Physical States and Transition Amplitudes in Piecewise Flat Quantum Gravity

Aleksandar Miković

Departamento de Informática e Sistemas de Informacão

Universidade Lusófona

Av. do Campo Grande, 376, 1749-024 Lisboa, Portugal

and

Grupo de Fisica Matemática

Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

E-mail<EMAIL_ADDRESS>

###### Abstract

We show how the path integral for gravity and matter on a piecewise flat
spacetime can be used to define the physical quantum gravity states and the
related transition amplitudes. The physical states are given by the path
integrals for open manifolds from a certain topological class, while the
corresponding transition amplitudes are obtained by gluing two such open
manifolds into a closed one and calculating the corresponding path integral.
We also solve the problem of how to associate a quantum gravity state to an
effective action. This is done by using the path integral for a closed
manifold from the transition-amplitude topological class so that the state
which corresponds to the effective action can be choosen to be the Hartle-
Hawking or the Vilenkin wavefunction for the vacuum manifold component of the
transition-amplitude manifold.

##  1 Introduction

A path integral (PI) quantization of General Relativity (GR) is the most
natural approach to quantization of gravity, because of its geometric nature
and conceptual simplicity, see [1, 2, 3, 4, 5]. Although it is not that
difficult to define a GR path integral on a spacetime triangulation, see [3,
4, 5], the main obstacle in the PI approach has been the definition of the
theory in the smooth-manifold limit. It was proposed in [6] that this problem
can be avoided by postulating that the short-distance structure of the
spacetime is given by a piecewise linear (PL) manifold which is piecewise flat
and which corresponds to a spacetime triangulation. We call such a quantum
gravity theory a picewise flat quantum gravity (PFQG) and one can take the
edge lengths of the triangulation to be the basic gravitational degrees of
freedom (DOF). A smooth manifold is then recovered as a PL manifold where the
number of edge lengths is large in a macroscopic spacetime region, while the
average edge length is microscopicaly small. In this case the corresponding
effective action (EA) can be approximated by the effective action for a
quantum field theory (QFT) with a cutoff, where the QFT is defined on the
corresponding smooth-manifold region [6, 5]. The QFT cutoff is then determined
by the average edge length [5], and the corresponding effective action is well
defined because of the existence of a physical cutoff. In this way one
recovers the correct semiclassical limit of the PFQG theory, since the
semiclassical expansion of the PFQG effective action is given by the usual
perturbative QFT effective action.

The PFQG theory is also well defined non-pertubatively, since the path
integral is finite for a certain class of PI measures [7, 8], so that the
effective action and the correlation functions are defined non-perturbatively.

In the standard quantum mechanics (QM) and in its direct generalization given
by a flat-spacetime QFT, it is clear how to express the various physical
amplitudes as path integrals, see [10]. However, when gravity is dynamical,
this relationship is less clear. The Hartle-Hawking (HH) construction [2], as
well as the Vilenkin construction [9], solve the conceptual problem of how to
construct a physical wavefunction of the Universe (WFU) by using a path
integral. However, due to the lack of the definition of the full QG path
integral, the HH and the Wilenkin constructions have been only implemented in
the context of minisuperspace models, see for example [11, 12, 13, 14]. Since
the PFQG theory gives a construction of a full QG path integral [7, 8], one
can now discuss the construction of a general WFU. In this paper we will show
how to construct a general WFU and also how to construct the related physical
amplitudes.

The fundamental object in PFQG theory is the effective action, since the
semiclassical limit is defined by using the effective action [7, 5]. However,
from the QM perspective, it is not clear what is the physical state which
corresponds to the PFQG effective action. Note that in the QFT case, the
effective action is defined with respect to the vacuum state, which is defined
as the ground state of the Hamiltonian. However, in the QG case, such a
definition cannot be used because there is no unique Hamiltonian111A
Hamiltonian in quantum gravity is defined with respect to the choice of a time
variable, see [5]. and every QG Hamiltonian is unbounded from bellow.
Fortunatelly, we have the HH and the Vilenkin constructions, and we will show
how these constructions can be used to associate a state to the QG effective
action.

Note that in [6, 7, 8] it was observed that the PFQG effective action can be
defined by using the path integral for a spacetime manifold without a
boundary, or by using the same manifold, but with two spatial boundaries. The
later case is dictated by the Cauchy problem for the classical action, but it
remained unclear what is the relationship between these two constructions.
Also, when using the manifolds with zero or two boundaries to construct the
effective action, it was not clear what are the QG states which correspond to
each of the QG effective actions. In this paper we will clarify these issues.

In section 2 we briefly review the PFQG path integral. In section 3 we briefly
review the QFT effective action and the construction of the PFQG effective
action. In section 4 we explain how to extend the basic QM path integral
formulas to the PFQG theory in order to give the path integral definitions of
the physical states and their transition amplitudes. In section 5 we show how
to construct a state that that can be associated to the PFQG effective action.
In section 6 we present our conclussions.

##  2 Quantum gravity path integral

In this section we review the basics of PFQG theory, see [5, 8]. Let $M$ be a
closed, compact and smooth 4-dimensional manifold and let $T(M)$ be a
piecewise linear manifold which corresponds to a regular finite
triangulation222We consider a triangulation to be regular if the dual one-
simplex is given by a connected 5-valent graph with more than 5 vertices. of
$M$. We will restrict the topology of $M$ to be

$M=M^{\prime}\sqcup\left(\Sigma\times I\right)\sqcup M^{\prime\prime}\,,$ (1)

where

$\partial M^{\prime}=\partial M^{\prime\prime}=\Sigma\,,$ (2)

and $I=[0,t]$, see Fig. 1.

Figure 1: Topology of a PFQG compact spacetime manifold

The 3-manifold $\Sigma$ and the 4-manifolds $M^{\prime}$ and
$M^{\prime\prime}$ can be of arbitrary topologies, with the restriction that
the manifolds $M^{\prime}$ and $M^{\prime\prime}$ should contain a submanifold
which is the 3-manifold $\Sigma$. In this paper we will be interested in the
case when $M^{\prime}=M^{\prime\prime}$.

Let us choose a triangulation $T(M)$, and let $\\{L_{\epsilon}|\,\epsilon\in
T_{1}(M)\\}$ be a set of the edge lengths such that $L_{\epsilon}^{2}\in{\bf
R}$, i.e. $L_{\epsilon}\in{\bf R}_{+}$ (a spacelike edge) or $L_{\epsilon}\in
i{\bf R}_{+}$ (a timelike edge). We do this in order to be able to construct
Lorentzian signature metrics on $T(M)$.

A metric on $T(M)$, which is flat in each 4-simplex $\sigma$ of $T(M)$, is
given by

$G_{\mu\nu}(\sigma)=L_{0\mu}^{2}+L_{0\nu}^{2}-L_{\mu\nu}^{2}\,,$ (3)

where the five vertices of $\sigma$ are labeled as $0,1,2,3,4$ and
$\mu,\nu=1,2,3,4$. This metric is known as the Cayley-Menger (CM) metric.

The CM metric is not dimensionless and hence it is not diffeomorphic to

$g_{\mu\nu}(\sigma)=\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}\,,$ (4)

see [5]. This can be corrected by using a dimensionless PL metric

$g_{\mu\nu}(\sigma)={G_{\mu\nu}(\sigma)\over|L_{0\mu}||L_{0\nu}|}\,.$ (5)

The Einstein-Hilbert (EH) action on $M$ is given by

$S_{EH}=\int_{M}\sqrt{|\textrm{det}g|}\,R(g)\,d^{4}x\,,$ (6)

where $R(g)$ is the scalar curvature associated to a metric $g$. On $T(M)$ the
EH action becomes the Regge action

$S_{R}(L)=\sum_{\Delta\in T(M)}A_{\Delta}(L)\,\delta_{\Delta}(L)\,,$ (7)

when the edge lengths correspond to a Eucledean PL geometry. $A_{\Delta}$ is
the area of a triangle $\Delta$, while the deficit angle $\delta_{\Delta}$ is
given by

$\delta_{\Delta}=2\pi-\sum_{\sigma\supset\Delta}\theta_{\Delta}^{(\sigma)}\,,$
(8)

where a dihedral angle $\theta_{\Delta}^{(\sigma)}$ is defined as the angle
between the 4-vector normals associated to the two tetrahedrons that share the
triangle $\Delta$.

In the case of a Lorentzian geometry, a dihedral angle can be a complex
number, so that it is necessary to modify the formula (7) such that the Regge
action takes only the real values. One can modify the Regge action as

$S^{*}_{R}(L)=Re\left(\sum_{\Delta(s)}A_{\Delta(s)}\,\frac{1}{\textrm{i}}\,\delta_{\Delta(s)}\right)+\sum_{\Delta(t)}A_{\Delta(t)}\,\delta_{\Delta(t)}\,,$
(9)

where $\Delta(s)$ denotes a spacelike triangle, while $\Delta(t)$ denotes a
timelike triangle. Consequently $S^{*}_{R}(L)$ is always a real number, and it
corresponds to the Einstein-Hilbert action on $T(M)$ [4, 5].

Therefore the PFQG path integral is defined as

$Z(T(M))=\int_{D}\prod_{\epsilon=1}^{N}dL_{\epsilon}\,\mu(L)\,e^{iS^{*}_{R}(L)/l_{P}^{2}}\,,$
(10)

where $dL_{\epsilon}=d|L_{\epsilon}|$, $l_{P}$ is the Planck length and
$\mu(L)$ is a mesure that ensures the convergence and gives the effective
action with a correct semiclassical expansion, see [6, 7]. The integration
region $D$ is a subset of ${\bf R}_{+}^{N}$, consistent with a choice of
spacelike and timelike edges.

One can then show that the path integral (10) is convergent for the measure

$\mu(L)=e^{-V_{4}(L)/L_{0}^{4}}\prod_{\epsilon=1}^{N}\left(1+{|L_{\epsilon}|^{2}\over
l_{0}^{2}}\right)^{-p}\,,$ (11)

and the corresponding effective action has a correct semiclassical limit,
where $V_{4}(L)$ is the 4-volume of $T(M)$, $L_{0}$ and $l_{0}$ are constants
and $p>1/2$, see [7].

When $\Sigma$ is a non-compact manifold, we maintain a finite number of DOF by
allowing non-zero $L$ vectors, i.e.
$(L_{1},L_{2},\cdots,L_{N})\neq(0,0,\cdots,0)$, only for a triangulation of
$B_{3}\times I$, where $B_{3}$ is a 3-ball in $\Sigma$. This subset of
$\Sigma\times I$ is then glued to two 4-balls $B_{4}^{\prime}$ and
$B_{4}^{\prime\prime}$ in $M^{\prime}$ and $M^{\prime\prime}$, see Fig. 2

Figure 2: Topology of a PFQG non-compact spacetime manifold

Therefore one has non-zero $L$ vectors in a triangulation of a compact region
of $M^{\prime}\sqcup(\Sigma\times I)\sqcup M^{\prime\prime}$ given by
$B_{4}^{\prime}\sqcup(B_{3}\times I)\sqcup B_{4}^{\prime\prime}$.

Note that in the 2d representation of $M$ in Fig. 2, the 3-ball $B$ is
represented by an interval $D$, while the 4-balls $B^{\prime}$ and
$B^{\prime\prime}$ in $M^{\prime}$ and $M^{\prime\prime}$ are represented by
half-discs.

We can include the matter DOF by giving the matter fields at the vertices of
the triangulation [8]. Let $\phi=\\{\phi_{\alpha}\,|\,\alpha=1,2,...,c\\}$ be
a set of $c$ matter field components at a spacetime point. The corresponding
gravity plus matter path integral will be given by

$Z=\int_{D}d^{N}L\,\mu(L)\,e^{iS^{*}_{R}(L)/l_{P}^{2}}\,Z_{m}(L)\,,$ (12)

where

$Z_{m}(L)=\int_{D_{m}}\prod_{\alpha=1}^{c}d^{n}\phi_{\alpha}\,e^{iS_{m}(\phi,L)/\hbar}\,,$
(13)

is the matter path integral. Here $S_{m}(\phi,L)$ denotes the matter action on
$T(M)$, $n$ is the number of vertices in $T(M)$ and $D_{m}$ is the range of
the matter fields components, given by

${\bf R}^{c^{\prime}n}\times\prod_{v=1}^{n}\Lambda_{c^{\prime\prime}}(v)\,,$
(14)

where $c^{\prime}$ is the number of bosonic fields components,
$\Lambda_{c^{\prime\prime}}(v)$ is a $c^{\prime\prime}$-dimensional Grassman
algebra at a vertex $v$ and $c^{\prime}+c^{\prime\prime}=c$.

Since the convergence of $Z_{m}$ is not guaranteed, we pass to a Eucledean
geometry defined by the edge lengths

$\tilde{L}_{\epsilon}=|L_{\epsilon}|\,,$

so that all the Eucledean edge lengths are positive real numbers. This is
equvalent to a Wick rotation where $\tilde{L}_{\epsilon}=L_{\epsilon}$ if
$\epsilon$ is a spacelike edge and
$\tilde{L}_{\epsilon}=(-\textrm{i})L_{\epsilon}$, if $\epsilon$ is a timelike
edge.

Then we will consider the integral

$\tilde{Z}_{m}(\tilde{L})=\int_{D_{m}}\prod_{\alpha}d^{n}\phi_{\alpha}\,e^{-\tilde{S}_{m}(\phi,\tilde{L})/\hbar}\,,$
(15)

where $\tilde{S}_{m}$ is the Euclidian matter action. Since
$\tilde{S}_{m}(\phi,\tilde{L})$ is a positive function of $\phi$, and

$\tilde{S}_{m}(\phi,\tilde{L})\to+\infty\,,\quad\textrm{for}\,\,|\phi_{\alpha}|\to+\infty\,,$

then the integral $\tilde{Z}_{m}$ will be convergent. Hence we will define the
Lorentzian matter path integral as

$Z_{m}(L)=\tilde{Z}_{m}(\tilde{L}){\Big{|}}_{\tilde{L}=w(L)}\,,$ (16)

where $w$ is the Wick rotation.

When the matter and its classical action are given by the Standard Model, we
can guarantee the absolute convergence of the PFQG path integral (12) if

$p>\textrm{52,5}\,,$ (17)

see [8]. Hence for the values of the parameter $p$ given by (17), we have a
finite path integral, and this makes it possible to construct the PFQG states,
the transition amplitudes and the effective action.

##  3 Quantum gravity effective action

The idea of the effective action was introduced in the context of a QFT as the
gnerating functional of the one-particle irreducible correlation functions

$\Gamma[\phi(x)]=\sum_{n=2}^{\infty}\int
d^{4n}x\,\Gamma_{n}(x_{1},...,x_{n})\phi(x_{1})\cdots\phi(x_{n})\,,$ (18)

see [10].

Note that

$\Gamma_{n}(x_{1},\cdots,x_{n})=\langle
0|\hat{\phi}(x_{1})\cdots\hat{\phi}(x_{n})|0\rangle{\Big{|}}_{1PI}\,,$ (19)

where $\langle...\rangle|_{1PI}$ means to take only the one particle
irreducible graphs in the Feynman diagram expansion of the $n$-point
correlation function. Also note that

$\langle 0|\hat{\phi}(x_{1})\cdots\hat{\phi}(x_{n})|0\rangle=\int{\cal
D}\phi\,\phi(x_{1})\cdots\phi(x_{n})e^{iS(\phi)/\hbar}\,,$ (20)

where $S(\phi)$ is the classical action. Hence the formulas (19) and (20)
establish a clear relationship between the effective action, the path integral
and the QFT vacuum state $|0\rangle$.

It is not difficult to show that the effective action satisfies the equation

$e^{i\Gamma[\phi]/\hbar}=\int{\cal
D}\phi^{\prime}\exp\left(\frac{i}{\hbar}S[\phi+\phi^{\prime}]-\frac{i}{\hbar}\int_{M}\frac{\delta\Gamma}{\delta\phi(x)}\,\phi^{\prime}(x)\,d^{4}x\right)\,,$
(21)

see [7, 15]. This equation gives a nonperturbative definition of the QFT
effective action, but also it can be directly applied to the PFQG case, where
it becomes an integro-differential equation, see [6, 7, 5].

More precisely, a non-perturbative effective action $\Gamma(L,\Phi)$ on
$T(M)$, where $\Phi=(\phi_{1},\phi_{2},...,\phi_{n})$, is determined by the
equation

$\displaystyle e^{i\Gamma(L,\Phi)/\hbar}$ $\displaystyle=$
$\displaystyle\int_{D(L)}d^{N}l\int_{D_{m}}d^{cn}\varphi\,\mu(L+l)\,e^{iS(L+l,\Phi+\varphi)/\hbar-i\sum_{\epsilon}\Gamma^{\prime}_{\epsilon}(L,\Phi)l_{\epsilon}/\hbar}$
(22) $\displaystyle\qquad\qquad\qquad\qquad\quad
e^{-i\sum_{\pi}{\Gamma}^{\prime}_{\pi}(L,\Phi)\varphi_{\pi}/\hbar}\,,$ (23)

where $D(L)$ is the PI integration region translated by a vector $-L$, see [5,
6], while $c$ is the number of components of the matter fields. Note that for
the Standard Model

$c=c_{f}+c_{gh}+c_{b}=96+24+52=172\,,$ (24)

where $c_{f}$ is the number of fermionic fields componets, $c_{gh}$ is the
number of Faddev-Popov ghost fields components and $c_{b}$ is the number of
bosonic fields components, $S=S^{*}_{R}/G_{N}+S_{m}$ and $G_{N}$ is the Newton
constant [8].

The equation (23) will be well defined if the gravity plus matter path
integral is finite, which is the case for $p>$ 52,5 [8]. This is a consequence
of

$|\tilde{Z}_{m}(\tilde{L},J)|\leq\tilde{Z}_{m}(\tilde{L})\,,$ (25)

where

$\tilde{Z}_{m}(\tilde{L},J)=\int_{D_{m}}\prod_{\alpha}d^{n}\phi_{\alpha}\,e^{[-\tilde{S}_{m}(\phi,\tilde{L})+iJ\phi]/\hbar}\,,$
(26)

is the matter path integral with sources. This is the discretized version of
the matter QFT generating functional.

The effective action defined by the equation (23) has an important property:
when the number of edges $N$ is large and the average edge length $\bar{L}$ in
$T(M)$ is microscopically small, then $\Gamma(L,\Phi)$ can be approximated by
a QFT effective action on $M$ where the QFT is given by the momentum cut-off
quantization of GR coupled to the Standard Model, while the momentum cutoff is
given by $2\pi\hbar/\bar{L}$, see [5]. In particular, when $|L_{\epsilon}|\gg
l_{P}$, $N$ is large and $\bar{L}$ is microscopically small, then one can show
that the perturbative solution of the EA equation (23) is very well
approximated by the classical GR action plus the classical SM action plus the
corresponding QFT loop corrections [6, 7]. These QFT loop corrections are all
finite, because we have a QFT with a momentum cutoff.

Hence the PFQG theory for the Standard Model defined by the path integral
(12), with the measure (11), is finite for $p>52,5$ and such a theory has the
correct classical limit, which is GR coupled to the Standard Model on $T(M)$.
When $N$ is large, while $\bar{L}$ is microscopically small in a macroscopic
spacetime region, one recovers GR coupled to the Standard Model on a smooth
spacetime.

Since the equation (23) for the effective action for gravity coupled to matter
is well defined when the PFQG path integral is finite, we can then determine
the non-perturbative effective action. Note that the EA equation (23) by
itself is not sufficient to determine the QG analog of the QFT vacuum state.
We will address this problem in the next section.

##  4 Path integral and physical states

In order to find a relationship between the path integral and a state in
quantum gravity, we start from the path-integral representation of the
propagator in quantum mechanics

$G(q_{f},q_{i};t)=\langle q_{f}|\hat{U}(t)|q_{i}\rangle=\left[\int{\cal
D}q\,e^{i\int_{0}^{t}d\tau
L(\dot{q},q)/\hbar}\right]_{q(0)=q_{i}}^{q(t)=q_{f}}\,,$ (27)

where $q$ denotes the set of generalized coordinates of the system,
$L(\dot{q},q)$ is the Lagrangian, $q_{i}$ denotes the set of initial $q$
values at time $t=0$, while $q_{f}$ denotes the set of final values at a time
instant $t$ and $\hat{U}(t)$ is the evolution operator.

In the QG case, when $M=\Sigma\times I$, then the path integral given by
$Z(T(M))$ corresponds to the formula (27), since

$G(l_{f},\varphi_{f};l_{i},\varphi_{i})=\langle
l_{f},\varphi_{f}|\hat{U}(t)|l_{i},\varphi_{i}\rangle=\left[Z(T(\Sigma\times
I))\right]_{L|_{i}=l_{i},\,\Phi|_{i}=\varphi_{i}}^{L|_{f}=l_{f},\,\Phi|_{f}=\varphi_{f}}\,,$
(28)

where the labels $l$ and $\varphi$ indicate the vectors $L$ and $\Phi$ on the
initial and the final triangulation of the spatial hypersurface $\Sigma$. The
label $t$ is defined by an embedding of $T(\Sigma\times I)$ in ${\bf R}^{5}$
such that the distance from $\Sigma_{i}$ and $\Sigma_{f}$ is $t$, i.e.
$I=[0,t]$, see Fig. 3.

Figure 3: Topology of the QM propagator manifold.

More precisely, the described embedding of $T(\Sigma\times I)$ in ${\bf
R}^{5}$ fixes the time-reparametrization symmetry, since the distance from the
initial $T(\Sigma)$ to the final $T(\Sigma)$ in ${\bf R}^{5}$ is fixed to be
$t$. In this way one fixes the residual time-reparametrization symmetry of the
reduced phase space Wheeler-de Witt (WdW) equation

$\left(i\hbar{\partial\over\partial\tau}-\hat{H}(p,q,\tau)\right)\Psi(q,\tau)=0\,,$
(29)

where $H$ is the reduced phase space Hamiltonian, $(p,q)$ is a point of the
reduced phase space and $\tau$ is the corresponding time variable, see [5].

The equation (29) still has the residual time reparametrization symmetry given
by $\tau\to\tau^{\prime}=f(\tau)$, and we fix it by setting the distance
between the inital and the final spatial hypersurface in ${\bf R}^{5}$ to be
$t$, so that $\tau=t$. Hence $(l,\phi)\cong(q,t)$, so that is the reason we
did not include the label $t$ for $G(l_{f},\varphi_{f};l_{i},\varphi_{i})$ in
the equation (28).

The QM propagator (27) does not say anything about the evolution of an
arbitrary state, so that we need to consider the time-evolution formula

$|\Psi(t)\rangle=\hat{U}(t)|\Psi(0)\rangle\,,$ (30)

where $|\Psi(0)\rangle$ is the initial state of the system, while the
evolution operator of the system $\hat{U}(t)$ obeys the Schrodinger equation

$i\hbar{d\hat{U}\over dt}=\hat{H}\hat{U}(t)\,,$ (31)

where $\hat{H}$ is the Hamiltonian operator for the quantum system.

Hence the state of the system obeys the usual Schrodinger equation

$i\hbar{\partial\Psi(q,t)\over\partial t}=\langle
q|\hat{H}|\Psi(t)\rangle=\hat{H}_{q}\Psi(q,t)\,,$ (32)

where $\hat{H}_{q}$ is the Hamiltonian operator in the $q$-representation.
Therefore the equation (30) describes the time evolution of a state from an
initial state $|\Psi(0)\rangle$, which can be freely chosen. In the QG
framework, the Schrodinger equation (32) corresponds to the reduced phase
space WdW equation (29), provided that one identifies the time variable, see
[5].

Hence the path integral $Z(T(M))$ where

$M=M^{\prime}\sqcup\left(\Sigma\times I\right)\,,$ (33)

see Fig. 4,

Figure 4: Topology of the time-evolution manifold.

would then correspond to a wavefunction

$\Psi(q,t)=\langle q|\Psi(t)\rangle=\langle q|\hat{U}(t)|\Psi(0)\rangle\,,$
(34)

where $(q,t)\cong(L|_{T(\Sigma)},\,\Phi|_{T(\Sigma)})=(l,\varphi)$, so that

$\Psi(l,\phi)=\left[Z(T(M))\right]_{L|_{T(\Sigma)}=l,\,\Phi|_{T(\Sigma)}=\varphi}\,,$
(35)

while $I=[0,t]$.

The state-sum formula (35) for $\Psi(l,\phi)$ on $\Sigma$ is the PFQG analog
of the Hartle-Hawking wavefunction [2]. Note that in the usual interpretation
of the HH wavefunction, one takes $t\to 0$, so that $M=M^{\prime}$, see Fig.
5,

Figure 5: Topology of the Hartle-Hawking manifold.

so that $\Psi(l,\phi)$ is identified with the initial wavefunction of the
Universe (WFU).

Also note that in the $t=0$ case of Fig. 4, which is given by Fig. 5, one
usually takes the Eucledean path integral, which seems odd from the path
integral perspective, since the full path integral for QG should be defined in
the Lorentzian signature (as is the case in PFQG theory). Hence, if one
constructs the HH state in the case of Fig. 5 by using the Eucledean path
integral, the corresponding $\Psi_{0}$ may not be well defined because the
factor $e^{-S(L,\Phi)}$ in the full Eucledean path integral is not bounded.
One may try to remedy this problem by using the $e^{iS(L,\Phi)}$ factor, but
then the corresponding wavefunction may not satisfy the WdW equation. However,
from the point of view of Quantum Mechanics, this is not a problem, since
$\Psi_{0}(l,\phi)$ can be an arbitrary wavefunction. But in order to obtain a
physical evolution, one should use the Lorentzian path integral in the
$\Sigma\times[0,t]$ component of the spacetime of Fig. 4. Hence the
wavefunction of the Universe (35) will certainly satisfy the WdW equation for
$t>0$. More precisely, it will satisfy the PL version of the WdW equation on
$T(\Sigma)$.

However, if one uses the Lorentzian path integral in $T(M^{\prime})$ in order
to construct $\Psi_{0}(l,\phi)$, then such a $\Psi_{0}$ will definitely obey
the PL version of the WdW equation provided that $M^{\prime}$ has a topology
of a 4-cone with a boundary given by the 3-manifold $\Sigma$ [5, 16]. This
type of quantum state is known as the Vilenkin wavefunction [9] and it is also
called the tunneling wavefunction.

##  5 Effective action and quantum gravity states

In order to find the relationship between the effective action and a state in
PFQG theory, we will now consider the following QM formula

$\langle\Psi_{2}|\hat{U}(t)|\Psi_{1}\rangle=\int_{D^{\prime}}d^{m}q_{f}\int_{D^{\prime}}d^{m}q_{i}\Psi_{2}^{*}(q_{f})G(q_{f},q_{i},t)\Psi_{1}(q_{i})\,,$
(36)

where $\Psi_{1}$ and $\Psi_{2}$ are the initial and the final wavefunction,
while $D^{\prime}\subseteq{\bf R}^{m}$, is the domain of the generalized
coordinates.

Let us consider the PI for the following case

$M=M^{\prime}\sqcup\left(\Sigma\times I\right)\sqcup M^{\prime}\,,$ (37)

see Fig. 6.

Figure 6: Topology of the spacetime manifold for the expectation value of the
evolution operator.

By comparing (36) to $Z(T(M))$ we conclude that $\Psi_{1}=\Psi_{2}=\Psi_{0}$
and $I=[0,t]$, so that

$Z(T(M))=\langle\Psi_{0}|\hat{U}(t)|\Psi_{0}\rangle\,.$ (38)

Note that if $\Psi_{0}$ is obtained by using a Eucledean path integral, it
will be real. However, the convergence of the Eucledean path integral will not
be always guaranteed, because the factor $e^{-S_{E}(L,\Phi)}$ in the integrand
can be arbitrarily large. For this reason one could take instead
$e^{iS_{E}(L,\Phi)}$, where $S_{E}$ is the Eucledean action, or to use the
Lorentzian PI factor $e^{iS(L,\Phi)}$. However, in any of these cases the
wavefunction wil not be real, so that in order to include the $\Psi_{2}^{*}$
factor from the formula (36), the corresponding path integral should be
calculated by using the $e^{-iS_{E}(L,\Phi)}$ or $e^{-iS(L,\Phi)}$ factor.

Let us now consider the manifold topology (37) with $I=[-t,t]$, see Fig. 7.

Figure 7: Topology of the effective action manifold

This case is relevant for the Heisenberg operator expectation values, because
of the QM formula

$q^{(H)}_{\alpha}(t)=\langle\Psi_{0}|\hat{U}(-t)\,\hat{q}_{\alpha}(0)\,\hat{U}(t)|\Psi_{0}\rangle\,,$
(39)

where $q_{\alpha}$ is a component of the vector $q$.

Note that there is also the following relation

$q^{(H)}_{\alpha}(t)=-\frac{i}{\hbar}\left[{\partial\over\partial
J_{\alpha}(0)}Z_{J}(T(M))\right]_{J=0}\,,$ (40)

where $Z_{J}$ is the state sum $Z$ with a source term. By comparing the
relations (39) and (40) and by taking into account the EA equation (23), we
conclude that the effective action defined by a manifold of Fig. 7, will
correspond to the state $\Psi_{0}$ which corresponds to the HH state for the
vacuum manifold $M^{\prime}$.

In order to have an exact correspondence with the relation (39), the path
integral $Z(T(M))$ has to be evaluated by using the action
$\tilde{S}(L,\Phi)=-S(L,\Phi)$ in the $\Sigma\times[0,t]$ region, while in the
$\Sigma\times[-t,0]$ region we use the plus sign.

##  6 Conclusions

A physical state of the PFQG theory is given by the path integral for an open
manifold of the topology (33), see Fig. 4. The corresponding wavefunction can
be explicitely calculated, given the triangulations $T(M^{\prime})$ and
$T(\Sigma\times[0,t])$, since the PFQG path integral (12) is finite. One can
also calculate the corresponding transition amplitudes by using the path
integrals for the closed manifolds of Fig. 6 and Fig. 7.

One can also calculate the full Hartle-Hawking wavefunction, provided that one
fixes the spatial manifold $\Sigma$ and the 4-manifold $M^{\prime}$, as well
as the corresponding triangulations $T(\Sigma)$ and $T(M^{\prime})$. Note that
the explicit expressions for the HH wavefunctions can be obtained in the case
of simple triangulations which are the analogs of the minisuperspace models,
see [5, 16].

Also note that for a fixed spatial hypersurface $\Sigma$ one can have many
different HH states, since each such HH state depends on the choice of the
vacuum manifold $M^{\prime}$ and its triangulation. If we define a HH state as
the path integral for the $t\to 0$ manifold topology of Fig. 6, which is the
manifold topology of Fig. 5, then the corresponding HH state does not have to
obey the WdW equation, because the HH state in this case is the analog of the
inital state in the QM time-evolution relation (30) and the initial state
$|\Psi_{0}\rangle$ does not have to obey the Schrodinger equation. However,
for $t>0$, the state $|\Psi(t)\rangle$ obeys the Schrodinger equation, so that
the corresponding HH wavefunction should obey the WdW equation. More
precisely, the wavefunction (35) will obey the PL version of the WdW equation
on $T(\Sigma)$.

Therefore there is no unique HH state in PFQG theory, and the question of how
to choose a HH state boils down to a choice of the manifolds $M^{\prime}$ and
$\Sigma$ and their triangulations. The question of whether the HH wavefunction
$\Psi_{0}$ for the vacuum manifold $M^{\prime}$ obeys the WdW equation,
remains open, although one can certainly enforce the WdW equation on
$\Psi_{0}$ by using the Lorentzian path integral and the conical topology for
the vacuum manifold, i.e. by constructing the Vilenkin wavefunction, see [5,
16] in the context of minisuperspace triangulations.

The problem of how to associate a QG state to the effective action defined as
a solution of the EA equation (23) is solved by using a closed manifold $M$ of
Fig. 7, so that the corresponding state is given by the HH/Vilenkin
wavefunction for the vacuum manifold $M^{\prime}$. This also resolves the
dilemma of choosing a closed or open manifold when constructing the effective
action, which was mentioned in the introduction. Hence we should always choose
a closed manifold of Fig. 7. However, the corresponding EA equations of motion
describe the quantum trajectories in the $\Sigma\times[0,t]$ component of that
manifold.

Note that the formula (40) suggests that $q_{\alpha}^{(H)}(t)$ can be
considered as a particular solution (or a very good approximation, see [17])
of the EA equations of motion in the $\Sigma\times[0,t]$ component the
spacetime associated with a manifold of Fig. 7. Hence the path integral
formula (40) provides a way to obtain the non-perturbative solutions of the EA
equations of motion. These solutions are important in the non-perturbative
regime, when $|L_{\epsilon}|\approx l_{P}$, since then one cannot use the
standard perturbative techniques.

Acknowledgements

Partially supported by by the Science Fund of the Republic of Serbia, grant
7745968, Quantum Gravity from Higher Gauge Theory 2021, QGHG-2021.

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