# Dynamics and Geometry of Entanglement in Many-Body Quantum Systems

Peyman Azodi<EMAIL_ADDRESS>Herschel A.Rabitz  Department of Chemistry,
Princeton University, Princeton, New Jersey 08544, USA

###### Abstract

A new framework is formulated to study entanglement dynamics in many-body
quantum systems along with an associated geometric description. In this
formulation, called the Quantum Correlation Transfer Function (QCTF), the
system’s wave function or density matrix is transformed into a new space of
complex functions with isolated singularities. Accordingly, entanglement
dynamics is encoded in specific residues of the QCTF, and importantly, the
explicit evaluation of the system’s time dependence is avoided. Notably, the
QCTF formulation allows for various algebraic simplifications and
approximations to address the normally encountered complications due to the
exponential growth of the many-body Hilbert space with the number of bodies.
These simplifications are facilitated through considering the patterns, in
lieu of the elements, lying within the system’s state. Consequently, a main
finding of this paper is the exterior (Grassmannian) algebraic expression of
many-body entanglement as the collective areas of regions in the Hilbert space
spanned by pairs of projections of the wave function onto an arbitrary basis.
This latter geometric measure is shown to be equivalent to the second-order
Rényi entropy. Additionally, the geometric description of the QCTF shows that
characterizing features of the reduced density matrix can be related to
experimentally observable quantities. The QCTF-based geometric description
offers the prospect of theoretically revealing aspects of many-body
entanglement, by drawing on the vast scope of methods from geometry.

††preprint: APS/123-QED

##  I Introduction

Entanglement, the unique and intriguing feature of the quantum realm, has been
subject to extensive theoretical and experimental studies in a variety of
fields. Entanglement is the cornerstone of quantum information science [1, 2,
3, 4] while also forming a basis to explain quantum thermalization [5, 6, 7,
8, 9, 10] as well as being closely related to the geometry of spacetime [11,
12, 13]. The evolution of entanglement in pure many-body quantum systems is
concealed in the cooperative dynamics amongst the exponentially large number
of modes of the system. Thus, except in rare cases, e.g., one-dimensional
integrable models with conformal symmetry [14, 15, 16], obtaining a
microscopic view of entanglement dynamics is generally not feasible in
interacting many-body quantum systems. The most common approximation tools to
explore many-body entanglement are variations of tensor-network-based
numerical simulations, including Matrix Product States (MPS), which are mainly
useful in the low-entanglement-density regimes [17, 18]. This paper will
introduce a new means to study multipartite entanglement dynamics in many-body
quantum systems, which we refer to as the Quantum Correlation Transfer
Function (QCTF) formulation. Additionally, we will use the QCTF to provide a
geometric interpretation of entanglement.

The QCTF provides a means to study correlation dynamics in many-body quantum
systems. We exploit the fact that correlations between the constituents of a
quantum system can also be obtained from the patterns in its wave function or
density matrix, rather than their elements. The investigation of such patterns
is enabled through the employment of the Z-transformation, wherein the unitary
time-evolution of a pure quantum system is mapped into a new complex-valued
function, that constitutes the QCTF. As a result, the dynamics of entanglement
is encoded in the analytic properties (e.g., poles, zeros, and residues) of
the QCTF. In this general framework, various mathematical tools can be used to
simplify the analysis. Most importantly, the QCTF permits bypassing the direct
evaluation of the time evolution while nevertheless permitting the study of
entanglement dynamics.

Employing the QCTF, can either lead to a formulation of the quantum system
which is fully dependent on the initially chosen basis, or alternatively, to a
completely basis-independent (i.e., tensorial) description, where tensor
indices represent the degrees of freedom in the Hilbert space. The basis-
dependent approach may be implemented to study entanglement, given the
Hamiltonian and the initial state of the system. In this case, by properly
choosing the basis used in the Z-transformation, a closed-form complex
function, or it’s Laurent expansion, can be obtained upon making the QCTF
transformation, from which entanglement dynamics is revealed by finding the
QCTF residues. For example, this approach has been used to obtain the exact
single-magnon entanglement dynamics in integrable Heisenberg chains [19].

The basis-independent approach will be employed to present an extrinsic
geometric portrayal of the QCTF-expressed entanglement (the second objective
of the paper). In particular, based on the tensorial description of
entanglement in the QCTF formulation, a natural geometrical description
follows. We show that the collective total squared areas spanned by the
marginal wave functions is a geometric measure of entanglement between parts
of a pure quantum system. A few clarifications on the terminology above are
called for. The word “natural” reflects that this description mirrors the
nature of entanglement between subsystems, which is, how different degrees of
freedom in the subsystem evolve distinctly due to interaction with the
remainder of the system. By “marginal”, we refer to the projection of the wave
function onto an arbitrary basis of the vector space spanning the subsystem of
interest. This formulation can enable employing various tools from geometry to
better understand entanglement. Since the focus of this paper is on constant
Hamiltonians, with the formulation being $U(1)$ gauge-invariant, the geometric
analysis is simply carried out on $\mathbb{C}^{n}$ with Euclidean metric
(which highly simplifies the theory), and not the complex projective space,
$\mathbb{C}\mathbf{P}^{n}$, with Fubini-Study metric [20, 21, 22]. We note
that viewing quantum systems through the lens of geometry has often been
fruitful in different areas of physics [23, 24, 25, 26, 27, 28, 29, 30, 31],
and it has led to links between the entropy of entanglement and the geometry
of the space wherein the system’s state lies [32, 33, 34, 35, 11, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45].

The paper is organized as follows. Section II introduces the transformation
producing the QCTF. In Section III a particular form of the QCTF
transformation is presented for two-level subsystems along with its resultant
entanglement measure formulation. Section IV starts with a basic introduction
to the geometry of the Hilbert space involved. Subsection IV.1 gives the main
results on the geometric structure of entanglement, and subsection IV.2,
demonstrates the relation between the QCTF entanglement measure and the
second-order Rényi entropy. Section V presents concluding remarks and
suggestions for future research building on the foundation set out in the
present work. The Appendices are referred to in the text for mathematical
details.

##  II Basics of the QCTF Formulation

The motivation behind the QCTF framework is to analyze a quantum system
through the analytic behavior of a corresponding complex-valued function,
referred to as QCTF. This function is obtained by employing the
Z-transformation (initially proposed by Ragazzini and Zadeh to study sampled
data [46]) to either the entire or a portion of the wave function
($\ket{\tilde{\psi}(s)}=\mathcal{L}\\{\ket{\psi(t)}\\}$) in the Laplace
domain; the density matrix (${\tilde{\rho}(s)}=\mathcal{L}\\{{\rho(t)}\\}$)
may be analogously treated, as will be explained. The general case of
transforming the wave function is given first to illustrate the use of the
QCTF. We consider the unitary evolution of the initial state $\ket{\psi_{0}}$
governed by the time-independent Hamiltonian $\mathbf{H}$. Given any arbitrary
basis set $\\{\ket{j}|j=0:d-1\\}$ that spans the Hilbert space associated with
$\ket{\psi}$, then one form of the QCTF transformation can be defined as:

$\begin{split}\bar{\mathcal{K}}(z,s)&=\sum_{j=0}^{d-1}z^{j}\innerproduct{j}{\tilde{\psi}(s)}\\\
&=\sum_{j=0}^{d-1}z^{j}\matrixelement{j}{\mathbf{G}(s)}{\psi_{0}},\end{split}$
(1)

where $\mathbf{G}(s)=(s+\frac{i}{\hbar}\mathbf{H})^{-1}$ is the resolvent of
$\mathbf{H}$ (the propagator of the Schrödinger’s equation in the Laplace
domain). The transformation (1) implicitly depends on the basis
$\\{\ket{j}\\}$ and how it is labeled (or numbered) due to the exponents
$z^{j}$; nevertheless, as will be shown later, one can rewrite the QCTF fully
independent of the choice of basis. We note that the transformation (1) is an
equivalent description of the quantum system’s wave function, therefore the
properties of the system are retained within the QCTF. The main purpose of
this paper is to show that the entanglement evolution of subsystems can be
effectively studied along with new insights via the QCTF formulation.

The density matrix of a pure quantum system can be equivalently described in
the QCTF formulation. The desired QCTF is a function of three complex
variables, $z_{d},z_{a}$ and $s$; Using (1), it can be defined as

${\mathcal{K}}(z_{d},z_{a},s)\doteq\bar{\mathcal{K}}(z_{d}z_{a},s)\star\bar{\mathcal{K}}^{*}\big{(}(z_{a}/z_{d})^{*},s^{*}\big{)},$
(2)

where $\star$ denotes the ordinary product operation in the $z_{d}$ and
$z_{a}$ domains along with the following convolution operation in the $s$
domain. If $\mathbf{F}_{1}(s)$ and $\mathbf{F}_{2}(s)$ are functions in the
Laplace domain, then

$\mathbf{F}_{1}(s)\star\mathbf{F}_{2}(s)\doteq\frac{1}{2\pi
i}\int_{-\infty}^{\infty}{\mathbf{F}_{1}(\sigma+i\omega)\mathbf{F}_{2}(s-\sigma-i\omega)d\omega},$
(3)

for some real $\sigma$ in the region of convergence of $\mathbf{F}_{1}(s)$.
This operation is the frequency-domain equivalent of the product of these
functions in the time domain. In the important special case of simple poles,
we have
$(s+i\omega_{1})^{-1}\star(s+i\omega_{2})^{-1}=(s+i(\omega_{1}+\omega_{2}))^{-1}$.
The QCTF (2) can be interpreted as a two-dimensional Z-transformation of the
density matrix in the Laplace domain (see Appendix A for details).

More generally, the dynamics of a pure quantum system evolving under the
Hamiltonian $\mathbf{H}$ can be represented by an operator function
$\mathcal{H}(\mathbf{H},z_{d},z_{a},s)$, which includes its dynamical
correlation properties, independent of its initial state. To see this feature,
using (2), the QCTF transformation can be rewritten as the expectation value
of $\mathcal{H}$ with respect to the arbitrary initial state
($\ket{\psi_{0}}$),

$\mathcal{K}(z_{d},z_{a},s)=\expectationvalue{\mathcal{H}(\mathbf{H},z_{d},z_{a},s)}{\psi_{0}},$
(4a)
$\mathcal{H}={\textbf{G}^{{\dagger}}(s^{*})\bigg{(}\sum_{i,j}{z_{a}}^{i+j}{z_{d}}^{i-j}\ket{j}{\bra{i}\bigg{)}\star\textbf{G}(s)}}.$
(4b)

Operator $\mathcal{H}$ captures the dynamical properties of the quantum
system, including the evolution of correlation between its constituent
subsystems. This property, which will be discussed in detail in the next
section, can be employed to understand how the correlation behavior varies for
different initial states of the quantum system.

Equation (4) is the dual Laurent series expansions of $\mathcal{K}$, in both
$z_{d}$ and $z_{a}$ variables, centered at the origins of the respective
spaces. Based on the labeling $j=0,\cdots,d-1$ for the Hilbert space basis,
the origin $z_{d}=0$ is a pole of order (at most) $d-1$, while the origin
$z_{a}=0$ is a removable singularity in the $z_{a}$ space. Also, for finite
$d$, the function $\mathcal{K}$ is holomorphic in any punctured neighborhood
of the origin, in both spaces.

The QCTF (2) provides an equivalent description for the evolution of a quantum
system in terms of its density matrix, $\rho(t)$. Therefore, desired matrix
operations can be equivalently carried out on either $\mathcal{K}$ or
functions of $\mathcal{K}$ by a variety of means from complex analysis (e.g.,
finding residues, etc.). As an elementary example, given another arbitrary
basis set $\\{\ket{j^{\prime}}\\}$, the element
$\matrixelement{i^{\prime}}{\tilde{\rho}}{j^{\prime}}$ can be obtained by
taking the inverse Laplace transform after utilizing the following residues
(see Appendix B for the proof)

$\begin{split}\underset{{\begin{subarray}{c}z_{d}=0\\\
z_{a}=0\end{subarray}}}{\mathbf{Res}}\Big{(}\sum_{i,j}\innerproduct{j}{j^{\prime}}\innerproduct{i^{\prime}}{i}z_{d}^{-(i-j)-1}z_{a}^{-(i+j)-1}{\mathcal{K}}(z_{d},z_{a},s)\Big{)}.\end{split}$
(5)

To summarize, in the QCTF formulation, the goal is to study aspects of a
quantum system from the patterns in the elements of the system’s state. These
patterns reveal themselves through the encoding in the analytical properties
of the QCTF (or functions of the QCTF) and can be found either by closed-
contour integration or through the properties of the dual Laurent series
expansion involved. The patterns arise through the Z-transformation’s inherent
summation over all elements of the system’s state. The main goal is to
simplify the analysis of entanglement and express the results in a new
perspective by either strategically choosing the QCTF basis labels or
constructing a basis-independent formulation.

##  III Entanglement Dynamics for Two-level Subsystems

The application of the generic QCTF many-body framework in Section II is now
focused on the entanglement dynamics of two-level subsystems when the overall
system is initialized in a pure state. Section IV will build upon this
scenario to consider entanglement dynamics for subsystems with more than two
energy levels. We first introduce the entanglement measure used in the
analysis and then its time evolution will be obtained from the system’s QCTF.

Here we consider a closed, discrete, bipartite quantum system, consisting of a
two-level subsystem (referred to as subsystem $\mathcal{M}$) interacting with
an accompanying $d$-dimensional quantum subsystem $\mathcal{R}$, that evolves
according to the Hamiltonian $\mathbf{H}$ from the initial state
$\ket{\psi_{0}}=\ket{\psi(t=0)}$. If we denote the reduced density matrix of
the subsystem $\mathcal{M}$ by
$\rho_{\mathcal{M}}(t)=\mathbf{\Tr}_{\mathcal{R}}\\{\outerproduct{{\psi(t)}}{{\psi(t)}}\\}$,
then $\mathcal{Q}_{\mathcal{M}}(t)=\det(\rho_{\mathcal{M}}(t))$ is a time-
dependent entanglement measure of subsystem $\mathcal{M}$, which is also
monotonically related (i.e., they concurrently increase, decrease, or don’t
change with an infinitesimal change in the density matrix) to the second-order
Rényi entanglement entropy through
$\mathcal{S}_{2}({\mathcal{M}})=-\log_{2}(1-2\mathcal{Q_{M}})$. Equivalently,
we will use its Laplace transformation
$\tilde{\mathcal{Q}}_{\mathcal{M}}(s)=\mathcal{L}\\{\mathcal{Q}_{\mathcal{M}}(t)\\}$
as the dynamical entanglement measure in the analysis.

The entanglement measure $\tilde{\mathcal{Q}}_{\mathcal{M}}(s)$ can be
obtained from the QCTF. Given any basis for the quantum system which is
constructed from the arbitrary basis vectors $\\{\ket{+},\ket{-}\\}$ for
$\mathcal{M}$ and $\\{\ket{j},j=0,...,d-1\\}$ ($d$ can be countably infinite)
for $\mathcal{R}$, we define the QCTF on the off-diagonal block
$\matrixelement{+}{\tilde{\rho}(s)}{-}$ (which fully describes the bipartite
entanglement between the subsystems) as,

$\mathcal{H}=\textbf{G}^{{\dagger}}(s^{*})\bigg{(}\sum_{i,j}{z_{a}}^{i+j}{z_{d}}^{i-j}\ket{-\otimes
j}{\bra{+\otimes i}\bigg{)}\star\textbf{G}(s)},$ (6a)
$\mathcal{K}(z_{d},z_{a},s)=\expectationvalue{\mathcal{H}}{\psi_{0}}.$ (6b)

Having introduced the form of QCTF for the purpose of this section, Appendix C
proves that the dynamical entanglement measure
($\tilde{\mathcal{Q}}_{\mathcal{M}}(s)$) can be obtained from the QCTF (6b)
as,

$\begin{split}\tilde{\mathcal{Q}}_{M}(s)=&\underset{{\begin{subarray}{c}z_{d}=0\\\
z_{a}=0\end{subarray}}}{\mathbf{Res}}\big{(}(z_{d}z_{a})^{-1}{\mathcal{K}(z_{d},z_{a},s)\star\mathcal{K}^{*}(1/z^{*}_{d},1/z^{*}_{a},s^{*})}\big{)}\\\
&-{\mathcal{K}_{d}(s)\star\mathcal{K}_{d}^{*}(s^{*})},\end{split}$ (7)

with
$\mathcal{K}_{d}(s)=\evaluated{\underset{{\begin{subarray}{c}z_{d}=0\end{subarray}}}{\mathbf{Res}}\big{(}z_{d}^{-1}{\mathcal{K}(z_{d},z_{a},s)}\big{)}}_{z_{a}=1}$,
where $\underset{{{z=a}}}{\mathbf{Res}}(f(z))$ denotes the residue of $f$ at
$z=a$. The first term in (7) corresponds to the Frobenius norm of the off-
diagonal sub-matrix $\matrixelement{+}{\tilde{\rho}(s)}{-}$, while the second
term corresponds to the summation of the cross-correlation of its diagonal.
These two quantities are identical when the subsystems $\mathcal{M}$ and
$\mathcal{R}$ are not entangled. In this formulation, direct evaluation of
system’s time evolution is avoided, and entanglement is directly obtained from
the Hamiltonian’s features and the initial wave function, which are reflected
in (6). Similar to the discussion in the previous section, the present QCTF
formulation can depend on the choice of basis sets $\\{j\\}$ and
$\\{\ket{+},\ket{-}\\}$. Nevertheless, as will be shown shortly, upon finding
the residues in (7), one is left with a basis-independent expression that is
geometrically meaningful. Concomitantly, strategically choosing the basis sets
and their labels can simplify the QCTF formulation, similar to the theoretical
analysis in [47].

Denoting $c_{k}\doteq\innerproduct{k}{\psi_{0}}$ and the eigenmodes of the
system’s Hamiltonian by $\\{\ket{k},E_{k}\\}$, the resolvent is

$\mathbf{G}(s)=\sum_{k}{(s+\frac{i}{\hbar}E_{k})^{-1}\ket{k}\bra{k}}.$ (8)

Using the QCTF transformation (6a), and then (7), the entanglement measure for
the two-level subsystem of interest ($\mathcal{M}$), is simplified to:

$\tilde{\mathcal{Q}}_{M}(s)=\sum_{k,l,k^{\prime},l^{\prime}}c_{l}c_{k}^{*}c_{k^{\prime}}c_{l^{\prime}}^{*}\frac{\innerproduct{+k}{+k^{\prime}}\innerproduct{-l^{\prime}}{-l}-\innerproduct{+k}{-l}\innerproduct{-l^{\prime}}{+k^{\prime}}}{s+\frac{i}{\hbar}(E_{l}-E_{l^{\prime}}-E_{k}+E_{k^{\prime}})},$
(9)

where the inner products in the numerator, referred to as local overlaps, are
defined through the following decomposition:

$\begin{split}\ket{k}=\ket{+k}\otimes\ket{+}+\ket{-k}\otimes\ket{-},\\\
(\text{equivalently})\ket{\pm
k}=\Big{(}\sum_{j}\big{(}\ket{j}(\bra{\pm}\otimes\bra{j})\big{)}\Big{)}\ket{k}.\end{split}$
(10)

The vectors $\ket{\pm k}$ and $\ket{\pm l}$, which are not necessarily
normalized, are in $\mathbb{C}^{{d}}$ and lie in the vector space underlying
subsystem $\mathcal{R}$. Equation (9) involves three different components
contributing to the entanglement dynamics. The first component is the product
of the wave-function components multiplying the fraction, which describes the
contribution from each eigenmode. The second component consists of the
products of local overlaps in the numerator of the fraction, which is related
to the geometric structure of the eigenstates and determines the strength of
each eigenmode. The third component consists of the denominator of the
fraction which determines the Laplace poles in the entanglement measure.

Equation (9) is a basis-invariant description of entanglement since the
dependence on labels $i,j$ is no longer present. Also, the summation is
independent of the choice of $\ket{\pm}$; to see this latter point, note that
the first and third components (i.e., the coefficients and the denominator of
the fraction) in equation (9) are invariant under the permutations
$k\leftrightarrow l^{\prime}$, $k^{\prime}\leftrightarrow l$ and
($k\leftrightarrow l^{\prime},k^{\prime}\leftrightarrow l$). Using this
symmetry, one can show the basis-invariance of (9) over the choice of
$\ket{\pm}$ (by considering a unitary change of basis
$\ket{\pm^{\prime}}=U\ket{\pm}$). This observation enables using tensor
algebra in the next section to give geometric insight into the entanglement
measure (9).

$\mathbb{C}^{d}$$|\bra{\hat{j}_{1}\psi}\wedge\bra{\hat{j}_{2}\psi}|$
$|\bra{\hat{j}_{3}\psi}\wedge\bra{\hat{j}_{2}\psi}|$$|\bra{\hat{j}_{3}\psi}\wedge\bra{\hat{j}_{1}\psi}|$$\ket{\hat{j}_{1}\psi}$$\ket{\hat{j}_{2}\psi}$$\ket{\hat{j}_{3}\psi}$
Figure 1: The area spanned by pairs of marginal (or projected) wave functions
is depicted. These vectors, given by (13), belong to $\mathbb{C}^{d}$, which
is the vector-space underlying subsystem $\mathcal{R}$. Equivalently, up to
normalization, marginal wave-function $\ket{\hat{j}\psi}$ is equal to the
post-measurement (i.e., the projected wave function of subsystem
$\mathcal{M}$, after projective measurement of subsystem $\mathcal{M}$, in the
$\\{\ket{\hat{j}}\\}$ basis) wave-function of subsystem $\mathcal{R}$. The
total squared areas (i.e., the three shaded regions in the Figure) gives the
entanglement between the subsystems.

##  IV  Geometry of the QCTF entanglement

In this section, a geometric interpretation of the QCTF entanglement measure
(9) is presented. Toward this goal, we will start by introducing the following
notation. Since the quantum system’s Hilbert space is isomorphic to
$\mathbb{C}^{2d}$, we will treat $\bra{z}$ as the (covariant) dual of
$\ket{z}=\sum_{i}z^{i}\ket{i}$ with the Euclidean metric $g_{ij}=\delta_{ij}$,
i.e. $\innerproduct{z}{z}=\delta_{ij}z^{i}{z^{*}}^{j}=z^{i}z^{*}_{i}$, where
in the last term we have used the Einstein summation convention. In this
treatment, $\bra{z}$ is a 1-form.

###  IV.1 Exterior algebraic description of entanglement

Using the above terminology, the numerator of the fraction in the QCTF
entanglement measure (9) can be written as a 2-form acting on the tensor
product of two contravariant (ket) vectors, all of which belong to
$\mathbb{C}^{d}$, as follows:

$\text{Numerator in
(\ref{enta})}=(\bra{+k}\wedge\bra{-l^{\prime}})(\ket{+k^{\prime}}\otimes\ket{-l}),$
(11)

where the wedge operator ($\wedge$) is the alternating product, i.e.,
$\bra{z_{1}}\wedge\bra{z_{2}}=(\bra{z_{1}}\otimes\bra{z_{2}}-\bra{z_{2}}\otimes\bra{z_{1}})$.
Note that in the remainder of the paper, any product between functions in the
Laplace domain is understood to be carried out using the $\star$ operator
defined in (3). Therefore, the evolution of entanglement between $\mathcal{M}$
and $\mathcal{R}$ is given by the exterior product between the components of
the Hamiltonian’s eigenstates in directions corresponding to an orthonormal
basis for subsystem $\mathcal{M}$. Building on (11), we can write the entire
entanglement measure $\tilde{\mathcal{Q}}_{\mathcal{M}}(s)$ in (9) as:

$\tilde{\mathcal{Q}}_{\mathcal{M}}(s)=(\bra{+\psi}\wedge\bra{-\psi})(\ket{+\psi}\otimes\ket{-\psi}),$
(12)

where $\ket{\psi}=\ket{+\psi}\otimes\ket{+}+\ket{-\psi}\otimes\ket{-}$, which
is
$\ket{\pm\psi}=\Big{(}\sum_{j}\big{(}\ket{j}(\bra{\pm}\otimes\bra{j})\big{)}\Big{)}\mathbf{G}(s)\ket{\psi_{0}}$.
This description of entanglement is fully basis-independent (with respect to
$\\{\ket{\pm}\\}$ and $\\{\ket{j}\\}$) and $U(1)$ gauge invariant, i.e.,
$\ket{\psi}\rightarrow\ket{\tilde{\psi}}=e^{i\theta}\ket{\psi}$ for any real
$\theta$. Geometrically, in the case of a two-level subsystem $\mathcal{M}$,
the QCTF entanglement measures the squared area of the parallelogram spanned
by $\ket{+\psi}$ and $\ket{-\psi}$ in $\mathbb{C}^{{d}}$.

This geometric observation can be extended to the QCTF entanglement measure
for subsystem $\mathcal{M}$ with more than two energy levels. In this case, we
denote an arbitrary basis for the sub-vector spaces underlying subsystem
$\mathcal{M}$ ($n$-dimensional) and $\mathcal{R}$, respectively by
$\\{\ket{\hat{j}};j=1,\cdots,n\\}$ and $\\{\ket{{i}}:i=1,\cdots,d\\}$.
Denoting the projected wavefunctions into its components along
$\\{\ket{\hat{j}}\\}$ as

$\begin{split}\ket{\hat{j}\psi}&=(\bra{\hat{j}}\otimes
I_{\mathcal{R}})\ket{\psi}\\\
&=\Big{(}\sum_{i}\big{(}\ket{i}(\bra{\hat{j}}\otimes\bra{i})\big{)}\Big{)}\mathbf{G}(s)\ket{\psi_{0}}=\sum_{i}\psi_{\hat{j}{i}}\ket{i},\end{split}$
(13)

the following expression in (14) is a measure of entanglement between
subsystems $\mathcal{M}$ and $\mathcal{R}$ : The total collective squared
areas spanned by pairs of projected wave functions (see Figure 1):

$\begin{split}\tilde{\mathcal{Q}}_{\mathcal{M}}(s)=\sum_{1\leq\hat{j}_{1}<\hat{j}_{2}\leq
n}{(\bra{\hat{j}_{1}\psi}\wedge\bra{\hat{j}_{2}\psi})(\ket{\hat{j}_{1}\psi}\otimes\ket{\hat{j}_{2}\psi})}.\end{split}$
(14)

In the next subsection, it is shown that this entanglement measure is
monotonically related to the second-order Rényi entropy.

Given this extension of $\tilde{\mathcal{Q}}_{\mathcal{M}}(s)$ to subsystems
with arbitrary energy levels, the QCTF transformation (6) can be modified by
considering an extra complex variable, $z_{c}$. Since the summation in (14)
goes over pairs of basis vectors for subsystem $\mathcal{M}$, one can use an
arbitrary labeling $h$ (i.e., by assigning a unique non-negative integer $h$
to each of the $\frac{n(n-1)}{2}$ pairs) for the set
$\\{(\hat{j}_{1},\hat{j}_{2})|1\leq\hat{j}_{1}<\hat{j}_{2}\leq n\\}$ (we
denote the pair labeled $h$ by $(h_{-},h_{+})$). Therefore, the QCTF can be
written as follows:

$\mathcal{H}=\textbf{G}^{{\dagger}}(s^{*})\bigg{(}\sum_{i,j,h}{z_{a}}^{i+j}{z_{d}}^{i-j}{z_{c}}^{h}\ket{h_{-}\otimes
j}{\bra{h_{+}\otimes i}\bigg{)}\star\textbf{G}(s)},$ (15a)
$\mathcal{K}(z_{d},z_{a},z_{c},s)=\expectationvalue{\mathcal{H}}{\psi_{0}}.$
(15b)

Given this transformation, the entanglement measure
$\tilde{\mathcal{Q}}_{M}(s)$ is:

$\begin{split}\tilde{\mathcal{Q}}_{M}(s)=&\underset{{\begin{subarray}{c}z_{d}=0\\\
z_{a}=0\\\
z_{c}=0\end{subarray}}}{\mathbf{Res}}\big{(}\frac{\mathcal{K}(z_{d},z_{a},z_{c},s)\star\mathcal{K}^{*}(1/z^{*}_{d},1/z^{*}_{a},1/z^{*}_{c},s^{*})}{z_{d}z_{a}z_{c}}\big{)}\\\
&-\underset{{\begin{subarray}{c}z_{c}=0\end{subarray}}}{\mathbf{Res}}\big{(}z_{c}^{-1}{\mathcal{K}_{d}(z_{c},s)\star\mathcal{K}_{d}^{*}(1/z^{*}_{c},s^{*})}\big{)},\end{split}$
(16)

with
$\mathcal{K}_{d}(z_{c},s)=\evaluated{\underset{{\begin{subarray}{c}z_{d}=0\end{subarray}}}{\mathbf{Res}}\big{(}z_{d}^{-1}{\mathcal{K}(z_{d},z_{a},z_{c},s)}\big{)}}_{z_{a}=1}$.
By employing the variable $z_{c}$, and finding the residue (at the origin) in
the last step, this process sums up the contribution from each pair of the
projected wave functions.

###  IV.2 Relation with the second-order Rényi entropy

The entanglement measure (14) is monotonically related to the second-order
Rényi entropy
(${\mathbf{S}}^{(2)}=-\log_{2}\big{(}\mathbf{\Tr({\tilde{\rho}}^{2})}\big{)}$)
of subsystem $\mathcal{M}$. To show this property, note that for each pair of
$(j_{1},j_{2})$ in (14), the argument of the summation,
($(\bra{\hat{j{{}_{1}\psi}}}\wedge\bra{\hat{j}_{2}\psi})(\ket{\hat{j{{}_{1}\psi}}}\otimes\ket{\hat{j}_{2}\psi})$),
is the collection of principal minors

$\begin{vmatrix}\innerproduct{\hat{j}_{1}\psi}{\hat{j}_{1}\psi}&\innerproduct{\hat{j}_{1}\psi}{\hat{j}_{2}\psi}\\\
\innerproduct{\hat{j}_{2}\psi}{\hat{j}_{1}\psi}&\innerproduct{\hat{j}_{2}\psi}{\hat{j}_{2}\psi}\end{vmatrix}$

of the reduced density matrix $\rho_{\mathcal{M}}$. Thus,

$\tilde{\mathcal{Q}}_{\mathcal{M}}(s)=\sum{\text{all {\hbox{2\crossproduct 2}}
principal minors of {\hbox{\rho_{\mathcal{M}}}} }}.$ (17)

Additionally, from matrix algebra, we have the following theorem [48]: The
total sum of $2\crossproduct 2$ principal minors of any square matrix (here,
the reduced density matrix $\rho_{\mathcal{M}}$) is equal to the second
elementary symmetric polynomial of its eigenvalues $\\{\lambda_{i}\\}$:

$s_{2}=\sum_{1\leq j_{1}<j_{1}\leq n}\lambda_{j_{1}}\lambda_{j_{2}}.$ (18)

Thus,

$\tilde{\mathcal{Q}}_{\mathcal{M}}(s)=s_{2}=\frac{1}{2}(1-\mathbf{\Tr({\tilde{\rho}}^{2})})=\frac{1}{2}(1-2^{-\mathbf{S}^{(2)})}).$
(19)

This equation relates the second-order Rényi entropy and the entanglement
measure used in this paper. We should note that the quantity
$1-\mathbf{\Tr({\tilde{\rho}}^{2})}$ is sometimes referred to as the linear
entropy of $\mathcal{M}$ [49].

##  V Summary and Conclusion

The new description of quantum many-body dynamics through the QCTF provides an
equivalent representation of the wave-function or the density matrix that
allows for obtaining entanglement dynamics in a quantum system directly from
its Hamiltonian. Therefore, the QCTF circumvents the bottleneck of evaluating
the many-body system’s evolution. The structure of the QCTF transformation
enables using various forms of simplifications and algebraic manipulations.
For example, the vector spaces’ basis used in the QCTF transformation (denoted
by $\\{\ket{j}\\}$ and $\\{\ket{i^{\prime}}\\}$ in the paper) can be picked
and labeled efficiently to give a closed-form function or a collection of them
as sub-series of the transformation, e.g., each series might correspond to a
perturbation order (similar to the application of QCTF in [47]). Another form
of such simplifications offered by the QCTF is given in [19], wherein basis
kets are labeled according to the single-magnon states that they are
representing.

Given the Hamiltonian and the initial state of the system, entanglement
dynamics of subsystems can be obtained by finding the residues of the QCTF.
Alternatively, the final QCTF entanglement can be described independently of
the basis sets for each subsystem. In this case, entanglement was shown to
have an exterior algebraic structure. More precisely, it is equal to the
collective addition of norms of 2-forms constructed by the projected (i.e.,
marginal) wave functions. These norms quantify the squared area spanned by the
corresponding marginal wave functions.

In this paper, a geometric portrayal of entanglement was presented as a
measure of this quantity by its direct and equivalent consequences on the
geometry of the system’s state. This feature opens up the prospect of
employing additional tools from geometry to understand and analyze
entanglement in different fields and settings in future research. In
particular, considering non-constant Hamiltonians (i.e., either with time-
dependence or with Hamiltonians described by independent parameters) should
lead to a differential geometric description of the QCTF entanglement
dynamics, which can be advantageous as it allows for using the generalized
Stokes theorem (given the 2-differential form structure of the QCTF
formulation) to study entanglement in these more complex scenarios.
Additionally, the link between the QCTF entanglement formulation and the
second-order Rényi entropy, discussed in subsection IV.2, which is established
through a relation between the principal minors of the density matrix and the
second elementary symmetric polynomials of its eigenvalues, may lead to novel
approaches to measure entanglement in experimental settings. This opportunity
arises because the elementary symmetric polynomials (18), fully describing the
Rényi entropy, can equivalently be obtained in terms of local overlaps of the
wavefunction, which are accessible experimentally [50, 51].

###### Acknowledgements.

P.A acknowledges support from the Princeton Program in Plasma Science and
Technology (PPST). H.R acknowledges support from the U.S Department Of Energy
(DOE) grant (DE-FG02-02ER15344).

## Appendix

##  Appendix A Derivation of equation (2)

Here, we show that the QCTF in (2) can be interpreted as a transformation of
the system’s density matrix by considering one chronological frequency, $s$,
and two structural complex variables, $z_{a}$ and $z_{d}$. Using (1) and the
definition of operation $\star$, we can rewrite (2) as

$\begin{split}&{\mathcal{K}}(z_{d},z_{a},s)\doteq\bar{\mathcal{K}}(z_{d}z_{a},s)\star\bar{\mathcal{K}}^{*}\big{(}(z_{a}/z_{d})^{*},s^{*}\big{)}\\\
&=\sum_{l,k=0}^{d-1}{\matrixelement{l}{\mathbf{G}(s)}{\psi_{0}}\star\matrixelement{\psi_{0}}{\mathbf{G}^{\dagger}(s^{*})}{k}z_{d}^{l-k}z_{a}^{l+k}}.\end{split}$
(20)

We define
$\tilde{c}_{l}(s)\doteq\matrixelement{l}{\mathbf{G}(s)}{\psi_{0}}=\mathcal{L}{\\{\innerproduct{l}{\psi(t)}\\}}=\mathcal{L}{\\{c_{l}(t)\\}}$.
Then using definition (3) we have:

$\begin{split}\tilde{c}_{l}(s)\star\tilde{c}_{k}^{*}(s^{*})&=\frac{1}{2\pi
i}\int_{-\infty}^{\infty}{d\omega}{\tilde{c}_{l}(\sigma+i\omega)\tilde{c}_{k}^{*}\big{(}(s-\sigma-i\omega)^{*}\big{)}}\\\
&=\int_{-\infty}^{\infty}{dt_{1}}\int_{-\infty}^{\infty}{dt_{2}}\frac{1}{2\pi
i}\int_{-\infty}^{\infty}{d\omega}{c_{l}(t_{1})c_{k}^{*}(t_{2})e^{-st_{2}-\sigma(t_{1}-t_{2})-i\omega(t_{1}-t_{2})}}\\\
&=\int_{-\infty}^{\infty}{dt_{1}}\int_{-\infty}^{\infty}{dt_{2}}{c_{l}(t_{1})c_{k}^{*}(t_{2})e^{-st_{2}-\sigma(t_{1}-t_{2})}\delta(t_{1}-t_{2})}\\\
&=\int_{-\infty}^{\infty}{dt}{c_{l}(t)c_{k}^{*}(t)e^{-st}}=\matrixelement{l}{\tilde{\rho}(s)}{k}.\end{split}$
(21)

Therefore, substituting
$\matrixelement{l}{\mathbf{G}(s)}{\psi_{0}}\star\matrixelement{\psi_{0}}{\mathbf{G}^{\dagger}(s^{*})}{k}$
with $\matrixelement{l}{\tilde{\rho}(s)}{k}$ in (20), gives:

${\mathcal{K}}(z_{d},z_{a},s)=\sum_{l,k=0}^{d-1}\matrixelement{l}{\tilde{\rho}(s)}{k}z_{d}^{l-k}z_{a}^{l+k}.$
(22)

##  Appendix B Derivation of equation (5)

Using equation (4), we can rewrite the R.H.S of equation (5) as follows

$\frac{1}{(2\pi i)^{3}}\oint_{\partial C_{s}}\,ds\oint_{\partial
C_{d}}\,dz_{d}\oint_{\partial
C_{a}}\,dz_{a}e^{st^{\prime}}\sum_{l,k}\sum_{l^{\prime\prime},k^{\prime\prime}}\innerproduct{k}{k^{\prime}}\innerproduct{l^{\prime}}{l}z_{d}^{(l^{\prime\prime}-k^{\prime\prime})-(l-k)-1}z_{a}^{(l^{\prime\prime}+k^{\prime\prime})-(l+k)-1}\matrixelement{l^{\prime\prime}}{\tilde{\rho}(s)}{k^{\prime\prime}}.$
(23)

Note that the three integrals are independent and can be evaluated
interchangeably. We consider the $z_{d}$ and $z_{a}$ integrals first. Using
the fact that the integrand is holomorphic inside the closed contours
$\partial C_{d}$ and $\partial C_{a}$, except at the origin, by employing
Cauchy’s residue theorem the value of these two integrations are equal to the
residue of the integrand at the origin $(z_{d}={0},z_{a}={0})$ times $(2\pi
i)^{2}$. Also note that since
$\matrixelement{l^{\prime\prime}}{\tilde{\rho}(s)}{k^{\prime\prime}}$ has no
$z_{d},z_{a}$ dependence, the residue of the integrand in the $\mathbb{C}^{2}$
space is equal to the coefficient of $(z_{a}z_{d})^{-1}$ in the summation,
which corresponds to the terms with $k^{\prime\prime}=k,l^{\prime\prime}=l$.
Therefore, after evaluating the structural integrations, equation (5) is

$(\ref{50})=\frac{1}{2\pi i}\oint_{\partial
C_{s}}\,dse^{st^{\prime}}\sum_{l,k}\innerproduct{l^{\prime}}{l}\matrixelement{l}{\tilde{\rho}(s)}{k}\innerproduct{k}{k^{\prime}}=\matrixelement{l^{\prime}}{\frac{1}{2\pi
i}\oint_{\partial
C_{s}}\,dse^{st^{\prime}}\tilde{\rho}(s)}{k^{\prime}}=\matrixelement{l^{\prime}}{\rho(t^{\prime})}{k^{\prime}},$
(24)

where we have used the definition of the inverse Laplace transform, and the
linearity of the transform.

##  Appendix C Proof of equation (7)

We start from the system’s state in the time domain, $\ket{\psi(t)}$, and
expand it in the product basis vectors of each subsystem,
$\\{\ket{+},\ket{-}\\}$, and $\\{\ket{l},l=0,...,{d-1}\\}$, as follows

$\begin{split}\ket{\psi(t)}=\sum_{a\in\\{+,-\\}}\sum_{l=0,...,{d-1}}{c_{al}(t){\ket{a\otimes
l}}};\\\ c_{al}(t)=\innerproduct{a\otimes l}{{\psi(t)}}.\end{split}$ (25)

Based on this expansion, we may construct the matrix $M_{2\times d}$ such that
its first and second rows consist of $\\{c_{+l}(t)\\}$ and $\\{c_{-l}(t)\\}$
respectively (the time dependence of the c’s is not shown below for
simplicity):

$M(t)\doteq\begin{pmatrix}c_{+0}&c_{+1}&\cdots&c_{+l}&\cdots&c_{+d-1}\\\
c_{-0}&c_{-1}&\cdots&c_{-l}&\cdots&c_{-d-1}\end{pmatrix}.$ (26)

If subsystem $\mathcal{M}$ is not entangled to subsystem $\mathcal{R}$ at
$t=t^{\prime}$, then $\ket{\psi(t^{\prime})}$ is a product state and the rows
of $M(t^{\prime})$ are linearly dependent, i.e.
$\rank\big(M(t^{\prime})\big{missing})=1$. This condition on the rank of $M$
is necessary and sufficient for the subsystem $\mathcal{M}$ to not be
entangled to subsystem $\mathcal{R}$. Thus, if $rank(M)=2$ these subsystems
are entangled. To construct a smooth indicator of entanglement, consider the
following square sub-matrices $M^{ij}_{2\times 2}$, formed from the $i$th and
$j$th columns of $M$,

$M^{ij}(t)\doteq\begin{pmatrix}c_{+i}&c_{+j}\\\ c_{-i}&c_{-j}\end{pmatrix},$
(27)

and use them to define the entanglement measure $\mathcal{Q}_{\mathcal{M}}(t)$
as follows:

$\mathcal{Q}_{\mathcal{M}}(t)\doteq\sum_{0\leq i<j\leq
d-1}{|\det\big(M^{ij}(t)\big{missing})|^{2}}.$ (28)

More generally, $\mathcal{Q}_{\mathcal{M}}(t)=0$ is a necessary and sufficient
condition for the subsystem $\mathcal{M}$ not to be entangled at $t$.
Expanding the summations in $\mathcal{Q}_{(}\mathcal{M})(t)$ will result in

$\displaystyle\mathcal{Q}_{\mathcal{M}}(t)=$ $\displaystyle\sum_{0\leq i<j\leq
d-1}{\det(M^{ij})\big{(}\det(M^{ij})\big{)}^{*}}$ (29a) $\displaystyle=$
$\displaystyle\sum_{0\leq i<j\leq
d-1}{(c_{+i}c_{-j}-c_{+j}c_{-i})(c^{*}_{+i}c^{*}_{-j}-c^{*}_{+j}c^{*}_{-i})}$
$\displaystyle=$ $\displaystyle\sum_{0\leq i\neq j\leq
d-1}{|c_{+i}|^{2}|c_{-j}|^{2}}-\sum_{0\leq i\neq j\leq
d-1}{c_{+i}c_{-j}c^{*}_{+j}c^{*}_{-i}}$ $\displaystyle=$
$\displaystyle\sum_{0\leq i,j\leq d-1}{|c_{+i}|^{2}|c_{-j}|^{2}}-\sum_{0\leq
i,j\leq d-1}{c_{+i}c_{-j}c^{*}_{+j}c^{*}_{-i}}$ $\displaystyle=$
$\displaystyle\big{(}\sum_{0\leq i\leq
d-1}{|c_{+i}|^{2}}\big{)}\big{(}\sum_{0\leq i\leq d-1}{|c_{-i}|^{2}}\big{)}$
(29b) $\displaystyle-$ $\displaystyle\big{(}\sum_{0\leq i\leq
d-1}{c_{+i}c^{*}_{-i}}\big{)}\big{(}\sum_{0\leq i\leq
d-1}{c^{*}_{+i}c_{-i}}\big{)}$ (29c) $\displaystyle=$
$\displaystyle\begin{vmatrix}\sum_{i}{|c_{+i}|^{2}}&\sum_{i}{\big{(}c_{+i}c^{*}_{-i}\big{)}}\\\
\sum_{i}{\big{(}c^{*}_{+i}c_{-i}\big{)}}&\sum_{i}{|c_{-i}|^{2}}\end{vmatrix}=\det\big(\rho_{\mathcal{M}}(t)\big{missing}),$

which is the determinant of the time dependent reduced density matrix of
subsystem $\mathcal{M}$.

As the second step, here we prove that the R.H.S of equation (7) is the
Laplace transform of $\mathcal{Q}_{\mathcal{M}}(t)$. We start from the first
term in (7). Before proceeding, we introduce the notation
$\tilde{c}_{al}(s)\doteq\mathcal{L}\\{c_{al}(t)\\}=\matrixelement{a\otimes
l}{\mathbf{G}(s)}{{\psi_{0}}}$. Therefore, by using the definition of
$\mathcal{K}(z_{a},z_{d},s)$ in (6b), the first term in (7) and the fact that
the $\star$ operation is associative and commutative , we have

$(2\pi i)^{-2}\sum_{l,k}\sum_{l^{\prime},k^{\prime}}\oint_{\partial
C_{d}}\textbf{d}z_{d}\oint_{\partial
C_{a}}\textbf{d}z_{a}{\tilde{c}_{+l}(s)\star\tilde{c}^{*}_{-k}(s^{*})\star{\tilde{c}^{*}_{+l^{\prime}}(s^{*})\star\tilde{c}_{-k^{\prime}}(s)z_{a}^{l+k-l^{\prime}-k^{\prime}-1}z_{d}^{l-k+k^{\prime}-l^{\prime}-1}}}.$
(30)

Upon employing Cauchy’s residue theorem, based on the definition of the closed
contours $\partial C_{d}$ and $\partial C_{a}$, then the value of the double
integral is equal to $(2\pi i)^{2}$ times the residue of the integrand at the
origin of the structural space $(z_{d},z_{a})$, which is the coefficient of
$z_{d}^{-1}z_{a}^{-1}$ in the double summation. Therefore, the only remaining
terms after the double integration must satisfy the following set of
conditions:

$\begin{cases}l+k-l^{\prime}-k^{\prime}-1=-1\\\
l-k+k^{\prime}-l^{\prime}-1=-1\end{cases}\iff\begin{cases}l=l^{\prime}\\\
k=k^{\prime}\end{cases}.$ (31)

Consequently, the first term in (7) is:

$\sum_{l,k}{\tilde{c}_{+l}(s)\star{\tilde{c}^{*}_{+l}(s^{*})\star\tilde{c}_{-k}(s)}\star\tilde{c}^{*}_{-k}(s^{*})}=\mathcal{L}\\{\sum_{l,k}{|c_{+l}|^{2}|c_{-k}|^{2}}\\},$
(32)

which is the Laplace-dual of (29). In the last step, we operated with $\star$
similar to our calculation in equation (21). Based on this equality, by
showing that the second term in (7) corresponds to (29c), the proof is
accomplished. To this end, we first consider $\mathcal{K}_{d}(z_{a},s)$. Based
on the definitions (6b) and (7) we have:

$\mathcal{K}_{d}(z_{a},s)=(2\pi i)^{-1}\sum_{l,k}\oint_{\partial
C_{d}}\textbf{d}z_{d}{{\tilde{c}_{+l}(s)\star{\tilde{c}^{*}_{-k}(s^{*})z_{a}^{l+k}z_{d}^{l-k-1}}}}.$
(33)

Applying Cauchy’s residue theorem, the integral is equal to $(2\pi i)$ times
the residue of the integrand at the origin of the $z_{d}$ space, which is the
coefficient of $z_{d}^{-1}$ in it’s Maclaurin series. Therefore, it is easy to
see that for this integrand, only the terms with $l=k$ remain after the
integration and we have

$\mathcal{K}_{d}(z_{a},s)=\sum_{l}{\tilde{c}_{+l}(s)\star\tilde{c}^{*}_{-l}(s^{*})z_{a}^{2l}}.$
(34)

Consequently, the second term in (7) is

$\evaluated{\sum_{l,k}{\tilde{c}_{+l}(s)\star{\tilde{c}^{*}_{-l}(s^{*})\star\tilde{c}_{+k}^{*}(s^{*})\star{\tilde{c}_{-k}(s)}z_{a}^{2(l-k)}}}}_{z_{a}=1}=\mathcal{L}\\{\sum_{l,k}{c_{+l}c_{-k}c^{*}_{+k}c^{*}_{-l}\\}},$
(35)

which is the Laplace-dual of the second term in (29c). Therefore, using the
linearity of the Laplace transformation, the R.H.S of (7) is equal to the
Laplace transformation of $\mathcal{Q}_{\mathcal{M}}(t)$, which proves the
assertion (7).

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# Canonical analysis of covariant unimodular gravity
and an extension of the Kodama state

Shinji Yamashita111<EMAIL_ADDRESS>
National Institute of Technology, Niihama College, Ehime, Japan

###### Abstract

We carry out the canonical analysis of a covariant version of unimodular
gravity in terms of the connection representation. We then proceed to quantize
this theory by implementing the Dirac procedure. We confirm whether and how
the Kodama state, which is a solution of quantum general relativity, can be
extended into covariant unimodular gravity. Finally, we discuss the difference
of quantum states between covariant unimodular gravity, the original
unimodular gravity, and general relativity.

##  1 Introduction

Unimodular gravity is a simple modification of general relativity (GR) in
which the determinant of the spacetime metric is restricted to be constant.
Due to this restriction, unimodular gravity does not preserve the full
diffeomorphism invariance. Nevertheless, the classical field equations in
unimodular gravity are almost the same as in GR. A subtle but crucial
difference from GR is that the cosmological constant is treated as an
arbitrary integration constant [1]. This arbitrariness brings a different
perspective on the cosmological constant problem [2, 3].

The Hamiltonian analysis of unimodular gravity in terms of the Arnowitt-Deser-
Misner (ADM) variables has been performed in Refs. [4, 5]. In contrast to
ordinary GR, the lapse function is not regarded as an independent variable due
to the unimodular condition, and the Hamiltonian constraint is a second-class
constraint. Additionally, the total Hamiltonian does not vanish on the
constraint surface. In canonical quantum theory, these differences from GR
cause the differences in the physical states. Specifically, the physical state
of unimodular gravity is constructed from the eigenstates of the cosmological
constant. Furthermore, unimodular gravity can have an appropriate time
variable, and the physical state obeys the Schrödinger-like equation rather
than the Wheeler-DeWitt one. In this sense, one can avoid the problem of time
in quantum gravity [2, 6].

In this paper, we perform the canonical analysis of unimodular gravity and its
quantization; however, the theory we will discuss has two different points
from the original unimodular gravity explained above. The first point is that
we employ a covariant version of unimodular gravity that was suggested in Ref.
[7]. In this framework, the square root of the determinant of the spacetime
metric is equal to the divergence of a densitized vector field, and the full
diffeomorphism invariance is retained. Moreover, one can introduce time as
spacetime volume [8]. This theory gives the same physics as the original
unimodular gravity at least at the classical level, while we can expect that
these two unimodular theories provide different quantum theories because of
the difference of the constraints.

The second point is that we describe the theory in terms of the connection
representation instead of the ADM one. Within the framework, one of the
configuration variables is the Ashtekar-Barbero connection with the Barbero-
Immirzi parameter $\beta$, and its conjugate momentum is the densitized triad
[9, 10, 11]. The advantage of this representation is that the constraints are
somewhat simpler than those in the ADM representation. Thanks to the
simplicity, several solutions that satisfy quantum first-class constraints of
GR have been found. In particular, the Kodama state, which is also called the
Chern-Simons state, is a well-known solution of quantum GR with a nonvanishing
cosmological constant in the case of $\beta=\text{i}$ (the imaginary unit)
[12, 13]. A generalization of the Kodama state for real values of $\beta$ was
also suggested in Ref. [14]. On the other hand, the Kodama state is not
regarded as a physical state in the original unimodular gravity [15].

The aim of this paper is to confirm the difference between covariant
unimodular gravity, the original unimodular gravity, and GR, especially at the
quantum level. The manuscript is organized as follows. In Sec. 2, we perform
the Hamiltonian analysis of covariant unimodular gravity in terms of the
connection representation. While some work along this line has been done for
$\beta=\text{i}$ [16, 17], we further develop the theory for real values of
$\beta$. Although this choice makes the Hamiltonian constraint more
complicated than in the case of $\beta=\text{i}$, it facilitates the
construction of the inner product in quantum theory. In Sec. 3, we quantize
the theory by applying the Dirac procedure [18, 19]. Then, we consider whether
and how the Kodama state can be extended into covariant unimodular gravity,
mainly following Ref. [14]. In Sec. 4, we summarize our results including the
comparison between covariant unimodular gravity, the original unimodular
gravity, and GR.

We use the following notation. Greek letters
$\mu,\nu,\dots\in\\{\tau,1,2,3\\}$ indicate four-dimensional spacetime indices
where $\tau$ is the time flow component. Capital letters
$I,J,\dots\in\\{0,1,2,3\\}$ are Lorentz indices. Letters
$a,b,\dots\in\\{1,2,3\\}$ are three-dimensional spatial indices, and
$i,j,\dots\in\\{1,2,3\\}$ are internal su(2) Lie algebra indices. We employ a
four-metric signature $(-,+,+,+)$, and use units in which the speed of light
is unity.

##  2 Canonical analysis

The simplest action of the original unimodular gravity without matter is
obtained by modifying the Einstein-Hilbert action

$\displaystyle S_{\text{UG}}(g_{\mu\nu},\Lambda)$
$\displaystyle=\frac{1}{2k}\int\text{d}^{4}x\ \left[\sqrt{-\det g}\
R^{(4)}-\Lambda\left(\sqrt{-\det g}-\alpha\right)\right],$ (1)

where $k$ is Newton’s constant times $8\pi$, $R^{(4)}$ is a scalar curvature
of four-dimensional spacetime, $\Lambda$ is a scalar field that plays the role
of a Lagrange multiplier, and $\alpha$ is a fixed scalar density. The
variation with respect to $\Lambda$ leads to the unimodular condition
$\sqrt{-\det g}-\alpha=0$. In this framework, the spacetime diffeomorphism is
restricted so that the value of the determinant of the four-metric remains
unchanged.

The reformulation of unimodular gravity that ensures full diffeomorphism
invariance was introduced by Henneaux and Teitelboim [7]. The action has the
form

$\displaystyle
S_{\text{HT}}(g_{\mu\nu},\Lambda,\phi^{\mu})=\frac{1}{2k}\int\text{d}^{4}x\
\left[\sqrt{-\det g}\ R^{(4)}-\Lambda\left(\sqrt{-\det
g}-\partial_{\mu}\phi^{\mu}\right)\right],$ (2)

where $\phi^{\mu}$ is a densitized vector field of weight one. The unimodular
condition is rewritten as $\sqrt{-\det g}\ -\partial_{\mu}\phi^{\mu}=0$. Let
us write the action corresponding to (2) in terms of the connection
representation. This can be done by modifying the Holst action [20]

$\displaystyle
S(\omega_{\mu}^{IJ},e_{I}^{\mu},\Lambda,\phi^{\mu})=\int\text{d}^{4}x\ L$
$\displaystyle=-\frac{1}{2k\beta}\int e^{I}\wedge
e^{J}\wedge\left(R_{IJ}^{(4)}-\frac{\beta}{2}\epsilon_{IJKL}R^{(4)KL}\right)-\frac{1}{48k}\int\Lambda\epsilon_{IJKL}e^{I}\wedge
e^{J}\wedge e^{K}\wedge e^{L}+\frac{1}{2k}\int\text{d}^{4}x\
\Lambda\partial_{\mu}\phi^{\mu},$ (3)

where $e^{I}$ is a cotetrad,
$R^{(4)IJ}=\text{d}\omega^{IJ}+\omega^{I}{}_{K}\wedge\omega^{KJ}$ is a
curvature of the spin connection $\omega_{\mu}^{IJ}$, and $\beta$ is the
Barbero-Immirzi parameter that takes nonvanishing real values. The $3+1$ form
of the above action under the time gauge $e_{a}^{0}=0$ is written as

$\displaystyle S=\frac{1}{k\beta}\int\text{d}^{4}x\
\left(E_{i}^{a}\dot{A}_{a}^{i}-A_{\tau}^{i}G_{i}-N^{a}V_{a}-NC\right)+\frac{1}{2k}\int\text{d}^{4}x\left(\Lambda\dot{\phi}^{\tau}-\Lambda
N\det e-\phi^{a}\partial_{a}\Lambda\right),$ (4)

where $\det e$ is a determinant of a cotriad $e_{a}^{i}$, $E_{i}^{a}=(\det
e)e_{i}^{a}$ is a densitized triad,
$A_{\mu}^{i}=-\frac{1}{2}\epsilon^{i}{}_{jk}\omega_{\mu}^{jk}-\beta\omega_{\mu}^{0i}$,
$N^{a}$ is a shift vector, and $N$ is a lapse function. The spatial component
of $A_{\mu}^{i}$ is usually expressed as $A_{a}^{i}=\Gamma_{a}^{i}+\beta
K_{a}^{i}$, where
$\Gamma_{a}^{i}=-\frac{1}{2}\epsilon^{i}{}_{jk}\omega_{a}^{jk}$ is a three-
dimensional spin connection compatible with $e_{i}^{a}$, and
$K_{a}^{i}=-\omega_{a}^{0i}=K_{ab}e_{j}^{b}\delta^{ij}$ is related with the
extrinsic curvature $K_{ab}$ and $e_{i}^{a}$. In addition,

$\displaystyle G_{i}$
$\displaystyle=-\left(D_{a}E^{a}\right)_{i}=-\left(\partial_{a}E_{i}^{a}+\epsilon_{ij}{}^{k}A_{a}^{j}E_{k}^{a}\right),$
(5) $\displaystyle V_{a}$ $\displaystyle=-E_{i}^{b}F_{ba}^{i},$ (6)
$\displaystyle C$ $\displaystyle=\frac{1}{2\beta\sqrt{\det
E}}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}\left[\left(1+\beta^{2}\right)R_{abk}(E)-F_{abk}\right],$
(7)

where
$F_{ab}^{i}=\partial_{a}A_{b}^{i}-\partial_{b}A_{a}^{i}+\epsilon^{i}{}_{jk}A_{a}^{j}A_{b}^{k}$
is a curvature of $A_{a}^{i}$,
$R_{ab}^{i}(E)=\partial_{a}\Gamma_{b}^{i}-\partial_{b}\Gamma_{a}^{i}+\epsilon^{i}{}_{jk}\Gamma_{a}^{j}\Gamma_{b}^{k}$
is a curvature of $\Gamma_{a}^{i}$ that is constructed from $E_{i}^{a}$, and
$\det E=(\det e)^{2}$ is a determinant of $E_{i}^{a}$. Note that while $C$ (7)
is often expressed as

$\displaystyle C=\frac{\beta}{2\sqrt{\det
E}}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}\left[F_{abk}-\left(1+\beta^{2}\right)\epsilon_{klm}K_{a}^{l}K_{b}^{m}\right],$
(8)

we use the former expression (7) for latter convenience.

The configuration variables of this theory are
$\left(A_{\tau}^{i},A_{a}^{i},N,N^{a},\Lambda,\phi^{\tau},\phi^{a}\right)$.
The canonical conjugate momenta (multiplied by $k\beta$ or $2k$) are given by

$\displaystyle\pi_{i}$ $\displaystyle=k\beta\frac{\partial
L}{\partial\dot{A}_{\tau}^{i}}=0,$ $\displaystyle E_{i}^{a}$
$\displaystyle=k\beta\frac{\partial L}{\partial\dot{A}_{a}^{i}},$ (9)
$\displaystyle\pi_{N}$ $\displaystyle=k\beta\frac{\partial
L}{\partial\dot{N}}=0,$ $\displaystyle\pi_{a}$
$\displaystyle=k\beta\frac{\partial L}{\partial\dot{N}^{a}}=0,$ (10)
$\displaystyle\pi_{\Lambda}$ $\displaystyle=2k\frac{\partial
L}{\partial\dot{\Lambda}}=0,$ (11) $\displaystyle p_{\tau}$
$\displaystyle=2k\frac{\partial L}{\partial\dot{\phi^{\tau}}}=\Lambda,$
$\displaystyle p_{a}$ $\displaystyle=2k\frac{\partial
L}{\partial\dot{\phi^{a}}}=0.$ (12)

These momenta yield primary constraints

$\displaystyle\pi_{i}\approx\pi_{N}\approx\pi_{a}\approx\pi_{\Lambda}\approx\Pi\approx
p_{a}\approx 0,$ (13)

where $\Pi=p_{\tau}-\Lambda$, and the symbol $\approx$ is weak equality, which
indicates that the equality holds on the constraint surface. The fundamental
Poisson bracket relations are

$\displaystyle\begin{aligned}
\left\\{A_{\tau}^{i}(x),\pi_{j}(y)\right\\}&=k\beta\delta_{j}^{i}\delta^{3}(x-y),&\left\\{A_{a}^{i}(x),E_{j}^{b}(y)\right\\}&=k\beta\delta_{a}^{b}\delta_{j}^{i}\delta^{3}(x-y),\\\
\left\\{N(x),\pi_{N}(y)\right\\}&=k\beta\delta^{3}(x-y),&\left\\{N^{a}(x),\pi_{b}(y)\right\\}&=k\beta\delta_{b}^{a}\delta^{3}(x-y),\\\
\left\\{\Lambda(x),\pi_{\Lambda}(y)\right\\}&=2k\delta^{3}(x-y),\\\
\left\\{\phi^{\tau}(x),p_{\tau}(y)\right\\}&=2k\delta^{3}(x-y),&\left\\{\phi^{a}(x),p_{b}(y)\right\\}&=2k\delta_{b}^{a}\delta^{3}(x-y).\end{aligned}$
(14)

The total Hamiltonian $H_{\text{T}}$ is a combination of the ordinary
Hamiltonian and the primary constraints with Lagrange multipliers
$v^{i},v^{a},v_{N},v_{\Lambda},\tilde{w},\tilde{w}^{a}$:

$\displaystyle
H_{\text{T}}(A_{\tau}^{i},\pi_{i},A_{a}^{i},E_{i}^{a},N,\pi_{N},N^{a},\pi_{a},\Lambda,\pi_{\Lambda},\phi^{\tau},p_{\tau},\phi^{a},p_{a})$
$\displaystyle=\int\text{d}^{3}x\biggl{[}\frac{1}{k\beta}\left(A_{\tau}^{i}G_{i}+N^{a}V_{a}+NC\right)+\frac{1}{2k}\left(\Lambda
N\det e+\phi^{a}\partial_{a}\Lambda\right)$ $\displaystyle\hskip
50.00008pt+\frac{1}{k\beta}\left(v^{i}\pi_{i}+v^{a}\pi_{a}+v_{N}\pi_{N}\right)+\frac{1}{2k}\left(v_{\Lambda}\pi_{\Lambda}+\tilde{w}\Pi+\tilde{w}^{a}p_{a}\right)\biggr{]},$
(15)

where $\tilde{w}$ and $\tilde{w}^{a}$ are densities of weight one. In general,
every constraint must satisfy the stability condition, that is, each
constraint must hold under time evolution on the constraint surface. Applying
this condition to the primary constraints (13), we have

$\displaystyle\left\\{\pi_{i},H_{\text{T}}\right\\}$
$\displaystyle=-G_{i}\approx 0,$ (16)
$\displaystyle\left\\{\pi_{a},H_{\text{T}}\right\\}$
$\displaystyle=-V_{a}\approx 0,$ (17)
$\displaystyle\left\\{\pi_{N},H_{\text{T}}\right\\}$
$\displaystyle=-\Phi\approx 0,$ (18)
$\displaystyle\left\\{\pi_{\Lambda},H_{\text{T}}\right\\}$
$\displaystyle=-\left(N\det e-\partial_{a}\phi^{a}-\tilde{w}\right)\approx 0,$
(19) $\displaystyle\left\\{\Pi,H_{\text{T}}\right\\}$
$\displaystyle=-v_{\Lambda}\approx 0,$ (20)
$\displaystyle\left\\{p_{a},H_{\text{T}}\right\\}$
$\displaystyle=-\Sigma_{a}\approx 0,$ (21)

where

$\displaystyle\Phi$ $\displaystyle=\frac{1}{2\beta\sqrt{\det
E}}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}\left[\left(1+\beta^{2}\right)R_{abk}-F_{abk}+\frac{\beta^{2}\Lambda}{6}\epsilon_{abc}E_{k}^{c}\right],$
(22) $\displaystyle\Sigma_{a}$ $\displaystyle=\partial_{a}\Lambda.$ (23)

While $v^{i},v^{a},v_{N}$, and $\tilde{w}^{a}$ remain unspecified,
$v_{\Lambda}$ and $\tilde{w}$ are determined by Eqs. (19) and (20) as

$\displaystyle v_{\Lambda}$ $\displaystyle=0,$ $\displaystyle\tilde{w}$
$\displaystyle=N\det e-\partial_{a}\phi^{a}.$ (24)

The secondary constraints $G_{i}\approx 0$ (16), $V_{a}\approx 0$ (17), and
$\Phi\approx 0$ (18) are the Gauss, vector, and Hamiltonian constraints,
respectively, which are the same as those in GR. The secondary constraint
$\Sigma_{a}\approx 0$ (21) implies that $\Lambda$ is a spatial constant. We
define the smeared versions of the secondary constraints:

$\displaystyle G[X^{i}]$ $\displaystyle=\frac{1}{k\beta}\int\text{d}^{3}x\
X^{i}G_{i}(x),$ (25) $\displaystyle V[X^{a}]$
$\displaystyle=\frac{1}{k\beta}\int\text{d}^{3}x\ X^{a}V_{a}(x),$ (26)
$\displaystyle\Phi[X]$ $\displaystyle=\frac{1}{k\beta}\int\text{d}^{3}x\
X\Phi(x),$ (27) $\displaystyle\Sigma[\tilde{X}^{a}]$
$\displaystyle=\frac{1}{2k}\int\text{d}^{3}x\ \tilde{X}^{a}\Sigma_{a}(x),$
(28)

where $X^{i},X^{a},X$ are test functions, and $\tilde{X}^{a}$ is a densitized
test function. One can check that every secondary constraint has a weakly
vanishing Poisson bracket with the total Hamiltonian (15). Then, every
secondary constraint automatically satisfies the stability condition, and no
more constraints arise.

Now, we can classify the primary constraints
$\left(\pi_{i},\pi_{a},\pi_{N},\pi_{\Lambda},\Pi,p_{a}\right)$ and the
secondary constraints
$\left(G[X^{i}],V[X^{a}],\Phi[X],\Sigma[\tilde{X}^{a}]\right)$ into the first-
and second-class constraints. The weakly nonvanishing Poisson brackets among
these constraints are

$\displaystyle\left\\{\pi_{\Lambda}(x),\Pi(y)\right\\}$
$\displaystyle=2k\delta^{3}(x-y),$ (29)
$\displaystyle\left\\{\pi_{\Lambda},\Phi[X]\right\\}$ $\displaystyle=-X\det
e,$ (30) $\displaystyle\left\\{\pi_{\Lambda},\Sigma[\tilde{X}^{a}]\right\\}$
$\displaystyle=\partial_{a}\tilde{X}^{a}.$ (31)

Hence constraints
$\left(\pi_{\Lambda},\Pi,\Phi[X],\Sigma[\tilde{X}^{a}]\right)$ are second
class, and the remaining constraints are first class.

The number of the second-class constraints can be reduced by replacing
$\Phi[X]$ and $\Sigma[\tilde{X}^{a}]$ with the modified constraints
$\Phi^{\prime}[X]$ and $\Sigma^{\prime}[\tilde{X}^{a}]$, respectively:

$\displaystyle\Phi^{\prime}[X]$
$\displaystyle=\Phi[X]+\frac{1}{2k}\int\text{d}^{3}x\ X\Pi\det e$
$\displaystyle=\frac{1}{2k\beta^{2}}\int\text{d}^{3}x\ \frac{X}{\sqrt{\det
E}}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}\left[\left(1+\beta^{2}\right)R_{abk}-F_{abk}+\frac{\beta^{2}}{6}\epsilon_{abc}p_{\tau}E_{k}^{c}\right],$
(32) $\displaystyle\Sigma^{\prime}[\tilde{X}^{a}]$
$\displaystyle=\Sigma[\tilde{X}^{a}]+\frac{1}{2k}\int\text{d}^{3}x\
\tilde{X}^{a}\partial_{a}\Pi=\frac{1}{2k}\int\text{d}^{3}x\
\tilde{X}^{a}\partial_{a}p_{\tau}.$ (33)

In fact, the weakly nonvanishing Poisson bracket among the primary constraints
$\left(\pi_{i},\pi_{a},\pi_{N},\pi_{\Lambda},\Pi,p_{a}\right)$ and the
secondary constraints
$\left(G[X^{i}],V[X^{a}],\Phi^{\prime}[X],\Sigma^{\prime}[\tilde{X}^{a}]\right)$
is only one:

$\displaystyle\left\\{\pi_{\Lambda}(x),\Pi(y)\right\\}=2k\delta^{3}(x-y).$
(34)

Then, constraints $\left(\pi_{\Lambda},\Pi\right)$ are second class, and the
remaining constraints are first class. Let us count the degrees of freedom of
this theory. Variables
$\left(A_{\tau}^{i},A_{a}^{i},N,N^{a},\Lambda,\phi^{\tau},\phi^{a}\right)$
have $3+9+1+3+1+1+3=21$ components. The first-class constraints
$\left(\pi_{i},\pi_{a},\pi_{N},p_{a},G[X^{i}],V[X^{a}],\Phi^{\prime}[X],\Sigma^{\prime}[\tilde{X}^{a}]\right)$
constrain $3+3+1+3+3+3+1+1=18$ components. Note that
$\Sigma^{\prime}[\tilde{X}^{a}]$ constrains only one component, because this
constraint is parametrized by $\partial_{a}\tilde{X}^{a}$ rather than
$\tilde{X}^{a}$. The second-class constraints $\left(\pi_{\Lambda},\Pi\right)$
constrain $(1+1)/2=1$ component. Therefore, the local degrees of freedom in
configuration space are $21-18-1=2$. This result is consistent with GR and the
original unimodular gravity [15].

Using the second-class constraints $\Pi\approx 0$ and $\pi_{\Lambda}\approx
0$, we can eliminate variables $\Lambda$ and $\pi_{\Lambda}$ by substituting

$\displaystyle\Lambda$ $\displaystyle=p_{\tau},$ $\displaystyle\pi_{\Lambda}$
$\displaystyle=0.$ (35)

After the elimination, the total Hamiltonian (15) is rewritten as

$\displaystyle
H_{\text{T}}(A_{\tau}^{i},\pi_{i},A_{a}^{i},E_{i}^{a},N,\pi_{N},N^{a},\pi_{a},\phi^{\tau},p_{\tau},\phi^{a},p_{a})$
$\displaystyle=G[A_{\tau}^{i}]+V[N^{a}]+\Phi^{\prime}[N]+\Sigma^{\prime}[\phi^{a}]+\int\text{d}^{3}x\
\left[\frac{1}{k\beta}\left(v^{i}\pi_{i}+v^{a}\pi_{a}+v_{N}\pi_{N}\right)+\frac{1}{2k}\tilde{w}^{a}p_{a}\right].$
(36)

The constraint $\Sigma^{\prime}[\tilde{X}^{a}]\approx 0$ (33) and the
evolution equation $\left\\{p_{\tau},H_{\text{T}}\right\\}\approx 0$ imply
that $p_{\tau}$ is a spacetime constant. In addition, the evolution equation
$\left\\{\phi^{\tau},H_{\text{T}}\right\\}=N\det e-\partial_{a}\phi^{a}$ leads
to the covariant version of the unimodular condition $N\det
e-\partial_{\mu}\phi^{\mu}=0$. Obviously, $p_{\tau}$ and $\Lambda$ correspond
to the cosmological constant (times two) in GR.

We can introduce the spatial diffeomorphism constraint $\mathcal{D}[X^{a}]$ by
extending the vector constraint $V[X^{a}]$:

$\displaystyle\mathcal{D}[X^{a}]=V[X^{a}]+G[X^{a}A_{a}^{i}]-\Sigma^{\prime}[X^{a}\phi^{\tau}].$
(37)

This constraint generates the spatial diffeomorphism for the dynamical
variables

$\displaystyle\left\\{A_{a}^{i},\mathcal{D}[X^{b}]\right\\}$
$\displaystyle=\mathcal{L}_{\vec{X}}A_{a}^{i},$
$\displaystyle\left\\{E_{i}^{a},\mathcal{D}[X^{b}]\right\\}$
$\displaystyle=\mathcal{L}_{\vec{X}}E_{i}^{a},$ (38)
$\displaystyle\left\\{\phi^{\tau},\mathcal{D}[X^{a}]\right\\}$
$\displaystyle=\mathcal{L}_{\vec{X}}\phi^{\tau},$
$\displaystyle\left\\{p_{\tau},\mathcal{D}[X^{a}]\right\\}$
$\displaystyle=\mathcal{L}_{\vec{X}}p_{\tau},$ (39)

where $\mathcal{L}_{\vec{X}}$ is a Lie derivative along $\vec{X}$. The
constraint $\mathcal{D}[X^{a}]$ is first class, and holds the stability
condition

$\displaystyle\left\\{\mathcal{D}[X^{a}],H_{\text{T}}\right\\}=G\left[\mathcal{L}_{\vec{X}}A_{\tau}^{i}\right]+V\left[\mathcal{L}_{\vec{X}}N^{a}\right]+\Phi^{\prime}\left[\mathcal{L}_{\vec{X}}N\right]+\Sigma^{\prime}\left[\mathcal{L}_{\vec{X}}\phi^{a}\right]\approx
0.$ (40)

We employ $\mathcal{D}[X^{a}]$ instead of $V[X^{a}]$ as an element of the
first-class constraints.

##  3 Quantization

After the reduction (35), all the remaining constraints
$\left(\pi_{i},\pi_{a},\pi_{N},p_{a},G[X^{i}],\mathcal{D}[X^{a}],\Phi^{\prime}[X],\Sigma^{\prime}[\tilde{X}^{a}]\right)$
belong to first class. Hence, we can proceed to quantize this theory by
replacing Poisson brackets $\left\\{\bullet,\bullet\right\\}$ with quantum
commutators $(\text{i}\hbar)^{-1}\left[\hat{\bullet},\hat{\bullet}\right]$.
The quantum operators corresponding to the canonical variables are given by

$\displaystyle\hat{A}_{\tau}^{i}$ $\displaystyle=A_{\tau}^{i},$
$\displaystyle\hat{\pi}_{i}$ $\displaystyle=-\text{i}\hbar
k\beta\frac{\delta}{\delta A_{\tau}^{i}},$ (41) $\displaystyle\hat{A}_{a}^{i}$
$\displaystyle=A_{a}^{i},$ $\displaystyle\hat{E}_{i}^{a}$
$\displaystyle=-\text{i}\hbar k\beta\frac{\delta}{\delta A_{a}^{i}},$ (42)
$\displaystyle\hat{N}$ $\displaystyle=N,$ $\displaystyle\hat{\pi}_{N}$
$\displaystyle=-\text{i}\hbar k\beta\frac{\delta}{\delta N},$ (43)
$\displaystyle\hat{N}^{a}$ $\displaystyle=N^{a},$ $\displaystyle\hat{\pi}_{a}$
$\displaystyle=-\text{i}\hbar k\beta\frac{\delta}{\delta N^{a}},$ (44)
$\displaystyle\hat{\phi}^{\tau}$ $\displaystyle=\phi^{\tau},$
$\displaystyle\hat{p}_{\tau}$ $\displaystyle=-2\text{i}\hbar
k\frac{\delta}{\delta\phi^{\tau}},$ (45) $\displaystyle\hat{\phi}^{a}$
$\displaystyle=\phi^{a},$ $\displaystyle\hat{p}_{a}$
$\displaystyle=-2\text{i}\hbar k\frac{\delta}{\delta\phi^{a}}.$ (46)

A physical state $\Psi$ must satisfy the quantized first-class constraints

$\displaystyle\hat{\pi}_{i}\Psi$
$\displaystyle=\hat{\pi}_{N}\Psi=\hat{\pi}_{a}\Psi=\hat{p}_{a}\Psi=0,$ (47)
$\displaystyle\hat{G}[X^{i}]\Psi$
$\displaystyle=\hat{\mathcal{D}}[X^{a}]\Psi=\hat{\Phi}^{\prime}[X]\Psi=\hat{\Sigma}^{\prime}[\tilde{X}^{a}]\Psi=0.$
(48)

Constraints (47) indicate that $\Psi$ should be independent from
$A_{\tau}^{i},N,N^{a}$, and $\phi^{a}$, that is,

$\displaystyle\Psi=\Psi[\phi^{\tau},A_{a}^{i}].$ (49)

We assume that the wave functional is variable-separable, namely,
$\Psi[\phi^{\tau},A_{a}^{i}]=\Psi[\phi^{\tau}]\Psi[A_{a}^{i}]$. Constraint
$\hat{\Sigma}^{\prime}[\tilde{X}^{a}]\Psi=(2k)^{-1}\int\text{d}^{3}x\
\tilde{X}^{a}\partial_{a}\hat{p}_{\tau}\Psi\approx 0$ implies that $\Psi$ has
the form

$\displaystyle\Psi[\phi^{\tau},A_{a}^{i}]=\exp\left[\frac{\lambda}{-2\text{i}\hbar
k}\int\text{d}^{3}x\ \phi^{\tau}\right]\Psi[A_{a}^{i}],$ (50)

that satisfies

$\displaystyle\hat{p}_{\tau}\Psi[\phi^{\tau},A_{a}^{i}]=\lambda\Psi[\phi^{\tau},A_{a}^{i}],$
(51)

where $\lambda$ is an unspecified constant. Note that $\hat{p}_{\tau}$ weakly
commutes with every quantum first-class constraint; therefore,
$\hat{p}_{\tau}$ is the physical observable in the sense of Dirac [19].

A possible solution of the constraints (47) and (48) is expressed as

$\displaystyle\Psi_{\lambda,R}[\phi^{\tau},A_{a}^{i}]=\Psi_{\lambda}[\phi^{\tau}]\Psi_{\lambda,R}[A_{a}^{i}]$
$\displaystyle=\exp\left[\frac{\lambda}{-2\text{i}\hbar k}\int\text{d}^{3}x\
\phi^{\tau}\right]\exp\left[\frac{6}{\text{i}\hbar
k\beta^{3}\lambda}\left(Y_{\text{CS}}[A_{a}^{i}]-2\left(1+\beta^{2}\right)\int\operatorname{Tr}(A\wedge
R)\right)\right],$ (52)

where

$\displaystyle Y_{\text{CS}}[A_{a}^{i}]$
$\displaystyle=\int\operatorname{Tr}\left(A\wedge\text{d}A+\frac{2}{3}A\wedge
A\wedge A\right)$ (53)

is the Chern-Simons functional, $\operatorname{Tr}$ indicates the trace of
SU(2) generators $T_{i}$, which is normalized as
$\operatorname{Tr}(T_{i}T_{j})=-\frac{1}{2}\delta_{ij}$, and $\lambda$ and
$R_{ab}^{i}$ are parameters associated with the cosmological constant and the
spatial curvature, respectively. We would like to emphasize that the form of
the second factor in (52), $\Psi_{\lambda,R}[A_{a}^{i}]$, was originally
proposed in Ref. [14] as a generalization of the Kodama state for real values
of $\beta$.

Since this state is a pure phase, we can define a naive inner product

$\displaystyle\left\langle\Psi_{\lambda^{\prime},R^{\prime}}\middle|\Psi_{\lambda,R}\right\rangle=\int\mathcal{D}\phi^{\tau}\mathcal{D}A\
\Psi_{\lambda^{\prime},R^{\prime}}^{\ast}[\phi^{\tau},A_{a}^{i}]\Psi_{\lambda,R}[\phi^{\tau},A_{a}^{i}]\sim\delta(\lambda-\lambda^{\prime})\delta(R-R^{\prime}),$
(54)

where

$\displaystyle\delta(R-R^{\prime})=\prod_{x}\prod_{a,b,i}\delta\left(R_{ab}^{i}(x)-R^{\prime}{}_{ab}^{i}(x)\right).$
(55)

The inner product (54) has an undesirable property. Due to the factor
$\delta(R-R^{\prime})$, when $R_{ab}^{i}$ and $R^{\prime}{}_{ab}^{i}$ have
different values, this inner product vanishes even if $R_{ab}^{i}$ and
$R^{\prime}{}_{ab}^{i}$ are connected by the gauge and spatial diffeomorphism
transformations. We can improve this inner product by using the group
averaging technique [14]

$\displaystyle\left(\Psi_{\lambda^{\prime},R^{\prime}}\middle|\Psi_{\lambda,R}\right)=\int\mathcal{D}g\
\left\langle\Psi_{\lambda^{\prime},\varphi_{g}R^{\prime}}\middle|\Psi_{\lambda,R}\right\rangle\sim\delta(\lambda-\lambda^{\prime})\int\mathcal{D}g\
\delta(R-\varphi_{g}R^{\prime}),$ (56)

where $\varphi_{g}$ is the gauge and spatial diffeomorphism transformations
parametrized by $g$, and $\int\mathcal{D}g$ is an integral over both
transformations. The inner product (56) does not vanish when
$\lambda=\lambda^{\prime}$ and $R^{\prime}{}_{ab}^{i}$ can reach $R_{ab}^{i}$
by these transformations. Note that these transformations do not affect
$\lambda$. We find that the state

$\displaystyle\left(\Psi_{\lambda,R}\right|$ $\displaystyle=\int\mathcal{D}g\
\left\langle\Psi_{\lambda,\varphi_{g}R}\right|$ (57)

is invariant under the gauge and spatial diffeomorphism transformations,
because

$\displaystyle\left(\Psi_{\lambda,R}\right|\hat{U}(\varphi_{g^{\prime}})=\int\mathcal{D}g\
\left\langle\Psi_{\lambda,\varphi_{g^{\prime}}\circ\varphi_{g}R}\right|=\left(\Psi_{\lambda,R}\right|,$
(58)

where $\hat{U}(\varphi_{g^{\prime}})$ is the operator corresponding to these
transformations. This is an analog of the strategy to obtain the gauge and
spatial diffeomorphism invariant state in loop quantum gravity [21, 22].

One can write the curvature operator $\hat{R}_{ab}^{i}$ by using the inner
product (56) as

$\displaystyle\int\text{d}^{3}x\
\tilde{X}_{i}^{ab}\hat{R}_{ab}^{i}=\int\text{d}^{3}x\
\tilde{X}_{i}^{ab}\int\mathcal{D}g\mathcal{D}R^{\prime}\mathcal{D}\lambda^{\prime}\
\varphi_{g}R^{\prime}{}_{ab}^{i}\left|\Psi_{\lambda^{\prime},\varphi_{g}R^{\prime}}\right\rangle\left\langle\Psi_{\lambda^{\prime},\varphi_{g}R^{\prime}}\right|,$
(59)

where $\tilde{X}_{i}^{ab}$ is a densitized test function, and
$\int\mathcal{D}R^{\prime}$ is an integral over the curvature parameter
$R^{\prime}$ modulo the gauge and spatial diffeomorphism transformations. From
Eq. (59), we have

$\displaystyle\int\text{d}^{3}x\
\tilde{X}_{i}^{ab}\hat{R}_{ab}^{i}\left|\Psi_{\lambda,R}\right\rangle=\int\text{d}^{3}x\
\tilde{X}_{i}^{ab}R_{ab}^{i}\left|\Psi_{\lambda,R}\right\rangle.$ (60)

Using (51), (60), and

$\displaystyle\hat{E}_{i}^{a}\Psi_{\lambda,R}[\phi^{\tau},A_{a}^{i}]=\frac{3}{\beta^{2}\lambda}\epsilon^{abc}\left[F_{bci}-\left(1+\beta^{2}\right)R_{bci}\right]\Psi_{\lambda,R}[\phi^{\tau},A_{a}^{i}],$
(61)

we find that the state $\Psi_{\lambda,R}[\phi^{\tau},A_{a}^{i}]$ satisfies the
Hamiltonian constraint:

$\displaystyle\hat{\Phi}^{\prime}[X]\Psi_{\lambda,R}[\phi^{\tau},A_{a}^{i}]$
$\displaystyle=\frac{1}{2k\beta^{2}}\int\text{d}^{3}x\
\frac{X}{\sqrt{\det\hat{E}}}\epsilon^{ijk}\hat{E}_{i}^{a}\hat{E}_{j}^{b}\left[\left(1+\beta^{2}\right)\hat{R}_{abk}-\hat{F}_{abk}+\frac{\beta^{2}}{6}\epsilon_{abc}\hat{p}_{\tau}\hat{E}_{k}^{c}\right]\Psi_{\lambda,R}[\phi^{\tau},A_{a}^{i}]=0.$
(62)

Thus, under the appropriate inner product (56), the state
$\left|\Psi_{\lambda,R}\right)$ is a solution of quantum covariant unimodular
gravity.

##  4 Conclusions

In this work, we have analyzed the full theory of covariant unimodular gravity
in terms of the connection representation with real values of the Barbero-
Immirzi parameter $\beta$. Unlike the original unimodular gravity, the
Hamiltonian constraint of covariant unimodular gravity (32) is first class.
Therefore, the constraint structure of covariant unimodular gravity is closer
to that of GR than that of the original unimodular gravity. The subtle
difference from GR is that the cosmological constant in GR is replaced with
the canonical momentum $p_{\tau}$ which is regarded as a constant of motion
and a Dirac observable.

In the original unimodular gravity, the Kodama state is not a solution of the
quantum constraints [15]. On the other hand, in covariant unimodular gravity,
the solution of the constraints can be obtained by extending the Kodama state.
The state (52) is regarded as a natural extension of the state proposed in
Ref. [14]. Since this state is a pure phase, it is delta-function
normalizable. In addition, since all the canonical variables are real, we can
avoid the reality condition problem. The main difference from the quantum
states of GR in Ref. [14] is that each state is labeled not only by the
spatial curvature $R_{ab}^{i}$ (modulo the gauge and spatial diffeomorphism
transformations) but also by the cosmological constant. This implies that a
general solution of the physical state can be written as a superposition of
the eigenstates of $\hat{p}_{\tau}$ and $\hat{R}_{ab}^{i}$. Thus, at least in
this framework, covariant unimodular gravity is different from GR at the
quantum level. Note that such a superposition of the different values of the
cosmological constant has already appeared in previous studies both within the
Hamiltonian formalism [6, 16] and the path integral formalism [23, 5, 3].

It is still unclear whether the original unimodular gravity and covariant
unimodular gravity differ at the quantum level. In the original unimodular
gravity, only the wave functional with zero cosmological constant has been
found [15]. On the other hand, in covariant unimodular gravity,
$\Psi_{\lambda,R}[\phi^{\tau},A_{a}^{i}]$ (52) is a state with a non-vanishing
cosmological constant. To compare the two theories, one needs to find a
solution with a non-vanishing cosmological constant in the original unimodular
gravity. Note that even if such a state is found, it is not so obvious whether
the difference in the physical states provides different physical predictions.

Another approach to confirm the difference between the two unimodular theories
is to compare the observables. In general, a physical observable must commute
with every first-class constraint [19], while the two unimodular theories have
different first-class constraints. If we can define appropriate observables,
it will be easier to compare the two theories. Let us consider
$V_{\tau}=\int\text{d}^{3}x\,sN\det e$ as an example of a candidate for the
physical observable. Here, $s=\text{sign}(\det e)$, and the integral is over
the spacelike surface parametrized by $\tau$. In covariant unimodular gravity,
the Poisson bracket between $V_{\tau}$ and the Hamiltonian constraint does not
weakly vanish:
$\left\\{V_{\tau},\Phi^{\prime}[X]\right\\}=\int\text{d}^{3}x\,sXNK_{a}^{i}E_{i}^{a}\not\approx
0$. This implies that, in quantum theory, not all the constraints commute with
$\hat{V}_{\tau}$, and a physical state is not an eigenstate of
$\hat{V}_{\tau}$. Hence, $V_{\tau}$ and the four-volume
$\int\text{d}\tau\,V_{\tau}$ are not observables in covariant unimodular
gravity. On the other hand, in the original unimodular gravity, the unimodular
constraint $N\det e-\alpha=0$ is a second-class constraint. This means that
$N$ is not an independent variable, and can be eliminated as $N=\alpha(\det
e)^{-1}$. Therefore, $V_{\tau}=\int\text{d}^{3}x\,s\alpha$. In this case, the
Poisson bracket between $V_{\tau}$ and every first-class constraint weakly
vanishes. However, $V_{\tau}$ and the four-volume in this theory should be
regarded as constants associated with the Hamiltonian and the Lagrangian
rather than observables. Thus, $V_{\tau}$ is not an appropriate quantity to
compare the two theories. To confirm the (in)equivalence of the two theories,
one needs to find some other quantities that can be physical observables.

It is worthwhile to investigate how covariant unimodular gravity is extended
into loop quantum gravity. Both covariant unimodular gravity discussed above
and loop quantum gravity employ real values of the Barbero-Immirzi parameter
$\beta$. Therefore, one can expect that this theory can be naturally extended
into a full theory of loop quantum gravity. Note that the symmetry-reduced
models of covariant unimodular gravity in the context of loop quantum
cosmology has been studied in Refs. [24, 25, 26].

Another direction for future research is to investigate the relation between
the extended Kodama state (52) and other physical states of unimodular
gravity. In Refs. [27, 28], the Hartle-Hawking state, which is the solution of
quantum GR in the ADM representation, is interpreted as the Fourier dual of
the Kodama state. On the other hand, there is also an argument that it is
difficult to translate the Hartle-Hawking state straightforwardly into the
connection representation [29]. The question of how these arguments are
modified in covariant unimodular gravity is left for future study.

## Acknowledgments

The author is grateful to Makoto Fukuda for useful comments on the manuscript.

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