# Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of
the Lorentz group

Mehraveh<EMAIL_ADDRESS>and Tom
<EMAIL_ADDRESS>
$(1)$ Institute of Theoretical Physics and Astrophysics, University of Gdańsk,
80-308 Gdańsk, Poland

###### Abstract

A number of approaches to gravitation have much in common with the gauge
theories of the standard model of particle physics. In this paper, we develop
the Hamiltonian formulation of a class of gravitational theories that may be
regarded as spontaneously-broken gauge theories of the complexified Lorentz
group $SO(1,3)_{C}$ with the gravitational field described entirely by a gauge
field valued in the Lie algebra of $SO(1,3)_{C}$ and a ‘Higgs field’ valued in
the group’s fundamental representation. The theories have one free parameter
$\beta$ which appears in a similar role to the inverse of the Barbero-Immirzi
parameter of Einstein-Cartan theory. However, contrary to that parameter, it
is shown that the number of degrees of freedom crucially depends on the value
of $\beta$. For non-zero values of $\beta$, it is shown the theories possesses
three complex degrees of freedom, and for the specific values $\beta=\pm i$ an
extension to General Relativity is recovered in a symmetry-broken regime. For
the value $\beta=0$, the theory possesses no local degrees of freedom. A non-
zero value of $\beta$ corresponds to the self-dual and anti-self-dual gauge
fields appearing asymmetrically in the action, therefore in these models, the
existence of gravitational degrees of freedom is tied to chiral asymmetry in
the gravitational sector.

##  1 Introduction

A great achievement of General Relativity has been the introduction of the
notion of spacetime diffeomorphism symmetry as a cornerstone of gravitational
physics. Less well known are formulations of non-gravitational physics which
nonetheless possess the same symmetry - these theories are named
_parameterized_ field theories. As an example, consider the action for degrees
of freedom $q^{i}(\tau)$ in Newtonian mechanics:

$\displaystyle S[q]$ $\displaystyle=\int
d\tau\bigg{(}\sum_{i}m_{i}\bigg{(}\frac{d}{d\tau}q^{i}\bigg{)}^{2}-V(q)\bigg{)}$
(1)

Alternatively, one can consider an action where the Newtonian time $\tau$ is
itself promoted to a dynamical field:

$\displaystyle\tau$ $\displaystyle\rightarrow\tau(\lambda)$ (2) $\displaystyle
S[q,\tau]$ $\displaystyle=\int
d\lambda\frac{d\tau}{d\lambda}\bigg{(}\sum_{i}m_{i}\bigg{(}\frac{d\tau}{d\lambda}\bigg{)}^{-2}\bigg{(}\frac{d}{d\lambda}q^{i}\bigg{)}^{2}-V(q)\bigg{)}$
(3)

Under a transformation generated by the infinitesimal vector
$\zeta=\epsilon\partial_{\lambda}$:

$\displaystyle\tau$ $\displaystyle\rightarrow\tau+{\cal L}_{\zeta}\tau,\quad
q^{i}\rightarrow q^{i}+{\cal L}_{\zeta}q^{i}$ (4)

\- where ${\cal L}$ is the Lie derivative - the action (3) changes by a
boundary term and hence the transformations (4), which represent
diffeomorphisms on the manifold coordinatized by $\lambda$, are a symmetry of
the theory. This is a symmetry which is not present for the action (1),
however the equations of motion following from (3) admit the same solutions as
those following from (1) if the gauge $\tau\overset{*}{=}\lambda$ is
accessible, with the $\tau$ equation of motion expressing conservation of
energy. The extension to parameterized field theory in higher dimensional
special-relativistic actions is via the replacement of the Minkowski metric
tensor $\eta_{\mu\nu}$ with

$\displaystyle\eta_{\mu\nu}\rightarrow\eta_{IJ}\partial_{\mu}\phi^{I}(x)\partial_{\nu}\phi^{J}(x)$
(5)

where $\eta_{IJ}=\mathrm{diag}(-1,1,1,1)$ and $x^{\mu}$ are coordinates in
spacetime. Analogously to the model (3), actions with the replacement (5) with
the promotion of $\phi^{I}$ to dynamical fields then possess a four-
dimensional spacetime diffeomorphism symmetry despite not including the
gravitational interaction. If the gauge $\phi^{I}\overset{*}{=}x^{I}(x^{\mu})$
is accessible, where $x^{I}$ are fields playing the role of Minkowski
coordinates in spacetime, then special-relativistic physics is recovered with
the $\phi^{I}$ equations of motion corresponding to equations expressing
conservation of stress energy.

It is possible then to recover a description of special-relativistic physics
which nonetheless possesses the symmetries associated with gravitational
theory. Can the gravitational interaction be recovered from this starting
point and, if so, is the resulting theory General Relativity? To take steps
towards this, we note that actions built using (5) have an additional symmetry
which corresponds to:

$\displaystyle\phi^{I}\rightarrow\Lambda^{I}_{\phantom{I}J}\phi^{J}+P^{I}$ (6)

where $\Lambda^{I}_{\phantom{I}J}\in SO(1,3)$ and $P^{I}$ are independent of
coordinates $x^{\mu}$ and hence (6) can be interpreted as a global Poincaré
transformation acting on the fields $\phi^{I}$. If some of
$\\{\Lambda^{I}_{\phantom{I}J},P^{I}\\}$ do depend on position then the
ordinary derivative $\partial_{\mu}\phi{{}^{I}}$ in (5) no longer transforms
homogeneously under the local generalization of (6) and so actions containing
(5) will not be invariant under such transformations. This can be remedied by
the introduction of fields
$\\{\omega^{I}_{\phantom{I}J\mu},\theta^{I}_{\mu}\\}$ such that an operator
${\cal D}_{\mu}$ can be constructed, acting on $\phi^{I}$ as:

$\displaystyle{\cal
D}_{\mu}\phi^{I}\equiv\partial_{\mu}\phi^{I}+\omega^{I}_{\phantom{I}J\mu}\phi^{J}+\theta^{I}_{\mu}$
(7)

It can be shown that (7) transforms homogeneously under the local
generalization of (6) if

$\displaystyle\omega^{I}_{\phantom{I}J\mu}$
$\displaystyle\rightarrow\Lambda^{I}_{\phantom{I}K}\omega^{K}_{\phantom{K}L\mu}(\Lambda^{-1})^{L}_{\phantom{L}J}-\partial_{\mu}\Lambda^{I}_{\phantom{I}K}(\Lambda^{-1})^{K}_{\phantom{K}J}$
(8) $\displaystyle\theta^{I}_{\mu}$
$\displaystyle\rightarrow\Lambda^{I}_{\phantom{I}J}\theta^{J}_{\mu}-\partial_{\mu}P^{I}$
(9)

It follows then that the tensor

$\displaystyle g_{\mu\nu}\equiv\eta_{IJ}{\cal D}_{\mu}\phi^{I}{\cal
D}_{\mu}\phi^{J}$ (10)

is invariant under the local Poincaré transformations and it is this composite
object that will play the role of the metric tensor. Equation (10) can be seen
as a definition of the metric tensor and is a composite object built from
$\\{\phi^{I},\omega^{I}_{\phantom{I}J\mu},\theta^{I}_{\mu}\\}$ which may be
regarded as the fields describing gravity. Indeed, it is straightforward to
build polynomial actions in these variables that correspond to the Einstein-
Cartan formulation of gravity [1]. One may additionally construct
parameterized field theories corresponding to matter fields in fixed
background metrics with non-vanishing curvature. Parameterized field theories
which correspond to background metrics for De Sitter space and anti-De Sitter
space have global symmetries and their gauging instead results in the
Macdowell-Mansouri formulation of gravity. This is discussed in more detail in
Appendix A. Returning to the Minkowski case, remarkably, other theories of
gravity may emerge if only a subgroup of the global Poincaré symmetry (6) is
promoted to a local one. If just the translational part is localized (hence
gravity is described entirely by $\\{\phi^{I},\theta^{I}_{\mu}\\}$), then the
resulting gravitational theory is teleparallel gravity [2]. On the other hand,
one can consider the case where only the global Lorentz symmetry is promoted
to a local one, hence gravity is to be described entirely by
$\\{\phi^{I},\omega^{I}_{\phantom{I}J\mu}\\}$. Remarkably, extensions of
General Relativity can be recovered from the following family of actions:

$\displaystyle S[\phi^{I},\omega^{IJ}_{\mu}]$ $\displaystyle=\frac{1}{2}\int
d^{4}x\,\tilde{\epsilon}^{\mu\nu\alpha\beta}\big{(}\epsilon_{IJKL}+2\beta\eta_{K[I}\eta_{J]L}\big{)}D_{\mu}\phi^{I}D_{\nu}\phi^{J}R^{KL}_{\phantom{KL}\alpha\beta}(\omega)$
(11)

when $\beta=\pm i$ [3, 4, 5, 6], where
$D_{\mu}\phi^{I}=\partial_{\mu}\phi^{I}+\omega^{I}_{\phantom{I}J\mu}\phi^{I}$
is the $SO(1,3)$-covariant derivative of $\phi^{I}$ and
$R^{IJ}_{\phantom{IJ}\alpha\beta}$ is the curvature two-form of the field
$\omega^{I}_{\phantom{I}J\mu}$. As such, the action (11) is manifestly
invariant under local $SO(1,3)$ transformations. As we are allowing for
complex $\beta$, the action is potentially complex as well as the fields
$(\phi^{I},\omega^{I}_{\phantom{I}J\mu})$, with the local Lorentz symmetry
being that of the _complexified_ Lorentz group $SO(1,3)_{C}$. Specifically, an
extension to General Relativity is recovered when $\phi^{I}\neq 0$ and with
the metric tensor $g_{\mu\nu}$ identified with
$\eta_{IJ}D_{\mu}\phi^{I}D_{\nu}\phi^{J}$; hence, formally this correspondence
arises in a spontaneously-broken gauge theory described by gauge field
$\omega^{I}_{\phantom{I}J\mu}$ and Higgs field $\phi^{I}$, with non-vanishing
values of the latter field breaking the symmetry $SO(1,3)_{C}$ to $SO(3)_{C}$
(for the case $\phi^{2}\equiv\eta_{IJ}\phi^{I}\phi^{J}<0$), $SO(1,2)_{C}$
($\phi^{2}>0$), or $ISO(2)_{C}$ ($\phi^{2}=0)$ for non-vanishing $\phi^{I}$.
The reader may note that upon replacement $D_{\mu}\phi^{I}\rightarrow
e^{I}_{\mu}$ (where $e^{I}_{\mu}$ is the spacetime co-tetrad), the action is
proportional to the Holst action [7]. The relation between these two actions
is discussed in more detail in Section 3.

This paper aims to analyze the canonical structure of the action (11)
corresponding to general complex values of $\beta$. The canonical/Hamiltonian
analysis is a very powerful tool to determine the dynamical structure of
theories (see for example [8, 9, 10, 11, 12, 13, 14] for applications to
different gravitational theories). It will be shown that the properties of the
theory such as the ability to describe real spacetime metrics and even the
number of degrees of freedom in the theory depend crucially on the value of
$\beta$.

The structure of the paper is as follows: In Section 2 we review several
mathematical preliminaries necessary to describe the models under
consideration here and their Hamiltonian form. In Section 3 we briefly survey
the action principles of some gravitational theories including the models that
we will develop. In Section 4 we construct the canonical formulation of the
models in question considering the propagation of constraints, the
classification of constraints, and the imposition of reality conditions on
complex fields. In Section 5 we show that some of the models correspond to an
extension of General Relativity and in Section 6 we briefly consider matter
couplings and possible additional terms in the gravitational action. In
Section 7 we discuss results and the potential for further study of these
models.

##  2 Mathematical Preliminaries

We now review a number of mathematical preliminaries that will be helpful for
the remainder of the analysis in this paper.

###  2.1 Review of the 3+1 Decomposition

Maintaining general covariance is a typical requirement in gravitational
theories. Dynamical fields are spacetime tensors and actions are coordinate-
independent functionals built entirely from these fields. As such, there is
a-priori no preferred notion of time in gravitational theory. However,
significant insights can be gained into the structure of generally covariant
theories by choosing a single coordinate by which to measure the change of
fields over the spacetime manifold. This is the 3+1 decomposition of physical
theories.

We assume that the spacetime manifold $M$ is topologically $R\times\Sigma$,
where $\Sigma$ is a three-dimensional submanifold of $M$ and $R$ may be
coordinatized by a number $t$ which can be thought of as ‘coordinate time’.
However, we emphasize that $t$ is to be regarded as a number labeling
different submanifolds of $M$ and may not be straightforwardly related to
proper time as determined by a spacetime metric (indeed we will encounter
solutions of the models (11) where no notion of proper time exists).
Furthermore, we are primarily interested in solutions to the theory in the
bulk spacetime and will neglect boundary conditions and surface integrals.
Hence our treatment will be exact only in cases where $\Sigma$ is a closed
manifold [15].

Alongside the label $t$ for coordinate time, we will use a set of three
coordinates $\\{x^{a}\\}$ ($a=1,2,3$) to cover the manifold $\Sigma$. As such,
a vector $V$ and one-form $\sigma$ can be decomposed as follows:

$\displaystyle V$
$\displaystyle=V^{\mu}\partial_{\mu}=V^{t}\partial_{t}+V^{a}\partial_{a}$ (12)
$\displaystyle\sigma$
$\displaystyle=\sigma_{\mu}dx^{\mu}=\sigma_{t}dt+\sigma_{a}dx^{a}$ (13)

By extension, if $M$ admits a metric tensor $g=g_{\mu\nu}dx^{\mu}\otimes
dx^{\nu}$ then the following decomposition can be used:

$\displaystyle g$ $\displaystyle=-N^{2}dt\otimes
dt+q_{ab}(N^{a}dt+dx^{a})\otimes(N^{b}dt+dx^{b})$ (14)

Additionally, we will adopt the notation that for a function $f(t,x^{a})$ then
$\dot{f}\equiv\partial_{t}f$ whilst $\partial_{a}f$ denotes the partial
derivative with respect to a coordinate $x^{a}$.

###  2.2 The Hamiltonian formalism

A classical field theory will be described by an action $S$ which is a
functional of fields $\chi^{{\cal A}}$ (which we use to denote a set of any
tensor fields such as $V,\sigma,g$ as defined in Section 2.1). The action can
be written as an integral of a spacetime density $\tilde{\cal L}(\chi^{{\cal
A}})$ called the Lagrangian density i.e.

$\displaystyle S[\chi^{{\cal A}}]$ $\displaystyle=\int\tilde{\cal L}dtd^{3}x$
(15)

Given the 3+1 decomposition of tensorial fields, the Lagrangian density
$\tilde{\cal L}$ can typically be written in the following form:

$\displaystyle\tilde{\cal L}$ $\displaystyle=\sum_{\cal B}a_{\cal
B}(\chi^{\cal A},\partial_{a}\chi^{\cal A})(\dot{\alpha}^{{\cal
B}})^{2}+\sum_{\cal C}b_{\cal C}(\chi^{\cal A},\partial_{a}\chi^{\cal
A})\dot{\beta}^{{\cal C}}-\tilde{\cal U}(\chi^{\cal A},\partial_{a}\chi^{\cal
A})$ (16)

i.e. the collection of fields $\chi^{\cal A}$ can be divided into those which
appear quadratically in time derivatives (the set $\\{\alpha^{\cal B}\\}$),
linear in time derivatives (the set $\\{\beta^{\cal C}\\}$), and without time
derivatives (the set which we will call $\\{\gamma^{\cal D}\\}$). By
introducing auxiliary ‘velocity’ fields ${\cal V}$ and Lagrange multiplier
fields ${\cal P}$, the following _extended_ Lagrangian density can be
constructed which yields identical equations of motion to (16):

$\displaystyle\tilde{\cal L}$ $\displaystyle=\sum_{\cal B}\tilde{\cal P}_{\cal
B}(\dot{\alpha}^{\cal B}-{\cal V}^{\cal B})+\sum_{\cal C}\tilde{\cal P}_{\cal
C}(\dot{\beta}^{\cal C}-{\cal V}^{\cal C})+\sum_{\cal D}\tilde{\cal P}_{\cal
D}(\dot{\gamma}^{\cal D}-{\cal V}^{\cal D})$ $\displaystyle+\sum_{\cal
B}a_{\cal B}(\chi^{\cal A},\partial_{a}\chi^{\cal A})({\cal V}^{\cal
B})^{2}+\sum_{\cal C}b_{\cal C}(\chi^{\cal A},\partial_{a}\chi^{\cal A}){\cal
V}^{\cal C}-\tilde{\cal U}(\chi^{\cal A},\partial_{a}\chi^{\cal A})$ (17)

For the fields ${\cal V}^{\cal B}$, the equation of motion for ${\cal V}^{\cal
B}$ allows for this field to be solved for in terms of the fields $(\chi^{\cal
A},{\cal P}^{\cal B})$ allowing it to be eliminated from the variational
principle. For fields ${\cal V}^{\cal C}$ and ${\cal V}^{\cal D}$, their
equations of motion do not allow for the fields to be solved for and
eliminated from the variational principle. The Lagrangian density then can be
reduced to the following form:

$\displaystyle\tilde{\cal L}$ $\displaystyle=\sum_{\cal A}\tilde{\cal P}_{\cal
A}\dot{\chi}^{\cal A}-\tilde{\cal H}(\tilde{\cal P}_{\cal A},\chi^{\cal
A},\partial_{a}\chi^{\cal A},{\cal V}^{\cal C},{\cal V}^{\cal D})$ (18)
$\displaystyle\tilde{\cal H}$ $\displaystyle={\cal H}_{0}(\tilde{\cal P}_{\cal
A},\chi^{\cal A},\partial_{a}\chi^{\cal A})+\sum_{{\cal C}}{\cal V}^{\cal
C}{\cal C}^{\cal C}(\tilde{\cal P}_{\cal A},\chi^{\cal
A},\partial_{a}\chi^{\cal A})+\sum_{{\cal D}}{\cal V}^{\cal D}{\cal C}^{\cal
D}(\tilde{\cal P}_{\cal A},\chi^{\cal A},\partial_{a}\chi^{\cal A})$ (19)

where the $({\cal V}^{C},{\cal V}^{D})$ are Lagrange multipliers (which
enforce via their equations of motion $({\cal C}^{\cal C}=0,{\cal C}^{\cal
D}=0)$) and $\tilde{\cal P}_{\cal A}$ consists of the collected fields
$(\tilde{\cal P}_{\cal B},\tilde{\cal P}_{\cal C},\tilde{\cal P}_{\cal D})$.
Equation (18) represents the Hamiltonian form of a theory, with stationarity
of the action with respect to small variations of $(\chi^{\cal A},\tilde{\cal
P}_{\cal A})$ yielding Hamilton’s equations:

$\displaystyle\dot{\chi}^{\cal A}$ $\displaystyle=\\{{\chi}^{\cal A},\int
d^{3}x\tilde{\cal H}\\}$ (20) $\displaystyle\dot{\tilde{\cal P}}_{\cal A}$
$\displaystyle=\\{\tilde{\cal P}_{\cal A},\int d^{3}x\tilde{\cal H}\\}$ (21)

where the Poisson bracket $\\{{\cal F},{\cal G}\\}$ between two functions
${\cal F}(\chi^{\cal A},\tilde{\cal P}_{\cal A})$ and ${\cal G}(\chi^{\cal
A},\tilde{\cal P}_{\cal A})$ is defined to be:

$\displaystyle\\{{\cal F},{\cal G}\\}$ $\displaystyle\equiv\int
d^{3}x\sum_{\cal A}\bigg{(}\frac{\delta{\cal F}}{\delta\chi^{\cal
A}}\frac{\delta{\cal G}}{\delta\tilde{\cal P}_{\cal A}}-\frac{\delta{\cal
G}}{\delta\chi^{\cal A}}\frac{\delta{\cal F}}{\delta\tilde{\cal P}_{\cal
A}}\bigg{)}$ (22)

Furthermore, it follows from (20) and (21) that for some function ${\cal
F}(\chi^{\cal A},\tilde{\cal P}_{\cal A})$ that

$\displaystyle\dot{\cal F}$ $\displaystyle=\\{{\cal F},\int d^{3}x\tilde{\cal
H}\\}$ (23)

The equations of motion that follow from the variation of fields $({\cal
V}^{\cal B},{\cal V}^{\cal C})$ are equations

$\displaystyle{\cal C}^{\cal B}(\tilde{\cal P}_{\cal A},\chi^{\cal
A},\partial_{a}\chi^{\cal A})$ $\displaystyle=0,\quad{\cal C}^{\cal
C}(\tilde{\cal P}_{\cal A},\chi^{\cal A},\partial_{a}\chi^{\cal A})=0$ (24)

These equations represent constraints that the fields $(\chi^{\cal
A},\tilde{\cal P}_{\cal A})$ must obey amongst themselves. If at some initial
moment $t=t_{0}$, the constraints are satisfied, then it must further be
required that the time derivative of these functions - defined via (23) - is
zero. This may imply additional constraints and, if so, their own time
derivatives must be ensured to be zero. The process continues until no further
constraints are generated.

###  2.3 Local Lorentz symmetry in gravitation and its complexification

A slight modification to the variables describing gravity is necessary to
couple gravity to fermionic fields. This requires the introduction of the co-
tetrad field $e^{I}_{\mu}$ from which the metric $g_{\mu\nu}$ is constructed
as

$\displaystyle g_{\mu\nu}=\eta_{IJ}e^{I}_{\mu}e^{J}_{\nu}$ (25)

Where $\eta_{IJ}=\mathrm{diag}(-1,1,1,1)$. Due to the appearance of the matrix
$\eta_{IJ}$, the expression (25) is invariant under transformations

$\displaystyle e_{\mu}^{I}\rightarrow\Lambda^{I}_{\phantom{I}J}e^{J}_{\mu}$
(26)

where $\Lambda^{I}_{\phantom{I}J}\in SO(1,3)$ i.e.
$\Lambda^{I}_{\phantom{I}J}$ are elements of the Lorentz group. The Weyl
spinors of the standard model transform in the fundamental representations of
the group $SL(2,C)$ and invariance under global $SL(2,C)$ transformations
necessitates coupling to $e^{I}_{\mu}$ in spinor Lagrangians and the
identification of $\Lambda^{I}_{\phantom{I}J}$ as the $SO(1,3)$ element
corresponding to that $SL(2,C)$ transformation. Note that (25) is invariant
under transformations associated with $\Lambda^{I}_{\phantom{I}J}$ which can
depend on spacetime position. For spinorial actions then to be invariant under
the associated _local_ $SL(2,C)$ transformation, it is necessary to introduce
a field $\bar{\omega}^{I}_{\phantom{I}J\mu}$ (where
$\bar{\omega}^{IJ}_{\mu}=-\bar{\omega}^{JI}_{\mu}$ when an index has been
raised with $\eta^{IJ}$, the matrix inverse of $\eta_{IJ}$) which transforms
as a connection under local $SO(1,3)$ transformations (indeed, it should
transform precisely as (8) does). In General Relativity, this field is defined
as the solution to the equation

$\displaystyle\partial_{[\mu}e^{I}_{\nu]}+\bar{\omega}^{I}_{\phantom{I}J[\mu}e^{J}_{\nu]}=0$
(27)

Therefore in General Relativity $\bar{\omega}^{I}_{\phantom{I}J\mu}$ is
determined by $e^{I}_{\mu}$ and its derivatives. A variation on General
Relativity is provided by instead introducing a field
$\omega^{I}_{\phantom{I}J\mu}$ \- called the spin connection - in place of
$\bar{\omega}^{I}_{\phantom{I}J\mu}(e,\partial e)$ which is to be regarded as
an independent field with its own equations of motion. This is the Einstein-
Cartan formulation of gravity. In its simplest form, the equation of motion
for $\omega^{I}_{\phantom{I}J\mu}$ yields a solution
$\omega^{I}_{\phantom{I}J\mu}=\bar{\omega}^{I}_{\phantom{I}J\mu}(e,\partial
e)+\dots$ where the dots denote terms linear in spinorial currents.

A further generalization of the Einstein-Cartan model is provided by the
Ashtekar chiral theory of gravity [16]. To motivate this, we note that it has
been up to now assumed that $\Lambda^{I}_{\phantom{I}J}$ are elements of the
real Lorentz group. However, the transformation (25) is invariant under
$\Lambda^{I}_{\phantom{I}J}$ belonging to the _complexified_ Lorentz group
$SO(1,3)_{C}$ 333Which in terms of properties of matrices
$\Lambda^{I}_{\phantom{I}J}\in SO(1,3)_{C}$ is defined to be the set of
complex-valued matrices that satisfy
$\eta_{IJ}=\eta_{KL}\Lambda^{K}_{\phantom{K}I}\Lambda^{L}_{\phantom{L}J}$ and
$\mathrm{det}(\Lambda)=1$. Can classical General Relativity also arise if the
theory possesses a complex Lorentz symmetry? To understand the answer to this,
it is first helpful to introduce self- and anti-self duality concepts for
representations of $SO(1,3)_{C}$.

We have seen that it is helpful to introduce a field $e^{I}_{\mu}$ where under
an $SO(1,3)$ transformation
$e^{I}_{\mu}\rightarrow\Lambda^{I}_{\phantom{I}J}e^{J}_{\mu}$. One can
consider more general ‘Lorentz tensors’ with a more complicated index
structure. Particularly useful will be antisymmetric Lorentz tensors
$F^{IJ}=-F^{JI}$ which transform as follows under Lorentz transformations

$\displaystyle F^{IJ}$
$\displaystyle\rightarrow\Lambda^{I}_{\phantom{I}K}\Lambda^{J}_{\phantom{J}L}F^{KL}$
(28)

When the transformations are complexified Lorentz transformations, further
decomposition of this (now complex-valued object) is possible. We can consider
the following decomposition of $F^{IJ}$:

$\displaystyle F^{IJ}$ $\displaystyle=F^{+IJ}+F^{-IJ}$ (29)

where

$\displaystyle F^{\pm IJ}$
$\displaystyle=\frac{1}{2}(F^{IJ}\mp\frac{i}{2}\epsilon^{IJ}_{\phantom{IJ}KL}F^{KL})$
(30) $\displaystyle\frac{1}{2}\epsilon^{IJ}_{\phantom{IJ}KL}F^{\pm KL}$
$\displaystyle=\pm iF^{\pm IJ}$ (31)

where recall that $\epsilon_{IJKL}$ is the four-dimensional Levi-Civita symbol
and indices are lowered or raised with $\eta_{IJ}$ and its matrix inverse
$\eta^{IJ}$ respectively. If follows, for example, that for some matrix
$Y_{IJ}$ that $Y_{IJ}F^{IJ\pm}=Y_{IJ}^{\pm}F^{\pm IJ}$. When the fields and
Lorentz transformations are real then $F^{+IJ}$ and $F^{-IJ}$ are simply
complex conjugates of one another. When the fields are complexified, they
become genuinely independent objects. Equation (31) defines the property of
self-dualness (here $F^{+IJ}$) or anti-self-dualness (here $F^{-IJ})$. It is
possible to parameterize a self-dual or anti-self-dual Lorentz tensor in terms
of a field $E^{I}$ as follows:

$\displaystyle F^{\pm IJ}$
$\displaystyle=\frac{1}{2}(n^{[I}E^{J]}\mp\frac{i}{2}\epsilon^{IJKL}n_{K}E_{L})$
(32) $\displaystyle=(n^{[I}E^{J]})^{\pm}$ (33)

where $n^{I}$ is an arbitrary Lorentz vector of non-vanishing norm i.e.
$\eta_{IJ}n^{I}n^{J}=\xi$ and $E_{I}n^{I}=0$, where $\xi<0$ for timelike
$n^{I}$ and $\xi>0$ for spacelike $n^{I}$, and furthermore, for example, for a
Lorentz tensor $W_{\dots[IJ]^{+}}$ which is self-dual in a pair of indices, we
have:

$\displaystyle W_{\dots[IJ]^{+}}F^{IJ^{+}}$
$\displaystyle=W_{\dots[IJ]^{+}}n^{I}E^{J}$ (34)

Finally, it is useful to define the following objects:

$\displaystyle{\cal K}^{\pm}_{IJKL}$
$\displaystyle=\frac{1}{2}(\epsilon_{IJKL}\pm 2i\eta_{I[K}\eta_{L]J})$ (35)

where it can be shown that

$\displaystyle{\cal K}^{\pm}_{IJKL}F^{KL}$
$\displaystyle=\epsilon_{IJKL}F^{\pm KL}$ (36)

i.e. the objects ${\cal K}^{\pm}_{IJKL}$ act to project out self- or anti-
self-dual parts of an antisymmetric Lorentz tensor.

The spin connection $\omega^{IJ}_{\mu}=-\omega^{JI}_{\mu}$ present in
Einstein-Cartan gravity is an antisymmetric Lorentz tensor and so can be
decomposed into self-dual and anti-self-dual parts:

$\displaystyle\omega^{I}_{\phantom{I}J\mu}=\omega^{+I}_{\phantom{+I}J\mu}+\omega^{-I}_{\phantom{-I}J\mu}$
(37)

Upon complexification of the fields (which results from complexification of
the $SO(1,3)$ gauge symmetry) then
$(\omega^{+I}_{\phantom{+I}J\mu},\omega^{-I}_{\phantom{-I}J\mu})$ become truly
independent fields and this independence will be shown in Section 3 to be
crucially important in the structure of gravitational fields based on this
complexified Lorentz symmetry.

###  2.4 Spacetime structure

It will be very useful to relate some fields appearing in the canonical
formalism to quantities appearing in the 3+1 metric formalism of gravity. To
this end, we can use the following general parameterization of $e^{I}_{\mu}$
[17]:

$\displaystyle e^{I}=e^{I}_{t}dt+e^{I}_{a}dx^{a}$ (38)

where

$\displaystyle e^{I}_{t}$ $\displaystyle=NN^{I}+N^{a}e^{I}_{a}$ (39)
$\displaystyle q_{ab}$ $\displaystyle=\eta_{IJ}e^{I}_{a}e^{J}_{b}$ (40)

where $N^{I}e_{Ia}=0$ and $N^{I}N_{I}=-1$. Computing the metric
$g_{\mu\nu}=\eta_{IJ}e^{I}_{\mu}e^{J}_{\nu}$ confirms that $(N,N^{a},q_{ab})$
should be identified with corresponding quantities appearing in (14).
Furthermore, it follows that

$\displaystyle\eta^{IJ}$ $\displaystyle=-N^{I}N^{J}+e^{aI}e_{a}^{J}$ (41)

where $e^{a}_{I}=q^{ab}e_{bI}$ where $q^{ab}$ is the matrix inverse of
$q_{ab}$.

In the present work, the basic variables describing the gravitational field
will be $(\phi^{I},\omega^{IJ}_{\mu})$ with the identification

$\displaystyle
D_{\mu}\phi^{I}=\partial_{\mu}\phi^{I}+\omega^{I}_{\phantom{I}J\mu}\phi^{J}=e^{I}_{\mu}$
(42)

and hence we will look to identify the spacetime metric with
$g_{\mu\nu}=\eta_{IJ}D_{\mu}\phi^{I}D_{\nu}\phi^{J}$. We may also usefully
decompose $\phi^{I}$ into parts parallel with and orthogonal to $N^{I}$:

$\displaystyle\phi^{I}$ $\displaystyle=\phi_{(N)}N^{I}+\varphi^{I}$ (43)

where $\varphi_{I}N^{I}=0$. It follows then that
$\phi_{I}e^{I}_{a}=\frac{1}{2}\partial_{a}\phi^{2}=\varphi_{I}e^{I}_{a}$ where
we’ve used the fact that $N_{I}e^{I}_{a}$. Therefore,

$\displaystyle\varphi_{I}=\frac{1}{2}q^{ab}e_{Ia}\partial_{b}\phi^{2}$ (44)

and hence

$\displaystyle\phi_{(N)}=-\xi\sqrt{-\phi^{2}+\frac{1}{4}q^{ab}\partial_{a}\phi^{2}\partial_{b}\phi^{2}}$
(45)

where $\xi=\mp 1$. There are therefore two distinct options for the sign of
$\phi_{(N)}$.

##  3 Gravitational actions

We now briefly survey several theories of gravitation and their symmetries.
The action for Einstein’s General Relativity can be written as:

$\displaystyle S_{GR}[g_{\mu\nu}]$ $\displaystyle=\frac{1}{16\pi
G}\int_{M}d^{4}x\sqrt{-g}R+\int_{\partial M}d^{3}y\,\tilde{\ell}_{GHY}$ (46)

Where the second term - the Gibbons-Hawking-York term - is a boundary action
necessary to provide a well-defined variational principle. A spacetime
diffeomorphism generated by a vector field $\xi^{\mu}$ transforms the
spacetime metric as $g_{\mu\nu}\rightarrow g_{\mu\nu}+{\cal
L}_{\xi}g_{\mu\nu}$ and it can readily shown that this changes the action by a
boundary term and hence such diffeomorphisms are symmetries of the theory.

As discussed in Section 2.3, the necessity to couple gravitation to fermions
motivates the introduction of the fields $(e^{I}_{\mu},\omega^{IJ}_{\mu})$ as
the descriptors of gravity. One of the simplest actions that can be
constructed that has a General-Relativistic limit is the Einstein-Cartan
Palatini action:

$\displaystyle S_{EC}[e^{I}_{\mu},\omega^{IJ}_{\mu}]$
$\displaystyle=\frac{1}{64\pi G}\int
d^{4}x\,\tilde{\epsilon}^{\mu\nu\alpha\beta}\epsilon_{IJKL}e^{I}_{\mu}e^{J}_{\nu}R^{KL}_{\phantom{KL}\alpha\beta}(\omega)$
(47)

where $\tilde{\epsilon}^{\mu\nu\alpha\beta}$ is the Levi-Civita density and

$\displaystyle R^{IJ}_{\phantom{KL}\alpha\beta}(\omega)$
$\displaystyle=2\partial_{[\mu}\omega^{IJ}_{\phantom{IJ}\nu]}+2\omega^{I}_{\phantom{I}K[\mu}\omega^{KJ}_{\phantom{KJ}\nu]}$
(48)

are the components of the curvature tensor associated with
$\omega^{I}_{\phantom{I}J\mu}$. The Einstein-Cartan Palatini action possesses
the same spacetime diffeomorphism symmetry as the action for General
Relativity and is additionally invariant under local Lorentz transformations
parameterized by matrices $\Lambda^{I}_{\phantom{I}J}(x)$. As the spin
connection is an antisymmetric tensor in its Lorentz indices, it decomposed
into self- and anti-self-dual parts. Upon complexification of the local
Lorentz symmetry, these two fields are in principle independent of one another
and remarkably the equations of motion from the following actions

$\displaystyle S_{EC\pm}$ $\displaystyle=\frac{1}{32\pi G}\int
d^{4}x\,\tilde{\epsilon}^{\mu\nu\alpha\beta}{\cal
K}^{\pm}_{IJKL}e^{I}_{\mu}e^{J}_{\nu}R^{KL}_{\phantom{KL}\alpha\beta}(\omega)=\frac{1}{32\pi
G}\int
d^{4}x\,\tilde{\epsilon}^{\mu\nu\alpha\beta}\epsilon_{IJKL}e^{I}_{\mu}e^{J}_{\nu}R^{KL\pm}_{\phantom{\pm
KL}\alpha\beta}(\omega)$ $\displaystyle=\frac{1}{32\pi G}\int
d^{4}x\,\tilde{\epsilon}^{\mu\nu\alpha\beta}\epsilon_{IJKL}e^{I}_{\mu}e^{J}_{\nu}R^{KL\pm}_{\phantom{\pm
KL}\alpha\beta}(\omega^{\pm})$ (49)

yield the (complexified) Einstein’s equations, where the solutions of real
General Relativity can be imposed after the imposition of appropriate reality
conditions on fields. The actions (49) form Ashtekar’s chiral formulation of
gravity in which only one of the $\omega^{+IJ}_{\phantom{+IJ}\mu}$ or
$\omega^{-IJ}_{\phantom{-IJ}\mu}$ appear in the action.

The models that we will look at are models where $g_{\mu\nu}$ is recovered
from the combination (10) with $\theta^{I}_{\mu}=0$ and the dynamical
variables of the theory will be
$\\{\phi^{I},\omega^{IJ}_{\phantom{IJ}\mu}\\}$. This suggests that ultimately
we should identify $e^{I}_{\mu}$ as being recovered from the object
$D_{\mu}\phi^{I}=\partial_{\mu}\phi^{I}+\omega^{I}_{\phantom{I}J\mu}\phi^{J}$
and so, as in the case of Einstein-Cartan theory and Ashtekar’s chiral theory
we can look to construct Lagrangian densities which are quadratic in this
field and linear in the curvature of $\omega^{IJ}_{\phantom{IJ}\mu}$,
anticipating that this may be the simplest action giving non-trivial
gravitational dynamics [5]:

$\displaystyle S$ $\displaystyle=\frac{\alpha}{2}\int
d^{4}x\,\tilde{\epsilon}^{\mu\nu\alpha\beta}\big{(}\epsilon_{IJKL}+\frac{2}{\gamma}\eta_{K[I}\eta_{J]L}\big{)}D_{\mu}\phi^{I}D_{\nu}\phi^{J}R^{KL}_{\phantom{KL}\alpha\beta}(\omega)$
(50)

where $\gamma=1/\beta$ and $\alpha$ is an overall multiplicative factor. With
the aid of the symbols (35) we can write (50) as:

$\displaystyle S[\phi^{I},\omega^{+IJ}_{\mu},\omega^{-IJ}_{\mu}]$
$\displaystyle=\int
d^{4}x\,\tilde{\epsilon}^{\mu\nu\alpha\beta}\epsilon_{IJKL}D_{\mu}\phi^{I}D_{\nu}\phi^{J}\bigg{(}g_{+}R^{KL}_{\phantom{KL}\alpha\beta}(\omega^{+})+g_{-}R^{KL}_{\phantom{KL}\alpha\beta}(\omega^{-})\bigg{)}$
(51)

where

$\displaystyle g_{\pm}$ $\displaystyle=\frac{\alpha}{2}\bigg{(}\frac{\gamma\mp
i}{\gamma}\bigg{)}$ (52)

Equivalently, we can express the two free constants $(\alpha,\gamma)$ in (50)
in terms of $(g_{+},g_{-})$ as:

$\displaystyle\alpha$ $\displaystyle=g_{+}+g_{-}$ (53) $\displaystyle\gamma$
$\displaystyle=i\bigg{(}\frac{g_{+}+g_{-}}{g_{-}-g{+}}\bigg{)}$ (54)

As in the case of (47) and (49), we will find that the overall multiplicative
constant $\alpha$ will be related to a multiple of the inverse of Newton’s
constant when a General-Relativistic limit of the model exists. We will
henceforth set $\alpha=1$ and consider this as a choice of units and not a
restriction on the space of theories.

By way of comparison the Palatini Lagrangian density (47) of Einstein-Cartan
theory can be generalized to:

$\displaystyle\tilde{\cal L}[e^{I}_{\mu},\omega^{\pm IJ}_{\mu}]$
$\displaystyle=\tilde{\epsilon}^{\mu\nu\alpha\beta}\epsilon_{IJKL}e_{\mu}^{I}e_{\nu}^{J}\bigg{(}\bar{g}_{+}R^{KL}_{\phantom{KL}\alpha\beta}(\omega^{+})+\bar{g}_{-}R^{KL}_{\phantom{KL}\alpha\beta}(\omega^{-})\bigg{)}$
(55)

With the identification $\bar{g}_{\pm}=(\bar{\alpha}/2)(\bar{\gamma}\mp
i)/\bar{\gamma}$ \- with $\bar{\gamma}$ being the Barbero-Immirzi parameter
and $\bar{\alpha}=1/(32\pi G)$\- this is equal to the Holst Lagrangian density
for gravity [7]. Formally the Lagrangian density (51) can be recovered from
(55) via the replacement $e^{I}_{\mu}\rightarrow
D_{\mu}\phi^{I}$444Interestingly, an approach based on an $SO(1,3)$ connection
$\omega^{IJ}_{\mu}$, a field $\phi^{I}$, and an _independent_ field
$e^{I}_{\mu}$ yields a variety of novel phenomenology [18].. The Palatini
action corresponds to
$(\bar{g}_{+}=\bar{\alpha}/2,\bar{g}_{-}=\bar{\alpha}/2)$
($\bar{\gamma}\rightarrow\infty)$ whilst the chiral Ashtekar theory
corresponds to cases $(\bar{g}_{+}=1,\bar{g}_{-}=0)$ ($\bar{\gamma}=i)$ and
$(\bar{g}_{+}=0,\bar{g}_{-}=1)$ ($\bar{\gamma}=-i)$ respectively. For general
values of $\bar{\gamma}$, Einstein’s theory can be recovered given suitable
reality conditions placed on the fields
$(e^{I}_{\mu},\omega^{IJ}_{\mu}=\omega^{+IJ}_{\mu}+\omega^{-IJ}_{\mu})$. In
contrast, we will find that the number of degrees of freedom for the models
(51) depends crucially on the values of $(g_{+},g_{-})$. The value
$\bar{\gamma}$ does affect the interaction between gravity and fermionic
matter [19, 20] and even in the absence of matter may be physically relevant
upon quantization of (55) [21].

The focus of this paper will be to develop the canonical formulation of the
action (51).

##  4 3+1 decomposition of Lagrangian density and canonical formulation

We now proceed to perform the 3+1 decomposition of the Lagrangian density for
the action (51). Motivated by the 3+1 decomposition of a spacetime one-form
introduced in (13), we introduce the following fields:

$\displaystyle\omega^{\pm IJ}_{\phantom{\pm IJ}\mu}dx^{\mu}$
$\displaystyle=\Omega^{\pm IJ}dt+\beta^{\pm IJ}_{\phantom{\pm IJ}a}dx^{a}$
(56)

Furthermore, for notational compactness, we introduce the following
quantities:

$\displaystyle R^{\pm IJ}_{ab}$ $\displaystyle\equiv
2\bigg{(}\partial_{[a}\beta^{\pm IJ}_{\phantom{IJ}b]}+\beta^{\pm
IK}_{\phantom{CE}[a|}\beta_{K\phantom{B}|b]}^{\pm J}\bigg{)}$ (57)
$\displaystyle e^{I}_{a}$
$\displaystyle\equiv\partial_{a}\phi^{I}+\beta^{I}_{\phantom{I}Ja}\phi^{J}$
(58)

where $\beta^{IJ}_{a}=\beta^{+IJ}_{a}+\beta^{-IJ}_{a}$. Using the
decomposition (56) in (51) we note that fields that appear with time
derivatives (the fields $(\phi^{I},\beta^{\pm IJ}_{\phantom{\pm IJ}a})$)
appear linearly in those time derivatives. Recalling the discussion in Section
2.2, the extended Lagrangian density can be constructed by auxiliary
introducing ‘velocity’ fields - here $(V^{I},V_{a}^{\pm IJ})$ \- which are
constrained to be equal to the time derivatives of $(\phi^{I},\beta_{a}^{\pm
IJ})$ on-shell, with this resulting as an equation of motion obtained by
varying Lagrange multiplier fields $(\tilde{P}_{I},\tilde{P}^{\pm a}_{IJ})$.
As the original time derivatives of fields appeared linearly, it follows that
$(V^{I},V_{a}^{\pm IJ})$ equations of motion cannot be used to determine
$(V^{I},V_{a}^{\pm IJ})$ and so these fields play the role of Lagrange
multipliers ensuring constraints amongst the collected fields
$(\phi^{I},\beta_{a}^{\pm IJ},\tilde{P}_{I},\tilde{P}^{\pm a}_{IJ})$. The
remaining fields in the variational problem - $\Omega^{\pm IJ}$ \- appear
without time derivatives and appear linearly in the action. The extended
Lagrangian density can be cast into the following form:

$\displaystyle\tilde{\cal L}$
$\displaystyle=\tilde{P}_{I}\dot{\phi}^{I}+\tilde{P}^{+c}_{IJ}\dot{\beta}^{+IJ}_{\phantom{CD}c}+\tilde{P}_{IJ}^{-c}\dot{\beta}^{-IJ}_{\phantom{IJ}c}-\tilde{\cal
H}+\partial_{a}\tilde{\ell}^{a}$ (59)

Where

$\displaystyle\tilde{\cal H}$ $\displaystyle=-\Omega^{+IJ}\tilde{\cal
G}^{+}_{IJ}-\Omega^{-IJ}\tilde{\cal
G}^{-}_{IJ}+V^{I}\tilde{C}_{I}+V^{+IJ}_{a}\tilde{C}^{+a}_{IJ}+V^{-IJ}_{a}\tilde{C}^{-a}_{IJ}$
(60)

$\displaystyle\tilde{\cal G}^{+}_{IJ}$ $\displaystyle\equiv
D^{(\beta^{+})}_{c}\tilde{P}^{+c}_{IJ}+\big{[}\tilde{P}_{[I}\phi_{J]}\big{]}^{+}$
(61) $\displaystyle\tilde{\cal G}^{-}_{IJ}$ $\displaystyle\equiv
D^{(\beta^{-})}_{c}\tilde{P}^{-c}_{IJ}+\big{[}\tilde{P}_{[I}\phi_{J]}\big{]}^{-}$
(62) $\displaystyle\tilde{C}_{I}$
$\displaystyle=\tilde{P}_{I}-2g_{+}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{J}_{a}R_{bc}^{+KL}-2g_{-}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{J}_{a}R^{-KL}_{bc}$
(63) $\displaystyle\tilde{C}^{+c}_{IJ}$
$\displaystyle=\tilde{P}_{IJ}^{+c}-2g_{+}\big{[}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{K}_{a}e^{L}_{b}\big{]}^{+}$
(64) $\displaystyle\tilde{C}^{-c}_{IJ}$
$\displaystyle=\tilde{P}_{IJ}^{-c}-2g_{-}\big{[}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e_{a}^{K}e_{b}^{L}\big{]}^{-}$
(65)

where we’ve introduced the ‘spatial covariant derivative’ with respect to any
of ${\cal B}=\\{\beta,\beta^{+},\beta^{-}\\}$:

$\displaystyle D^{(\cal B)}_{a}Y^{AB\dots}_{\phantom{AB\dots}CD\dots}$
$\displaystyle\equiv\partial_{a}Y^{AB\dots}_{\phantom{AB\dots}CD\dots}+{\cal
B}^{A}_{\phantom{A}Ea}Y^{EB\dots}_{\phantom{AB\dots}CD\dots}+{\cal
B}^{B}_{\phantom{A}Ea}Y^{AE\dots}_{\phantom{AB\dots}CD\dots}+\dots$
$\displaystyle-{\cal
B}^{E}_{\phantom{E}Ca}Y^{AB\dots}_{\phantom{AB\dots}ED\dots}-{\cal
B}^{E}_{\phantom{E}Da}Y^{AB\dots}_{\phantom{AB\dots}CE\dots}+\dots$ (66)

The total derivative $\partial_{a}\tilde{\ell}^{a}$ will be neglected as an
ignorable boundary term. Recall that the fields $(e^{I}_{a},R^{\pm IJ}_{ab})$
are themselves composed of $(\phi^{I},\beta^{\pm IJ}_{a})$ and their spatial
derivatives and so the Lagrangian density (59) can be interpreted as an action
in canonical form with a phase space coordinatized by canonical pairs
$(\phi^{I},\tilde{P}_{I})$ and $(\beta^{\pm IJ}_{a},\tilde{P}^{\pm a}_{IJ})$
with a primary Hamiltonian density $\tilde{\cal H}$ comprised entirely of
constraints $(\tilde{\cal G}^{\pm}_{IJ},\tilde{\cal C}_{I},\tilde{\cal C}^{\pm
a}_{IJ})$ enforced by stationarity of the action with respect to small
variations of Lagrange multiplier fields $(\Omega^{\pm IJ},V^{I},V^{\pm
IJ}_{a})$.

###  4.1 The evolution of constraints

For functions $F$ and $G$ depending on phase space variables we can define the
Poisson bracket as follows:

$\displaystyle\\{F,G\\}\equiv\int d^{3}x\bigg{[}\frac{\delta
F}{\delta\beta^{+IJ}_{d}}\frac{\delta
G}{\delta\tilde{P}^{+d}_{IJ}}+\frac{\delta
F}{\delta\beta^{-IJ}_{d}}\frac{\delta
G}{\delta\tilde{P}^{-d}_{IJ}}+\frac{\delta F}{\delta\phi^{I}}\frac{\delta
G}{\delta\tilde{P}_{I}}\bigg{]}-(F\leftrightarrow G)$ (67)

Explicit forms for functional derivatives with respect to the phase space
fields are given in Appendix B. Time evolution of fields
$(\phi^{I},\beta_{a}^{\pm IJ},\tilde{P}_{I},\tilde{P}^{\pm a}_{IJ})$ are
obtained from the Euler-Lagrange equations following from variation of (59).
Therefore for a function $F$ of these fields then

$\displaystyle\dot{F}$ $\displaystyle=\\{F,\int d^{3}x\tilde{\cal H}\\}$ (68)

Finally, it will be useful to introduce the notion of smearing of phase space
functions. For a phase space function $F$, its smearing with a test function
$\xi(x)$ (i.e. a function which does not depend on the phase space fields) is
defined as:

$\displaystyle F[\xi]$ $\displaystyle\equiv\int d^{3}x\xi F$ (69)

We require that time evolution according to (68) preserves the set of
constraints $(\tilde{\cal G}^{\pm}_{IJ},\tilde{\cal C}_{I},\tilde{\cal C}^{\pm
a}_{IJ})$. For illustrative purposes, a detailed example of the evaluation of
the Poisson bracket of two constraints is presented in Appendix C. It turns
out that preservation of ${\cal G}^{\pm}_{IJ}$ under time evolution is ensured
if the primary constraints are satisfied - indeed these constraints generate
self and anti-self dual Lorentz transformations, and hence the Poisson bracket
of these constraints with any other constraint (when both constraints are
smeared) will be proportional to constraints up to boundary terms that we do
not consider in this analysis. For the remaining constraints $(\tilde{\cal
C}_{I},\tilde{\cal C}^{\pm a}_{IJ})$ we recover the following equations:

$\displaystyle\partial_{t}\tilde{C}_{I}$
$\displaystyle\approx-V^{+KL}_{a}W^{a+}_{KLI}-V^{-KL}_{a}W^{a-}_{KLI}$ (70)
$\displaystyle\partial_{t}\tilde{C}_{IJ}^{+e}$
$\displaystyle\approx\bigg{[}g_{+}Y^{de}_{IJKL}+g_{+}Y^{de}_{KLIJ}\bigg{]}^{+}V_{d}^{+KL}+\bigg{[}g_{+}Y^{de}_{IJKL}+g_{-}Y^{de}_{KLIJ}\bigg{]}^{+}V_{d}^{-KL}$
$\displaystyle+W^{e+}_{IJM}V^{M}$ (71)
$\displaystyle\partial_{t}\tilde{C}_{IJ}^{-e}$
$\displaystyle\approx\bigg{[}g_{-}Y^{de}_{IJKL}+g_{-}Y^{de}_{KLIJ}\bigg{]}^{-}V_{d}^{-KL}+\bigg{[}g_{-}Y^{de}_{IJKL}+g_{+}Y^{de}_{KLIJ}\bigg{]}^{-}V^{+KL}_{d}$
$\displaystyle+W^{e-}_{IJM}V^{M}$ (72)

where $\approx$ denotes weak equality (i.e. equality up to the addition of
constraints) and where we’ve defined the following:

$\displaystyle Y^{de}_{IJKL}$ $\displaystyle=Y^{de}_{[IJ][KL]}\equiv
4\tilde{\varepsilon}^{dbe}\epsilon_{MIJ[K}\phi_{L]}e^{M}_{b}$ (73)
$\displaystyle W^{e\pm}_{IJK}$
$\displaystyle\equiv\bigg{[}2(g_{\mp}-g_{\pm})\tilde{\varepsilon}^{ebc}\epsilon_{KMN[I}\phi_{J]}R^{\mp
MN}_{bc}+g_{\pm}\tilde{\varepsilon}^{ebc}\phi_{K}\epsilon_{IJMN}R^{\pm
MN}_{bc}\bigg{]}^{\pm}$ (74)

This object is a tensor in the space coordinatized by antisymmetric Lorentz
tensors. Now, adding the projection of (71) along $V^{+IJ}_{e}$ to the
projection of (72) along $V^{-IJ}_{e}$ we obtain:

$\displaystyle
V^{+IJ}_{e}\partial_{t}\tilde{C}^{+e}_{IJ}+V^{-IJ}_{e}\partial_{t}\tilde{C}^{-e}_{IJ}$
$\displaystyle=-V^{I}\partial_{t}\tilde{C}_{I}$ (75)

It is shown in Appendix D that $V^{I}$ contains information about the lapse
($N$) and shift ($N^{a}$) functions from the 3+1 decomposition of the
spacetime metric (14) and whose functional forms are arbitrary insofar as they
reflect the freedom to foliate spacetime in different ways. As such, (75) can
be taken to imply that generally the preservation of constraints
$\tilde{C}^{\pm a}_{IJ}$ under time evolution implies the preservation of
$\tilde{C}_{I}$.

To proceed, it will be useful to explicitly work out self-dual and anti-self
dual projections of indices of the object (73), for which explicit expressions
are given in equation (157) in Appendix E. Using the decomposition of self and
anti-self-dual Lorentz tensors defined in (33) and making use of the vector
$N^{I}$ introduced in (39) we can define the objects ${\cal V}^{\pm I}_{a}$ as
follows

$\displaystyle V_{a}^{\pm IJ}$ $\displaystyle=\big{[}N^{[I}{\cal V}^{\pm
J]}_{a}\big{]}^{\pm}$ (76)

where ${\cal V}^{\pm I}_{a}N_{I}=0$. We would like to find out whether the
constraint propagation equations (71) and (72) for vanishing left-hand side
amount to equations which uniquely determine ${\cal V}^{\pm I}_{a}$ Projecting
these equations along $N^{I}$ we have:

$\displaystyle 0$
$\displaystyle\approx\phantom{-}ig_{+}\tilde{\varepsilon}^{dbe}\frac{1}{2}\partial_{b}\phi^{2}\eta_{IJ}{\cal
V}_{d}^{+J}+(g_{+}-g_{-})N^{L}N^{J}Y^{de}_{[LI]^{+}[JK]^{-}}{\cal
V}_{d}^{-K}+W^{e+}_{KIJ}N^{K}V^{J}$ (77) $\displaystyle 0$
$\displaystyle\approx-
ig_{-}\tilde{\varepsilon}^{dbe}\frac{1}{2}\partial_{b}\phi^{2}\eta_{IJ}{\cal
V}_{d}^{-J}+(g_{-}-g_{+})N^{L}N^{J}Y^{de}_{[LI]^{-}[JK]^{+}}{\cal
V}_{d}^{+K}+W^{e-}_{KIJ}N^{K}V^{J}$ (78)

where we’ve used the fact that
$\phi_{K}e^{K}_{b}=\frac{1}{2}\partial_{b}\phi^{2}$.

####  4.1.1 The special case $g_{+}=g_{-}=1/2$

A number of terms in (80) and (81) vanish when $g_{+}=g_{-}$, substantially
simplifying the equations. If we further define

$\displaystyle R^{\pm IJ}$ $\displaystyle=\big{[}N^{[I}{\cal R}_{bc}^{\pm
J]}\big{]}^{\pm}$ (79)

where $N_{I}{\cal R}^{\pm I}_{bc}=0$ then (80) and (81) take the form:

$\displaystyle 0$
$\displaystyle\approx-\tilde{\varepsilon}^{dbe}\partial_{b}\phi^{2}\delta_{IJ}{\cal
V}_{d}^{+J}+\tilde{\varepsilon}^{ebc}{\cal R}^{+}_{Ibc}\phi_{J}V^{J}$ (80)
$\displaystyle 0$
$\displaystyle\approx-\tilde{\varepsilon}^{dbe}\partial_{b}\phi^{2}\delta_{IJ}{\cal
V}_{d}^{-J}+\tilde{\varepsilon}^{ebc}{\cal R}^{-}_{Ibc}\phi_{J}V^{J}$ (81)

where $\delta_{IJ}=\eta_{IJ}+N_{I}N_{J}$ and recall that
$\phi^{2}\equiv\phi_{I}\phi^{I}$. Equations (80) and (81) can be regarded as a
pair of linear inhomogeneous equations, involving either ${\cal V}^{+I}_{a}$
or ${\cal V}^{-I}_{a}$, each of which can be regarded as a 9 dimensional
vector. The quantity
$M^{de}_{\phantom{de}IJ}=\tilde{\varepsilon}^{dbe}\partial_{b}\phi^{2}\delta_{IJ}$
that multiplies these vectors in each equation can be thought of as a $9\times
9$ matrix. If this matrix is invertible then equations (80) and (81) uniquely
determine ${\cal V}^{\pm I}_{a}$. However, the matrix has the following three
null-eigenvectors

$\displaystyle\partial_{a}\phi^{2}S^{I(i)}$ (82)

where $i=1,2,3$ and $S^{I(i)}N_{I}=0$. This suggests that the matrix
$M^{de}_{\phantom{de}IJ}$ is not invertible and not all ${\cal V}^{\pm I}_{a}$
can be determined from these equations. Acting on (80) and (81) with these
null eigenvectors we obtain

$\displaystyle 0=\tilde{\varepsilon}^{ebc}R^{\pm IJ}_{bc}\partial_{e}\phi^{2}$
(83)

However, these are not new constraints on the phase space. This is due to the
following identity:

$\displaystyle D^{(\beta^{\pm})}_{c}\tilde{C}^{\pm c}_{IJ}$
$\displaystyle\approx\tilde{\cal
G}^{\pm}_{IJ}+ig_{\pm}\tilde{\epsilon}^{bca}R^{\pm}_{IJca}\partial_{b}\phi^{2}$
(84)

i.e. the primary constraints imply (83). The existence of null eigenvectors of
$M^{ab}_{\phantom{ab}IJ}$ shows that not all components of ${\cal V}^{\pm
I}_{a}$ are determined by the propagation constraint equations; however, some
can be solved for. Acting on the propagation equations with
$\utilde{\epsilon}_{fea}\partial^{a}\phi^{2}\eta^{KI}$ (where
$\partial^{a}\phi^{2}\equiv q^{ab}\partial_{b}\phi^{2}$) and introducing the
projector: ${\cal
P}^{a}_{\phantom{a}b}=\delta^{a}_{b}-\frac{\partial_{b}\phi^{2}\partial^{a}\phi^{2}}{(\partial_{c}\phi^{2}\partial^{c}\phi^{2})}$,
the equations can be solved to yield:

$\displaystyle\bar{\cal V}^{+I}_{a}$ $\displaystyle\equiv{\cal
P}^{b}_{\phantom{d}a}{\cal
V}_{b}^{+I}\approx\frac{2}{(\partial_{b}\phi^{2}\partial^{b}\phi^{2})}\partial^{c}\phi^{2}\phi_{J}{\cal
R}^{+I}_{ca}V^{J}$ $\displaystyle\bar{\cal V}^{-I}_{a}$
$\displaystyle\equiv{\cal P}^{b}_{\phantom{d}a}{\cal
V}_{b}^{-I}\approx\frac{2}{(\partial_{b}\phi^{2}\partial^{b}\phi^{2})}\partial^{c}\phi^{2}\phi_{J}{\cal
R}^{-I}_{ca}V^{J}$ (85)

where we have introduced the convention that barred Lagrange multipliers
denote multipliers that have been solved for in terms of fields in phase
space. It will further be useful to introduce the symbols

$\displaystyle\bar{V}^{\pm IJ}_{a}$ $\displaystyle=\big{[}N^{[I}{\cal V}^{\pm
J]}_{a}\big{]}^{\pm}=\bar{U}^{\pm IJ}_{\phantom{\pm IJ}Ka}V^{K}$ (86)

where by inspection $\bar{U}^{\pm IJ}_{\phantom{\pm
IJ}Ka}=\frac{2}{(\partial_{b}\phi^{2}\partial^{b}\phi^{2})}\partial^{c}\phi^{2}\phi_{K}R^{\pm
IJ}_{\phantom{\pm IJ}ca}$.

####  4.1.2 General case

We now look at the case where $g_{+}\neq g_{-}$. It can be seen that in the
general case, all components of the ${\cal V}^{\pm I}_{a}$ can be solved for
if the following $9\times 9$ complex matrix is invertible:

$\displaystyle Y^{de}_{[KI]^{(1)+}[LJ]^{(2)-}}N^{K}N^{L}$
$\displaystyle=\frac{1}{2}\tilde{\varepsilon}^{dbe}(e^{L}_{b}\phi^{M}+e_{b}^{M}\phi^{L})N_{M}N^{K}\epsilon_{IJKL}-\frac{i}{2}\tilde{\varepsilon}^{dbe}e^{K}_{b}(\phi_{L}N^{L}N_{K}\delta_{IJ}-\phi_{J}\delta_{IK})$
$\displaystyle-\frac{i}{2}\tilde{\varepsilon}^{dbe}e^{K}_{b}((\phi_{L}N^{L}N_{K}+\phi_{K})\delta_{IJ}-\phi_{I}\delta_{JK})$
(87)

Calculation shows that the determinant of the matrix (87) is generally non-
zero if $\mathrm{det}[q_{ab}]\neq 0$ and $\phi_{I}\phi^{I}\neq 0$. These
quantities can generally be assumed to be non-zero, hence in the case
$g_{+}\neq g_{-}$ the matrix inverse exists, and all components of ${\cal
V}^{\pm I}_{a}$ can be determined from the propagation of primary constraints
in terms of $(R^{+IJ}_{ab},R^{-IJ}_{ab},\phi^{I})$. We will again use the
convention that barred Lagrange multipliers denote their form when expressed
in terms of these phase space fields and their spatial derivatives. It will
further be useful to introduce the symbols

$\displaystyle\bar{V}^{\pm IJ}_{a}$ $\displaystyle=\bar{Z}^{\pm
IJ}_{\phantom{\pm IJ}Ka}V^{K}$ (88)

This expresses the formal solution for $\bar{V}^{\pm IJ}_{a}$ but we note that
a closed expression for $\bar{Z}^{\pm IJ}_{\phantom{\pm IJ}Ka}$ for general
values of $(g_{+},g_{-})$ may be difficult to obtain. This completes the
constraint analysis in the general case as no more constraints have been
generated. We can now determine Hamilton’s equations of motion, which for any
values of $(g_{+},g_{-})$ are given by:

$\displaystyle\dot{\phi}^{I}$
$\displaystyle=V^{I}-(\Omega^{+IJ}+\Omega^{-IJ})\phi_{J}$ (89)
$\displaystyle\dot{\beta}^{+IJ}_{c}$ $\displaystyle\approx
D^{(\beta^{+})}_{c}\Omega^{+IJ}+V^{+IJ}_{c}$ (90)
$\displaystyle\dot{\beta}^{-IJ}_{c}$ $\displaystyle\approx
D^{(\beta^{-})}_{c}\Omega^{-IJ}+V^{-IJ}_{c}$ (91)
$\displaystyle\dot{\tilde{P}}_{K}$ $\displaystyle\approx
D^{(\beta)}_{a}\leftmoon^{a}_{K}+(\Omega^{+I}_{\phantom{+I}K}+\Omega^{-I}_{\phantom{-I}K})\tilde{P}_{I}$
(92) $\displaystyle\dot{\tilde{P}}^{+a}_{KL}$
$\displaystyle\approx-4D_{b}^{(\beta^{+})}(V^{I}g_{+}\epsilon_{IJKL}\tilde{\epsilon}^{eba}e^{J}_{e})^{+}-(2\Omega^{+\phantom{L}J}_{\phantom{+}[L}\tilde{P}^{+a}_{K]J}+\leftmoon^{a}_{[K}\phi_{L]})^{+}$
(93) $\displaystyle\dot{\tilde{P}}^{-a}_{KL}$
$\displaystyle\approx-4D_{b}^{(\beta^{-})}(V^{I}g_{-}\epsilon_{IJKL}\tilde{\epsilon}^{eba}e^{J}_{e})^{-}-(2\Omega^{-\phantom{L}J}_{\phantom{-}[L}\tilde{P}^{-a}_{K]J}+\leftmoon^{a}_{[K}\phi_{L]})^{-}$
(94)

where we have defined

$\displaystyle\leftmoon_{K}^{a}$
$\displaystyle=2\big{(}V^{I}(g_{+}R^{+JP}_{bc}+g_{-}R^{-JP}_{bc})+2(V^{+IJ}_{b}g_{+}+V^{-IJ}_{b}g_{-})e^{P}_{c}\big{)}\epsilon_{IJKP}\tilde{\epsilon}^{abc}$
(95)

Weak equalities have been used to allow for the fact that the constraint
propagation analysis fixes some or all of $V^{\pm IJ}_{a}$ (depending on
whether $g_{+}=g_{-}$ or not) to explicitly depend on phase space fields.
Hence, additional terms involving derivatives of $V^{\pm IJ}_{a}$ with respect
to these fields will appear in Hamilton’s equations but they will all be
proportional to constraints $\tilde{C}^{\pm a}_{IJ}$ and so vanish on the
constraint surface.

We finally point out an exotic solution that has not been covered by the prior
analysis: that in which $\phi^{I}=0$ throughout spacetime. From (89) we see
that if $\phi^{I}=0$ initially then it will remain so only if $V^{I}=0$. From
the results of Appendix D we see that this implies that the function $N=0$ and
furthermore if $\phi^{I}=0$ then $q_{ab}=0$ and hence from equation (14) the
spacetime metric $g_{\mu\nu}=0$. Furthermore, if $\phi^{I}=0$ then
$Y^{de}_{IJKL}$ and $W^{e\pm}_{IJK}$ are zero and hence $V^{\pm KL}$ are
completely undetermined by the constraint propagation equations, thus implying
from (90) and (91) that the time evolution of fields $\beta^{\pm IJ}_{a}$ is
undetermined. It is unclear whether such solutions play any phenomenological
role.

###  4.2 The Algebra of Constraints

Having completed the calculation of propagation of constraints, we can now
classify the primary constraints in terms of whether they are first-class
constraints (i.e. their Poisson bracket with all other constraints weakly
vanishes) or second-class constraints (i.e. their Poisson bracket with some
constraints does not weakly vanish). The character of the constraints depends
on the values of $(g_{+},g_{-})$. Given a phase space with dimensionality $P$
per spatial point and a number of first-class constraints $F$ and second class
constraints $S$ per spatial point, the number of degrees of freedom per
spatial point is defined to be:

$\displaystyle DOF=\frac{1}{2}(P-2F-S)$ (96)

This number ultimately represents how many numbers per spatial point at an
initial time $t_{0}$ may be freely chosen as to generate a unique, physically
distinctive solution (as opposed to a solution that is equal to another one up
to a gauge transformation) to the equations of motion. As the theory is
generally complex, this may be a complex number - and we will refer to this
number as the number of complex degrees of freedom that the model possesses.
The choice of complex numbers within the initial data of fields may result in
complex fields in spacetime such as a complex spacetime metric $g_{\mu\nu}$.
The physical interpretation of an imaginary component to the spacetime metric
is not clear and so one path is to look to determine what choice of initial
data ensures the reality of fields such as the metric tensor. This question is
considered in detail in Sections 4.3 and 4.4.

####  4.2.1 Case $g_{+}\neq g_{-}$

For the general case $g_{+}\neq g_{-}$, the classification of constraints is
illustrated in Figure 1. Given the classification of constraints, we can now
count how many complex degrees of freedom the model possesses. The
dimensionality of the phase space per spatial point is

$P=8\,(\phi^{I},\tilde{P}_{I})+18\,(\beta^{+IJ}_{a},\tilde{P}^{+a}_{IJ})+18\,(\beta^{-IJ}_{a},\tilde{P}^{-a}_{IJ})=44$

The number of first-class constraints is

$F=3\,(\tilde{G}^{+IJ})+3\,(\tilde{G}^{-IJ})+4\,(\tilde{\mathscr{H}}^{I})=10$

and the number of second-class constraints is

$S=9\,(\tilde{C}^{+a}_{IJ})+9\,(\tilde{C}^{-a}_{IJ})=18$

The number of degrees of freedom per spatial point is therefore

$\displaystyle DOF$ $\displaystyle=\frac{1}{2}(P-2F-S)$ $\displaystyle=3$ (97)

pt$\tilde{C}^{I}$pt$\tilde{\cal{G}}^{+IJ}$pt$\tilde{\cal{G}}^{-IJ}$pt$\tilde{C}^{+a}_{IJ}$pt$\tilde{C}^{-a}_{IJ}$pt$\tilde{\mathscr{H}}^{I}=\tilde{C}^{I}+\bar{Z}^{+JKI}_{\phantom{+JKI}a}\tilde{C}^{+a}_{JK}+\bar{Z}^{-JKI}_{\phantom{-JKI}a}\tilde{C}^{-a}_{JK}$ptptpt
Figure 1: The structure of constraints in the case $g_{+}\neq g_{-}$. First-
class constraints are blue whilst constraints that are individually second-
class are shown as green. The constraint analysis reveals that a linear
combination of second-class constraints yields the first-class constraints
$\tilde{\mathscr{H}}^{I}$.

####  4.2.2 Case $g_{+}=g_{-}$

For the special case $g_{+}=g_{-}$, the classification of constraints is
illustrated in Figure 2. As in the previous case, the dimensionality of the
phase space per spatial point is

$P=8\,(\phi^{I},\tilde{P}_{I})+18\,(\beta^{+IJ}_{a},\tilde{P}^{+a}_{IJ})+18\,(\beta^{-IJ}_{a},\tilde{P}^{-a}_{IJ})=44$

However, now the number of first-class constraints is

$F=3\,(\tilde{G}^{+IJ})+3\,(\tilde{G}^{-IJ})+4\,(\tilde{\mathscr{H}}^{I})+3(\partial_{a}\phi^{2}\tilde{C}^{+a}_{IJ})+3\,(\partial_{a}\phi^{2}\tilde{C}^{-a}_{IJ})=16$

and the number of second-class constraints is

$S=6\,({\cal P}^{a}_{\phantom{a}b}\tilde{C}^{+b}_{IJ})+6\,({\cal
P}^{a}_{\phantom{a}b}\tilde{C}^{-b}_{IJ})=12$

The number of degrees of freedom per spatial point is therefore

$\displaystyle DOF$ $\displaystyle=\frac{1}{2}(P-2F-S)$ $\displaystyle=0$ (98)

pt$\tilde{C}^{I}$pt$\tilde{\cal{G}}^{+IJ}$pt$\tilde{\cal{G}}^{-IJ}$pt${\cal
P}^{a}_{\phantom{a}b}\tilde{C}^{+b}_{IJ}$pt$\partial_{a}\phi^{2}\tilde{C}^{+a}_{IJ}$pt${\cal
P}^{a}_{\phantom{a}b}\tilde{C}^{-b}_{IJ}$pt$\partial_{a}\phi^{2}\tilde{C}^{-a}_{IJ}$pt$\tilde{\mathscr{H}}^{I}=\tilde{C}^{I}+\bar{U}^{+JKI}_{\phantom{+JKI}a}\tilde{C}^{+a}_{JK}+\bar{U}^{-JKI}_{\phantom{-JKI}a}\tilde{C}^{-a}_{JK}$ptptpt
Figure 2: The structure of constraints in the case $g_{+}=g_{-}$. First class
constraints are blue whilst second class constraints are green. Unlike in the
case $g_{+}\neq g_{-}$, a subset of the individual $\tilde{C}^{\pm a}_{IJ}$
constraints are first class. As in the $g_{+}\neq g_{-}$ constraint analysis
reveals that a linear combination of individually second class constraints
yields the first class constraints $\tilde{\mathscr{H}}^{I}$.

Therefore the theory with $g_{+}=g_{-}$ possesses no degrees of freedom.
Indeed, this case has more first-class constraints than the case $g_{+}\neq
g_{-}$ and so it is to be expected that this specific case has more symmetry
than the general case. The precise additional symmetry that the theory
possesses compared to the case $g_{+}\neq g_{-}$ can be demonstrated in the
Lagrangian formalism. It is useful to write the action for general
$(g_{+},g_{-})$ in the language of differential forms:

$\displaystyle S$ $\displaystyle=2\int\epsilon_{IJKL}D\phi^{I}\wedge
D\phi^{J}\wedge\big{(}g_{+}R^{+KL}+g_{-}R^{-KL}\big{)}$ (99)

where we use $D$ to denote the covariant derivative $d+\omega$ according to
the entire spin connection $\omega=\omega^{+}+\omega^{-}$. Now consider the
following field transformation:

$\displaystyle\phi^{I}\rightarrow\phi^{I},\quad\omega^{IJ\pm}_{\mu}\rightarrow\omega^{IJ\pm}_{\mu}+\partial_{\mu}\phi^{2}\xi^{\pm
IJ}$ (100)

Under (100) the action changes as:

$\displaystyle\delta S$ $\displaystyle=\int
4(g_{+}-g_{-})\epsilon_{IJKL}d\phi^{2}\wedge
D\phi^{J}\wedge\bigg{(}\xi^{-IM}R^{+KL}+\xi^{+IM}R^{-KL}\bigg{)}\phi_{M}$
(101) $\displaystyle+2D\bigg{(}\epsilon_{IJKL}d\phi^{2}\wedge D\phi^{I}\wedge
D\phi^{J}(g_{+}\xi^{+KL}+g_{-}\xi^{-KL})\bigg{)}$ (102)

Therefore in the case $g_{+}=g_{-}$ and only in this case does the action
change by a total derivative - and hence boundary term - under the
transformation of fields (100). This result holds even ‘off-shell’ and
therefore the field transformation is a symmetry of the theory [22]. Note that
the transformation (100) when applied to the pullback of $\omega^{\pm
IJ}_{\mu}$ to surfaces of constant time $\beta^{\pm IJ}_{a}$ agrees with the
transformation generated by the first class constraints
$\partial_{a}\phi^{2}\tilde{C}^{\pm a}_{IJ}$ i.e.

$\displaystyle\delta\beta^{\pm IJ}_{a}$ $\displaystyle=\\{\beta^{\pm
IJ}_{a},\partial_{b}\phi^{2}\tilde{C}^{\pm b}_{KL}[\xi^{\pm KL}]\\}$
$\displaystyle=\partial_{a}\phi^{2}\xi^{\pm IJ}$ (103)

###  4.3 Reality conditions

We have seen that models with $g_{+}\neq g_{-}$ possess three complex degrees
of freedom according to the definition of Section 4.2. Because of the inherent
complexity of the theory, in principle, the Hamiltonian may generate classical
time evolution of fields to become complex even if initially real at some
moment $t=t_{0}$. A standard requirement is that the spacetime metric be real.
From (14) this is ensured if fields $(N,N^{a},q_{ab})$ are real. From Appendix
D it’s clear that $(N,N^{a})$ are real if $V^{I}$ and $e^{I}_{a}$ are real
555The complex-valued fields that coordinatize the phase space may combine
with complex $V^{I}$ to produce real four-dimensional metrics of Euclidean
signature but we do not explore that possibility in this work. We will require
that $V^{I}$ is real and that $q_{ab}=\eta_{IJ}e^{I}_{a}e^{J}_{b}$ be
initially real and for this realness to be preserved by time evolution.
Additionally, anticipating that the norm $\phi^{2}=\phi_{I}\phi^{I}$ will have
physical significance, this quantity should also be required to be real. Time
evolution is generated by the Hamiltonian

$\displaystyle H=-\tilde{\cal G}^{+}_{IJ}[\Omega^{+IJ}]-\tilde{\cal
G}^{-}_{IJ}[\Omega^{-IJ}]+\tilde{\cal C}_{I}[V^{I}]+\tilde{\cal
C}^{+d}_{IJ}[V_{d}^{+IJ}]+\tilde{\cal C}^{-d}_{IJ}[V_{d}^{-IJ}]$ (104)

Then, recalling the definition (58) of $e^{I}_{b}$ we have for the general
case $g_{+}\neq g_{-}$ that

$\displaystyle\partial_{t}q_{ab}$
$\displaystyle=2\eta_{IJ}e^{I}_{a}\\{\partial_{b}\phi^{J}+\beta^{J}_{\phantom{J}Kb}\phi^{K},H\\}$
$\displaystyle=2\eta_{IJ}e^{I}_{a}\big{(}-\partial_{b}V^{J}+(\bar{Z}^{+JKL}_{b}+\bar{Z}^{-JKL}_{b})V_{L}\phi_{K}+\beta^{J}_{\phantom{J}Kb}V^{K}\big{)}$
(105) $\displaystyle\partial_{t}\phi^{2}$
$\displaystyle=2\phi_{I}\\{\phi^{I},H\\}$ $\displaystyle=2\phi_{I}V^{I}$ (106)

From (106), we see that given our assumption that $V^{I}$ is real then an
initially real $\phi^{2}$ remains real if $\phi_{I}$ is real. In the general
case, it’s likely not possible to find closed expressions for
$(\bar{Z}^{+JKL}_{b},\bar{Z}^{-JKL}_{b})$. However, they will depend on the
generally complex $(g_{+},g_{-})$ which may create an imaginary part to
$\partial_{t}q_{ab}$ even if initial data for the phase space fields is real.
It is challenging to determine in the case of general $(g_{+},g_{-})$ whether
maintaining the reality of $q_{ab}$ generates further constraints on the
complex phase space. We will see, however, that in the special cases
$(g_{+}=1,g_{-}=0)$ and $(g_{+}=0,g_{-}=1)$ that contact with familiar results
from the Ashtekar model is possible. First, it is helpful to illustrate a
manifestation of the challenge of finding reality conditions in the general
case in a simple physical example: the propagation of linear metric
perturbations on a Minkowski space background.

###  4.4 The propagation of metric perturbations on Minkowski space

The Euler-Lagrange equations following from the Lagrangian density (50) have a
solution $R^{IJ}_{\phantom{IJ}\mu\nu}(\omega)=0$ [5]. Thus a gauge can be
found where $\omega^{IJ}_{\mu}\overset{*}{=}0$ and
$g_{\mu\nu}\overset{*}{=}\eta_{IJ}\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{J}$.
If $R^{IJ}_{\phantom{IJ}\mu\nu}(\omega)=0$ then it turns out that $\phi^{I}$
is otherwise undetermined by the field equations and hence $\phi^{I}$ can take
a profile where it forms a set of Minkowski coordinate such that
$g_{\mu\nu}=\eta_{\mu\nu}$ i.e. Minkowski space is a solution to the theory
for general values of $\gamma$. Now we restrict ourselves to a wedge where
$\phi^{2}=\eta_{IJ}\phi^{I}\phi^{J}<0$ and adopt a Lorentz gauge where
$\bar{\phi}^{I}\overset{*}{=}T\delta^{I}_{0}$. Then, using $T$ as a time
coordinate and denoting $x^{a}$ as spatial coordinates on the surface of
constant time we have

$\displaystyle ds^{2}$ $\displaystyle=-dT\otimes
dT+T^{2}\delta_{ij}E^{i}_{a}E^{j}_{b}dx^{a}\otimes dx^{b}$ (107)

where $i,j=1,2,3$ index spatial coordinates,
$E^{1}_{a}dx^{a}=d\chi,E^{2}_{a}dx^{a}=\sinh(\chi)d\theta,E^{3}_{a}dx^{a}=\sinh(\chi)\sin(\theta)d\phi$.
Therefore we can identify the region $\phi_{I}\phi^{I}<0$ in Minkowski
spacetime coordinatized by $(t,x^{a})$ as an open Friedmann-Robertson-Walker
universe with scale factor $a=T$. This is a Milne wedge and without loss of
generality, we choose the upper Milne wedge of Minkowski spacetime. Now
consider the following small perturbations to the spin connection

$\displaystyle\delta\omega^{0i}$
$\displaystyle=\frac{1}{2}H^{i}_{\phantom{i}j}E^{j}$ (108)
$\displaystyle\delta\omega^{ij}$
$\displaystyle=a^{ij}dt+\epsilon^{ijl}W_{l}^{\phantom{g}k}E_{k}$ (109)

where $(H^{ij},a^{ij},W^{ij})$ are symmetric and traceless. These
perturbations produce a perturbed spatial metric:

$\displaystyle\delta g_{ab}=H^{jk}E_{ja}E_{kb}$ (110)

It can be shown that the perturbation $(a_{ij},W_{ij})$ can be solved for
algebraically in the field equations in terms of first derivatives of $H_{ij}$
and so eliminated from the variational principle to recover the following
Lagrangian density

$\displaystyle\frac{1}{2}\tilde{\cal L}_{(H)}$
$\displaystyle=a^{3}\bigg{(}-\frac{1}{\gamma^{2}}\dot{H}_{ij}\dot{H}^{ij}-\frac{1}{a^{2}}h^{ab}{\cal
D}_{a}{H}^{ij}{\cal D}_{b}{H}_{ij}-\frac{2k}{a^{2}}H_{ij}H^{ij}\bigg{)}$ (111)

where here $h^{ab}$ is the unperturbed inverse spatial metric on the surface
of constant $T$ \- with spatial curvature constant $k=-1$, $a=T$ \- is the
‘cosmological’ scale factor, and where $H_{ij}$ is assumed to have support in
the upper Milne wedge. In the case $\gamma^{2}=-1$ (which corresponds to
either $(g_{+}=1,g_{-}=0)$ or $(g_{+}=0,g_{-}=1)$) the Lagrangian density
reduces to that of General Relativity, describing the lightlike propagation of
the spin-2 perturbation $H_{ij}$ on this background. For all other values of
$\gamma$, the propagation of $H_{ij}$ is at a different speed, and it is
readily seen that if $\gamma^{2}$ has an imaginary component then an initially
real perturbation $H_{ij}(t=t_{0})$ will evolve by the equations of motion to
become complex and hence in this simple case the only way to preserve reality
of the spatial metric would be to constrain $H_{ij}(t=t_{0})=0$, showing that
the reality conditions can reduce the number of propagating degrees of
freedom.

##  5 An extension of General Relativity

We see from (111) that only for the case $\gamma^{2}=-1$ does the propagation
of gravitational waves correspond to that of General Relativity and we will
now focus on this case. The condition $\gamma^{2}=-1$ encompasses two
independent possibilities: $(g_{+}=1,g_{-}=0)$ and $(g_{+}=0,g_{-}=1)$. We
will first focus on the former case and later demonstrate how the latter case
can be straightforwardly recovered. When $(g_{+}=1,g_{-}=0)$, the primary
Hamiltonian density simplifies considerably. Recalling its general form

$\displaystyle\tilde{\cal H}$ $\displaystyle=-\Omega^{+IJ}\tilde{\cal
G}^{+}_{IJ}-\Omega^{-IJ}\tilde{\cal
G}^{-}_{IJ}+V^{I}\tilde{C}_{I}+V^{+IJ}_{a}\tilde{C}^{+a}_{IJ}+V^{-IJ}_{a}\tilde{C}^{-a}_{IJ},$
(112)

the constraints now simplify to:

$\displaystyle\tilde{\cal G}^{+}_{IJ}$ $\displaystyle\equiv
D^{(\beta^{+})}_{c}\tilde{P}^{+c}_{IJ}+\big{[}\tilde{P}_{[I}\phi_{J]}\big{]}^{+}$
(113) $\displaystyle\tilde{\cal G}^{-}_{IJ}$ $\displaystyle\equiv
D^{(\beta^{-})}_{c}\tilde{P}^{-c}_{IJ}+\big{[}\tilde{P}_{[I}\phi_{J]}\big{]}^{-}$
(114) $\displaystyle\tilde{C}_{I}$
$\displaystyle=\tilde{P}_{I}-2\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{J}_{a}R_{bc}^{+KL}$
(115) $\displaystyle\tilde{C}^{+c}_{IJ}$
$\displaystyle=\tilde{P}_{IJ}^{+c}-2\big{[}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{K}_{a}e^{L}_{b}\big{]}^{+}$
(116) $\displaystyle\tilde{C}^{-c}_{IJ}$ $\displaystyle=\tilde{P}_{IJ}^{-c}$
(117)

Given these simplifications, it is possible to algebraically solve for the
$(\beta^{-IJ}_{a},\tilde{P}^{-a}_{IJ})$ and eliminate them from the
variational problem. Firstly, from (117) we have $\tilde{P}_{IJ}^{-a}=0$.
Then, recalling the definition (58), the constraint $\tilde{C}^{+c}_{IJ}$ can
be regarded as an equation for which one can solve for
$\beta^{-IJ}_{a}=\beta^{-IJ}_{a}(\beta^{+},\tilde{P}^{+},\phi,\partial\phi)$.
Therefore the second-class constraints can be solved. Given these solutions,
the constraint $\tilde{\cal G}^{-}_{IJ}$ simplifies to
$\big{[}\tilde{P}_{[I}\phi_{J]}\big{]}^{-}=0$; if we decompose
$\tilde{P}^{I}=\tilde{\Pi}\phi^{I}+\tilde{P}_{\perp}^{I}$, where
$\tilde{P}_{\perp}^{I}\phi_{I}=0$, then $\tilde{\cal G}^{-}_{IJ}=0$ can be
taken to imply the solution $\tilde{P}_{\perp}^{I}=0$ which we now adopt.
Additionally the quantity $V^{I}\tilde{C}_{I}$ can be expressed in terms of
quantities $(\utilde{N}\equiv N/\sqrt{q},N^{a})$ and the remaining phase space
variables using a number of useful results detailed in Appendix E, so that the
gravitational Lagrangian density can be written:

$\displaystyle\tilde{\cal
L}[\tilde{P}^{+a}_{IJ},\beta^{+IJ}_{a},\tilde{\Pi},\phi^{2},\Omega^{+IJ},\utilde{N},N^{a}]$
$\displaystyle=\frac{1}{2}\tilde{\Pi}\dot{\phi^{2}}+\tilde{P}^{+a}_{IJ}\dot{\beta}^{+IJ}_{\phantom{CD}a}+\Omega^{+IJ}\bigg{(}D^{(\beta^{+})}_{a}\tilde{P}^{+a}_{IJ}\bigg{)}$
$\displaystyle-\utilde{N}\bigg{(}\xi\tilde{\Pi}\sqrt{q}\sqrt{-\phi^{2}+\frac{1}{4}q^{ab}\partial_{a}\phi^{2}\partial_{b}\phi^{2}}-\frac{1}{4}\tilde{P}^{+a}_{IK}\tilde{P}^{+bK}_{\phantom{+cK}J}R^{+IJ}_{\phantom{+IJ}ab}\bigg{)}$
$\displaystyle-N^{a}\bigg{(}\frac{1}{2}\tilde{\Pi}\partial_{a}\phi^{2}+\tilde{P}^{+b}_{IJ}F^{+IJ}_{ab}-\beta^{+IJ}_{a}D^{(\beta^{+})}_{b}\tilde{P}^{+b}_{IJ}\bigg{)}$
(118)

where $\sqrt{q}$ and $q^{ab}$ can be expressed in terms of
$\tilde{P}^{+a}_{IJ}$ using (159) and we have redefined
$\Omega^{+IJ}\rightarrow\Omega^{+IJ}+N^{a}\beta_{a}^{+IJ}$ so that the
constraint obtained from the $N^{a}$ equation of motion when smeared with a
field $\zeta^{a}$ generates (non-Lorentz covariant) spatial diffeomorphisms
$f\rightarrow f+{\cal L}_{\zeta}f$ on fields $f$ in the phase space
coordinatized by $(\phi^{2},\tilde{\Pi},\beta^{+IJ}_{a},\tilde{P}^{+a}_{IJ})$.
The Lagrangian density that we have recovered corresponds to the canonical
formulation of Ashtekar’s theory of gravity [15] coupled to a field
$\phi^{2}=\eta_{IJ}\phi^{I}\phi^{J}$ whose dynamics is classically that of a
pressureless perfect fluid when $\phi^{2}<0$ [23, 5]. If $\phi^{2}<0$ and
local coordinates are selected so that $\partial_{a}\phi^{2}$ for some region
of spacetime then the energy density of the fluid is of sign $\xi\tilde{\Pi}$
and so the choice of the sign of $\xi=\mp 1$ reflects the relative sign of the
energy density and $\tilde{\Pi}$. More generally, $\phi^{2}$ by definition is
not positive-definite and its equation of motion is:

$\displaystyle\frac{1}{2}\partial_{t}\phi^{2}$
$\displaystyle=N\xi\sqrt{-\phi^{2}+\frac{1}{4}q^{ab}\partial_{a}\phi^{2}\partial_{b}\phi^{2}}+\frac{1}{2}N^{a}\partial_{a}\phi^{2}$
(119)

The equation of motion for $\phi^{2}$ differs from the equation of motion for,
for example, the Higgs boson of the standard model; unlike that case,
$\partial_{t}\phi^{2}$ is independent of the field’s momentum $\tilde{\Pi}$
and generally the right-hand side of (119) will be non-zero whenever the
spacetime metric is non-degenerate meaning that generally the magnitude of
$\phi^{2}$ will vary throughout spacetime. Notably there do exist solutions to
the theory’s field equations where, furthermore, the sign of $\phi^{2}$ varies
throughout spacetime [5].

If instead, we had chosen the parameters $(g_{+}=0,g_{-}=1)$ we would have
instead recovered the _anti-_ self-dual formulation of Ashtekar’s theory
coupled to an effective matter component described by
$(\phi^{2},\tilde{\Pi})$, the Lagrangian density of which can be recovered
from (118) by the replacement of $(\beta^{+IJ}_{a},\tilde{P}^{+a}_{IJ})$ with
$(\beta^{-IJ}_{a},\tilde{P}^{-a}_{IJ})$.

###  5.1 The field $\phi^{2}$ as a standard of time

Can the degree of freedom $\phi^{2}=\eta_{IJ}\phi^{I}\phi^{J}$ act as a useful
‘clock field’ in physics? If we choose initial data such that
$\partial_{a}\phi^{2}|_{t=t_{0}}=0$ then the equation of motion for $\phi^{2}$
at the initial moment becomes:

$\displaystyle\frac{1}{2}\partial_{t}\phi^{2}$ $\displaystyle=\xi
N\sqrt{-\phi^{2}}$ (120)

The condition that $\partial_{a}\phi=0$ will be preserved if $N=N(t)$. Given
this condition, we can regard the integration of equation (120) as providing a
functional relation between $\phi^{2}$ and $t$. If $N=1/\xi$ (which, recall
from (14) implies that $t$ corresponds to proper time) then $\phi^{2}=-t^{2}$
is a solution. If instead $N=1/(2\xi\sqrt{t})$ then $\phi^{2}=-t$ is a
solution. Therefore we will associate a choice of $N(t)$ with a partial
spacetime gauge fixing ($N^{a}$ is left undetermined) and denote the
associated time coordinate as $t_{\phi}$. We may additionally solve the
Hamiltonian constraint for $\tilde{\Pi}$ in this gauge to yield the following
Lagrangian density:

$\displaystyle\tilde{\cal
L}[\tilde{P}^{+a}_{IJ},\beta^{+IJ}_{a},\Omega^{+IJ},N^{a}]$
$\displaystyle\overset{*}{=}\tilde{P}^{+a}_{IJ}\frac{\partial}{\partial
t_{\phi}}{\beta}^{+IJ}_{\phantom{CD}a}-{\cal H}_{Phys}$
$\displaystyle+\Omega^{+IJ}\bigg{(}D^{(\beta^{+})}_{a}\tilde{P}^{+a}_{IJ}\bigg{)}-N^{a}\bigg{(}\tilde{P}^{+b}_{IJ}F^{+IJ}_{ab}-\beta^{+IJ}_{a}D^{(\beta^{+})}_{b}\tilde{P}^{+b}_{IJ}\bigg{)}$
(121) $\displaystyle{\cal H}_{Phys}$
$\displaystyle=-\frac{N(t_{\phi})}{4\sqrt{q}}\tilde{P}^{+a}_{IK}\tilde{P}^{+bK}_{\phantom{+cK}J}R^{+IJ}_{\phantom{+IJ}ab}$
(122)

Therefore we recover an action principle with a phase space coordinatized by
fields $(\tilde{P}^{+a}_{IJ},\beta_{a}^{+IJ})$, with a physical Hamiltonian
density ${\cal H}_{Phys}$ and phase space constraints implemented by
stationarity of the action under variation of $(\Omega^{+IJ},N^{a})$. Note
that now $N$ appears not as an independent field but as a fixed function of
time and only in gauges $\partial_{t_{\phi}}N(t_{\phi})=0$ does the physical
Hamiltonian not have explicit time-dependence (as opposed to the implicit
time-dependence it possesses via its dependence on time-varying phase space
fields), which corresponds to the case where $\sqrt{-\phi^{2}}$ measures
metric proper time. The extension to General Relativity is encoded in the fact
that ${\cal H}_{Phys}$ is not constrained to vanish. Non-vanishing values
would be interpreted as an effective energy density of an additional dust-like
gravitating component in physics [5].

It is instructive to determine the reality conditions on fields evolving
according to the equations of motion that follow from (121). We can require
the spacetime metric to be initially real. This implies that $N$ should be
real, and we can furthermore impose that $N^{a}$ is real throughout spacetime.
Additionally, the spatial metric $q_{ab}$ should be required to be initially
real, and its reality should be preserved by time evolution. Equivalently, we
can require that the densitized inverse spatial metric remain real, which from
(159) amounts to requiring that $\tilde{P}^{+c}_{IJ}\tilde{P}^{+dIJ}$ remains
real. From Hamilton’s equations, we have that

$\displaystyle\frac{\partial}{\partial t_{\phi}}\tilde{P}^{a}_{IJ}$
$\displaystyle=-\frac{N(t_{\phi})}{2}D_{b}^{(\beta^{+})}\bigg{(}\frac{1}{\sqrt{q}}\tilde{P}_{[I|K}^{[b}\tilde{P}^{a]K}_{\phantom{a]K}|J]}\bigg{)}+2D_{b}\bigg{(}N^{[b}\tilde{P}^{a]}_{IJ}\bigg{)}$
(123)

and hence given the assumed reality of $(N,N^{a},q_{ab})$ at the initial time,
it follows that the following term must be real so that
$\tilde{P}^{+(a}_{IJ}\tilde{P}^{b)IJ}$ remains real:

$\displaystyle\tilde{P}^{+(a|IJ}D_{c}^{(\beta^{+})}\bigg{(}\frac{1}{\sqrt{q}}\bigg{(}\tilde{P}_{IK}^{+c}\tilde{P}^{+|b)K}_{\phantom{+aK}J}-\tilde{P}_{IK}^{+|b)}\tilde{P}^{+cK}_{\phantom{+bK}J}\bigg{)}\bigg{)}$
(124)

This condition is equivalent to the corresponding condition on the momenta of
the self-dual spin connection $\beta^{+IJ}_{a}$ in Ashtekar’s chiral
formulation of General Relativity [24].

The presence of degrees of freedom which would classically correspond to a
pressureless perfect fluid have been proposed as solutions to the ‘problem of
time’ in quantum gravity [25, 26, 27, 23]. In particular in ([23]) a
pressureless perfect fluid Lagrangian described by phase space fields
$(T(x^{a}),P_{T}(x^{2}))$ was proposed, with it argued that the canonical
gauge choice $T=t$ may be imposed prior to quantization. Notably, the
classical equations of motion generally do not allow $T$ to be used as a
global time variable due to the generic formation of caustics on surfaces of
constant $T$ [28]. This suggests that if the flow of time $T$ in the quantum
theory remains unimpeded, then quantum corrections to the classical equations
of motion may become important near would-be caustic formation. An interesting
illustration of this is how the classical Big Bang singularity of a dust-
dominated universe can be replaced by an effective bounce when dust-time is
used in a quantum cosmological description of the system [29]. If degrees of
freedom described by fields $(\phi^{I},\tilde{P}_{I})$ or $(T,\tilde{P}_{T})$
are to play the role of dark matter, then these corrections may have an
experimental signature via how they affect the distribution and evolution of
dark matter density.

##  6 Additional terms in the gravitational action and coupling to matter

We now briefly consider additional terms in the gravitational action, the
coupling of matter to gravitation, and the degree to which they affect the
above results regarding how many local degrees of freedom are present in each.
A straightforward extension of the models considered here would be to add a
term with the following Lagrangian density:

$\displaystyle\tilde{\cal L}$
$\displaystyle=-\frac{\Lambda}{12}\tilde{\epsilon}^{\mu\nu\alpha\beta}\epsilon_{IJKL}D_{\mu}\phi^{I}D_{\nu}\phi^{J}D_{\alpha}\phi^{K}D_{\beta}\phi^{L}$
(125)

Where $\Lambda$ is a constant. It can be shown that this term modifies only
the constraint (63), contributing a term
$(\Lambda/3!)\epsilon_{IJKL}\tilde{\epsilon}^{abc}e^{J}_{a}e^{K}_{b}e^{L}_{c}$,
and that for general values of $(g_{+},g_{-})$ this term does not modify the
character of the constraints and so does not affect the number of degrees of
freedom of the theory. Indeed it can be shown that (125) is invariant under
the transformation (100). In the cases $(g_{+}=1,g_{-}=0)$ and
$(g_{+}=0,g_{-}=1)$ the effect of (125) is that of a cosmological constant
term.

We also now briefly consider other possible terms in the gravitational action.
After integration by parts, the action (99) can be expressed as a linear
combination of integrated Lagrangian four-forms
$\phi^{M}\phi_{M}\epsilon_{IJKL}R^{IJ}\wedge R^{KL}$ and
$\phi_{J}\phi_{L}\eta_{IK}R^{IJ}\wedge R^{KL}$ [5]. There is one additional,
independent four-form which is quadratic in $\phi^{I}$ and in $R^{IJ}$:
$\phi^{K}\phi_{K}R_{IJ}\wedge R^{IJ}$. This term - and generally other terms
involving $\phi_{I}\phi^{I}$ \- may be excluded by the requirement that the
gravitational action have a symmetry under ‘translations’
$\phi^{I}\rightarrow\phi^{I}+p^{I}$ subject to $Dp^{I}=0$ (the action (99) is
manifestly invariant under this transformation).

The coupling of fields $(\phi^{I},\omega^{\pm IJ})$ to matter fields depends
on the representation of the $SL(2,C)_{+}\times SL(2,C)_{-}$ gravitational
symmetry that the matter field belongs to. For fields in the trivial
representation, such as spacetime scalar fields $\varphi$ or one-forms
$A_{\mu}$, coupling to gravity is expected to be entirely via the spacetime
metric $g_{\mu\nu}=\eta_{IJ}D_{\mu}\phi^{I}D_{\nu}\phi^{J}$. In which case, in
the canonical formalism, time derivatives of $\phi^{I}$ but not of
$\omega^{\pm IJ}_{\mu}$ will appear in matter actions, leading to a
modification of the definition of the momentum $\tilde{P}_{I}$ via additional
terms appearing in (63). Additional couplings between gravity and matter
fields are necessary when the fields belong to non-trivial representations of
the gravitational symmetry group, such as left and right-handed fermions
$\psi^{\pm}$. Here some of the gravitational gauge fields $\omega^{\pm
IJ}_{\mu}$ must couple to $\psi^{\pm}$ to create covariant derivative terms
for these fields; these couplings will introduce no new time derivatives of
fields $(\phi^{I},\omega^{\pm IJ}_{\mu})$ but will result in additional terms
in the constraints $\tilde{\cal G}^{\pm IJ}$.

##  7 Conclusions

We now briefly summarize the main results of the paper and discuss the
potential for future work. In Sections 1 to 4 we introduced the models looked
at in this paper and produced the Hamiltonian form of the models, focusing on
an analysis of the propagation and nature of the phase space constraints
present. The Lagrangian density of the class of models considered is:

$\displaystyle\tilde{\cal L}[\phi^{I},\omega^{\pm IJ}_{\mu}]$
$\displaystyle=\tilde{\epsilon}^{\mu\nu\alpha\beta}\epsilon_{IJKL}D_{\mu}\phi^{I}D_{\nu}\phi^{J}\bigg{(}g_{+}R^{KL}_{\phantom{KL}\alpha\beta}(\omega^{+})+g_{-}R^{KL}_{\phantom{KL}\alpha\beta}(\omega^{-})\bigg{)}$
(126)

It was found that in the general case $g_{+}\neq g_{-}$ that the theory
possesses three complex degrees of freedom whereas for the particular case
$g_{+}=g_{-}$ there are no local degrees of freedom. Furthermore, it was shown
that the cases $(g_{+}=1,g_{-}=0)$ and $(g_{+}=0,g_{-}=1)$ correspond to an
extension of General Relativity that includes solutions corresponding to an
additional effective pressureless perfect fluid matter source. Interestingly,
such a matter source has been of prior interest as a possible solution to the
problem of time in quantum gravity [25, 27, 30, 31, 32]. We find it
encouraging that in the present models, the perfect fluid arises ‘naturally’
from the theory and does not have to be independently posited.

The extension to General Relativity (118) looks potentially promising: the
requirement that a General Relativistic limit only occurs when the self-dual
parts of the spatial pullback of the spin connection and its momenta represent
the gravitational degrees of freedom implies that in this limit the
gravitational Hamiltonian takes a simple, polynomial form. The presence of the
new degree of freedom $\phi^{2}$ in this model produces a gravitational effect
equivalent to that of a pressureless perfect fluid in the classical equations
of motion. Given that an extremely wide array of cosmological and
astrophysical data points towards the presence of an additional, unknown
gravitational component that behaves as such a fluid on large scales [33], it
is tempting to speculate whether the new degree of freedom may be responsible
for at least some of this effect.

However, the model (11) should be understood first as a quantum theory.
Clearly, non-perturbative quantization of gravitational theories represents a
considerable technical challenge so one approach is to first develop
incremental results. As briefly discussed in Section 5.1, it is possible
$\phi^{2}$ playing a putative role in determining a privileged and global time
in quantum may have observable consequences. Furthermore, the case of
Ashtekar’s chiral formulation of gravity (whose canonical formulation arises
from (118) in the limit $\tilde{\Pi}\rightarrow 0$), it has been argued [34]
that whatever the ultimate form that the quantized theory takes, it should
possess a regime which is mappable to a classical cosmological background with
metric perturbations describable in terms of the usual inflationary
calculation of tensor vacuum quantum fluctuations. Given this assumption, it
was found that the primordial spectrum of tensor modes of $+$ and $-$
chirality differed [34, 35], in contrast to the case of tensor modes in
standard inflationary cosmology where gravity is described by metric General
Relativity and no such effect exists. Remarkably, such effects in primordial
tensor modes may be observable via their effect on the cross-correlation
between CMB temperature and polarization fluctuations [36]. Given this, a
first step that could be taken in the case of (118) would be to allow for the
effect of additional degrees of freedom $(\tilde{\Pi},\phi^{2})$ in the
quantization of cosmological perturbations and see how the above picture is
affected.

An alternative approach is to look to describe the behavior of the model as a
quantum theory in situations of high symmetry. This has been carried out for
Ashtekar’s chiral formulation of gravity in the context of loop quantum
cosmology [37, 38, 39] and this approach could be generalized to the case of
(118).

Finally, it is interesting to note that despite the classical equivalence of
parameterized field theory to the theory of matter fields propagating in fixed
Minkowski space background, the Dirac quantization of the former theory faced
technical obstacles [40], the resolution of which required the use of
techniques originally developed in the loop quantum gravity paradigm [41].

A natural generalization of (126) would be the introduction of fields
$(\psi_{+},\psi_{-})$ (potentially with non-trivial Lorentz index structure)
such that the hitherto constant $(g_{+},g_{-})$ are reflective of expectation
values of these fields. The General-Relativistic limits $(g_{+}=1,g_{-}=0)$
and $(g_{+}=0,g_{-}=1)$ would then potentially arise from spontaneous symmetry
breaking (with the action formally symmetric under the transformations (100)
and accompanying transformation of $(\psi_{+},\psi_{-})$) and with time
variation of the new dynamical fields being of significance in the early
universe [42].

## Acknowledgements

We thank William Barker, Amel Durakovic, and Hans Westman for helpful
discussions. MJ and TZ are supported by the project No. 2021/43/P/ST2/02141
co-funded by the Polish National Science Centre and the European Union
Framework Programme for Research and Innovation Horizon 2020 under the Marie
Skłodowska-Curie grant agreement No. 945339.

##  Appendix A Comparison to MacDowell-Mansouri gravity

In Section 1 we discussed how the gravitational model considered in this paper
could be recovered from the gauging of global symmetries present in a
parameterized field theory whose solutions corresponded to fields propagating
in a background Minkowski space. In the resulting gravitational theory,
gravity is described by an $SO(1,3)_{C}$ connection
$\omega^{I}_{\phantom{I}J\mu}$ and a field $\phi^{I}$ in the group’s
fundamental representation.

This bears an immediate resemblance to the description of gravity as a
spontaneously broken gauge theory of the De Sitter ($SO(1,4)$) or anti-De
Sitter groups ($SO(2,3)$). Initially proposed by MacDowell and Mansouri [43],
a manifestly locally $SO(1,4)$ or $SO(2,3)$\- invariant action for gravity was
proposed by Stelle and West [44], with the gravitational field described by an
$SO(1,4)$ or $SO(2,3)$ connection $A^{A}_{\phantom{A}B\mu}$ and a field
$V^{A}$ in the group’s fundamental representation.

The Stelle-West theory can additionally be considered to arise from the
gauging of the parameterized field theory whose solutions correspond to fields
propagating in a background De Sitter or anti-De Sitter space. For
concreteness, we will focus on the De Sitter case, with the generalization to
anti-De Sitter space straightforward. For this case, the non-gravitational
parameterized field theory is obtained by replacing the fixed background De
Sitter metric $\hat{\eta}_{\mu\nu}$ by a composite object
$\eta_{AB}\partial_{\mu}X^{A}\partial_{\nu}X^{B}$ where
$\eta_{AB}=\mathrm{diag}(-1,1,1,1,1)$ and the five dynamical fields $X^{A}$
are subject to the constraint $\eta_{AB}X^{A}X^{B}=L^{2}$. Then, the De Sitter
metric is recovered if the equations of motion for matter fields and for
$X^{A}$ allow for solutions where $X^{A}$ take the form of Minkowski
coordinates in $\mathbb{R}^{(1,4)}$ and the condition
$\eta_{AB}X^{A}X^{B}=L^{2}$ defines a De Sitter space submanifold. This
submanifold can be covered with coordinates $x^{\mu}$, with the metric on this
surface being the pullback of the metric $\eta_{AB}$ of $\mathbb{R}^{(1,4)}$
to this surface: $\eta_{AB}\partial_{\mu}X^{A}\partial_{\nu}X^{B}$. A
parameterized field theory can be constructed by replacing the fixed De Sitter
background metric with $\eta_{AB}\partial_{\mu}X^{A}\partial_{\nu}X^{B}$ and
adding a Lagrangian constraint $\eta_{AB}X^{A}X^{B}-L^{2}$ into the action.
The resulting action has four-dimensional diffeomorphism symmetry as well as
symmetry under the global De Sitter transformation

$\displaystyle X^{A}\rightarrow M^{A}_{\phantom{A}B}X^{B}$ (127)

with $M^{A}_{\phantom{A}B}\in SO(1,4)$. This global symmetry can be promoted
to a local one by the introduction of a De Sitter group connection
$A^{A}_{\phantom{A}B\mu}$ so that the covariant derivative ${\cal
D}^{(A)}_{\mu}X^{A}=\partial_{\mu}X^{A}+A^{A}_{\phantom{A}B\mu}X^{B}$ can be
constructed. This suggests that the metric tensor that matter fields couple to
should be identified with $\eta_{AB}{\cal D}^{(A)}X^{A}{\cal D}^{(A)}X^{B}$.

It is then necessary to introduce an action involving
$A^{A}_{\phantom{A}B\mu}$ so that equations of motion for this field don’t
produce artificial constraints on any matter fields it couples to. Stelle and
West suggested the following action:

$\displaystyle S[A,X,\lambda]=-\frac{\alpha}{4}\int d^{4}x\bigg{(}$
$\displaystyle\epsilon_{ABCDE}\epsilon^{\alpha\beta\mu\nu}X^{E}F^{AB}_{\phantom{AB}\alpha\beta}F^{CD}_{\phantom{CD}\mu\nu}-\lambda(\eta_{AB}X^{A}X^{B}-L^{2})\bigg{)}$
(128)

where $F^{AB}_{\phantom{AB}\alpha\beta}$ is the curvature two-form of the
connection $A^{A}_{\phantom{A}B\mu}$ and the Lagrangian constraint on the norm
of $X^{A}$ is retained from the non-gravitational theory. The $\lambda$
equation of motion $\eta_{AB}X^{A}X^{B}-L^{2}=0$ enforces a non-vanishing
vacuum expectation value of $X^{A}$, spontaneously breaking the $SO(1,4)$
symmetry down to $SO(1,3)$ It is useful to gauge-fix at the level of the
action. Choosing a gauge where $X^{A}\overset{*}{=}L\delta^{A}_{-1}$ (with
$\eta_{-1-1}=1$ and indices $I,J,\dots$ used to label remaining indices), a
useful decomposition of the connection $A^{A}_{\phantom{A}B\mu}$ is:

$\displaystyle
A^{AB}_{\phantom{AB}\mu}\overset{*}{=}\begin{pmatrix}0&\frac{1}{L}e^{I}_{\mu}\\\
-\frac{1}{L}e^{I}_{\mu}&\omega^{IJ}_{\phantom{IJ}\mu}\end{pmatrix}$ (129)

From which it follows that:

$\displaystyle g_{\mu\nu}$
$\displaystyle\overset{*}{=}\eta_{IJ}e^{I}_{\mu}e^{J}_{\nu}$ (130)
$\displaystyle F^{IJ}_{\phantom{IJ}\mu\nu}$
$\displaystyle\overset{*}{=}R^{IJ}_{\phantom{IJ}\mu\nu}(\omega)-\frac{2}{L^{2}}e^{I}_{[\mu}e^{J}_{\nu]}$
(131) $\displaystyle F^{-1I}_{\phantom{-1I}\mu\nu}$
$\displaystyle\overset{*}{=}\frac{2}{L}{\cal D}^{(\omega)}_{[\mu}e^{I}_{\nu]}$
(132)

where $R^{IJ}_{\phantom{IJ}\mu\nu}(\omega)$ is to be identified with the
curvature of the $SO(1,3)$ connection $\omega^{IJ}_{\mu}$ with ${\cal
D}^{(\omega)}_{\mu}$ being the associated covariant derivative. Inserting
(131) and (132) in (128), three independent contributions to the action are
recovered: a term quadratic in $e^{I}_{\mu}$ and linear in $R^{IJ}_{\mu\nu}$
which is the Palatini action of Einstein-Cartan gravity, a term quartic in
$e^{I}_{\mu}$ which yields a positive cosmological constant, and a term
quadratic in $R^{IJ}_{\phantom{IJ}\mu\nu}$ which is a topological term.
Indeed, the gauged-fixed action corresponds to the action considered by
MacDowell and Mansouri.

An independent interpretation of the recovery of gravity as a spontaneously
broken gauge theory is via the framework of Cartan geometry [45, 46], where
the connection (129) can be interpreted as a prescription for ‘rolling without
slipping’ of a model De Sitter space on a local patch of physical spacetime,
taken to be a four-dimensional surface embedded in $\mathbb{R}^{(1,4)}$. By
comparison, for the $SO(1,3)_{C}$ model with a General Relativistic limit
(without loss of generality we choose $\beta=i$), consider a patch of
spacetime where $\eta_{IJ}\phi^{I}\phi^{J}<0$. It’s helpful to choose a
partial gauge fixing where $\phi^{I}\overset{*}{=}T(x^{\mu})\delta^{I}_{0}$
(where $\eta_{00}=-1$). Then from the definition $e_{\mu}^{I}\equiv{\cal
D}_{\mu}X^{I}$ we have $e_{\mu}^{0}\overset{*}{=}\partial_{\mu}T$,
$e_{\mu}^{i}\overset{*}{=}{\cal A}^{i}_{\phantom{i}0\mu}$ it can then be shown
from the field equations that if $T$ is adopted as a time coordinate then the
pullback of $\omega^{IJ}_{\mu}$ to a patch of constant $T$ takes the form:

$\displaystyle\omega^{IJ}_{\phantom{IJ}a}$
$\displaystyle\overset{*}{=}\begin{pmatrix}0&\frac{1}{T}e_{a}^{i}\\\
-\frac{1}{T}e^{i}_{a}&\Gamma^{ij}_{\phantom{ij}a}-i\epsilon^{ij}_{\phantom{ij}k}(\Gamma^{0k}_{\phantom{0k}a}-\frac{1}{T}e_{a}^{k})\\\
\end{pmatrix}$ (133)

where $\Gamma^{ij}_{\phantom{ij}a}$ is the torsion-free spin connection
associated with the spatial triad field $e^{i}_{a}$ and
$\Gamma^{0k}_{\phantom{0k}a}$ is proportional the extrinsic curvature of the
patch. This illustrates how $\omega^{IJ}_{\phantom{IJ}a}$ encodes information
about distances on the patch at constant $T$ (via $e^{i}_{a}$), the intrinsic
curvature of the patch (via $\Gamma^{ij}_{\phantom{ij}a}$), and its extrinsic
curvature in spacetime (via $\Gamma^{0k}_{\phantom{0k}a}$). The first two of
these parts have much in common with the Cartan geometrodynamical approach to
the Hamiltonian formulation of gravity [47], whereas the incorporation of
information about the time evolution of space via complexification of
$\omega^{IJ}_{\phantom{IJ}a}$ is to our knowledge novel.

##  Appendix B Useful functional derivatives

Useful non-zero functional derivatives are as follows:

$\displaystyle\frac{\delta\beta^{+IJ}_{a}(x)}{\delta\beta^{+KL}_{b}(x^{\prime})}$
$\displaystyle=\frac{1}{2}\bigg{(}\delta^{[I}_{K}\delta^{J]}_{L}-\frac{i}{2}\epsilon_{KL}^{\phantom{KL}IJ}\bigg{)}\delta^{b}_{a}\delta(x-x^{\prime})$
(134)
$\displaystyle\frac{\delta\tilde{P}^{+a}_{IJ}(x)}{\delta\tilde{P}^{+b}_{KL}(x^{\prime})}$
$\displaystyle=\frac{1}{2}\bigg{(}\delta^{[K}_{I}\delta^{L]}_{J}-\frac{i}{2}\epsilon_{IJ}^{\phantom{IJ}KL}\bigg{)}\delta^{a}_{b}\delta(x-x^{\prime})$
(135)
$\displaystyle\frac{\delta\beta^{-IJ}_{a}(x)}{\delta\beta^{-KL}_{b}(x^{\prime})}$
$\displaystyle=\frac{1}{2}\bigg{(}\delta^{[I}_{K}\delta^{J]}_{L}+\frac{i}{2}\epsilon_{KL}^{\phantom{KL}IJ}\bigg{)}\delta^{b}_{a}\delta(x-x^{\prime})$
(136)
$\displaystyle\frac{\delta\tilde{P}^{-a}_{IJ}(x)}{\delta\tilde{P}^{-b}_{KL}(x^{\prime})}$
$\displaystyle=\frac{1}{2}\bigg{(}\delta^{[K}_{I}\delta^{L]}_{J}+\frac{i}{2}\epsilon_{IJ}^{\phantom{IJ}KL}\bigg{)}\delta^{a}_{b}\delta(x-x^{\prime})$
(137)
$\displaystyle\frac{\delta\tilde{P}_{I}(x)}{\delta\tilde{P}_{J}(x^{\prime})}$
$\displaystyle=\delta^{J}_{I}\delta(x-x^{\prime})$ (138)
$\displaystyle\frac{\delta\phi^{I}(x)}{\delta\phi^{J}(x^{\prime})}$
$\displaystyle=\delta^{I}_{J}\delta(x-x^{\prime})$ (139)

##  Appendix C Detailed example of the evaluation of the Poisson bracket
between two constraints

Making use of the functional derivatives defined in the previous section and
the definition of the Poisson bracket (67), as an illustrative example we
consider a detailed calculation of the Poisson bracket
$\\{\tilde{C}^{-a}_{IJ}[A_{a}^{IJ}],\tilde{C}_{K}[V^{K}]\\}$. To do so, we
first calculate the functional derivative of each smeared constraint as
follows:

###  C.1 Functional derivatives of $\tilde{C}^{-a}_{IJ}[A_{a}^{IJ}]$

Given a test function $A_{a}^{IJ}=A_{a}^{-IJ}$ (i.e. a Lorentz tensor which
depends on of $x^{a}$ and is considered independent of phase space fields), we
can consider the following smeared constraint:

$\displaystyle\tilde{C}^{-a}_{IJ}[A_{a}^{IJ}]$ $\displaystyle=\int
d^{3}xA_{c}^{IJ}\bigg{(}\tilde{P}_{IJ}^{-c}-2g_{-}\bigg{[}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{K}_{a}e^{L}_{b}\bigg{]}^{-}\bigg{)}$
(140)

where recall that
$e^{I}_{a}=\partial_{a}\phi^{I}+\beta^{I}_{\phantom{I}Ja}\phi^{J}$. Using the
results of Section B we have:

$\displaystyle\frac{\delta\tilde{C}^{-a}_{KL}[A_{a}^{KL}]}{\delta\tilde{P}^{+d}_{IJ}}$
$\displaystyle=0$ (141)
$\displaystyle\frac{\delta\tilde{C}^{-a}_{KL}[A_{a}^{KL}]}{\delta\beta^{+IJ}_{d}}$
$\displaystyle=4g_{-}\bigg{[}A_{c}^{-KL}\epsilon_{MKL[I}\tilde{\varepsilon}^{dbc}\phi_{J]}e_{b}^{M}\bigg{]}^{+}$
(142)

$\displaystyle\frac{\delta\tilde{C}^{-a}_{KL}[A_{a}^{KL}]}{\delta\tilde{P}^{-d}_{IJ}}$
$\displaystyle=A^{IJ}_{d}$ (143)
$\displaystyle\frac{\delta\tilde{C}^{-a}_{KL}[A_{a}^{KL}]}{\delta\beta^{-IJ}_{d}}$
$\displaystyle=4g_{-}\bigg{[}A_{c}^{-KL}\epsilon_{MKL[I}\tilde{\varepsilon}^{dbc}\phi_{J]}e^{M}_{b}\bigg{]}^{-}$
(144)

$\displaystyle\frac{\delta\tilde{C}^{-a}_{KL}[A_{a}^{KL}]}{\delta\phi^{I}}$
$\displaystyle=4g_{-}D^{(\beta)}_{a}\bigg{(}A_{c}^{-KL}\epsilon_{IKLM}\tilde{\varepsilon}^{abc}e^{M}_{b}\bigg{)}$
(145)
$\displaystyle\frac{\delta\tilde{C}^{-a}_{KL}[A_{a}^{KL}]}{\delta\tilde{P}_{I}}$
$\displaystyle=0$ (146)

###  C.2 Functional derivatives of $\tilde{C}_{I}[A^{I}]$

Similarly for the constraint $\tilde{C}_{I}$ we may consider a test function
$A^{I}$ and smear the constraint as follows:

$\displaystyle\tilde{C}_{I}[A^{I}]$ $\displaystyle=\int
d^{3}xA^{I}\bigg{[}\tilde{P}_{I}-2g_{+}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{J}_{a}R_{bc}^{+KL}-2g_{-}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{J}_{a}R^{-KL}_{bc}\bigg{]}$
(147)

Again using the results of Section B we have:

$\displaystyle\frac{\delta\tilde{C}_{K}[A^{K}]}{\delta\tilde{P}^{+d}_{IJ}}$
$\displaystyle=0$ (148)
$\displaystyle\frac{\delta\tilde{C}_{K}[A^{K}]}{\delta\beta^{+IJ}_{d}}$
$\displaystyle=\bigg{[}-2A^{M}\epsilon_{MKL[I}\tilde{\varepsilon}^{dbc}\phi_{J]}(g_{+}R^{+KL}_{bc}+g_{-}R^{-KL}_{bc})-4g_{+}\tilde{\varepsilon}^{bad}D^{+}_{b}\bigg{(}\epsilon_{IJKL}A^{K}e_{a}^{L}\bigg{)}\bigg{]}^{+}$
(149)
$\displaystyle\frac{\delta\tilde{C}_{K}[A^{K}]}{\delta\tilde{P}^{-d}_{IJ}}$
$\displaystyle=0$ (150)
$\displaystyle\frac{\delta\tilde{C}_{K}[A^{K}]}{\delta\beta^{-IJ}_{d}}$
$\displaystyle=\bigg{[}-2A^{M}\epsilon_{MKL[I}\tilde{\varepsilon}^{dbc}\phi_{J]}(g_{+}R^{+KL}_{bc}+g_{-}R^{-KL}_{bc})-4g_{-}\tilde{\varepsilon}^{bad}D^{-}_{b}\bigg{(}\epsilon_{IJKL}A^{K}e_{a}^{L}\bigg{)}\bigg{]}^{-}$
(151)

$\displaystyle\frac{\delta\tilde{C}_{K}[A^{K}]}{\delta\phi^{I}}$
$\displaystyle=2g_{+}D^{(\beta)}_{a}\bigg{(}A^{M}\epsilon_{MIKL}\tilde{\varepsilon}^{abc}R^{+KL}_{bc}\bigg{)}+2g_{-}D^{(\beta)}_{a}\bigg{(}A^{L}\epsilon_{LIJK}\tilde{\varepsilon}^{abc}R^{-JK}_{bc}\bigg{)}$
(152) $\displaystyle\frac{\delta\tilde{C}_{J}[A^{J}]}{\delta\tilde{P}_{I}}$
$\displaystyle=A^{I}$ (153)

###  C.3 Poisson bracket
$\\{\tilde{C}^{-a}_{IJ}[A_{a}^{IJ}],\tilde{C}_{K}[V^{K}]\\}$

Applying the results from Sections C.1 and Sections C.2 we have that:

$\displaystyle\\{\tilde{C}^{-c}_{IJ}[A_{c}^{IJ}],\tilde{C}_{K}[V^{K}]\\}$
$\displaystyle\overset{b}{=}\int
d^{3}xA^{IJ}_{d}V^{K}\bigg{[}2(g_{+}-g_{-})\tilde{\varepsilon}^{dbc}\epsilon_{KLM[I}\phi_{J]}R_{bc}^{+LM}+g_{-}\tilde{\varepsilon}^{dbc}\phi_{K}\epsilon_{IJMN}R^{-MN}_{bc}\bigg{]}^{-}$
(154)

where $\overset{b}{=}$ denotes equality up to a total derivative (and hence
boundary) term.

##  Appendix D Interpretation of the Lagrange multiplier $V^{I}$

From Hamilton’s equations we have that

$\displaystyle\dot{\phi}^{I}$
$\displaystyle=V^{I}-(\Omega^{+IJ}+\Omega^{-IJ})\phi_{J}$ (155)

Therefore, using the results of Section 2.4 we have that:

$\displaystyle V^{I}=D_{t}^{(\omega)}\phi^{I}$
$\displaystyle=e^{I}_{t}=NN^{I}+N^{a}e_{a}^{I}$ (156)

and hence $V^{I}$ is straightforwardly related to parts of the spacetime
metric structure.

##  Appendix E Some useful results

###  E.1 Self and anti-self dual parts of the Lorentz tensor $Y^{de}_{IJPR}$

Calculation shows self and anti-self dual parts of the Lorentz tensor
$Y^{de}_{IJKL}$ are given by:

$\displaystyle Y^{de}_{[IJ]^{(1)\pm}[KL]^{(2)\pm}}$
$\displaystyle=\tilde{\varepsilon}^{dbe}e^{M}_{b}\epsilon_{MIJ[K}\phi_{L]}+(\pm)^{(1)}(\pm)^{(2)}\tilde{\varepsilon}^{dbe}\phi^{M}e_{b[I}\epsilon_{J]MKL}$
$\displaystyle+2i\tilde{\varepsilon}^{dbe}e^{M}_{b}((\pm)^{1}\eta_{M[I}\eta_{J][K}\phi_{L]}+(\pm)^{(2)}\phi_{[I}\eta_{J][L}\eta_{K]M})$
$\displaystyle-(\pm)^{(2)}\tilde{\varepsilon}^{dbe}e^{M}_{b}\phi_{M}\eta_{K[I}\eta_{J]L}$
(157)

###  E.2 Development of the constraint $\tilde{C}^{I}$ in the case
$(g_{+}=1,g_{-}=0)$

Using (156) we see that the constraint $\tilde{C}_{I}$ contributes to the
Hamiltonian density via

$\displaystyle V^{I}\tilde{C}_{I}$
$\displaystyle=(NN^{I}+N^{a}e_{a}^{I})\bigg{(}\tilde{P}_{I}-2\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{J}_{a}R_{bc}^{+KL}\bigg{)}$
(158)

Furthermore, when the constraints hold we have
$\tilde{P}_{IJ}^{+c}=-2\big{[}\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e^{K}_{a}e^{L}_{b}\big{]}^{+}$.
We can therefore recover the following useful results. Firstly, the product of
the determinant of $q_{ab}$ and its matrix inverse can be expressed in terms
of momenta $\tilde{P}^{+c}_{IJ}$ as:

$\displaystyle-16qq^{cd}$ $\displaystyle=\tilde{P}^{+c}_{IJ}\tilde{P}^{+dIJ}$
(159)

Furthermore we can express individual parts of $\tilde{C}^{I}$ as:

$\displaystyle\epsilon_{IJKL}\tilde{\varepsilon}^{abc}e_{d}^{I}e_{a}^{J}R^{+KL}_{bc}$
$\displaystyle=-\frac{1}{2}\tilde{P}^{+e}_{IJ}R^{+IJ}_{de}$ (160)
$\displaystyle-2\sqrt{q}N^{I}\epsilon_{IJKL}e_{a}^{J}\tilde{\epsilon}^{abc}R_{bc}^{+KL}$
$\displaystyle=-\frac{1}{4}\tilde{P}^{+b}_{IK}\tilde{P}^{+cK}_{\phantom{+cK}J}R^{+IJ}_{\phantom{+IJ}bc}$
(161) $\displaystyle N^{I}\tilde{P}_{I}$
$\displaystyle=\xi\tilde{\Pi}\sqrt{-\phi^{2}+\frac{1}{4}q^{ab}\partial_{a}\phi^{2}\partial_{b}\phi^{2}}$
(162) $\displaystyle e_{a}^{I}\tilde{P}_{I}$
$\displaystyle=\frac{1}{2}\tilde{\Pi}\partial_{a}\phi^{2}$ (163)

where we have used the result that $\tilde{P}^{I}\propto\phi^{I}$ in the cases
where $(g_{+}=1,g_{-}=0)$ and $(g_{-}=0,g_{+}=1)$.

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