilon}N-W\delta^{\text{diff}}_{C\epsilon}\theta_{1}}\,,&\delta_{\Omega}P_{N}=0\,,\\\\[8.61108pt]
\delta_{\Omega}N^{i}=\mathcal{P}^{i}\epsilon\,,&\delta_{\Omega}\pi_{i}=0\,,\\\\[4.30554pt]
{\displaystyle\delta_{\Omega}C^{i}=\frac{1}{2}\delta^{\text{diff}}_{C\epsilon}C^{i}}\,,&\delta_{\Omega}\bar{\mathcal{P}}_{i}=\delta^{\text{diff}}_{C\epsilon}\bar{\mathcal{P}}_{i}+\mathcal{H}_{i}\epsilon\,,\\\\[4.30554pt]
\delta_{\Omega}\mathcal{P}^{i}=0\,,&\delta_{\Omega}\bar{C}_{i}=\pi_{i}\epsilon\,.\end{array}$
(3.12)

Now we study the explicit operation of these transformations on the quantum
action. To manage this, it is convenient to separate the quantum action in
four sectors: the kinetic terms, the primary Hamiltonian $\mathcal{H}_{0}$,
the gauge-fixing terms that come from $\\{\Psi\,,\Omega\\}_{\text{D}}$, and
the measure of the second-class constraints (2.21). We start with the
transformation of the primary Hamiltonian $\mathcal{H}_{0}$. One may use the
fact that its integral is equivalent to the integral of the second-class
constraint $\theta_{1}$, Eq. (2.13). By doing the functional calculus
explicitly, we obtain

$\begin{split}\delta_{\Omega}\int d^{2}x\mathcal{H}_{0}\,=\,&\int
d^{2}x\left(\frac{\delta\theta_{1}}{\delta
g_{ij}}\delta_{\Omega}g_{ij}+\frac{\delta\theta_{1}}{\delta\pi^{ij}}\delta_{\Omega}\pi^{ij}+\frac{\delta\theta_{1}}{\delta
N}\delta_{\Omega}N\right)\\\ =\,&\int
d^{2}x\left(\frac{\delta\theta_{1}}{\delta
g_{ij}}\delta^{\text{diff}}_{C\epsilon}g_{ij}+\frac{\delta\theta_{1}}{\delta\pi^{ij}}\delta^{\text{diff}}_{C\epsilon}\pi^{ij}+\frac{\delta\theta_{1}}{\delta
N}\delta^{\text{diff}}_{C\epsilon}N-\delta^{\text{diff}}_{C\epsilon}\theta_{1}\right)\\\
=\,&0\,,\end{split}$ (3.13)

where we have used that $\theta_{1}$ depends only on $g_{ij}$, $\pi^{ij}$, and
$N$. Note that the exact cancellation holds in the whole phase space,
including the region where the constraints are not zero. This procedure can be
generalized for any functional $\mathcal{F}$ that depends only on the
canonical variables $g_{ij}$, $\pi^{ij}$ and $N$,

$\delta_{\Omega}\mathcal{F}=\frac{\delta\mathcal{F}}{\delta
g_{ij}}\delta_{\Omega}g_{ij}+\frac{\delta\mathcal{F}}{\delta\pi^{ij}}\delta_{\Omega}\pi^{ij}+\frac{\delta\mathcal{F}}{\delta
N}\delta_{\Omega}N=\delta^{\text{diff}}_{C\epsilon}\mathcal{F}-\frac{\delta\mathcal{F}}{\delta
N}W\delta^{\text{diff}}_{C\epsilon}\theta_{1}\,.$ (3.14)

Next, we check the transformation of the kinetic terms. In the process we
discard total time derivatives since the action of the field theory has
initial and final boundaries at infinity. Thus, the transformation

$\delta_{\Omega}\left(\pi_{i}\dot{N}^{i}+\mathcal{P}^{i}\dot{\bar{C}}_{i}\right)$
(3.15)

vanishes since it forms a total time derivative. The transformation of the
other BFV ghosts can be grouped in two parts,

$\delta_{\Omega}\left(\bar{\mathcal{P}}_{i}\dot{C}^{i}\right)=\left(\delta^{\text{diff}}_{C\epsilon}\bar{\mathcal{P}}_{i}\dot{C}^{i}-\frac{1}{2}\dot{\bar{\mathcal{P}}}_{i}\delta^{\text{diff}}_{C\epsilon}C^{i}\right)-\mathcal{H}_{i}\dot{C}^{i}\epsilon\,.$
(3.16)

By using the explicit form of the diffeomorphisms, we obtain that the terms
between brackets cancel out exactly. Similarly, and using the form of
$\mathcal{H}_{i}$ explicitly, the last term combines with the transformation

$\delta_{\Omega}\left(\pi^{ij}\dot{g}_{ij}\right)=\delta^{\text{diff}}_{C\epsilon}\pi^{ij}\dot{g}_{ij}-\dot{\pi}^{ij}\delta^{\text{diff}}_{C\epsilon}g_{ij}$
(3.17)

to cancel completely between them. The last kinetic term to consider is

$\delta_{\Omega}\left(P_{N}\dot{N}\right)=\theta_{2}\partial_{t}\left(\delta_{\Omega}N\right)\,.$
(3.18)

This transformation does not vanish outside the constrained surface. In the
following we show that this nonzero transformation can be compensated by the
transformation of a Lagrange multiplier, hence the BRST invariance is kept in
the full phase space.

We study the sector of the quantum Lagrangian associated with the measure of
the second-class constraints, Eq. (2.21). $\mathcal{A}$ is the Lagrange
multiplier of $\theta_{1}$. This constraint is left invariant by the BRST
transformation since it is a second-class constraint. Consequently, we impose

$\delta_{\Omega}\mathcal{A}=0\,.$ (3.19)

$\mathcal{B}$ is also a multiplier of a second-class constraint, but there is
still the nonzero transformation (3.18) proportional to $\theta_{2}$. Hence we
require

$\delta_{\Omega}\mathcal{B}=-\partial_{t}(\delta_{\Omega}N)=-\partial_{t}\left(\delta^{\text{diff}}_{C\epsilon}N\right)+\partial_{t}\left(W\delta^{\text{diff}}_{C\epsilon}\theta_{1}\right)\,.$
(3.20)

Finally, the transformation of the ghosts $\eta,\bar{\eta}$ must be defined in
such a way that the integral of the last term in (2.21) remains invariant.
Notice that the bracket $\\{\theta_{1},\theta_{2}\\}$, given in (3.4), is a
functional of $g_{ij}$, $\pi^{ij}$ and $N$. Thus, by using formula (3.14), we
obtain

$\delta_{\Omega}\\{\theta_{1},\theta_{2}\\}=\delta^{\text{diff}}_{C\epsilon}\left(\frac{\delta\theta_{1}}{\delta
N}\right)-\frac{\delta^{2}\theta_{1}}{\delta N\delta
N}W\delta^{\text{diff}}_{C\epsilon}\theta_{1}\,.$ (3.21)

We may get the combination $\bar{\eta}\\{\theta_{1},\theta_{2}\\}\eta$ to
transform with a diffeomorphisms along $C^{i}\epsilon$, hence its integral is
invariant, if we define the transformation of the ghosts as follows:

$\displaystyle\delta_{\Omega}\eta$ $\displaystyle=$
$\displaystyle\delta^{\text{diff}}_{C\epsilon}\eta+\frac{1}{2}\frac{\delta^{2}\theta_{1}}{\delta
N\delta N}W^{2}\eta\,\delta^{\text{diff}}_{C\epsilon}\theta_{1}\,,$ (3.22)
$\displaystyle\delta_{\Omega}\bar{\eta}$ $\displaystyle=$
$\displaystyle\delta^{\text{diff}}_{C\epsilon}\bar{\eta}+\frac{1}{2}\frac{\delta^{2}\theta_{1}}{\delta
N\delta N}W^{2}\bar{\eta}\delta^{\text{diff}}_{C\epsilon}\theta_{1}\,.$ (3.23)

Finally, we study the BRST transformation of the gauge-fixing terms in (2.28).
Since the gauge-fixing condition is imposed on perturbative variables, we
study the linearized transformations. The linearized version of the BRST
transformations (3.12) of the fields involved in the gauge-fixing are

$\begin{array}[]{ll}\delta_{\Omega}h_{ij}=2\partial_{(i}C_{j)}\epsilon\,,&\delta_{\Omega}\pi^{ij}=0\,,\\\\[4.30554pt]
\delta_{\Omega}n^{i}=\mathcal{P}^{i}\epsilon\,,&\delta_{\Omega}\pi_{i}=0\,,\\\\[4.30554pt]
\delta_{\Omega}C^{i}=0\,,&\delta_{\Omega}\bar{\mathcal{P}}_{i}=\mathcal{H}_{i}\epsilon\,,\\\\[4.30554pt]
\delta_{\Omega}\mathcal{P}^{i}=0\,,&\delta_{\Omega}\bar{C}_{i}=\pi_{i}\epsilon\,.\end{array}$
(3.24)

In the quantum Lagrangian (2.28), the quadratic-order terms coming from the
gauge-fixing are

$-\mathcal{H}_{i}n^{i}-\bar{\mathcal{P}}_{i}\mathcal{P}^{i}-\pi_{i}\mathfrak{D}^{ij}\pi_{j}+2\pi_{i}(\Delta\partial_{j}h_{ij}-\rho\Delta\partial_{i}h+\kappa\partial_{ijk}h_{jk})+2\bar{C}^{i}\left(\delta_{ij}\Delta^{2}+\tilde{\rho}\Delta\partial_{ij}\right)C_{j}\,.$
(3.25)

Cancellations between these terms work as follows:

$\begin{split}&\delta_{\Omega}(\mathcal{H}_{i}n^{i})=-\delta_{\Omega}(\bar{\mathcal{P}}_{i}\mathcal{P}^{i})\,,\\\
&\delta_{\Omega}(\pi_{i}\mathfrak{D}^{ij}\pi_{j})=0\,,\\\
&\delta_{\Omega}\left[\pi_{i}(\Delta\partial_{j}h_{ij}-\rho\Delta\partial_{i}h+\kappa\partial_{ijk}h_{jk})\right]=-\delta_{\Omega}\left[\bar{C}^{i}\left(\delta_{ij}\Delta^{2}+\tilde{\rho}\Delta\partial_{ij}\right)C_{j}\right]\,.\end{split}$
(3.26)

A further question about the BRST transformations (3.12) is their locality.
This is affected by the presence of the second-class constraints (we recall
that in the BFV formalism only the second-class constraints are imposed
directly). Dirac brackets are intrinsically nonlocal operations when applied
to a field theory (unlike the case of a mechanical system). Therefore,
nonlocal effects on the BRST transformations are to be expected. Specifically,
nonlocal effects arise in the last term of $\delta_{\Omega}N$ in (3.12). We
remark that this is not an obstruction for the consistency of the BRST
transformations in the whole phase space, as we have proven. Moreover, by
evaluating the transformation of $N$ on the surface where the second-class
constraints are satisfied, hence $\theta_{1}=0$ and
$\delta^{\text{diff}}_{C\epsilon}\theta_{1}=0$, it becomes a pure
diffeomorphism on $N$,

$\delta_{\Omega}N=\delta^{\text{diff}}_{C\epsilon}N\,.$ (3.27)

This transformation is a local operation. The rest of the transformations in
(3.12) are local. Therefore, we find that the BRST symmetry transformations
are strictly local operations when the theory is restricted to the surface
where the second-class constraints are satisfied. The Hamiltonian, with the
gauge-fixing condition, is also local.

##  4 Nilpotence of the BRST transformations

It is illustrative to see explicitly how the nilpotence of the BRST
transformations is based largely on the diffeomorphism along $C^{i}\epsilon$.
We can study this on nonperturbative variables and on the whole phase space.

We introduce the BRST operator $s$, such that the BRST transformation of a
field $\phi$ is $\delta_{\Omega}\phi=\epsilon s\phi$. Let us start with the
case of the spatial metric,

$sg_{ij}=-\partial_{k}g_{ij}C^{k}-2g_{k(i}\partial_{j)}C^{k}\,.$ (4.1)

We compute the square action of the BRST transformation in the form
$\delta_{\Omega}(sg_{ij})$ [19]. By operating this transformation on $sg_{ij}$
and substituting (3.2) and (3.10) into it, we get that it cancels completely.
The square transformation $\delta_{\Omega}(s\pi^{ij})$ is a computation
parallel to this one. From the transformation of $N$ in (3.12) we obtain

$\displaystyle sN=-\partial_{i}NC^{i}+W\delta^{\text{diff}}_{C}\theta_{1}\,,$
(4.2)
$\displaystyle\delta^{\text{diff}}_{C}\theta_{1}\equiv\partial_{i}\theta_{1}C^{i}+\theta_{1}\partial_{i}C^{i}\,.$
(4.3)

Note that the diffeomorphism $\delta^{\text{diff}}_{C}\theta_{1}$ is of
Grassmann nature and does not involve the parameter $\epsilon$. When computing
the square $\delta_{\Omega}(sN)$, a required term is
$\delta_{\Omega}W\delta^{\text{diff}}_{C}\theta_{1}$. $W$, given in (3.6),
depends only on $g_{ij}$, $\pi^{ij}$ and $N$; hence formula (3.14) applies:

$\delta_{\Omega}W\delta^{\text{diff}}_{C}\theta_{1}=\left(\delta^{\text{diff}}_{C\epsilon}W-\frac{\delta
W}{\delta
N}W\delta^{\text{diff}}_{C\epsilon}\theta_{1}\right)\delta^{\text{diff}}_{C}\theta_{1}=\delta^{\text{diff}}_{C\epsilon}W\delta^{\text{diff}}_{C}\theta_{1}\,.$
(4.4)

In the last equality we have used
$\delta^{\text{diff}}_{C\epsilon}\theta_{1}\times\delta^{\text{diff}}_{C}\theta_{1}=0$,
due to the Grassmann nature of $C^{i}$. Transformation (4.4) requires a
diffeomorphism on $W$, which is a scalar density of weight $-1$. Another term
that is required is

$W\delta_{\Omega}(\delta^{\text{diff}}_{C}\theta_{1})=-W\left(\partial_{i}\theta_{1}\delta_{\Omega}C^{i}+\theta_{1}\partial_{i}\delta_{\Omega}C^{i}\right)\,,$
(4.5)

where we have used $\delta_{\Omega}\theta_{1}=0$ since $\theta_{1}$ is a
second-class constraint. After using (4.4) and (4.5), the vanishing of the
square transformation $\delta_{\Omega}(sN)$ becomes a straightforward
computation. The vanishing of $\delta_{\Omega}(sC^{i})$ is a direct result
from (3.10). The transformation $s\bar{\mathcal{P}}_{i}$ is extracted from
(3.11). When computing the square $\delta_{\Omega}(s\bar{\mathcal{P}}_{i})$, a
useful fact is
$\delta_{\Omega}\mathcal{H}_{i}=\delta^{\text{diff}}_{C\epsilon}\mathcal{H}_{i}$,
since $\mathcal{H}_{i}$ is a functional only of $g_{ij}$ and $\pi^{ij}$ and
the formula (3.14) can be applied for it. In addition, the transformation of
the terms linear in $\bar{\mathcal{P}}_{i}$ in (3.11) generates terms linear
in $\mathcal{H}_{i}$; they form another
$\delta^{\text{diff}}_{C\epsilon}\mathcal{H}_{i}$ factor that cancels the
previous one. The rest is a straightforward computation to arrive at the
vanishing of $\delta_{\Omega}(s\bar{\mathcal{P}}_{i})$. The remaining fields
in (3.12) have an obvious nilpotent action of the BRST transformation.

##  5 Unitarity of the nonprojectable case

The unitarity of the $S$ matrix holds if the path integral can be formulated
as an integral over physical or reduced phase space with measure 1 (that is,
the measure only contains the differentials of the independent physical
fields). Previously we discussed that the $2+1$ theory must describe the
physics of a scalar mode (the so-called extra scalar mode of the Hořava
theory). We first check that the global balance in the quantum theory matches
with this: in (2.14) the bosonic canonical fields
$\\{g_{ij},\pi^{ij},N,P_{N},N^{i},\pi_{i}\\}$ sum up 12 independent degrees of
freedom. The BFV ghosts
$\\{C^{i},\bar{\mathcal{P}}_{i},\bar{C}_{i},\mathcal{P}^{i}\\}$ are introduced
to reduce the dynamics on the phase space, and they sum up 8 degrees of
freedom. There are two Dirac deltas for the second-class constraints; hence
there remains 2 degrees of freedom in the physical phase space. This is the
quantum scalar mode.

The reduction of the phase space requires us to solve the second-class
constraints. Below we shall see that the measure leads naturally to solve the
constraint $\theta_{1}$ for $N$; hence this is a requisite for unitarity. We
discuss first the existence of the solution. By considering the complete $2+1$
theory with the terms of order $z=2$, we have that the $\theta_{1}=0$
constraint (LABEL:theta1) can be posed as a fourth-order nonlinear
differential equation for $N$:

$\displaystyle
2\alpha_{6}N^{-1}\nabla^{i}\nabla^{j}(N\nabla_{j}a_{i})+2\alpha_{7}N^{-1}\nabla^{2}(N\nabla_{i}a^{i})+\alpha_{4}\left(N^{-1}\nabla^{2}(Na^{2})-2\nabla_{i}(a^{i}\nabla_{j}a^{j})-a^{2}\nabla_{i}a^{i}\right)$
$\displaystyle+3\alpha_{6}\nabla^{i}a^{j}\nabla_{i}a_{j}+\alpha_{7}(\nabla_{i}a^{i})^{2}-4\alpha_{2}\nabla_{i}(a^{2}a^{i})+\alpha_{5}\left(R\nabla_{i}a^{i}+N^{-1}\nabla^{2}(NR)\right)$
$\displaystyle-\alpha_{3}\left(2\nabla_{i}(Ra^{i})+Ra^{2}\right)+\alpha\left(2\nabla_{i}a^{i}+a^{2}\right)-3\alpha_{2}a^{4}$
$\displaystyle=-\alpha_{1}R^{2}+\beta
R-g^{-1}\left(\pi^{ij}\pi_{ij}+\bar{\sigma}\pi^{2}\right)\,.$ (5.1)

Its highest-order derivative comes from the first two terms, resulting in

$2(\alpha_{6}+\alpha_{7})g^{ij}g^{kl}\partial_{ijkl}\ln N\,.$ (5.2)

Since $g_{ij}$ is a Riemannian metric, this term is an elliptic operator
acting on $\ln N$. The linearized version of Eq. (5.1) is a fourth-order
elliptic equation,

$2(\alpha_{6}+\alpha_{7})\Delta^{2}n+2\alpha\Delta
n=\left(-\alpha_{5}\Delta+\beta\right)\left(-\Delta
h+\partial_{ij}h_{ij}\right)\,.$ (5.3)

To hold this feature, we require

$\alpha_{6}+\alpha_{7}\neq 0\,.$ (5.4)

To talk about the existence of the solution of Eq. (5.3), we adopt the
exposition of Ref. [20], which uses the Hilbert space approach on linear
elliptic equations of higher order. According to the theorems collected in
this book [20], if a Dirichlet problem is posed with Eq. (5.3), the weak
solution exists if the associate homogeneous equation
$2(\alpha_{6}+\alpha_{7})\Delta^{2}n+2\alpha\Delta n=0$ has $n=0$ as its only
solution.

In the high-energy regime, the low-order derivatives in Eq. (5.3) can be
neglected, such that the resulting equation has only the squared Laplacian
operator on $n$,

$2(\alpha_{6}+\alpha_{7})\Delta^{2}n=-\alpha_{5}\Delta\left(-\Delta
h+\partial_{ij}h_{ij}\right)\,.$ (5.5)

In this case the solution can be given in a closed way by means of the
convolution with the fundamental solution of $\Delta^{2}$ in two dimensions,
which is

$\Delta^{2}K(r)=\delta(r)\,,\qquad K(r)=\frac{1}{8\pi}r^{2}\ln r\,,$ (5.6)

where $r\equiv\sqrt{x_{1}^{2}+x_{2}^{2}}$. For $d=3$ and other dimensions, the
fundamental solution of the operator $\Delta^{z}$ can be found in Ref. [21].
Moreover, in Ref. [21] the fundamental solution for a more general elliptic
operator, having nonconstant coefficients or low-order derivatives as (5.3),
is presented in terms of plane waves (see also [20]).

Further evidence for the existence of the solution of (5.1) comes from a
different limit: the large-distance limit on nonperturbative variables. Under
this approximation, discarding all spatial derivatives of fourth order, Eq.
(5.1) becomes a linear elliptic equation of second-order for $\sqrt{N}$ (see
[17]),

$4\alpha\nabla^{2}\sqrt{N}-\left(\beta
R-g^{-1}(\pi^{ij}\pi_{ij}+\bar{\sigma}\pi^{2})\right)\sqrt{N}=0$ (5.7)

(whenever $\alpha\neq 0$). Summarizing, the two limiting cases given in Eqs.
(5.5) and (5.7) have solutions, for the linearized equation (5.3) there is a
theory under which the weak solution can be addressed, whereas other
approaches are available, and the complete Eq. (5.1) remains as a nonlinear
equation with a highest-order operator that is elliptic.

To make the reduction of the path integral, we return to an undefined gauge-
fixing function $\chi^{i}$; condition (2.26) is not imposed. The path integral
is taken in its form (2.14), with the Hamiltonian $\mathcal{H}_{\Psi}$ given
in (2.24). We show that the unitarity of the $S$ matrix naturally leads to the
necessity of the solution of $\theta_{1}$ in terms of $N$: the delta
$\delta(\theta_{2})$ sets $P_{N}=0$, eliminating the kinetic term
$P_{N}\dot{N}$. According to (3.4), the measure associated with the second-
class constraints becomes

$\delta(\theta_{1})\det\left(\frac{\delta\theta_{1}}{\delta
N}\right)=\delta(N-\hat{N})\,,$ (5.8)

where we have posed the constraint $\theta_{1}=0$ as an equation for $N$, and
we have denoted by $\hat{N}$ its solution, assuming that it exists at
arbitrary order or at nonperturbative level. Thus, the measure of the second-
class constraints naturally leads to fixing $N$ after the reduction of the
phase space. The path integral takes the form

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}C^{i}\mathcal{D}\bar{\mathcal{P}}_{i}\mathcal{D}\bar{C}_{i}\mathcal{D}\mathcal{P}^{i}\\\
&\times\exp\left[i\int
dtd^{2}x\left(\pi^{ij}\dot{g_{ij}}+\pi_{i}\dot{N}^{i}+\bar{\mathcal{P}}_{i}\dot{C}^{i}+\mathcal{P}^{i}\dot{\bar{C}}_{i}-\mathcal{H}_{\Psi}|_{N=\hat{N}}\right)\right]\,.\end{split}$
(5.9)

$\mathcal{H}_{\Psi}$ depends functionally on $N$ only through
$\mathcal{H}_{0}$ and the gauge-fixing condition $\chi^{i}$. The second-class
constraints have been eliminated, but the phase space still contains several
unphysical variables.

To arrive at the final unitary form, we apply the procedure developed in Refs.
[22, 23]. The first step is to integrate the BFV ghosts. Suppose that
$\Phi^{i}(g_{ij},\pi^{ij})=0$ is a gauge-fixing condition of interest. To
adapt the BFV quantization to this gauge we set

$\chi^{i}=\frac{1}{\varepsilon}\Phi^{i}(g_{ij},\pi^{ij})\,,$ (5.10)

where $\varepsilon$ is an arbitrary numerical parameter. The BFV theorem
ensures that the resulting path integral is independent of $\varepsilon$.
Hence, by taking at the end the limit $\varepsilon\rightarrow 0$ we recover
the desired gauge-fixing condition $\Phi^{i}=0$. Simultaneously, the following
rescaling on $\pi_{i}$ and $\bar{C}_{i}$ is done,

$\pi_{i}\rightarrow\varepsilon\pi_{i}\,,\quad\bar{C}_{i}\rightarrow\varepsilon\bar{C}_{i}\,.$
(5.11)

Since $\bar{C}_{i}$ is a Grassmann variable, the Jacobian of this rescaling is
one. The Hamiltonian (2.24) takes the form

$\mathcal{H}_{\Psi}|_{N=\hat{N}}=\mathcal{H}_{0}|_{N=\hat{N}}+\mathcal{H}_{i}N^{i}+\bar{\mathcal{P}}_{i}\mathcal{P}^{i}-\bar{\mathcal{P}}_{i}\left(N^{j}\partial_{j}C^{i}+N^{i}\partial_{j}C^{j}\right)+\pi_{i}\Phi^{i}+\bar{C}_{i}\\{\Phi^{i}\,,\mathcal{H}_{j}\\}C^{j}\,.$
(5.12)

$\mathcal{H}_{0}|_{N=\hat{N}}$ depends exclusively on the canonical pair
$(g_{ij},\pi^{ij})$. Note that the last term of this Hamiltonian has a Poisson
bracket indicated, instead of a Dirac bracket. This is consistent with the
fact that we have solved the second-class constraints. The path integral takes
the form

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}C^{i}\mathcal{D}\bar{\mathcal{P}}_{i}\mathcal{D}\bar{C}_{i}\mathcal{D}\mathcal{P}^{i}\\\
&\times\exp\left[i\int
dtd^{2}x\left(\pi^{ij}\dot{g_{ij}}+\varepsilon\pi_{i}\dot{N}^{i}+\bar{\mathcal{P}}_{i}\dot{C}^{i}+\varepsilon\mathcal{P}^{i}\dot{\bar{C}}_{i}-\mathcal{H}_{\Psi}|_{N=\hat{N}}\right)\right]\,.\end{split}$
(5.13)

Now we may take the limit $\varepsilon\rightarrow 0$, obtaining

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}C^{i}\mathcal{D}\bar{\mathcal{P}}_{i}\mathcal{D}\bar{C}_{i}\mathcal{D}\mathcal{P}^{i}\\\
&\times\exp\left[i\int
dtd^{2}x\left(\pi^{ij}\dot{g_{ij}}-\mathcal{H}_{0}|_{N=\hat{N}}-\mathcal{H}_{i}N^{i}-\pi_{i}\Phi^{i}\right.\right.\\\
&\left.\left.-\bar{\mathcal{P}}_{i}\left(\mathcal{P}^{i}-\dot{C}^{i}-N^{j}\partial_{j}C^{i}-N^{i}\partial_{j}C^{j}\right)-\bar{C}_{i}\\{\Phi^{i}\,,\mathcal{H}_{j}\\}C^{j}\right)\right]\,.\end{split}$
(5.14)

A shift on the $\mathcal{P}^{i}$ field with unit Jacobian leads us to the
quadratic form $-\bar{\mathcal{P}}_{i}\mathcal{P}^{i}$; hence the integration
on these momenta can be done with no consequences on the path integral. Next,
the integration on $N^{i}$, $\pi_{i}$, $\bar{C}_{i}$, and $C^{i}$ leads to the
form of the path integral,

$Z=\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\delta(\mathcal{H}_{i})\delta(\Phi^{i})\det\\{\Phi^{i}\,,\mathcal{H}_{j}\\}\exp\left[i\int
dtd^{2}x\left(\pi^{ij}\dot{g_{ij}}-\mathcal{H}_{0}|_{N=\hat{N}}\right)\right]\,.$
(5.15)

This is the Faddeev formula for the path integral of a system with first-class
constraints only [22]. The first-class constraint is
$\mathcal{H}_{i}(g_{ij},\pi^{ij})=0$, and the canonical gauge-fixing condition
is $\Phi^{i}(g_{ij},\pi^{ij})=0$. The last step is to show that this path
integral can be formulated strictly as an integral over the physical phase
space with measure 1, yielding a unitary $S$ matrix. We apply Faddeev’s
procedure to achieve this [22]. It consists of making a canonical
transformation on the phase space with the aim of identifying a part of the
new canonical variables directly with the unphysical degrees of freedom and
the rest with the physical ones. The consistent formulation of a system with
first-class constraints requires that the gauge-fixing condition $\Phi^{i}$
satisfies

$\displaystyle\det\\{\Phi^{i}\,,\mathcal{H}_{j}\\}\neq 0\,,$ (5.16)
$\displaystyle\\{\Phi^{i}\,,\Phi^{j}\\}=0\,.$ (5.17)

We perform a canonical transformation on the coordinates $(g_{ij},\pi^{ij})$,
where the new coordinates are labeled by the two sets of canonical fields:
$(q_{i},p^{i})$ and $(q^{*},p^{*})$.555In arbitrary spatial dimensions $d$,
the splitting is $(q_{i},p^{i})$ and $(q^{*A},p^{*A})$, with $i=1,\dots,d$ and
$A=1,\ldots,\frac{1}{2}d(d-1)$. The canonical transformation preserves the
kinetic terms in (5.15) and has unit Jacobian since it is done on bosonic
variables. By virtue of condition (5.17), we can use the canonical
transformation to make the identification

$\Phi^{i}(g_{kl},\pi^{kl})=p^{i}\,.$ (5.18)

With this setting the condition (5.16) takes the form

$\det\left(\frac{\partial\mathcal{H}_{j}}{\partial q_{i}}\right)\neq 0\,.$
(5.19)

This implies that the constraint $\mathcal{H}_{j}=0$ can be solved for the
$q_{i}$ variables. On the basis of (5.18) and the solution of
$\mathcal{H}_{j}=0$, the physical phase space is defined by the equations,

$p^{i}=0\,,\quad q_{i}=q_{i}(q^{*},p^{*})\,,$ (5.20)

such that $(q^{*},p^{*})$ are free coordinates on the physical phase space. We
can transform the factors of the measure in (5.15) as follows:

$\delta(\mathcal{H}_{i})\det\\{\Phi^{i}\,,\mathcal{H}_{j}\\}=\delta(\mathcal{H}_{i})\det\left(\frac{\partial\mathcal{H}_{j}}{\partial
q_{i}}\right)=\delta(q_{i}-q_{i}(q^{*},p^{*}))\,,$ (5.21)

where $q_{i}(q^{*},p^{*})$ identifies the solution of $\mathcal{H}_{i}=0$
(5.20). Therefore, the integration on $q_{i}$ sets $q_{i}=q_{i}(q^{*},p^{*})$
and the integration on $p^{i}$ sets $p^{i}=0$. There are no more factors
remaining in the measure. The path integral (5.15) becomes

$Z=\int\mathcal{D}q^{*}\mathcal{D}p^{*}\exp\left[i\int
dtd^{2}x\left(p^{*}\dot{q^{*}}-\mathcal{H}_{0}|_{\Gamma^{*}}\right)\right]\,.$
(5.22)

where $\Gamma^{*}$ is the subset of the phase space defined by

$N=\hat{N}\,,\quad q_{i}=q_{i}(q^{*},p^{*})\,,\quad p^{i}=0\,.$ (5.23)

$\Gamma^{*}$ is the physical phase space, since any unphysical variable has
been eliminated to define it. In the present theory, $(q^{*},p^{*})$ are the
coordinates of the physical scalar mode, and the physical Hamiltonian
$\mathcal{H}_{0}|_{\Gamma^{*}}$ depends exclusively on them.

##  6 Projectable case

The projectable version of the Hořava theory is defined by the condition that
the lapse function is a function only of time, $N=N(t)$. This condition is
preserved by the gauge-symmetry group of the theory, the foliation-preserving
diffeomorphisms (see Eq. (2.1)). Therefore, this case constitutes an
independent formulation of the theory. In this case $N(t)$ is a pure gauge
degree of freedom; we may use the symmetry transformation $\delta t=f(t)$ to
set $N=1$, a condition that we assume throughout this section. The Lagrangian
of the projectable case is given by (2.4), where the potential
$\mathcal{V}(g_{ij})$ depends only on the spatial metric. In $2+1$ dimensions,
excluding the term linear in the spatial Ricci scalar since it leads to a
topological invariant, the $z=2$ potential is given by

$\displaystyle\mathcal{V}=\alpha_{0}R^{2}.$ (6.1)

In the classical canonical formulation [24], the canonical pair is
$(g_{ij},\pi^{ij})$. The central difference between the dynamics of the
nonprojectable and projectable case is that the latter has only first-class
constraints, which is given by the momentum constraint (2.10). An analog of
the local “Hamiltonian” constraint is absent. Instead, here there arises the
global (integrated) constraint,

$\int\,d^{2}x\mathcal{H}=0,\qquad\mathcal{H}=\frac{1}{\sqrt{g}}\left(\pi^{ij}\pi_{ij}+\frac{\lambda}{1-d\lambda}\pi^{2}\right)+\sqrt{g}\mathcal{V}\,.$
(6.2)

The primary Hamiltonian density can be determined by

$H_{0}=\int d^{2}x\mathcal{H}_{0}=\int d^{2}x\mathcal{H}.$ (6.3)

Thus, the primary Hamiltonian density is equivalent to $\mathcal{H}$. The
classical functional variables are $\\{g_{ij},\pi^{ij}\\}$, which sum six. The
constraint $\mathcal{H}_{i}$ has $2$ degrees and there are $2$ more degrees in
the spatial diffeomorphisms. The resulting physical phase space has dimension
two, the same as the nonprojectable case.

We show a summary of the BFV quantization of the projectable theory. Details
can be found in Ref. [4]. The BFV path integral is

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}C^{i}\mathcal{D}\bar{\mathcal{P}}_{i}\mathcal{D}\mathcal{P}^{i}\mathcal{D}\bar{C}_{i}\\\
&\exp\left[i\int\,dt\,d^{2}x\Big{(}\pi^{ij}\dot{g_{ij}}+\pi_{k}\dot{N}^{k}+\bar{\mathcal{P}}_{i}\dot{C}^{i}+\mathcal{P}^{i}\dot{\bar{C}}_{i}-\mathcal{H}_{\Psi}\Big{)}\right]\,.\end{split}$
(6.4)

The BRST generator $\Omega$ has the same form of the nonprojectable case, Eq.
(2.19). By adopting the fermionic gauge-fixing function (2.23), we obtain the
gauge-fixed Hamiltonian

$\mathcal{H}_{\Psi}=\mathcal{H}+\mathcal{H}_{k}N^{k}+\bar{\mathcal{P}}_{k}\mathcal{P}^{k}-\bar{\mathcal{P}}_{i}\left(N^{j}\partial_{j}C^{i}+N^{i}\partial_{j}C^{j}\right)+\pi_{k}\chi^{k}+\bar{C}_{i}\\{\chi^{i}\,,\mathcal{H}_{k}\\}C^{k}+\bar{C}_{i}\dfrac{\delta\chi^{i}}{\delta
N^{l}}\mathcal{P}^{l}\,,$ (6.5)

where the gauge-fixing condition is independent of the BFV ghost.

Since this is a theory with first-class constraints only, the BRST symmetry
transformations with parameter $\epsilon$ are defined by
$\delta_{\Omega}\varphi=\\{\varphi\,,\Omega\\}\epsilon$. The transformations
result to be the same as the nonprojectable case (3.12), if the pair $N,P_{N}$
is excluded. Thus, we have again that the spatial diffeomorphism along the
vector $C^{i}\epsilon$ plays a prominent role. The primary Hamiltonian is BRST
invariant:

$\delta_{\Omega}H_{0}=\int d^{2}x\left(\frac{\delta\mathcal{H}}{\delta
g_{ij}}\delta_{\Omega}g_{ij}+\frac{\delta\mathcal{H}}{\delta\pi^{ij}}\delta_{\Omega}\pi^{ij}\right)=\int
d^{2}x\>\delta^{\text{diff}}_{C\epsilon}\mathcal{H}=0\,.$ (6.6)

Since the BRST transformations are the same, the invariance of the kinetic
terms of the action follows the same steps from (3.15) to (3.17). If we adopt
the gauge-fixing form (2.26), then the proof of the invariance of the terms
associated with the gauge fixing is parallel to the nonprojectable case. This
completes the invariance of the action.

To prove the unitarity of the $S$ matrix, we manage the integration on the
ghost fields. We assume that the gauge-fixing condition has the form (5.10)
and we rescale the fields $\pi^{i}$ and $\bar{C}_{i}$ as in (5.11). We take
the limit $\varepsilon\rightarrow 0$ and then we integrate on
$\mathcal{P}^{i},\bar{\mathcal{P}}_{i}$. The path integral takes the form

$\begin{split}Z=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\mathcal{D}N^{k}\mathcal{D}\pi_{k}\mathcal{D}C^{i}\mathcal{D}\bar{C}_{i}\\\
&\times\exp\left[i\int\,dt\,d^{2}x\Big{(}\pi^{ij}\dot{g_{ij}}-\mathcal{H}-N^{k}\mathcal{H}_{k}-\pi_{k}\Phi^{k}-\bar{C}_{i}\\{\Phi^{i},\mathcal{H}_{k}\\}C^{k}\Big{)}\right]\,.\\\
=&\int\mathcal{D}g_{ij}\mathcal{D}\pi^{ij}\delta(\Phi^{k})\delta(\mathcal{H}_{i})\det\\{\Phi^{j},\mathcal{H}_{j}\\}\exp\left[i\int\,dt\,d^{2}x\Big{(}\pi^{ij}\dot{g_{ij}}-\mathcal{H}\Big{)}\right]\,.\end{split}$
(6.7)

We have arrived at the Faddeev form of the $S$ matrix [22]. The action has a
canonical form on the variables $(g_{ij},\pi^{ij})$; $\mathcal{H}$ is a
functional of $(g_{ij},\pi^{ij})$ from the beginning. By repeating the same
steps of section 5, the unitary form of the $S$ matrix can be proven,
describing the quantum dynamics of a scalar mode.

## Conclusions

We have succeed in finding explicit expressions for the BRST symmetry
transformations of the Hořava theory, both in its nonprojectable and
projectable versions, under the BFV formalism. These expressions have allowed
us to prove the BRST invariance of the quantum action explicitly. The
consistency of the BRST symmetry is a fundamental aspect for the quantization
of the theory. This may be an input for a future proof of the renormalization
of the nonprojectable case. We find a very useful result in the fact that
these transformations can be cast in terms of a diffeomorphism along the ghost
vector field $C^{i}$, including the transformation of the ghost itself and its
conjugate momentum. The transformation of various objects that are tensors or
tensor densities can be managed in terms of the spatial diffeomorphism. We
have done the analysis in the original nonperturbative variables and in the
whole phase space, including the region where the second-class constraints are
not satisfied. Regarding this, the BRST transformations are strictly local
when evaluated on the constrained surface, where the quantum theory is
defined. Outside this surface, there are nonlocal contributions.

We have proved that, if we assume that the solution of the $\theta_{1}=0$
constraint in terms of the lapse function $N$ exists, then the unitarity of
the theory holds, including the case when the theory is quantized under the
gauge required for renormalization, thanks to the independence of the BFV
formalism on the gauge chosen. The necessity of the solution for $N$ arises
naturally in the reduction to the physical degrees of freedom, since the
quantum measure leads to solve the constraint $\theta_{1}=0$ for $N$, a
condition that is not evident at the level of the classical theory. This
constraint is a nonlinear equation for $N$. We have appealed to its linearized
version to arrive at a linear elliptic equation of fourth order (in the $2+1$
case). According to the results in the literature, existence in a general
higher-order linear equation has been proven under certain conditions. In the
high-energy limit of the linearized theory, the solution can be given in a
closed way by the convolution with the fundamental solution, which is known.
Another limit we have presented is the large-distance limit. In this case,
$\theta_{1}=0$ becomes a linear elliptic equation of second order, which is a
well-known case, and it is of nonperturbative character. These evidences lead
us to believe that the solution for $N$ of the $\theta_{1}=0$ constraint
exists at any order.

## Acknowledgments

C.B. is partially supported by Grant No. CONICYT PFCHA/DOCTORADO BECAS CHILE
/2019 – 21190960. C.B. is partially supported as a graduate student in the
“Doctorado en Física Mención Física-Matemática” Ph.D. program at the
Universidad de Antofagasta.

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