# Connection Dynamics of Reduced 5-dimensional Kaluza-Klein Theory and Its
Deparametrization

Haida Li<EMAIL_ADDRESS>Department of Physics, Beijing Normal
University, Beijing 100875, China  Shengzhi Li<EMAIL_ADDRESS>Department of Physics, Beijing Normal University, Beijing 100875, China
Yongge Ma<EMAIL_ADDRESS>Department of Physics, Beijing Normal University,
Beijing 100875, China

###### Abstract

The connection dynamics of the 5-dimensional Kaluza-Klein theory reduced on
4-dimensional spacetime is obtained by performing the Hamiltonian analysis and
canonical transformations. Deparametrization is achieved in the spherically
symmetric model of the theory without introducing additional matter fields
beyond the 5-dimensional gravity. Thus the physical time evolution and the
physical spatial coordinate can be provided by the geometrical degrees of
freedom in the higher dimensional spacetime.

##  I Introduction

In the past three decades, the problem of defining physical time evolution in
canonical loop quantum gravity (LQG) has been tackled. The main problem begins
with the Hamiltonian formalism of general relativity (GR), where the physical
time evolution of gauge invariant Dirac observables is difficult to be
realized due to their Poisson commutativity with the Hamiltonian of the
system, i.e., $\\{O_{D},H\\}=0$. This implies a trivial physical time
evolution for those Dirac observables. Early works on this topic are mainly
based on Rovelli’s relational framework rovelli1 ; rovelli2 ; rovelli3 ;
rovelli4  as well as the Brown-Kuchař mechanism brown1995dust  where dust
fields are introduced to the Hamiltonian system of GR to serve as reference
fields of both space and time. The relational framework was later on improved
and applied to LQG in Refs. dittrich2006partial ; dittrich2007partial , and in
a series of works the detailed construction of gauge invariant Dirac
observables was achieved for GR couple to dust fields giesel2010algebraic ;
giesel2010manifestly ; giesel2015scalar , where some dust fields have been
considered as reference fields to achieve classical deparametrization of both
Hamitonian and diffeomorphism constraints of GR, providing a viable approach
to define physical time evolution and dynamical spatial coordinates.

It is then natually expected that the relational framework, which solves the
problem of defining physical time evolution in the Hamiltonian formalism of
GR, will also solve the problem in its quantum theory. In fact, there are at
least two approaches developed to derive quantum dynamics from deparametrized
canonical LQG. The first approach is the so-called reduce phase space
quantization approach proposed in Refs. thiemann2006reduced ;
giesel2010algebraic . In this program, Dirac observables related to phase
space variables are first constructed at classical level by deparametrizing
the first-class constraints using the degrees of freedom provided by some dust
fields, leading to the reduction of phase space where only physical degrees of
freedom remain for the classical theory. Meanwhile, physical time evolution of
the constructed Dirac observables can be given as their relational evolution
with respect to the dust fields. Thus the anomaly arising from quantum
constraint algebra is no longer an issue since all constraints are imposed at
classical level. Also, it has been demonstrated that both the Hamiltonian and
diffeomorphism constraints can be deparametrized. This leads to a fully
deparametrized system, where not only physical time evolution are defined for
Dirac observables, but also their physical spatial coordinates. The second
approach towards defining physical time evolution in LQG is introduced in
Refs. domagala2010gravity ; lewandowski2016loop , in which the relational
evolution of gravity is defined with respect to a massless scalar field at
quantum level. Being different from the first approach, where constraints are
imposed at classical level, in the second approach the quantization is first
performed to the Hamiltonian constrained system of gravity coupled to the
massless scalar field. Solutions of the quantum constraint equation restrict
the kinematical Hilbert space to the physical Hilbert space, and in the
Heisenberg picture the physical time evolution of operators in the kinematical
Hilbert space is obtained by the construction of their corresponding quantum
Dirac observables using quantum evolution operator defined on the physical
Hilbert space. The models of canonical LQG following this approach have
produced some very interesting results, including the bouncing behaviour at
full quantum level zhang2019bouncing , which reveals the resemblance between
full LQG and its symmetry-reduced theories such as loop quantum cosmology
(LQC), where the quantum bounce is predicted as the resolution of the big bang
singularities in GR ashtekar2006quantum ; ashtekar2006quantum123 ;
ding2009effective ; yang2009alternative . However, one of the challanges to
this approach is that at quantum level the diffeomorphism constaint is solved
by using the group-averaging technique while the Hamiltonian constraint is
deparametrized. To obtain a consistent treatment to both constraints, how to
implement physical spatial coordinates in this approach remains an open issue.

Despite the fact that there has been much success in solving the problem of
time in LQG, in current results one only considered the relational evolution
with respect to certain matter fields. In this paper, our main purpose is to
consider the systems of higher dimensional gravity, where the degrees of
freedom (DoFs) chosen to fully deparametrize the system are provided by
spacetime geometry itself. Conceptually speaking, in such a system physical
time evolution is not given by some additionally introduced matter fields,
instead it is depicted by spacetime degrees of freedom including the extra
dimensions.

Higher dimensional gravity represents a series of efforts to unify
4-dimensional gravity and other interactions. We will consider the
5-dimensional Kaluza-Klein theory, which can be reduced on 4-dimensional
spacetime as gravity coupled with an electromagnetic (EM) field and a scalar
field kaluza1921unitatsproblem ; klein1926quantum ; bailin1987kaluza . Note
that if the extra dimension is compactified as a circle with a microscopic
radius, a killing vector field may arise naturally in the low energy regime
blagojevic2002gravitation . The killing reduction of the 5-dimensional gravity
has been studied previously yang68killing , where the 5-dimensional spacetime
is reduced along the trajectory of the killing vector along the fifth
dimension so that a 4-dimensional gravity with some extra DoFs from the
reduced fifth dimension is generated. Meanwhile, following the same idea on
the extension of loop quantization to some modified gravity theories
zhang2011loop ; zhang2011extension ; zhang2011nonperturbative , the killing
reduced theory is for the first time recast in connection formalism in this
paper such that the study of its loop quantization becomes possible. However,
it is rather difficult in the full 4-dimensional theory to deparametrize both
Hamiltonian and diffeomorphism constraints simultaneously. As a simplified
model, spherically symmetric reduction will be applied to the theory in the
connection formalism such that a fully deparametrized theory where both
physical time evolution and physical spatial coordinates for gauge invariant
complete observables can be well defined.

The structure of this paper is organised as the following. In section II we
briefly review the classical framework of defining relational evolution in
Hamiltonian constrained systems. In section III we perform Hamiltonian
analysis of the 5-dimensional Kaluza-Klein theory reduced on 4-dimensional
spacetime, obtain the connection dynamics and then perform spherically
symmetric reduction to the theory. In section IV we will perform
deparametrization to the spherically symmetric model, and both Hamiltonian
constraint and diffeomorphism cosntraint will be deparametrized, providing
relational physical time evolution and physical spatial coordinates to the
system. In the last section we will summarize the results and provide some
outlook towards future prospects in this direction.

##  II Relational Framework of GR coupled to matter fields

In the scheme to establish the relational framework for GR couple to dust
fields and massless scalar field, one usually begins with identifying the DoFs
used for representing relational evolution. Then the constrained system is
reformulated into its deparametrization form, enabling the construction of
complete observables which are invariant along the gauge orbit induced by the
first-class constraints. The exact procedure of deparametrization is closely
related to the specific form of constraints.

Consider GR coupled to matter fields in general cases on a manifold $M$ with
proper foliation $\Sigma\times\mathcal{R}$. Suppose that one could identify
two sets of canonical pairs $(P^{I},T_{I})$ and $(q^{a},p_{a})$ representing
the reference fields and remaining DoFs respectively, such that the
constraints can be written in the equivalent form:

$\begin{split}C_{I}(x)=P_{I}(x)+h_{I}[T^{I},q^{a},p_{a}](x),\end{split}$ (1)

where $C_{I}$ denotes all first-class constraints including the Hamiltonian
and spatial diffeomorphism constraints. Note that in comparison to the cases
of GR coupled to the dust fields and the massless scalar field treated before,
we admit the physical coordinates $T^{I}$ appearing in the expression of
$h_{I}$ in more general cases. Gauge invariant Dirac observables can be
defined as follows. Let $F$ be an arbitrary function on the phase space which
depends only on $(q^{a},p_{a})$. Then an observable which is invariant under
the gauge transformation of a first-class constraint $C_{i}$ can be defined as
giesel2010algebraic :

$O_{F}(\tau_{i}):=\left(e^{\\{C_{i}[\beta],.\\}}\cdot
F\right)_{\beta=\tau^{i}-T^{i}},$ (2)

where $C_{i}[\beta]$ represents the smeared version of $C_{i}$ defined as
$C_{i}[\beta]:=\int d^{3}x\beta(x)C_{i}(x)$, and the action of
$e^{\\{C_{i}[\beta],.\\}}$ on $F$ is defined by:

$\begin{split}e^{\\{C_{i}[\beta],.\\}}\cdot
F&:=F+\sum\limits^{\infty}_{n=1}\int d^{3}x_{1}...\int
d^{3}x_{n}[\beta(x_{1})]..[\beta(x_{n})]\\{C_{i}(x_{1}),\\{..\\{C_{i}(x_{n}),F\\}..\\}\\}.\\\
\end{split}$ (3)

Now let us consider a complete gauge invariant observable of both Hamiltonian
and diffeomorphism constraints, which can be constructed as
dittrich2006partial :

$\begin{split}O_{F}(\tau,\sigma_{j})=O_{O_{F}(\sigma_{j})}(\tau).\end{split}$
(4)

The following properties can be derived as is the case for GR coupled to dust:

$\begin{split}&\\{O_{F}[\tau,\sigma_{j}],O_{F^{\prime}}[\tau,\sigma_{j}]\\}=\\{O_{F}[\tau,\sigma_{j}],O_{F^{\prime}}[\tau,\sigma_{j}]\\}^{*}=O_{\\{F,F^{\prime}\\}^{*}}[\tau,\sigma_{j}],\\\
&O_{F+F^{\prime}}[\tau,\sigma_{j}]=O_{F}[\tau,\sigma_{j}]+O_{F^{\prime}}[\tau,\sigma_{j}],\qquad
O_{F\cdot F^{\prime}}[\tau,\sigma_{j}]=O_{F}[\tau,\sigma_{j}]\cdot
O_{F^{\prime}}[\tau,\sigma_{j}],\end{split}$ (5)

where the Dirac bracket is defined as:

$\\{f,g\\}^{*}:=\\{f,g\\}-\sum\limits_{a,b}\\{f,\tilde{\phi}_{a}\\}M^{-1}_{ab}\\{\tilde{\phi}_{b},g\\},$
(6)

where $\tilde{\phi}_{a}$, $\tilde{\phi}_{b}$ are second-class constraints and
$M_{ab}=\\{\tilde{\phi}_{a},\tilde{\phi}_{b}\\}$. The relational evolution of
Dirac observable $O_{F}(\tau,\sigma_{j})$, namely the first order derivative
of $O_{F}(\tau,\sigma_{j})$ with respect to the physical time parameter
$\tau$, can thus be calculated as:

$\begin{split}\frac{d}{d\tau}O{}_{F}(\tau,\sigma_{j})&=\sum\limits^{\infty}_{n=1}\int
d^{3}\sigma_{1}^{\prime}...\int
d^{3}\sigma_{n}^{\prime}[\tau-T(\sigma_{1}^{\prime})]..[\tau-T(\sigma_{n-1}^{\prime})]\\{C(\sigma_{1}^{\prime}),\\{..\\{C(\sigma_{n}^{\prime}),O_{F}(\sigma_{j})\\}..\\}\\}\\\
&=O{}_{\\{h[1],F\\}}(\tau,\sigma_{j})\\\
&=\\{O{}_{(h[1])}(\tau,\sigma_{j}),O{}_{F}(\tau,\sigma_{j})\\},\end{split}$
(7)

where $T(x)$ is a scalar field and $T(\sigma):=O_{T}(\sigma_{j})$. Here it is
worth noting that if the function $F$ depended not only on $(q,p)$ but also on
$T(x)$. The second step of (7) may not hold.

From Eq.(5) one can see that the multi-parameter family of maps
$O^{(\tau,\sigma)}:F\mapsto O_{F}(\tau,\sigma)$ are actually homomorphisms
from the Poisson algebra of functions on phase space to the Poisson algebra of
weak Dirac observables. Thus for a general functional:

$F=F[q_{a}(x),p^{a}(x),T^{I}(x),P_{I}(x)],$ (8)

we have the following identity:

$O_{F}(\tau,\sigma_{j})=F[O_{q_{a}}(\tau,\sigma_{j}),O_{p^{a}}(\tau,\sigma_{j}),\tau,\sigma_{j},-O_{h}(\tau,\sigma_{j}),-O_{h_{j}}(\tau,\sigma_{j})].$
(9)

Using this identity, phase space functions can be easily constructed as gauge
invariant Dirac observables on the reduced phase space and the relational
framework is thus established.

##  III Hamiltonian Formulation of 5D Kaluza-Klein theory

As is the purpose of this work, we will consider 5-dimensional Kaluza-Klein
gravity which in itself can provide degrees of freedom for deparametrization,
since the refernece fields can be provided without introducing additional
fields. In this section, we will first perform the Hamiltonian analysis to the
Kaluza-Klein theory reduced on 4-dimensional spacetime. Then, we will
transform the Hamiltonian theory canonically into connection formalism
suitable for loop quantization. Finally, to carry out the deparametrization,
the spherically symmetric model of the theory will also be studied.

###  III.1 Hamiltonian Analysis of 5D Kaluza-Klein Theory

Since the extra dimension is compactified as a circle with a microscopic
radius in 5-dimensional Kaluza-Klein theory, a killing vector field naturally
arise in the low energy regime. Hence one may parametrize the 5-dimensional
metric by the 4-dimensional fields as bailin1987kaluza :

$\begin{split}&g_{55}=\phi^{2}\\\ &g_{5\mu}=g_{\mu
5}=\phi^{2}\mathcal{A}_{\mu}\\\
&g_{\mu\nu}=h_{\mu\nu}+\phi^{2}\mathcal{A}_{\mu}\mathcal{A}_{\nu},\end{split}$
(10)

where $\mu$, $\nu=0,1,2,3$, $h_{\mu\nu}$ represents a 4-dimentioanl metric,
$\mathcal{A}_{\mu}$ can be identified as an EM 4-potential, and $\phi$ is a
scalar field. The action for 5-dimensional Kaluza-Klein theory reads:

$S=\frac{1}{\kappa_{5}}\int_{\mathcal{M}^{5}}d^{5}x\sqrt{|g|}({}^{5}R),$ (11)

where ${}^{5}R$ denotes the Ricci scalar in five dimensions. This action can
be reduced to four dimensions up to a boundary term as:

$S=\frac{1}{\kappa}\int_{\mathcal{M}^{4}}d^{4}x\phi\sqrt{-h}(R+\frac{1}{4}\phi^{2}\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}),$
(12)

where $\kappa$ is a constant defined as
$\frac{1}{\kappa}:=\frac{1}{\kappa_{5}}\int dx^{5}$ such that the integration
in the fifth dimension is carried out, $\phi$ is set to be positive, $R$ is
the 4-dimentional scalar curvature determined by $h_{\mu\nu}$, and
$\mathcal{F}_{\mu\nu}=(d\mathcal{A})_{\mu\nu}$ is the EM tensor.

By the standard 3+1 decomposition of the reduced 4-dimensional spacetime, the
components of the 4-metric can be written as:

$\left(h_{\mu\nu}\right)=\left(\begin{array}[]{cc}-N^{2}+q_{ab}N^{a}N^{b}&N_{a}\\\
N_{a}&q_{ab}\\\ \end{array}\right),\quad a,b=1,2,3$ (13)

where $N_{a}=h_{ab}N^{b}$. It’s inverse reads:

$\left(h^{\mu\nu}\right)=\left(\begin{array}[]{cc}-\frac{1}{N^{2}}&\frac{N^{a}}{N^{2}}\\\
\frac{N^{b}}{N^{2}}&q^{ab}-\frac{N^{a}N^{b}}{N^{2}}\\\ \end{array}\right).$
(14)

Following the treatment in lacquaniti2006adm , the 4-dimensional scalar
curvature $R$, the extrinsic curvature $K_{ab}$ of the 3-dimensional spatial
hypersurface and the EM tensor take the following form:

$\begin{split}R&=(^{3}R+K_{ab}K^{ab}-K^{2})+2\nabla_{a}(-n^{b}\nabla_{b}n^{a}+n^{a}K),\\\
K_{ab}&=\frac{1}{2N}(\dot{q}_{ab}-D_{a}N_{b}-D_{b}N_{a}),\\\
\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}&=\mathcal{F}_{ab}\mathcal{F}^{ab}-\frac{2}{N^{2}}q^{ab}M_{a}M_{b},\\\
\end{split}$ (15)

where $M_{a}:=\mathcal{F}_{a0}-N^{b}\mathcal{F}_{ab}$, $\nabla_{\mu}$ is the
covariant derivative for the 4-dimensional spacetime, $n^{\mu}$ and $D_{a}$
are respectively the normal vector and the covariant derivative of the
3-dimensinal hypersurface. Furthermore, it is worth noting that the total
derivative terms contained in scalar curvature can not be neglected due to the
non-minimally coupled scalar field $\phi$ in the action (12). This term can be
written as lacquaniti2006adm :

$\begin{split}&\sqrt{-h}\phi\nabla_{a}(n^{a}K-n^{b}\nabla_{b}n^{a})=-N\sqrt{q}K\partial_{n}\phi-\sqrt{q}ND^{a}D_{a}\phi,\end{split}$
(16)

where we used the notation
$\partial_{n}\phi:=n^{a}\nabla_{a}\phi=(t^{a}-N^{a})/N\nabla_{a}\phi=\frac{1}{N}(\partial_{0}\phi-N^{a}\partial_{a}\phi)$.
The Langrangian density in (12) can thus be rewritten in terms of the spatial
variables up to boundary terms as:

$\begin{split}\mathcal{L}=\frac{N}{\kappa}\sqrt{q}\phi(^{3}R+K_{ab}K^{ab}-K^{2})-\frac{2}{\kappa}N\sqrt{q}(q^{ab}D_{a}\partial_{b}\phi+K\partial_{n}\phi)+\frac{1}{4\kappa}N\sqrt{q}\phi^{3}(\mathcal{F}_{ab}\mathcal{F}^{ab}-\frac{2}{N^{2}}q^{ab}M_{a}M_{b}).\end{split}$
(17)

By the Legendre transformation, there are six pairs of conjugate variables
$(N,\pi_{N})$, $(N^{a},\pi_{N^{a}})$,
$(\mathcal{A}_{0},\pi_{\mathcal{A}_{0}})$, $(q_{ab},\Sigma^{ab})$,
$(\mathcal{A}_{a},\pi^{a})$ and $(\phi,\pi_{\phi})$, satisfying:

$\begin{split}&\pi_{N}=\frac{\partial\mathcal{L}}{\partial\dot{N}}=0,\pi_{N^{a}}=\frac{\partial\mathcal{L}}{\partial\dot{N}^{a}}=0,\pi_{A_{0}}=\frac{\partial\mathcal{L}}{\partial\dot{A}_{0}}=0,\\\
&\Sigma^{ab}=\frac{\partial\mathcal{L}}{\partial\dot{q}_{ab}}=\frac{1}{\kappa}(\phi\sqrt{q}(K^{ab}-Kq^{ab})-\sqrt{q}q^{ab}\partial_{n}\phi),\\\
&\pi^{a}=\frac{\partial\mathcal{L}}{\partial\dot{A}_{a}}=\frac{1}{\kappa
N}\sqrt{q}\phi^{3}q^{ab}M_{b},\\\
&\pi_{\phi}=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=-\frac{2}{\kappa}\sqrt{q}K.\end{split}$
(18)

It can be seen from (18) that $\pi_{N}=0$, $\pi_{N^{a}}=0$ and
$\pi_{\mathcal{A}_{0}}=0$ are three sets of primary constraints, and their
corresponding configuration variables are merely Lagrangian multipliers. The
non-vanishing Poisson brackets of phase space variables read:

$\begin{split}&\\{q_{ab}(x),\Sigma^{cd}(y)\\}=\delta^{(c}_{a}\delta^{d)}_{b}\delta^{(3)}(x,y),\\\
&\\{\phi(x),\pi_{\phi}(y)\\}=\delta^{(3)}(x,y),\\\
&\\{\mathcal{A}_{a},\pi^{a}\\}=\delta^{(3)}(x,y).\end{split}$ (19)

The Hamiltonian density can be derived as
$\mathcal{H}=\dot{q}^{i}p_{i}-\mathcal{L}=NH+N^{a}V_{a}+\mathcal{A}_{0}J$,
with the constraints lacquaniti2006adm :

$\begin{split}H=&-\frac{\phi}{\kappa}\sqrt{q}R+\frac{2}{\kappa}\sqrt{q}q^{ab}D_{a}\partial_{b}\phi+\frac{\kappa
N}{\sqrt{q}\phi}(\Sigma_{ab}\Sigma^{ab}-\frac{\Sigma^{2}}{2}+\frac{(\Sigma-\phi\pi^{\phi})^{2}}{6})\\\
&-\frac{1}{4\kappa}\phi^{3}\sqrt{q}\mathcal{F}_{ab}\mathcal{F}^{ab}-\frac{\kappa}{2\phi^{3}\sqrt{q}}\pi^{a}\pi^{b}q_{ab},\\\
V_{a}=&-2q_{ab}D_{c}\Sigma^{cb}+\pi_{\phi}\partial_{a}\phi-\pi^{b}\mathcal{F}_{ab}+\mathcal{A}_{a}\partial_{b}\pi^{b},\\\
J=&-\partial_{a}\pi^{a},\end{split}$ (20)

where the term $\mathcal{A}_{a}\partial_{b}\pi^{b}$ proportional to the
constraint $J$ has been added into $V_{a}$ so that it generates the correct
spatial diffeomorphism trnasformations for $(\mathcal{A}_{a},\pi^{a})$. Thus
$H$ is the Hamitonian constraint associated with time evolution, $V_{a}$ is
the spatial diffeomorphism constraint which generates spatial diffeomorphism
transformations, $J$ is the Gaussian constraint for the EM field. It can be
further verified that all the secondary constraints in (20) are of first
class. Especially, for the smeared secondary constraints one has:

$\begin{split}&\\{J[\mathcal{A}_{0}],J[\mathcal{B}_{0}]\\}=0,\quad\\{J[\mathcal{A}_{0}],V[\vec{N}]\\}=-J(\mathscr{L}_{\vec{N}}\mathcal{A}_{0}),\quad\\{J[\mathcal{A}_{0}],H[N]\\}=0,\\\
&\\{V[\vec{N}],V[\vec{N}^{\prime}]\\}=V([\vec{N},\vec{N}^{\prime}]),\quad\\{H[M],V[\vec{N}]\\}=-H(\mathscr{L}_{\vec{N}}M),\\\
&\\{H[N],H[M]\\}=V(ND^{a}M-MD^{a}N),\end{split}$ (21)

where $J[\mathcal{A}_{0}]=\int_{\Sigma}d^{3}x\mathcal{A}_{0}(x)J(x)$,
$H(N)=\int_{\Sigma}d^{3}xN(x)H(x)$ and
$V(\vec{N})=\int_{\Sigma}d^{3}xN^{a}(x)V_{a}(x)$.

###  III.2 Connecton Dynamics

Recall that the non-perturbative loop quantization of GR is based on its
connection dynamical formalism. In this subsection we will derive a connection
dynamical formalism for the 5D Kaluza Klein theory in 4-dimensional spacetime.
In order to achieve this goal, we first define:

$\tilde{K}^{ab}:=\phi
K^{ab}+\frac{q^{ab}}{2N}(\dot{\phi}-N^{c}\partial_{c}\phi).$ (22)

This leads to the simplification of the conjugate momentum of $q_{ab}$ as:

$\begin{split}\Sigma^{ab}=\frac{1}{\kappa}(\tilde{K}^{a}_{i}E^{bi}-\frac{1}{q}\tilde{K}^{i}_{c}E^{c}_{i}E^{a}_{j}E^{b}_{j}),\end{split}$
(23)

where $\tilde{K}^{a}_{i}:=\tilde{K}^{ab}e^{i}_{b}$, and
$E^{a}_{i}=\sqrt{q}e^{a}_{i}$ is the densitized triad. Then the following
property is guaranteed:

$\begin{split}\int d^{4}\Sigma^{ab}\dot{q}_{ab}=\frac{2}{\kappa}\int
d^{4}E^{a}_{i}\frac{d\tilde{K}^{i}_{a}}{dt},\end{split}$ (24)

and the Poisson structure of $(\tilde{K}^{i}_{a},E^{b}_{j})$ can be calculated
using the original Poisson structure as:

$\begin{split}\\{K^{i}_{a}(x),K^{j}_{b}(y)\\}&=0,\\\
\\{E^{a}_{i}(x),E^{b}_{j}(y)\\}&=0,\\\
\\{K^{i}_{a}(x),E^{b}_{j}(y)\\}&=\frac{\kappa}{2}\delta^{b}_{a}\delta^{i}_{j}\delta(x,y).\end{split}$
(25)

We can canonically transform the Hamiltonian system using the new sets of
variables $(\tilde{K}^{i}_{a},E^{b}_{j})$, $(A_{a},\pi^{b})$,
$(\phi,\pi_{\phi})$. Since $\tilde{K}^{ab}$ is symmetric, i.e.
$\tilde{K}^{ab}=\tilde{K}^{ba}$, an additional set of constraints are
introduced:

$\begin{split}&G_{jk}:=\tilde{K}_{a[j}E^{a}_{k]}=0,\qquad j,k=1,2,3\\\
\end{split}$ (26)

which is equivalent to the following constraint:

$G_{j}(x)=\epsilon_{jkl}\tilde{K}^{k}_{a}E^{a}_{l}.$ (27)

The smeared version of this constraint can be written as:

$G[\Lambda]=\int_{\Sigma}d^{3}xG_{j}(x)\Lambda^{j}(x).$ (28)

Now we define:

$\begin{split}&A^{i}_{a}:=\Gamma^{i}_{a}+\gamma\tilde{K}^{i}_{a},\\\
\end{split}$ (29)

where $\Gamma^{i}_{a}$ is the spin-connection determined by $E^{a}_{i}$ and
$\gamma$ is a non-zero real number known as the Barbero-Imirzi parameter in
LQG han2007fundamental . A second canonical transformation can be performed by
changing the variables depicting the gravitational degrees of freedom from
$(\tilde{K}^{i}_{a},E^{b}_{j})$ to $(A^{i}_{a},E^{b}_{j})$. The Poisson
brackets between the new sets of variables can be calculated as:

$\begin{split}&\\{E^{a}_{i}(x),E^{b}_{j}(y)\\}=0,\\\
&\\{A^{i}_{a}(x),A^{j}_{b}(y)\\}=0,\\\
&\\{A^{i}_{a}(x),E^{b}_{j}(y)\\}=\frac{\kappa\gamma}{2}\delta^{(b}_{a}\delta^{i)}_{j}\delta^{(3)}(x,y).\\\
\end{split}$ (30)

The compatibility condition between the covariant derivative and the
densitized triad implies:

$\partial_{a}E^{a}_{j}+\epsilon_{jkl}\Gamma^{k}_{a}E^{a}_{l}=0.$ (31)

Then taking account of (27), the standard Gaussian constraint can be obtained
as:

$\begin{split}G_{j}&=\mathcal{D}_{a}E^{a}_{j}/\gamma\equiv(\partial_{a}E^{a}_{j}+\epsilon_{jkl}A^{k}_{a}E^{al})/\gamma=0,\end{split}$
(32)

which suggests that $A^{i}_{a}$ is a $su(2)$-connection. Thus our canonical
transformations share the same structure as that for the connection formalism
of GR despite the fact that we used $\tilde{K}^{i}_{a}$ instead of the
extrinsic curvature during the canonical transformation from
$(q_{ab},\Sigma^{ab})$ to $(\tilde{K}^{i}_{a},E^{a}_{i})$.

We can then rewrite all of the constraints of our system by using the
connection variables as:

$\begin{split}H&=\phi[F^{j}_{ab}-(\gamma^{2}+\frac{1}{\phi^{2}})\epsilon_{jmn}\tilde{K}^{m}_{a}\tilde{K^{n}_{b}}]\frac{\epsilon_{jkl}E^{a}_{k}E^{b}_{l}}{\sqrt{q}\kappa}\\\
&+\frac{2}{3\kappa\phi\sqrt{q}}(\tilde{K}^{i}_{a}E^{a}_{i})^{2}+\frac{2\sqrt{q}}{\kappa}D^{a}D_{a}\phi+\frac{\kappa}{6\sqrt{q}}(\pi_{\phi})^{2}\phi+\frac{2}{3\sqrt{q}}\pi_{\phi}(\tilde{K}^{i}_{a}E^{a}_{i})\\\
&-\frac{1}{4\kappa\sqrt{q}^{3}}\phi^{3}\mathcal{F}_{ab}\mathcal{F}_{cd}E^{a}_{i}E^{c}_{i}E^{b}_{j}E^{d}_{j}-\frac{2\kappa\sqrt{q}}{\phi^{3}}\pi^{a}\pi^{b}E^{i}_{a}E^{i}_{b},\\\
V_{a}&=\frac{2}{\kappa\gamma}F^{j}_{ab}E^{b}_{j}+\pi_{\phi}\partial_{a}\phi-\pi^{b}\mathcal{F}_{ab}+\mathcal{A}_{a}\partial_{b}\pi^{b},\\\
G_{j}&=\frac{2}{\kappa\gamma}\mathcal{D}_{a}E^{a}_{j}\equiv\frac{2}{\kappa\gamma}(\partial_{a}E^{a}_{j}+\epsilon_{jkl}A^{k}_{a}E^{al}),\\\
J&=-\partial_{a}\pi^{a},\end{split}$ (33)

where $F^{j}_{ab}:=2\partial_{[a}A^{j}_{b]}+\epsilon_{jkl}A^{k}_{a}A^{l}_{b}$
is the curvature of the connection $A_{b}^{j}$. As is shown in (33), the
expressions of the constraints are complicated. For simplicity we will perform
spherically symmetric reduction to the 5-dimensional Kaluza-Klein theory on
4-dimensional spacetime and then fully deparametrize the symmetry-reduced
model.

###  III.3 Spherically Symmetric Reduction

Previous works on obtaining the spherically symmetric form of the connection
and desitized triad were described in several articles,
e.g.bojowald2000symmetry ; kastrup1994spherically . Later on, there has been a
few works discussing both classical simplification and loop quantization of
the spherically symmetric systemcampiglia2007loop ; gambini2013loop ;
gambini2020spherically ; chiou2012loop ; zhang2020loop ; zhang2022loop . We
start by recalling the spherically symmetry-reduced version of the connection
campiglia2007loop ; chiou2012loop :

$\begin{split}A=A^{i}_{a}\tau_{i}dx^{a}=&A_{x}(x)\tau_{3}dx+(A_{1}(x)\tau_{1}+A_{2}(x)\tau_{2})d\theta\\\
&+[(A_{1}(x)\tau_{2}-A_{2}(x)\tau_{1})\sin\theta+\tau_{3}\cos\theta]d\varphi,\end{split}$
(34)

where $x$ represents the radial coordinate of the spherical symmetric space,
$\theta$ and $\varphi$ are the angular coordinates, $A_{x}$, $A_{1}$ and
$A_{2}$ are functions of $x$ and $\tau_{i}=-i\sigma_{i}/2$ with the Pauli
matrices $\sigma_{i}$ being orthonormal generators of $SU(2)$.
Correspondingly, the densitized triad takes the form:

$\begin{split}E=E^{a}_{i}\tau^{i}\partial_{a}=&E^{x}(x)\tau_{3}\sin\theta\frac{\partial}{\partial
x}+(E^{1}(x)\tau_{1}+E^{2}(x)\tau_{2})\sin\theta\frac{\partial}{\partial\theta}\\\
&+(E^{1}(x)\tau_{2}-E^{2}(x)\tau_{1})\frac{\partial}{\partial\varphi},\end{split}$
(35)

where $E^{x}$, $E^{1}$ and $E^{2}$ are functions of $x$. In the symmetry-
reduced theory, under the coordinate transformation $x\rightarrow\bar{x}(x)$,
$A_{x}$ transforms as a scalar density of weight 1, i.e.,
$\bar{A}_{x}(\bar{x})=(\partial x/\partial\bar{x})A_{x}(x)$, while $A_{1}$ and
$A_{2}$ are scalars. $E^{x}$ tranforms as a scalar, while $E^{1}$ and $E^{2}$
are scalar densties of weight 1. Meanwhile, The conjugate pairs of the EM
field $(\mathcal{A}_{a},\pi^{a})$ and the scalar field $(\phi,\pi_{\phi})$
take the following form in the spherically symmetric model:

$\begin{split}&\mathcal{A}=\mathcal{A}_{a}dx^{a}=\mathcal{A}(x)dx,\quad\pi=\pi^{a}(\frac{\partial}{\partial
x^{a}})=\pi^{x}(x)\sin\theta\frac{\partial}{\partial x},\\\
&\phi=\phi(x),\quad\pi_{\phi}=P(x)\sin\theta.\\\ \end{split}$ (36)

Also, it is worth noting that the EM field tensor $F_{\mu\nu}$ only has
$F_{01}=-F_{10}$ as non-zero components so that $F_{ab}=0$.

In terms of the symmetry-reduced variables, the symplectic structure is given
by:

$\begin{split}\Omega&=\int_{\Sigma}d^{3}x\frac{2}{\kappa\gamma}\delta
E^{a}_{i}\wedge\delta A^{i}_{a}+\delta\phi\wedge\delta\pi_{\phi}+\delta
A_{a}\wedge\delta\pi^{a}\\\ &=\int_{I}dx(\frac{8\pi}{\kappa\gamma}\delta
E^{x}\wedge\delta A_{x}+\frac{16\pi}{\kappa\gamma}(\delta E^{1}\wedge\delta
A_{1}+\delta E^{2}\wedge\delta A_{2})+4\pi\delta\phi\wedge\delta
P+4\pi\delta\mathcal{A}\wedge\delta\pi^{x}),\end{split}$ (37)

which implies:

$\begin{split}&\\{A_{x}(x),E^{x}(x^{\prime})\\}=\frac{\kappa\gamma}{8\pi}\delta(x,x^{\prime}),\quad\\{A_{1}(x),E^{1}(x^{\prime})\\}=\frac{\kappa\gamma}{16\pi}\delta(x,x^{\prime}),\\\
&\\{A_{2}(x),E^{2}(x^{\prime})\\}=\frac{\kappa\gamma}{16\pi}\delta(x,x^{\prime}),\quad\\{\phi(x),P(x)\\}=\frac{1}{4\pi}\delta(x,x^{\prime}),\\\
&\\{\mathcal{A}(x),\pi^{x}(x^{\prime})\\}=\frac{1}{4\pi}\delta(x,x^{\prime}),\\\
\end{split}$ (38)

and all the other Poisson brackets among the basic variables vanish. The field
strength 2-form can be written as:

$\begin{split}F&=(A^{2}_{1}+A^{2}_{2}-1)\tau_{3}\sin\theta d\theta\wedge
d\varphi-[(A_{1}^{\prime}\tau_{2}-A_{2}^{\prime}\tau_{1})-A_{x}(A_{1}\tau_{1}+A_{2}\tau_{2})]\sin\theta
d\varphi\wedge dx\\\
&+[(A_{1}^{\prime}\tau_{1}+A_{2}^{\prime}\tau_{2})+A_{x}(A_{1}\tau_{2}-A_{2}\tau_{1})]dx\wedge
d\theta,\end{split}$ (39)

where $A^{\prime}_{1}$ denotes the first-order derivative of $A_{1}$ along the
radial direction. In the spherically symmetric case, the Gaussian constraint
for the 4-dimensional gravity reduces to:

$G[\lambda]=\frac{8\pi}{\kappa\gamma}\int_{I}dx\lambda(E^{x}{}^{\prime}+2A_{1}E^{2}-2A_{2}E^{1})=:\int_{I}dx\lambda(x)\overline{G}_{3}(x),$
(40)

where $\lambda(x):=\Lambda^{3}(x)$. It is not difficult to verify that the
constraint (40) no longer generates full SU(2) gauge transformations, instead
it only generates U(1) transformation. Based on this fact, we will perform a
change of variables to further simplify the theory. First we define $A_{\rho}$
and $E^{\rho}$ by:

$\begin{split}&A_{1}=A_{\rho}\cos\beta,\qquad A_{2}=A_{\rho}\sin\beta,\\\
&E^{1}=E^{\rho}\cos(\alpha+\beta),\qquad
E^{2}=E^{\rho}\sin(\alpha+\beta).\end{split}$ (41)

Note that $A_{\rho}$, $E^{\rho}$ and the inner product
$(A_{1},A_{2})\cdot(E^{1},E^{2})=A_{\rho}E^{\rho}\cos\alpha$ are invariant
under the $U(1)$ rotations generated by the constraint (40). Note also that
the angle $\alpha(x)$ is gauge independent and $\beta(x)$ is a pure gauge
angle. Then we can canonically transform the set of basic variables
$(A_{x},A_{1},A_{2},A^{e},\phi;E^{x},E^{1},E^{2},\pi^{x},\pi^{\phi})$ to the
new set of
$(A_{x},\bar{A}_{\rho},\eta,\mathcal{A},\phi;E^{x},E^{\rho},P^{\eta},\pi^{x},P)$
by defining:

$\begin{split}\bar{A}_{\rho}&:=2A_{\rho}\cos\alpha,\\\ \eta&:=\alpha+\beta,\\\
P^{\eta}&:=2A_{\rho}E^{\rho}\sin\alpha=2A_{1}E^{2}-2A_{2}E^{1}.\end{split}$
(42)

With those new variables, the Gaussian constraint (40) can be written as:

$\overline{G}_{3}=\frac{8\pi}{\kappa\gamma}(E^{x}{}^{\prime}+P^{\eta}).$ (43)

Moreover, another change of variables can be achieved to further simplify the
theory. Following the same approach introduced in campiglia2007loop ;
chiou2012loop , let:

$\begin{split}&\bar{A}_{x}:=A_{x}+\eta^{\prime}=A_{x}+(\alpha+\beta)^{\prime},\\\
&\bar{P}^{\eta}:=P^{\eta}+E^{x}{}^{\prime}.\end{split}$ (44)

Under this transformation, the Gaussian constraint becomes:

$\overline{G}_{3}=\frac{1}{2G\gamma}(\bar{P}^{\eta}),$ (45)

which can be easily solved by setting $\bar{P}^{\eta}=0$ and gauge fixing
$\eta=0$. In terms of the new canonical variables
$(\bar{A}_{x},\bar{A}_{\rho},\eta;E^{x},E^{\rho},P_{\eta})$, we can compute
the Poisson brackets between phase space variables as:

$\begin{split}&\\{\bar{A}_{x}(x),E^{x}(x^{\prime})\\}=\frac{\kappa\gamma}{8\pi}\gamma\delta(x,x^{\prime}),\quad\\{\bar{A}_{\rho}(x),E^{\rho}(x^{\prime})\\}=\frac{\kappa\gamma}{8\pi}\gamma\delta(x,x^{\prime}),\\\
&\\{\phi(x),P(x^{\prime})\\}=\frac{1}{4\pi}\delta(x,x^{\prime}),\quad\\{\mathcal{A}(x),\pi^{x}(x^{\prime})\\}=\frac{1}{4\pi}\delta(x,x^{\prime}).\\\
\end{split}$ (46)

Then our constrained system is simplified to the following three constraints:

$\begin{split}H&=8\pi[\frac{\kappa}{12E^{\rho}\sqrt{|E^{x}|}}P^{2}\phi+\frac{1}{3E^{\rho}\sqrt{|E^{x}|}}P\tilde{K}-\frac{\phi
E^{\rho}}{\kappa\sqrt{|E^{x}|}}-\frac{\bar{A}^{2}_{\rho}E^{\rho}}{4\kappa\gamma^{2}\phi\sqrt{|E^{x}|}}\\\
&\qquad-
sgn(E^{x})\frac{\sqrt{|E^{x}|}\bar{A}_{\rho}\bar{A}_{x}}{\kappa\gamma^{2}\phi}+sgn(E^{x})\frac{\phi\sqrt{|E^{x}|}}{\kappa}\partial_{x}(\frac{E^{x}{}^{\prime}}{E^{\rho}})+\frac{\phi(E^{x}{}^{\prime})^{2}}{4\kappa\sqrt{|E^{x}|}E^{\rho}}\\\
&\qquad+\frac{\tilde{K}^{2}}{3\kappa\phi
E^{\rho}\sqrt{|E^{x}|}}-\frac{\kappa\pi^{x}\pi^{x}E^{\rho}}{\phi^{3}\sqrt{|E^{x}|}|E^{x}|}+\frac{E^{\rho}\sqrt{|E^{x}|}}{\kappa}D^{a}D_{a}\phi],\\\
V_{x}&=\frac{8\pi}{\kappa\gamma}(\bar{A}^{\prime}_{\rho}E^{\rho}-\bar{A}_{x}E^{x}{}^{\prime})+4\pi
P\partial_{x}\phi+4\pi\mathcal{A}\partial_{x}\pi^{x},\\\
J&=-4\pi\partial_{x}\pi^{x},\end{split}$ (47)

where
$\tilde{K}:=K^{i}_{a}E^{a}_{i}=(E^{x}\bar{A}_{x}+E^{\rho}\bar{A}_{\rho})\sin\theta/\gamma$.
The Poisson brackets between constraints can be calculated as:

$\begin{split}&\\{J[A_{0}],J[A_{0}^{\prime}]\\}=0,\quad\\{J[A_{0}],H_{x}[N^{x}]\\}=-J[\mathcal{L}_{N^{x}}A_{0}],\quad\\{J[A_{e}],H_{N}[N]\\}=0,\\\
&\\{V_{x}[M^{x}],V_{x}[N^{x}]\\}=-V_{x}[\mathcal{L}_{N^{x}}M^{x}],\quad\\{H[N],V_{x}[N^{x}]\\}=-H[\mathcal{L}_{N^{x}}N],\\\
&\\{H[N],H^{N}[M]\\}=V_{x}[(NM^{\prime}-MN^{\prime})E^{2}_{x}/|q|].\end{split}$
(48)

It is thus verified that after a number of canonical tranformations and
simplifications of the original phase space, the theory remains an first-class
constrained system.

##  IV Deparametrization of The Symmetry-Reduced Theory

In the previous section, we have obatined the 4-dimensional connection-
dynamics of the spherically symmetric 5D Kaluza-Klein theory, and the
constraints of the system are given in (47). In this section, we will solve
the Gaussian constraint of the EM field and deparametrize the diffeomorphism
constraint and the Hamiltionian constraint subsequently. First, by the
deparametrization of the diffeomorphism constraint, the Hamiltionian
constraint can be rewritten in terms of diffeomorphism invariant observables.
We shall then deparametrize the Hamiltonian constraint and define relational
evolution by explicitly calculating the physical Hamiltonian.

###  IV.1 Solving the Gaussian Constraint of the EM Field

Since we have already imposed the Gaussian constraint for the 4-dimensional
gravity during the spherically symmetric reduction, the remaining Hamiltonian
constrained system, as is given in (47), contains three secondary first-class
constraints. Since the Gaussian constraint of the EM field is not in
deparametrizable form, we will first solve it before deparametrizing the
remaining constraints.

It is obvious that the Gaussian constraint $J=-\partial_{x}\pi^{x}$ requires
$\pi^{x}=constant$, and the only non-trivial gauge transformation
corresponding to this constraint is:

$\begin{split}e^{\\{G_{EM}[N_{0}],\cdot\\}}\mathcal{A}(x)=\mathcal{A}(x)-\partial_{x}N_{0}(x).\end{split}$
(49)

Let $\mathcal{A}(x)$ be an integrable function of $x$. Eq.(49) implies that
one can always choose the parameter function $N_{0}(x)$ of the gauge
transformation such that $\mathcal{A}(x)=0$ by the gauge transformation. Note
that, although the EM connection $\mathcal{A}(x)$ can be gauge fixed to be
zero in the spherically symmetric case, the electric field $\mathcal{E}$ is
nonzero. Using the definition for momentum $\pi^{a}$, the electric field can
be expressed as:

$\mathcal{E}:=n^{\mu}F_{\mu x}=-\frac{\kappa
E^{\rho}\pi^{x}}{\phi^{3}|E^{x}|^{\frac{3}{2}}}.$ (50)

After solving the Gaussian constraint for EM field, the only two remaining
constraints are the following Hamiltonian constraint and diffeomorphism
constraint in the radial direction:

$\begin{split}H&=8\pi\\{\frac{\kappa}{12E^{\rho}\sqrt{|E^{x}|}}P^{2}\phi+\frac{1}{3E^{\rho}\sqrt{|E^{x}|}}P\tilde{K}-\frac{\phi
E^{\rho}}{\kappa\sqrt{|E^{x}|}}-\frac{\bar{A}^{2}_{\rho}E^{\rho}}{4\kappa\gamma^{2}\phi\sqrt{|E^{x}|}}-sgn(E^{x})\frac{\sqrt{|E^{x}|}\bar{A}_{\rho}\bar{A}_{x}}{\kappa\gamma^{2}\phi}\\\
&\qquad+sgn(E^{x})\frac{\phi\sqrt{|E^{x}|}}{\kappa}\partial_{x}(\frac{E^{x}{}^{\prime}}{E^{\rho}})+\frac{\phi(E^{x}{}^{\prime})^{2}}{4\kappa\sqrt{|E^{x}|}E^{\rho}}+\frac{\tilde{K}^{2}}{3\kappa\phi
E^{\rho}\sqrt{|E^{x}|}}-\frac{\kappa\pi^{x}\pi^{x}E^{\rho}}{\phi^{3}\sqrt{|E^{x}|}|E^{x}|}+\frac{E^{\rho}\sqrt{|E^{x}|}}{\kappa}D^{a}D_{a}\phi\\}=0,\\\
V_{x}&=\frac{8\pi}{\kappa\gamma}(\bar{A}^{\prime}_{\rho}E^{\rho}-\bar{A}_{x}E^{x}{}^{\prime})+4\pi
P\partial_{x}\phi=0.\\\ \end{split}$ (51)

###  IV.2 Deparametrization of the Diffeomorphism Constraint

On the constraint surface, by assuming that $E^{x}{}^{\prime}$ is nonzero the
diffeomorphism constraint can be written as:

$\bar{V}_{x}{}=\frac{8\pi}{\kappa\gamma}(\bar{A}_{x}-\frac{\bar{A}^{\prime}_{\rho}E^{\rho}+\frac{\kappa\gamma}{2}P\partial_{x}\phi}{E^{x}{}^{\prime}})=0,$
(52)

where $E^{x}$ is chosen as the reference field for the deparametrization. Then
for a phase space function $f$, the corresponding diffeomorphism invariant
observables can be constructed as:

$\begin{split}O_{f}(\sigma)=e^{\\{\bar{V}_{x}{}[\beta^{x}],\cdot\\}}f|_{E^{x}=l_{p}\sigma},\end{split}$
(53)

where $l_{p}$ is the Planck length. Here we have taken into account of the
fact that the dimension of $E^{x}$ is $L^{2}$ chiou2012loop ;
campiglia2007loop , while we want to choose the coordinate $\sigma$ to be of
dimension $L$.

Now in the original phase space, before deparametrizing the diffeomorphism
constraint, we consider the following phase space functions:

$\begin{split}\tilde{E}^{\rho}:=\frac{E^{\rho}}{E^{x}{}^{\prime}},\quad\tilde{P}:=\frac{P}{E^{x}{}^{\prime}}.\end{split}$
(54)

It is easy to see that $\tilde{E^{\rho}}(x)$ and $\tilde{P}(x)$ are spatial
scalar fields under the coordinate transformation along $x$ direction
chiou2012loop . Since $\bar{A}(x)$ and $\phi(x)$ are also spatial scalar
fields, the diffeomorphism invariant observables of $\bar{A}_{\rho}(x)$,
$\tilde{E}^{\rho}(x)$, $\phi(x)$ and $\tilde{P}(x)$ can be directly obtained
by performing the coordinate transformation $x\mapsto\sigma$, i.e.

$\begin{split}&O_{\bar{A}_{\rho}}(\sigma)=\bar{A}_{\rho}(\sigma),\quad
O_{\tilde{E}^{\rho}}(\sigma)=\tilde{E}^{\rho}(\sigma),\quad
O_{\phi}(\sigma)=\phi(\sigma),\quad
O_{\tilde{P}}(\sigma)=\tilde{P}(\sigma).\end{split}$ (55)

Then the following Poisson brackets between diffeomorphism-invariant
observables can be obtained by direct calculations:

$\begin{split}&\\{O_{\bar{A}_{\rho}}(\sigma),O_{\tilde{E}_{\rho}}(\sigma^{\prime})\\}=\frac{\kappa\gamma}{8\pi
l_{p}}\delta(\sigma,\sigma^{\prime})\\\
&\\{O_{\phi}(\sigma),O_{\tilde{P}_{\phi}}(\sigma^{\prime})\\}=\frac{1}{4\pi
l_{p}}\delta(\sigma,\sigma^{\prime}).\end{split}$ (56)

Therefore, in the reduced phase space where the diffeomorphism constraint has
been deparametrized, $\tilde{E}^{\rho}(\sigma)$ and $\tilde{P}(\sigma)$ are
the conjugate momenta of $\bar{A}_{\rho}(\sigma)$ and $\phi(\sigma)$
respectively.

Recall that in the spherically reduced theory the components of 3-metric can
be expressed by the densitized triad as:

$q_{xx}=\frac{(E^{\rho})^{2}}{|E^{x}|},\qquad q_{\theta\theta}=|E^{x}|,\qquad
q_{\phi\phi}=|E^{x}|\sin^{2}\theta.$ (57)

Since $|E^{x}|$ is associated with the square of curvature radius at point
$x$, for the Schwarzchild spacetime it changes monotonically along the radial
directon. This motivates us to choose $E^{x}$ as the reference field for the
spatial direction.

###  IV.3 Deparametrization of the Hamiltionian Constraint

Consider the intermediate Hamiltonian $\bar{H}:=\frac{H}{E^{x}{}^{\prime}}$,
which transforms as a spatial scalar field in the spherically symmetric model.
Then its diffeomorphism invariant observable $O_{\bar{H}}(\sigma)$, just like
$O_{\bar{A}_{\rho}}(\sigma)$ and $O_{\tilde{E}^{\rho}}(\sigma)$ in (55), can
also be viewed as a coordinate transformation of $\bar{H}(x)$ from $x$ to
$\sigma$. Given a suitable function $\tilde{N}(\sigma)$, the smeared version
of the diffeomorphism invariant intermediate Hamiltonian can be defined as:

$\begin{split}\bar{H}[\tilde{N}]&:=\frac{8\pi}{\kappa}\int
d\sigma\tilde{N}(\sigma)\bar{H}(\sigma).\end{split}$ (58)

Note that, because of Eq.(9), from now on we use $\bar{H}(\sigma)$ to
represent $O_{\bar{H}}(\sigma)$ and the same notation applies for all
variables in the diffeomorphism invariant phase space as well.

Recall that the scalar $\phi$ in the original Kaluza-Klein theory represents
the circumference of the fifth dimension. Therefore, in order for $\phi$ to be
chosen as the reference field for time evolution, we shall assume that $\phi$
changes monotoniacally in time. Thus we consider the cases where the fifth
dimension is either expanding or shrinking.

Using Eqs.(51) and (54), the intermidiate Hamitonian can be written as:

$\begin{split}\bar{H}(\sigma)&=8\pi\\{[\frac{\kappa\phi}{12\sqrt{|l_{p}\sigma|}\tilde{E}^{\rho}}+\frac{\kappa|\sigma|^{\frac{3}{2}}(\partial_{\sigma}\phi)^{2}}{12\sqrt{l_{p}}\phi\tilde{E}^{\rho}}+\frac{\kappa
sgn(\sigma)\sqrt{|\sigma|}\partial_{\sigma}\phi}{6\sqrt{l_{p}}\tilde{E}^{\rho}}]\tilde{P}^{2}\\\
&+[\frac{|\sigma|^{\frac{3}{2}}\partial_{\sigma}\bar{A}_{\rho}\partial_{\sigma}\phi}{3\sqrt{l_{p}}\gamma\phi}+\frac{sgn(\sigma)\sqrt{|\sigma|}\partial_{\sigma}\bar{A}_{\rho}}{3\sqrt{l_{p}}\gamma}-\frac{sgn(\sigma)\sqrt{|\sigma|}\bar{A}_{\rho}\partial_{\sigma}\phi}{6\sqrt{l_{p}}\phi\gamma}+\frac{\bar{A}_{\rho}}{3\gamma\sqrt{l_{p}|\sigma|}}]\tilde{P}\\\
&+\frac{\phi
sgn(\sigma)\sqrt{|\sigma|}}{\sqrt{l_{p}}\kappa}\partial_{\sigma}(\frac{1}{\tilde{E}^{\rho}})+\frac{\phi}{4\kappa\sqrt{l_{p}|\sigma|}\tilde{E}^{\rho}}-sgn(\sigma)\frac{\sqrt{|\sigma|}\bar{A}_{\rho}\tilde{E}^{\rho}\partial_{\sigma}\bar{A}^{\rho}}{\sqrt{l_{p}}\phi\kappa\gamma^{2}}-\frac{\kappa\pi^{x}\pi^{x}\tilde{E}^{\rho}}{\phi^{3}(l_{p}\sigma)^{2}}\\\
&+\frac{|\sigma|^{\frac{3}{2}}(\partial_{\sigma}\bar{A}_{\rho})^{2}\tilde{E}^{\rho}}{3\sqrt{l_{p}}\kappa\gamma^{2}\phi}+(\frac{3sgn(\sigma)\sqrt{|\sigma|}}{2\sqrt{l_{p}}\tilde{E}^{\rho}}-\frac{|\sigma|{}^{\frac{3}{2}}}{\sqrt{l_{p}}(\tilde{E}^{\rho})^{2}}\partial_{\sigma}\tilde{E}^{\rho})\partial_{\sigma}\phi\\\
&+\frac{|\sigma|{}^{\frac{3}{2}}}{\sqrt{l_{p}}\tilde{E}^{\rho}}\partial_{\sigma}\partial_{\sigma}\phi-\frac{\phi\tilde{E}^{\rho}}{\kappa\sqrt{l_{p}|\sigma|}}-\frac{\bar{A}^{2}_{\rho}\tilde{E}^{\rho}}{4\kappa\phi\gamma^{2}\sqrt{l_{p}|\sigma|}}+\frac{\tilde{E}^{\rho}\bar{A}_{\rho}^{2}+2\bar{A}_{\rho}\sigma\partial_{\sigma}\bar{A}_{\rho}\tilde{E}^{\rho}}{3\kappa\gamma^{2}\phi\sqrt{l_{p}|\sigma|}}\\}.\\\
\end{split}$ (59)

Thus the constraint $\bar{H}=0$ can be seen as a quadratic equation with
respect to $\tilde{P}$, which takes the form $a\tilde{P}^{2}+b\tilde{P}+c=0$,
where the three coefficients can be written as:

$\begin{split}a&:=2\pi\frac{\kappa(\phi+\sigma\partial_{\sigma}\phi)^{2}}{3\sqrt{|l_{p}\sigma|}\tilde{E}^{\rho}\phi},\\\
b&:=\frac{4\pi}{3\gamma\phi}(2\frac{|\sigma|^{\frac{3}{2}}}{\sqrt{l_{p}}}\partial_{\sigma}\bar{A}_{\rho}\partial_{\sigma}\phi+2sgn(\sigma)\phi\sqrt{\frac{|\sigma|}{l_{p}}}\partial_{\sigma}\bar{A}_{\rho}-sgn(\sigma)\sqrt{\frac{|\sigma|}{l_{p}}}\bar{A}_{\rho}\partial_{\sigma}\phi+\frac{2\phi\bar{A}_{\rho}}{\sqrt{l_{p}|\sigma|}}),\\\
c&:=\bar{H}-a(\pi_{\phi})^{2}-b\pi_{\phi}.\\\ \end{split}$ (60)

In the special case when $a=0$, one has
$\partial_{\sigma}\phi=-\frac{\phi}{\sigma}$ and
$b=\frac{4\pi\bar{A}_{\rho}}{\gamma\sqrt{l_{p}|\sigma|}}$. Then we have the
following two special cases. First, if $b=0$, one gets $\bar{A}_{\rho}=0$ and
$c=0$. Since the only remaining phase space variable present in $c$ is
$\tilde{E}^{\rho}$, this case corresponds to a special solution of the theory
after imposing the Hamiltonian constriant. Second, if $b\neq 0$, the model is
deparametrizable. The Hamiltonian constraint becomes:

$\bar{H}(\sigma)=b\tilde{P}+c=0.$ (61)

Hence one obtains

$\tilde{P}+\frac{\gamma\sqrt{l_{p}|\sigma|}c}{4\pi\bar{A}_{\rho}}=0,$ (62)

which is suitable for deparametrization by using $\phi$ as the reference field
for time. The construction of gauge-invariant observables can be done
following the framework introduced in section II. We will show the
deparametrization process explicitly below for the more general case when
$a\neq 0$. In this case, on the constraint surface the condition
$b^{2}-4ac\geq 0$ is automatically satisfied. Therefore, we can reformulate
the Hamiltonian constraint in the following form:

$\tilde{H}=\tilde{P}+(H_{1}\pm H_{2}),$ (63)

where:

$\begin{split}H_{1}(\sigma)&=\frac{2\tilde{E}^{\rho}\sigma^{2}\partial_{\sigma}\bar{A}_{\rho}\partial_{\sigma}\phi+2\phi\tilde{E}^{\rho}\sigma\partial_{\sigma}\bar{A}_{\rho}-\tilde{E}^{\rho}\sigma\bar{A}_{\rho}\partial_{\sigma}\phi+2\phi\tilde{E}_{\rho}\bar{A}_{\rho}}{\kappa\gamma(\phi+2\sigma\partial_{\sigma}\phi)^{2}},\\\
H_{2}(\sigma)&=-\frac{\sqrt{l_{p}|\sigma|}\tilde{E}^{\rho}}{\kappa\gamma(\phi+2\sigma\partial_{\sigma}\phi)^{2}}\sqrt{H_{s}},\\\
\end{split}$ (64)

with:

$\begin{split}H_{s}&=\frac{4}{l_{p}}|\sigma|^{3}(\partial_{\sigma}\bar{A}_{\rho}\partial_{\sigma}\phi)^{2}+\frac{4}{l_{p}}|\sigma|(\partial_{\sigma}\bar{A}_{\rho})^{2}\phi^{2}+\frac{|\sigma|}{l_{p}}\bar{A}_{\rho}^{2}(\partial_{\sigma}\phi)^{2}+\frac{4\phi^{2}\bar{A}_{\rho}{}^{2}}{l_{p}|\sigma|}\\\
&+\frac{8}{l_{p}}\sigma|\sigma|(\partial_{\sigma}\bar{A}_{\rho})^{2}\phi\partial_{\sigma}\phi-\frac{4}{l_{p}}|\sigma|\bar{A}_{\rho}\partial_{\sigma}\bar{A}_{\rho}\partial_{\sigma}\phi-\frac{4}{l_{p}}\sigma|\sigma|\bar{A}_{\rho}\phi\partial_{\sigma}\bar{A}_{\rho}(\partial_{\sigma}\phi)^{2}\\\
&+\frac{8}{l_{p}}\phi|\sigma|\bar{A}_{\rho}\partial_{\sigma}\bar{A}_{\rho}\partial_{\sigma}\phi+\frac{8}{l_{p}}sgn(\sigma)\phi^{2}\bar{A}_{\rho}\partial_{\sigma}\bar{A}_{\rho}-\frac{4}{l_{p}}\phi
sgn(\sigma)\bar{A}_{\rho}{}^{2}\partial_{\sigma}\phi\\\
&-12\kappa\phi\gamma^{2}(\phi+2\sigma\partial_{\sigma}\phi)^{2}\\\
&\times(\frac{l_{p}\phi
sgn(\sigma)}{\kappa\tilde{E}^{\rho}}\partial_{\sigma}(\frac{1}{\tilde{E}^{\rho}})+\frac{\phi}{4l_{p}\kappa|\sigma|(\tilde{E}^{\rho})^{2}}-sgn(\sigma)\frac{\bar{A}_{\rho}\partial_{\sigma}\bar{A}^{\rho}}{l_{p}\phi\kappa\gamma^{2}}-\frac{\kappa\pi^{x}\pi^{x}}{l_{p}\phi^{3}\sigma^{2}\sqrt{|\sigma|}}\\\
&+\frac{|\sigma|(\partial_{\sigma}\bar{A}_{\rho})^{2}}{3l_{p}\kappa\gamma^{2}\phi}+(\frac{3sgn(\sigma)}{2l_{p}(\tilde{E}^{\rho})^{2}}-\frac{|\sigma|}{l_{p}(\tilde{E}^{\rho})^{3}}\partial_{\sigma}\tilde{E}^{\rho})\partial_{\sigma}\phi\\\
&+\frac{|\sigma|}{l_{p}(\tilde{E}^{\rho})^{2}}\partial_{\sigma}\partial_{\sigma}\phi-\frac{\phi}{l_{p}\kappa|\sigma|}-\frac{\bar{A}^{2}_{\rho}}{4l_{p}\kappa\phi\gamma^{2}|\sigma|}+\frac{\bar{A}_{\rho}^{2}+2\bar{A}_{\rho}\sigma\partial_{\sigma}\bar{A}_{\rho}}{3l_{p}\kappa\gamma^{2}\phi|\sigma|}).\\\
\end{split}$ (65)

Given an arbitrary phase space function $f$, its gauge-invariant Dirac
observable invariant with respect to the Hamiltonain constraint can be
constructed as:

$\begin{split}f(\tau,\sigma):=O_{O_{f}(\sigma)}(\tau)=e^{\\{\tilde{H}{}[\beta],\cdot\\}}(O_{f}(\sigma))|_{\phi=\tau},\end{split}$
(66)

where $\tau$ is chosen to be independent of $\sigma$, i.e.,
$\partial_{\sigma}\tau=0$, so that we have completely separated spatial and
time coordinates. Physically, this setting implies that the physical clocks on
different points on each spatial hypersurface are synchronized to have a
single value $\tau$. On the final reduced phase space, only one pair of
physical DoFs remains and they satisfy:

$\begin{split}&\\{O_{\bar{A}_{\rho}}(\tau,\sigma),O_{\tilde{E}_{\rho}}(\tau,\sigma^{\prime})\\}=\frac{\kappa\gamma}{8\pi
l_{p}}\delta(\sigma,\sigma^{\prime}).\\\ \end{split}$ (67)

After deparametrizing the Hamiltonian constraint, we are left with the
following smeared physical Hamiltonian which determines the physical time
evolution of the system:

$O_{H[1]}(\tau,\sigma)=H_{1}(\tau,\sigma)\pm H_{2}(\tau,\sigma),$ (68)

where:

$\begin{split}H_{1}(\tau,\sigma)&=\frac{2\tilde{E}^{\rho}\sigma\partial_{\sigma}\bar{A}_{\rho}+2\tilde{E}_{\rho}\bar{A}_{\rho}}{\kappa\gamma\tau},\\\
H_{2}(\tau,\sigma)&=-\frac{\sqrt{|\sigma|}\tilde{E}^{\rho}}{l_{p}\kappa\gamma\tau^{2}}\sqrt{H_{s}(\tau,\sigma)},\\\
H_{s}(\tau,\sigma)&=-12\kappa\gamma^{2}\tau^{3}\times(\frac{\tau
sgn(\sigma)}{l_{p}\kappa\tilde{E}^{\rho}}\partial_{\sigma}(\frac{1}{\tilde{E}^{\rho}})+\frac{\tau}{4l_{p}\kappa|\sigma|(\tilde{E}^{\rho})^{2}}-sgn(\sigma)\frac{\bar{A}_{\rho}\partial_{\sigma}\bar{A}^{\rho}}{l_{p}\tau\kappa\gamma^{2}}-\frac{\kappa\pi^{x}\pi^{x}}{l_{p}^{2}\tau^{3}\sigma^{2}\sqrt{l_{p}|\sigma|}}\\\
&-\frac{\tau}{l_{p}\kappa|\sigma|}-\frac{\bar{A}^{2}_{\rho}}{4l_{p}\kappa\tau\gamma^{2}|\sigma|}).\end{split}$
(69)

Finally, the evolution equation of basic variables can be obtained by using
Eq.(7) as:

$\begin{split}\frac{\partial\tilde{E}^{\rho}(\tau,\sigma)}{\partial\tau}&=\\{O_{h[1]}(\tau),\tilde{E}^{\rho}(\tau,\sigma)\\}\\\
&=-\frac{\kappa\gamma}{8\pi}[\frac{2\sigma\partial_{\sigma}\tilde{E}^{\rho}}{\kappa\gamma\tau}\pm(-\frac{6}{l_{p}\kappa\gamma}\partial_{\sigma}(\frac{sgn(\sigma)\sqrt{l_{p}|\sigma|}\tilde{E}^{\rho}}{\sqrt{H_{s}}})+\frac{3\tilde{E}^{\rho}}{l_{p}\kappa\gamma\sqrt{|\sigma|}\sqrt{H_{s}}})\bar{A}_{\rho}],\\\
\frac{\partial\bar{A}_{\rho}(\tau,\sigma)}{\partial\tau}&=\\{O_{h[1]}(\tau),\bar{A}_{\rho}(\tau,\sigma)\\}\\\
&=-\frac{\kappa\gamma}{8\pi}[\frac{2\sigma\partial_{\sigma}\bar{A}_{\rho}+2\bar{A}_{\rho}}{\kappa\gamma\tau}\mp(\frac{\sqrt{l_{p}|\sigma|}\sqrt{H_{s}}}{\kappa\gamma\tau^{2}}+\frac{3\gamma\tau^{2}}{l_{p}\kappa}\partial_{\sigma}(\frac{sgn(\sigma)\sqrt{l_{p}|\sigma|}\tilde{E}^{\rho}}{\sqrt{H_{s}}})\frac{2}{(\tilde{E}^{\rho})^{3}}+\frac{3\gamma\tau^{2}}{\kappa\sqrt{l_{p}|\sigma|}\sqrt{H_{s}}(\tilde{E}^{\rho})^{2}})].\end{split}$
(70)

###  IV.4 The $\pm$ Sign Problem

One ambiguity still remains for our deparametrization scheme: A $\pm$ sign
appears when we rewrite the intermediate Hamiltonian into its deparametrized
form. Now we clarify this ambiguity by considering the homogeneous and
isotropic cosmology in the Kaluza-Klein theory. Recall that the induced line
element of the spherically symmetric metric on an 3-dimensional spacelike
hypersurface can be written in terms of the spherical coordinates
$x^{a}=(x,\theta,\varphi)$ by two functions $\Lambda(t,x)$ and $R(t,x)$ as:

$ds^{2}=q_{ab}dx^{a}dx^{b}=\Lambda^{2}(t,x)dx^{2}+R^{2}(t,x)d\Omega^{2},$ (71)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$, and the metric
components are related to those of the densitized triad by:

$|E^{x}|=R^{2},\qquad E^{\rho}=R\Lambda.$ (72)

Since the line element of the Friedman-Lemaître-Robertson-Walker (FLRW) metric

$\begin{split}ds=-dt^{2}+a^{2}(t)[dx^{2}+x^{2}(d\theta^{2}+\sin\theta
d\varphi^{2})]\end{split}$ (73)

is a special case of the spherically symmetric one, we can obtain the
expressions of the basic variables in the FLRW model as:

$\begin{split}&N=1,\quad N^{a}=0,\quad\Lambda=a,\quad R=xa,\\\
&E^{\rho}=xa^{2},\quad E^{x}=a^{2}x^{2},\quad\bar{A}_{\rho}=2\gamma
x\dot{a},\quad\bar{A}_{x}=\gamma\dot{a},\\\
&\tilde{K}=3x^{2}a^{2}\dot{a},\quad
P=\frac{3x^{2}a^{3}}{\kappa}\frac{\dot{\phi}}{\phi}-\frac{6x^{2}a^{2}\dot{a}}{\kappa\phi}.\end{split}$
(74)

Note that the expressions in (74) coincide with those in the equivalent model
of Brans-Dicke theory zhang2013loop . While the diffeomorphism constraint is
satisfied automatically in the homogeneous model, the Hamiltonian constraint
remains to be deparametrized. It can be recast into the following form:

$\begin{split}\tilde{H}:=&P+(H_{1}\pm H_{2})=0,\\\ \end{split}$ (75)

where:

$\begin{split}H_{1}&=\frac{2\tilde{K}}{\kappa\phi},\quad
H_{2}=-\sqrt{H_{s}},\\\
H_{s}&=\frac{12E^{\rho}{}^{2}}{\kappa^{2}}+\frac{3\bar{A}_{\rho}^{2}E^{\rho}{}^{2}}{\gamma^{2}\kappa^{2}\phi^{2}}+\frac{12\bar{A}_{\rho}\bar{A}_{x}E^{\rho}{}E^{x}}{\gamma^{2}\kappa^{2}\phi^{2}}-\frac{12E^{\rho}{}E^{x}}{\kappa^{2}}\partial_{x}(\frac{E^{x}{}^{\prime}}{E^{\rho}})-\frac{3(E^{x}{}^{\prime})^{2}}{\kappa^{2}}.\end{split}$
(76)

By Eqs. (74), the following equation can be derived from (75):

$\frac{\dot{\phi}}{\phi}=\pm\sqrt{(\frac{2\dot{a}}{a}+\frac{\dot{\phi}}{\phi})^{2}}.$
(77)

Let us first consider the case of taking the "plus" sign in Eq.(77). Then if
$\frac{\dot{\phi}}{\phi}<-\frac{2\dot{a}}{a}$ the equation becomes
$\frac{\dot{a}}{a}+\frac{\dot{\phi}}{\phi}=0$, which is exactly the Friedman
equation for the homogeneous and isotropic spacetime in 5-dimensional Kaluza-
Klein theory. If $\frac{\dot{\phi}}{\phi}\geq-\frac{2\dot{a}}{a}$, only the
trivial solution $\dot{a}=0$ can be obtained. Another case is to take "minus"
sign in Eq.(77). Then if $\frac{\dot{\phi}}{\phi}<-\frac{2\dot{a}}{a}$, one
gets the trivial solution $\dot{a}=0$. If
$\frac{\dot{\phi}}{\phi}\geq-\frac{2\dot{a}}{a}$, we have again
$\frac{\dot{a}}{a}+\frac{\dot{\phi}}{\phi}=0$. Hence the two non-trivial
sectors shares the same equation of motion. Moreover, it is straightforward to
check that, by performing the deparametrization of the Hamiltonian constraint,
the physical Hamitonians obtained in both cases also take the same form.
Therefore, both the "plus" sign theory and the "minus" sign theory contain
non-trivial solutions belonging to different sectors in the phase space. Hence
both cases should be taken into account when the deparametrized theory is
considered.

##  V Concluding Remark

In the previous sections, the Hamiltonian analysis of the 5-dimensional
Kaluza-Klein theory reduced on 4-dimensional spacetime has been performed. Its
connection dynamics has been obtained by canonical transformations. Then the
spherically symmetric model of the theory is studied. We have successfully
defined relational evolution for this higher dimensional gravity theory in its
spherically symmetric model by using only the gravitational degrees of
freedom. Conceptually speaking, in such a system physical time evolution and
spatial coordinates are depicted by the geometrical degrees of freedom of the
higher dimensional spacetime, rather than being depicted by matter fields
introduced additionally to the gravity theory. The fields we used to
deparametrize the Hamiltionian constraint and diffeomorphism constraint are
the scalar field $\phi$ and the radial component $E^{x}$ of the densitized
triad respectively, where $\phi$ is obtained via killing reduction of the
scale of the fifth dimension in 5-dimensional Kaluza-Klein theory.

It should be noted that the full deparametrization of the general
4-dimensional cosntrained system of the 5-dimensional Kaluza-Klein theory has
not been achieved in this work, while the deparametrization of the spherically
symmetric model has been done. In the full theory it is very difficult to
recast the cosntrained system into a form suitable for the deparametrization
by its own degrees of freedom. Nevertheless, the deparametrization of the
model indicates the scheme to that of the full theory. It is worth studying
further the quantization of this deparametrized model, which we leave for our
next work. It is also expected that our classical deparametrization of the
theory by its geometrical degrees of freedom could provide some hints on
understanding the issues of quantum time and quantum space in quantum gravity.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 11875006, and 11961131013). We thank Hongguang Liu, Gaoping Long,
Shupeng Song, Faqiang Yuan and Cong Zhang for helpful discussions.

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