# Fermions on Quantum Geometry and Resolution of Doubling Problem

Cong Zhang  czhang(AT)fuw.edu.pl Faculty of Physics, University of Warsaw,
Pasteura 5, 02-093 Warsaw, Poland  Hongguang Liu
hongguang.liu(AT)gravity.fau.de Institut für Quantengravitation, Friedrich-
Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7/B2, 91058 Erlangen,
Germany  Muxin Han  hanm(AT)fau.edu Department of Physics, Florida Atlantic
University, 777 Glades Road, Boca Raton, FL 33431, USA  Institut für
Quantengravitation, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Staudtstr. 7/B2, 91058 Erlangen, Germany

###### Abstract

The fermion doubling problem has an important impact on quantum gravity, by
revealing the tension between fermion and the fundamental discreteness of
quantum spacetime. In this work, we discover that in Loop Quantum Gravity, the
quantum geometry involving superposition of states associated with lattice
refinements provides a resolution to the fermion doubling problem. We
construct and analyze the fermion propagator on the quantum geometry, and we
show that all fermion doubler modes are suppressed in the propagator. Our
result suggests that the superposition nature of quantum geometry should
resolve the tension between fermion and the fundamental discreteness, and
relate to the continuum limit of quantum gravity.

The physical world is made of gravity and matters. The quantum theory of
matters has been well-developed by the standard model, where gravity is,
however, classical. It has been suggested that a consistent theory of gravity
coupled to quantized matter should also have the gravitational field quantized
Page and Geilker (1981). On the other hand, Quantum Gravity (QG) also needs
matter couplings, not only because matter fields are dominant in the universe,
but also due to the universal gravity-matter interaction, which makes matters
serve as probes for empirically testing QG effects. The matter couplings
provide a toolbox for early studies of the ultraviolet (UV) behavior of QG
Deser and van Nieuwenhuizen (1974); ’t Hooft and Veltman (1974); Goroff and
Sagnotti (1986), and play important roles in cosmology, black holes,
asymptotic safety, QG phenomenology, etc Ashtekar  _et al._ (2021); Hawking
(1975); Donà  _et al._ (2014); Perez and Sudarsky (2019).

Loop Quantum Gravity (LQG) is a promising candidate for the background-
independent and non-perturbative QG. Matter couplings in LQG have been
extensively explored (see e.g. Ashtekar  _et al._ (1989); Thiemann (1998a);
Sahlmann and Thiemann (2006); Lewandowski and Zhang (2021); Bodendorfer  _et
al._ (2013); Bojowald and Das (2008); Domagala  _et al._ (2010); Kisielowski
and Lewandowski (2019); Bianchi  _et al._ (2013); Oriti and Ryan (2006);
Mansuroglu and Sahlmann (2021); Zhang and Ma (2011); Kamiński  _et al._
(2006)). A framework of coupling standard model to LQG is developed with the
interesting feature of the ultraviolet regularity Thiemann (1998a).

This work focuses on chiral fermions in LQG. Due to the fundamental
discreteness of LQG, the fermion coupling resembles the Lattice Field Theory
(LFT) to some extent, on any given discrete spacetime in LQG. It is well-known
that chiral fermions in LFT suffers the doubling problem, i.e. each fermion
results in $2^{m}$ fermion species on $m$-dimensional lattice Montvay and
Münster (1994); Nielsen and Ninomiya (1981). This fact leads to the suspicion
of the fermion doubling problem in LQG Barnett and Smolin (2015). However,
there also exists the opposite opinion arguing that the fermion doubling
problem may not exist in LQG due to the superposition of quantum geometries
Gambini and Pullin (2015); Bianchi  _et al._ (2013); Han and Rovelli (2013).
The confusion on fermion doubling has been long-standing in the LQG community
since the first paper on LQG-fermion in 1997. The similar issue should exist
in all QG approaches with discrete spacetimes. This issue is crucial because
it reveals the tension between fermion and the fundamental discreteness of
quantum spacetime, given that the fundamental discreteness is believed to be a
key feature of QG ’t Hooft (2016); Wheeler (1955); Hawking (1978).

A resolution of this tension is proposed in this paper from the perspective of
LQG. We construct and compute the chiral fermion propagator of LQG. Due to the
superposition of lattices given by the quantum geometry in LQG, the resulting
propagator averages the lattice fermion propagators over a sequence of lattice
refinements. We show that the average results in suppression of all fermion
doubling modes, while the physical mode is kept unchanged. Our result implies
that the LQG fermion propagator has the correct continuum limit, and suggests
the nature of superposition in quantum geometries be the key to resolve the
tension between fermion and the fundamental discreteness. Our result also
provides an interesting example for understanding the continuum limit of LQG.

_Fermion doubling and a resolution._ — Let us firstly consider LFT on 4d
Minkowski spacetime, where the time is continuous and the space is discretized
by a cubic lattice $\gamma$ Kogut and Susskind (1975); Creutz  _et al._
(2002). The propagator of chiral fermion has the following expression:

$\displaystyle G_{\omega,\vec{k}}(\gamma,a)=\frac{-\omega\mathbf{1}_{2\times
2}+\frac{1}{a}\sum_{I=1}^{3}\bm{\sigma}^{I}\sin\left(ak_{I}\right)}{\omega^{2}-\frac{1}{a^{2}}\sum_{I=1}^{3}\sin^{2}\left(ak_{I}\right)+i\epsilon}$
(1)

where $I=1,2,3$ labels the directions on $\gamma$. $a$ is the lattice spacing,
and $\epsilon$ is the Feynman regulator. $G_{\omega,\vec{k}}(\gamma,a)$ is a
$2\times 2$ matrix whose matrix indices are Weyl spinor indices, denoted by
$A,B=\pm$. The range of momentum $k^{I}$ is the fundamental Brillouin zone
(FBZ) $-\pi/a\leq k^{I}<\pi/a$. $G_{\omega,\vec{k}}(\gamma,a)$ has the
doubling problem: A physical pole at e.g. $(\omega,\vec{k})=(\omega,k,0,0)$
(satisfying $\omega=\frac{1}{a^{2}}\sin\left(ak\right)>0$) implies another
spurious doubler pole at $(\omega,\frac{\pi}{a}-k,0,0)$, so the fermion
species is doubled in each direction on the lattice. The fermion doubling
problem causes the trouble of the continuum limit of fermions on the lattice,
and is intractably linked to chirality by the Nielsen-Ninomiya no-go theorem
Nielsen and Ninomiya (1981).

LQG gives quantum geometry states as superposition of lattices, and it
indicates a resolution to the doubling problem by taking into account the
average of the propagators over the refinements of the lattice. Namely we
consider the following quantity, whose derivation from LQG is given in a
moment

$\displaystyle{G}_{\omega,\vec{k}}=\frac{1}{L}\sum_{n=1}^{L}\chi_{n,\vec{k}}G_{\omega,\vec{k}}\left(\gamma_{n},a_{n}\right),$
(2)

where $L$ is large and $\gamma_{n}$ are refinements of $\gamma_{1}=\gamma$ and
has the spacing $a_{n}=a_{1}/n$ (see FIG.1). We introduce the short-hand
notation $\mathrm{FBZ}(\gamma_{n})\equiv[-\pi/a_{n},\pi/a_{n})^{3}$ for the
FBZ on $\gamma_{n}$ with the spacing $a_{n}$, and we have
$\mathrm{FBZ}(\gamma_{n})\subset\mathrm{FBZ}(\gamma_{n+1})$.
$\chi_{n,\vec{k}}=1$ if $\vec{k}$ belongs to $\mathrm{FBZ}(\gamma_{n})$,
otherwise $\chi_{n,k}=0$.

Figure 1:  Lattice refinement of a cell of $\gamma_{1}$, to $\gamma_{2}$ and
$\gamma_{3}$. Along each direction on $\gamma_{n}$, the number of vertices
$N_{n}$ satisfies $N_{n}=nN_{1}$.

The assumptions of Nielsen-Ninomiya theorem is clearly violated by summing
over lattices. Although apparently (2) adds to ${G}_{\omega,\vec{k}}$ the
spurious doubler poles from all $\gamma_{n}$, the contribution from each
doubler pole are suppressed, thanks to the refinement and averaging. To
explain the mechanism, we firstly note that for $\epsilon>0$, the poles of
${G}_{\omega,\vec{k}}$ are away from real $(\omega,\vec{k})$-space, so
${G}_{\omega,\vec{k}}$ is finite for real $(\omega,\vec{k})$’s. Given an
$\omega_{o}>0$ such that the physical mode, e.g. $\vec{k}_{o}=(k_{o},0,0)$, is
in $\mathrm{FBZ}(\gamma_{1})$, and assuming $a_{1}$ is sufficiently small so
that $k_{o}\simeq\omega_{o}$, $(\omega_{o},\vec{k}_{o})$ is close to a pole of
every $G_{\omega,\vec{k}}\left(\gamma_{n},a_{n}\right)$ and gives equally
large contributions to all terms in (2), while the contributions are averaged
over $\\{\gamma_{n}\\}$. In contrast, each doubler mode
$\vec{k}_{n}=({\pi}/{a_{n}}-k_{o},0,0)$ associated to $\gamma_{n}$ only gives
a single large contribution to the $n$-th term in (2), because different terms
in (2) has different doubler poles and $\vec{k}_{n}$ is only close to one of
them. Then $G_{\omega_{o},\vec{k}_{n}}$ only receives the dominant
contribution from one term, and thus it is suppressed by the overall $1/L$.
The numerical experiment with $\omega_{o}=50,a_{1}=2^{-50},L=1.2\times
10^{5},\epsilon=10^{-4}$ shows that $|G^{++}_{\omega_{o},\vec{k}}|<4.2$ at the
doubler modes on all $\\{\gamma_{n}\\}_{n=1}^{L}$, while
$|G^{++}_{\omega_{o},\vec{k}_{o}}|\simeq 5\times 10^{5}$ (equals
$\omega_{o}/\epsilon$) at the physical mode. FIG.2 demonstrates that as $L$
goes large, $G_{\omega_{o},\vec{k}}$ remains large and constant at the
physical mode $k_{o}$, while $G_{\omega_{o},\vec{k}}$ is suppressed at the
doubler mode. The coincidence between $G_{\omega_{o},\vec{k}_{n}}$ and
$\omega_{o}/(\epsilon L)$ shown in FIG.2 indicates that
$G_{\omega_{o},\vec{k}}$ at the doubler mode indeed only receives the dominant
contribution from one term in the sum.

Figure 2: Log-log plots of $|G^{++}_{\omega_{o},\vec{k}_{o}}|$ at the physical
mode $k_{o}\simeq\omega_{o}$ (blue dots), and $|G^{++}_{\omega_{o},\vec{k}}|$
at a typical doubler mode on $\gamma_{n}$ with $n=2360$ (green circles). The
parameters are $\omega_{o}=50,a_{1}=2^{-50}$ $\epsilon=10^{-4}$. The red line
draws $\omega_{o}/(\epsilon L)$ as a function of $L$

In absence of doubling mode, $G_{\omega_{o},\vec{k}}$ is peaked at the
physical mode $\vec{k}=\vec{k}_{o}$. In any neighborhood of $\vec{k}_{o}$ and
with sufficiently large $L$, $G_{\omega_{o},\vec{k}}$ approximates the
continuum fermion propagator arbitrarily well, due to the well-known result
$\lim_{L\to\infty}\frac{1}{L}\sum_{n=1}^{L}f_{n}=\lim_{n\to\infty}f_{n}$ for
any sequence $\\{f_{n}\\}$.

We can generalize (2) and consider

$\displaystyle
G_{\omega,\vec{k}}^{(w)}=\sum_{n=1}^{L}|w_{n}|^{2}\chi_{n,\vec{k}}G_{\omega,\vec{k}}(\gamma_{n},a_{n}),$
(3)

with the generic weight $|w_{n}|^{2}$ satisfying
$\sum_{n=1}^{L}|w_{n}|^{2}=1$. $G_{\omega,\vec{k}}^{(w)}$ reduces to
$G_{\omega,\vec{k}}$ when the constant weight $|w_{n}|^{2}=1/L$ is chosen. The
mechanism of suppressing the doubler modes does not rely on the specific
choice of $w_{n}$. Thus our result also applies to $G_{\omega,\vec{k}}^{(w)}$
with generic $w_{n}$.

_LQG coupled to chiral fermion._ — In the following, we show that the
propagator (2) can be derived from LQG with fermion coupling. The 4d spacetime
topology is assumed to be $\mathbb{R}\times\Sigma$. A graph $\gamma$ in the
spatial slice $\Sigma$ consists of oriented edges $e$ and vertices $v$ as
sources and targets of the edges. The LQG Hilbert space $\mathcal{H}$ is given
by the direct sum over all graphs
$\mathcal{H}=\oplus_{\gamma}\mathcal{H}_{\gamma}/\mathcal{G}$, where
$\mathcal{H}_{\gamma}$ is given by
$\mathcal{H}_{\gamma}=\mathcal{H}_{\gamma}^{G}\otimes\mathcal{H}_{\gamma}^{F}$,
where $\mathcal{H}_{\gamma}^{G}$ is spanned by spin-networks on $\gamma$ with
nonzero spins, and $\mathcal{H}_{\gamma}^{F}=\otimes_{v}\mathcal{H}_{v}^{F}$
($\mathcal{H}_{v}^{F}\simeq\mathbb{C}^{4}$) is the Hilbert space of fermions.
$\oplus_{\gamma}\mathcal{H}_{\gamma}$ carries the representation of the
holonomy-flux algebra Lewandowski  _et al._ (2006); Ashtekar and Lewandowski
(1994) $[\hat{h}_{e},\hat{h}_{e^{\prime}}]=0$,
$[\hat{h}_{e},\hat{p}^{a}_{e^{\prime}}]=\beta\ell_{P}^{2}\delta_{e,e^{\prime}}(\bm{\sigma}^{a}/2)\hat{h}_{e}$,
$[\hat{p}^{a}_{e},\hat{p}^{b}_{e^{\prime}}]=i\beta\ell_{P}^{2}\delta_{e,e^{\prime}}\varepsilon^{abc}\hat{p}^{c}_{e}$
($\ell_{P}^{2}=8\pi G\hbar$ and $\beta$ is the Barbero-Immirzi parameter) and
the Weyl fermion oscillators
$[\hat{\zeta}_{v},\hat{\zeta}_{v^{\prime}}^{\dagger}]_{+}=\delta_{v,v^{\prime}}\mathbf{1}_{2\times
2}$. $\mathcal{G}$ are SU(2) gauge transformations, and $\mathcal{H}$ only
contains SU(2) gauge-invariant states. We assume the spatial slice
$\Sigma\simeq\mathbb{T}^{3}$ and the periodic boundary condition.

There is a physical Hamiltonian acting on $\mathcal{H}$ Thiemann (2020);
Giesel and Thiemann (2015, 2010)

$\displaystyle\hat{\bf H}=\oplus_{\gamma}\hat{\bf H}_{\gamma},\quad\hat{\bf
H}_{\gamma}=\hat{\bf
P}_{\gamma}(\hat{H}^{G}_{\gamma}+\hat{H}^{F}_{\gamma})\hat{\bf P}_{\gamma},$
(4)

where each $\hat{\bf P}_{\gamma}$ is the projection onto
$\mathcal{H}_{\gamma}$. $\hat{H}^{G}_{\gamma}$ is identical to the graph-
preserving LQG Hamiltonian constraint operator with unit lapse Giesel and
Thiemann (2007a, b). $\hat{H}^{F}_{\gamma}$ is the graph-preserving fermion
Hamiltonian Lewandowski and Zhang (2021); Thiemann (1998a); Sahlmann and
Thiemann (2006); Giesel and Thiemann (2007a). $\hat{H}^{F}_{\gamma}$ is
quadratic in $\hat{\zeta}_{v}$: $\hat{H}^{F}_{\gamma}/\hbar=\sum_{\left\langle
v,v^{\prime}\right\rangle}\hat{\zeta}_{v}^{\dagger}M_{v,v^{\prime}}(\hat{h},\hat{p})\hat{\zeta}_{v^{\prime}}$
where $M_{v,v^{\prime}}$ is a $2\times 2$ matrix with entries involving only
quantum-geometry operators. The detailed expression of
$\hat{H}^{G}_{\gamma}+\hat{H}^{F}_{\gamma}$ is reviewed in Appendix A. The
theory quantizes Einstein gravity coupling to the Weyl fermion and Gaussian
dust in the reduced phase space formulation (relational framework) Kuchar and
Torre (1991); Giesel and Thiemann (2015); Han and Liu (2020a); Dittrich
(2006); Thiemann (2006); Rovelli (2002, 1991). Semiclassically
$h_{e},p_{e}^{a}$ are Dirac observables being the holonomy and flux of the
Ashtekar-Barbero variables in the reference frame defined by the Gaussian
dust. As a promising aspect, the theory is free of the complications from the
quantum Hamiltonian and diffeomorphism constraints, because it quantizes the
reduced phase space, where both constraints are resolved at the classical
level. All quantities in the theory are Dirac observables from the start.

_Initial state._ — Based on the above framework of fermion coupling in LQG,
the fermion propagator are going to be constructed, and shown to recovers (3)
in the semiclassical approximation. As an important ingredient in the fermion
propagator, $|\Omega\rangle\in\mathcal{H}$ coupling quantum geometry and
fermions needs to be constructed as the LQG analog of the fermion ground state
on the semiclassical flat spacetime. Here we define $|\Omega\rangle$ to be an
entangled state being the superposition of
$|\Omega^{\prime}_{\gamma}\rangle\equiv\hat{\mathbf{P}}_{\gamma}|\psi_{\gamma}\rangle\otimes|\omega_{\gamma}\rangle\in\mathcal{H}_{\gamma}^{G}\otimes\mathcal{H}_{\gamma}^{F}$
over many different graphs $\gamma$, modulo SU(2) gauge transformations. On
each $\gamma$, the quantum geometry state $|\psi_{\gamma}\rangle$ is
Thiemann’s coherent state peaked at the flat spacetime geometry Thiemann and
Winkler (2001a, b). $|\psi_{\gamma}\rangle$ endows $\gamma$ with the
semiclassical flat geometry and vanishing extrinsic curvature.
$|\omega_{\gamma}\rangle$ is the normalized fermion ground state on the
semiclassical flat spacetime, and associates to the lowest energy level of the
_effective_ fermion Hamiltonian
$\hat{H}_{\gamma}^{F}(\psi_{\gamma})/\hbar=\|\psi_{\gamma}\|^{-2}\sum_{\left\langle
v,v^{\prime}\right\rangle}\hat{\zeta}_{v}^{\dagger}\langle\psi_{\gamma}|M_{v,v^{\prime}}(\hat{h},\hat{p})|\psi_{\gamma}\rangle\hat{\zeta}_{v^{\prime}}$.
$\hat{H}_{\gamma}^{F}(\psi_{\gamma})$ equals the usual fermion Hamiltonian on
the flat lattice plus $O(\ell_{P}^{2})$ corrections (see Appendix B for more
details, see also Sahlmann and Thiemann (2006) for the early study of the
similar idea).

Any smooth geometry admits many different discretizations based on different
graphs. We propose that the semiclassical state of the smooth geometry should
be a superposition of states on different graphs, where each state is
semiclassical on one graph and relate to the discretization of the smooth
geometry on the graph. Here, we consider the smooth geometry to be flat and
choose cubic graphs for the satisfactory semiclassical properties at the
discrete level Giesel and Thiemann (2007b); Flori and Thiemann (2008). We
consider a set of cubic graphs $\\{\gamma_{n}\\}_{n=1}^{L}$, where
$\gamma_{n}$ is a refinement of $\gamma_{1}$ by subdividing every cube into
$n^{3}$ cubes. $N_{n}=nN_{1}$ is the number of vertices on $\gamma_{n}$ in
every direction. $N_{1}$ is assumed to be even. We define
$|\Omega^{\prime}\rangle=\sum_{n=1}^{L}w_{n}|\Omega^{\prime}_{\gamma_{n}}\rangle$,
where
$|\Omega^{\prime}_{\gamma_{n}}\rangle=\hat{\mathbf{P}}_{\gamma_{n}}|\psi_{\gamma_{n}}\rangle\otimes|\omega_{\gamma_{n}}\rangle$,
and the gauge invariant projection

$\displaystyle|\Omega\rangle=\sum_{n=1}^{L}w_{n}|\Omega_{\gamma_{n}}\rangle,\
\ |\Omega_{\gamma_{n}}\rangle=\hat{\bf
P}_{\mathcal{G}}\left(\hat{\mathbf{P}}_{\gamma_{n}}|\psi_{\gamma_{n}}\rangle\otimes|\omega_{\gamma_{n}}\rangle\right).$

where $\hat{\bf P}_{\mathcal{G}}$ is the (group-averaging) projection onto
$\mathcal{H}$. $|\Omega_{\gamma_{n}}\rangle$ are mutually orthogonal and
$\|\Omega_{\gamma_{n}}\|=1$. $L$ is finite and large. The weight $w_{n}$
satisfies $\sum_{n=1}^{L}|w_{n}|^{2}=1$ so that $\|\Omega\|=1$. The discrete
geometries on $\\{\gamma_{n}\\}_{n=1}^{L}$ given by
$\\{|\psi_{\gamma_{n}}\rangle\\}_{n=1}^{L}$ are discretizations of the smooth
flat geometry, and give the lattice spacings $\\{a_{n}\\}_{n=1}^{L}$
satisfying $a_{n}=a_{1}/n$. Indeed, the discrete geometry on $\gamma_{n}$ are
implied by the expectation values
$\langle\psi_{\gamma_{n}}|\hat{h}_{e}|\psi_{\gamma_{n}}\rangle=\mathbf{1}_{2\times
2}+O(\ell_{P}^{2})$ and
$\langle\psi_{\gamma_{n}}|\hat{p}^{a}_{e(I)}|\psi_{\gamma_{n}}\rangle=a^{2}_{n}\delta^{a}_{I}+O(\ell_{P}^{2})$,
where $e(I)$ is the edge along the $I$-direction. The total spatial volume
$V_{\rm tot}=a_{n}N_{n}$ is $n$-independent.

_Fermion propagator._ — We need the local field operator $\hat{\Psi}(v)$ in
order to construct the fermion propagator. However, classically the fermion
field $\Psi$ and $\zeta$ are not the same but related by
$\zeta=\sqrt[4]{\det(q)}\Psi$, where $\sqrt{\det(q)}$ is the spatial volume
density Thiemann (1998b, a). Motivated by the classical relation, we define
the fermion field operator by
$\hat{\Psi}(v)=\hat{\zeta}_{v}\hat{V}_{v}^{-\frac{1}{2}}$, where
$\hat{V}_{v}^{-\frac{1}{2}}$ is the square-root inverse volume (see Appendix A
or Giesel and Thiemann (2006); Yang and Ma (2016)). We define the Heisenberg
operator $\hat{\Psi}(\tau,v):=e^{\frac{i}{\hbar}\hat{\bf H}(\tau-
T_{i})}\hat{\Psi}(v)e^{-\frac{i}{\hbar}\hat{\bf H}(\tau-T_{i})}$ and similarly
for $\hat{\Psi}^{\dagger}(\tau,v)$, where $T_{i}$ is the initial time. We
prepare the initial state $|\Omega\rangle$ at $T_{i}$ and define the fermion
propagator by

$\displaystyle\mathscr{G}^{AB}(\tau_{1},\tau_{2};v_{1},v_{2})=\langle\Omega^{\prime}|\mathcal{T}[\hat{\Psi}^{A}(\tau_{1},v_{1})\hat{\Psi}^{B}{}^{\dagger}(\tau_{2},v_{2})]|\Omega\rangle,$

where $\mathcal{T}$ is the time-ordering. Here $v_{1},v_{2}$ are vertices in
$\gamma_{1}$, and thus they belong to all the refinements $\gamma_{n}$.
$\mathscr{G}^{AB}$ is not SU(2) gauge invariant, as the usual situation in
gauge theories. An example of the gauge invariant observable is
$\mathscr{G}_{h}(\tau_{1},\tau_{2};v_{1},v_{2})=\langle\Omega|(\hat{h}_{c})_{BA}\mathcal{T}[\hat{\Psi}^{A}(\tau_{1},v_{1})\hat{\Psi}^{B}{}^{\dagger}(\tau_{2},v_{2})]|\Omega\rangle$,
where $\hat{h}_{c}$ is the holonomy operator along a path $c\subset\gamma_{1}$
connecting $v_{1},v_{2}$ 111In $\mathscr{G}_{h}$, $\langle\Omega|$ can be
replaced by $\langle\Omega^{\prime}|$ for free, because the operator is gauge
invariant and
$\hat{\mathbf{P}}_{\mathcal{G}}^{2}=\hat{\mathbf{P}}_{\mathcal{G}}$.. In the
following, we proceed to compute $\mathscr{G}^{AB}$, then we show
$\mathscr{G}_{h}\simeq\mathscr{G}^{AB}\delta_{AB}$ up to $O(\ell_{P}^{2})$.

Since $\hat{\bf H}$ is a direct sum and $|\Omega_{\gamma_{n}}\rangle$ are
mutually orthonormal, $\mathscr{G}^{AB}$ is a sum of the fermion propagators
on graphs $\\{\gamma_{n}\\}$

$\displaystyle\mathscr{G}^{AB}(\tau_{1},\tau_{2};v_{1},v_{2})$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{L}|w_{n}|^{2}\mathscr{G}_{n}^{AB}(\tau_{1},\tau_{2};v_{1},v_{2})$
(5) $\displaystyle\mathscr{G}_{n}^{AB}(\tau_{1},\tau_{2};v_{1},v_{2})$
$\displaystyle=$
$\displaystyle\langle\Omega^{\prime}_{\gamma_{n}}|\mathcal{T}[\hat{\Psi}^{A}(\tau_{1},v_{1})\hat{\Psi}^{B}{}^{\dagger}(\tau_{2},v_{2})]|\Omega_{\gamma_{n}}\rangle$

We compute $\mathscr{G}_{n}^{AB}$ with the time-order $\tau_{1}>\tau_{2}$,
while the computation for $\tau_{1}<\tau_{2}$ is similar. We apply the
coherent-state path integral method by discretizing the time-evolution
operator $e^{-\frac{i}{\hbar}\hat{\bf H}_{\gamma}(\tau-
T_{i})}\simeq[1{-\frac{i}{\hbar}\hat{\bf H}_{\gamma}\delta\tau}]^{j_{\tau}}$
with $j_{\tau}\delta\tau={\tau-T_{i}}$ and large $j_{\tau}$, and inserting at
each step the over-completeness relation of the coherent state
$\int\mathrm{d}Z_{j}(\gamma_{n})|Z_{j}(\gamma_{n})\rangle\langle{Z_{j}(\gamma_{n})}|=1$
($j=0,\cdots,j_{\tau}$) with
$|Z(\gamma_{n})\rangle=|\psi_{(h,p)(\gamma_{n})}\rangle\otimes|\phi_{\zeta({\gamma_{n}})}\rangle$,
where $|\psi_{(h,p)(\gamma_{n})}\rangle$ is Thiemann’s coherent state of
quantum geometry, and $|\phi_{\zeta({\gamma_{n}})}\rangle$ satisfying
$\hat{\zeta}_{v}|\phi_{\zeta({\gamma_{n}})}\rangle=\zeta_{v}|\phi_{\zeta({\gamma_{n}})}\rangle$
is the standard coherent state of the fermion oscillators. $Z(\gamma_{n})$ is
a short-hand notation of the data $\\{h_{e},p_{e}^{a},\zeta_{v}\\}_{e,v}$
where the coherent state is peaked.
$\mathrm{d}Z(\gamma_{n})=\mathrm{d}g(\gamma_{n})\prod_{v,A}\mathrm{d}\zeta^{A}_{v}{}^{*}\mathrm{d}\zeta^{A}_{v}$
and $\mathrm{d}g(\gamma_{n})$ denotes the measure of
$\\{h_{e},p_{e}^{a}\\}_{e}$ Thiemann and Winkler (2001a). Following the
standard derivation, we obtain the path integral:

$\displaystyle\mathscr{G}_{n}^{AB}$
$\displaystyle=\int\prod_{j=0}^{N}\mathrm{d}Z_{j}(\gamma_{n})\,e^{S^{G}_{\gamma_{n}}(h,p)+S_{\gamma_{n}}^{F}(h,p,\zeta,\zeta^{*})}$
$\displaystyle(V^{-\frac{1}{2}}_{v_{1}}{\zeta}_{v_{1}}^{A})_{j_{1}}(V^{-\frac{1}{2}}_{v_{2}}{\zeta}^{B*}_{v_{2}})_{j_{2}}\langle\Omega^{\prime}_{\gamma_{n}}|Z_{N}(\gamma_{n})\rangle\langle
Z_{0}(\gamma_{n})|\Omega_{\gamma_{n}}\rangle$

where $j_{1,2}=j_{\tau_{1,2}}=(\tau_{1,2}-T_{i})/\delta\tau$ and $N=2j_{1}$.
$\langle\Omega^{\prime}_{\gamma_{n}}|Z_{N}(\gamma_{n})\rangle$ and $\langle
Z_{0}(\gamma_{n})|\Omega_{\gamma_{n}}\rangle$ are linear in $\zeta_{v}$ and
$\zeta_{v}^{*}$ respectively , and the fermion action $S^{F}_{\gamma_{n}}$ is
quadratic in $\zeta_{v}$

$\displaystyle S_{\gamma_{n}}^{F}$ $\displaystyle=$
$\displaystyle\sum_{j,j^{\prime},v,v^{\prime}}\zeta^{\dagger}_{v,j}\mathscr{H}_{v,j;v^{\prime},j^{\prime}}(h,p)\zeta_{v^{\prime},j^{\prime}}$
$\displaystyle\mathscr{H}_{v,j;v^{\prime},j^{\prime}}$ $\displaystyle=$
$\displaystyle\delta_{v^{\prime},v}\delta_{j,j^{\prime}+1}-\delta_{j^{\prime},j}\delta_{v^{\prime},v}$
$\displaystyle-i\rho_{j}\delta_{j,j^{\prime}+1}{\delta\tau}\left\langle\psi_{g_{j}(\gamma_{n})}\left|M_{v,v^{\prime}}(\hat{h},\hat{p})\right|\psi_{g_{j^{\prime}}(\gamma_{n})}\right\rangle,$

where $\rho_{j}=1$ or $-1$ at instances before or after $\tau_{1}$. We perform
the Gaussian integral of the fermion and obtain the following expression:

$\displaystyle\\!\\!\\!\\!\mathscr{G}_{n}^{AB}=\int\limits_{\mathcal{G}}\mathrm{d}u\int\prod_{j=0}^{N}\mathrm{d}g_{j}(\gamma_{n})\,e^{S^{G}_{\gamma_{n}}}(V^{-\frac{1}{2}}_{v_{1}})_{j_{1}}(V^{-\frac{1}{2}}_{v_{2}})_{j_{2}}\,\mathcal{I}^{AB}_{n}$
(7)

where $\mathcal{I}^{AB}_{n}=\mathcal{I}^{AB}_{n}(p,h,u)$, resulting from the
fermion Gaussian integral, is independent of $\hbar$. The integral over SU(2)
gauge transformations $u$ comes from $\mathbf{P}_{\mathcal{G}}$ in
$|\Omega_{\gamma_{n}}\rangle$. The effective action of geometry
$S^{G}_{\gamma_{n}}$ scales as $1/\ell_{P}^{2}$ as $\ell_{P}\to 0$ Han and Liu
(2020b) (see also Appendix C). So the stationary phase approximation can be
employed to study the semiclassical approximation of (7). It is shown in Han
and Liu (2020b, a) that the equations of motion (EOMs) $\delta
S^{G}_{\gamma_{n}}=0$ reproduce the classical Hamilton’s equation of holonomy
and fluxes ${\mathrm{d}h_{e}}/{\mathrm{d}\tau}=\\{h_{e},H^{G}\\}$,
${\mathrm{d}p^{a}_{e}}/{\mathrm{d}\tau}=\\{p^{a}_{e},H^{G}\\}$, where $H^{G}$
is the discrete gravity Hamiltonian on $\gamma_{n}$ with unit lapse and
vanishing shift. The only solution $(\mathring{h},\mathring{p},\mathring{u})$
satisfying the EOMs and compatible to the initial state
$|\Omega_{\gamma_{n}}\rangle$ is $\mathring{u}=\mathbf{1}$ and the flat
spacetime geometry, where $\mathring{h}_{e,j}=\mathbf{1}_{2\times 2}$ and
$\mathring{p}_{e(I),j}^{a}=a_{n}^{2}\delta^{a}_{I}$, i.e. the lattice geometry
is constantly flat with spacing $a_{n}$ at all time. Eq.(7) has the following
semiclassical approximation

$\displaystyle\mathscr{G}_{n}^{AB}$ $\displaystyle=$ $\displaystyle
a_{n}^{-3}\mathcal{I}_{n}^{AB}(\mathring{p},\mathring{h},\mathring{u})\int\limits_{\mathcal{G}}\mathrm{d}u\int\prod_{j=0}^{N}\\!\mathrm{d}g_{j}(\gamma_{n})\,e^{S^{G}_{\gamma_{n}}}\left[1+O(\ell_{P}^{2})\right]$
(8) $\displaystyle=$
$\displaystyle\,a_{n}^{-3}\mathcal{I}_{n}^{AB}(\mathring{p},\mathring{h},\mathring{u})\left[1+O(\ell_{P}^{2})\right].$

Here the integral removes all fermions degrees of freedom in
$\mathscr{G}_{n}$, so it equals
$\langle\tilde{\psi}_{\gamma_{n}}|\tilde{\psi}_{\gamma_{n}}\rangle$, where
$|\tilde{\psi}_{\gamma_{n}}\rangle\equiv\mathbf{P}_{\mathcal{G}}\mathbf{P}_{\gamma_{n}}|\psi_{\gamma_{n}}\rangle$.
$\langle\tilde{\psi}_{\gamma_{n}}|\tilde{\psi}_{\gamma_{n}}\rangle=1+O(\ell_{P}^{2})$
is implied by $\|\Omega_{\gamma_{n}}\|=1$ and $\|\omega_{\gamma_{n}}\|=1$.
More details about the semiclassical analysis of $\mathscr{G}_{n}^{AB}$ can be
found in Appendix C.

The gauge invariant observable $\mathscr{G}_{h}$ inserts an $({h}_{c})_{AB}$
in the above path integral. Then the stationary phase approximation is (8)
contracted by $(\mathring{h}_{c})_{AB}=\delta_{AB}$.

The geometry is freezed to be flat in (8), and $I_{n}^{AB}$ reduces to the
standard fermion propagator on the flat lattice with spacing $a_{n}$:

$\displaystyle\mathscr{G}_{n}$ $\displaystyle=$ $\displaystyle\frac{1}{V_{\rm
tot}}\sum_{\vec{k}\in\text{FBZ}(\gamma_{n})}\int_{\mathbb{R}}\frac{\mathrm{d}\omega}{2\pi
i}\,e^{-i\omega(\tau_{1}-\tau_{2})+ia_{n}\vec{k}\cdot\left(\vec{v}_{1}-\vec{v}_{2}\right)}$
(9) $\displaystyle
G_{\omega,\vec{k}}(\gamma_{n},a_{n})\left[1+O(\ell_{P}^{2})\right],$

where the continuous limit of time has been taken. The lattice vertices
$\vec{v}_{i}\in\mathbb{Z}^{3}$ and $\vec{v}_{i}=\vec{v}_{i}+N_{n}$ ($i=1,2$).
The lattice Fourier mode ${k}^{I}\in 2\pi\mathbb{Z}/(N_{1}a_{1})$ is summed
over the FBZ on $\gamma_{n}$ with the spacing $a_{n}$.

Insert (9) in (5), and extend the sum of $\vec{k}$ to FBZ on the finest
lattice ($\text{FBZ}(\gamma_{n})\subset\text{FBZ}(\gamma_{L})$ for all $n<L$).
Notice that $\vec{x}_{1}-\vec{x}_{2}\equiv a_{n}(\vec{v}_{1}-\vec{v}_{2})$ is
$n$-independent. The LQG fermion propagator $\mathscr{G}$ becomes

$\displaystyle\mathscr{G}$ $\displaystyle=$ $\displaystyle\frac{1}{V_{\rm
tot}}\sum_{k^{I}\in\text{FBZ}(\gamma_{L})}\int_{\mathbb{R}}\frac{\mathrm{d}\omega}{2\pi
i}\,e^{-i\omega(\tau_{1}-\tau_{2})+i\vec{k}\cdot\left(\vec{x}_{1}-\vec{x}_{2}\right)}\mathscr{G}_{\omega,\vec{k}},$
$\displaystyle\mathscr{G}_{\omega,\vec{k}}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{L}|{w}_{n}|^{2}\chi_{n,\vec{k}}G_{\omega,\vec{k}}(\gamma_{n},a_{n})\left[1+O(\ell_{P}^{2})\right].$
(10)

Up to $O(\ell_{P}^{2})$ corrections, $\mathscr{G}_{\omega,k}$ recovers (3) in
which the fermion doublers are suppressed for large $L$.

_Discussion._ — Our analysis shows that the quantum geometry as the
superposition of states with different lattices results in that the fermion
propagator from LQG averages the fermion propagators on the lattices. In the
resulting fermion propagator, all doubler modes on the lattices are suppressed
by the average, while the physical mode are kept unchanged. Our result
indicates that the fermion in LQG is free of the doubling problem. It also
suggests that the superposition nature of quantum geometry should be the key
to resolve the tension between fermion and the fundamental discreteness in QG.

The fact that the superposition of lattices brings the fermion propagator
close to the continuum limit suggests that the continuum limit of the full LQG
should also relate to the superposition of lattices. This is similar to the
approach of group field theory Finocchiaro and Oriti (2021).

Interestingly, the propagator $\mathscr{G}_{\omega,\vec{k}}$ suggests that the
quantum geometry provide a soft UV cut-off to fermions. Let us scale $\vec{k}$
large such that $\vec{k}$ is outside $\mathrm{FBZ}(\gamma_{m})$ for certain
$m$, then all terms with $n\leq m$ in (10) vanish because of
$\chi_{n,\vec{k}}$. Hence, as we scale $\vec{k}$ large,
$|\mathscr{G}^{AB}_{\omega,\vec{k}}|$ is suppressed because less and less
terms survive in the sum. This result is consistent with the expectation that
QG should regularize the UV behavior of matter fields.

We have set the upper bound $L$ of the sum to be finite, so that the lattice
refinement gives maximally $(LN_{1})^{3}$ vertices. $L$ actually correlates to
$\ell_{P}$, and $L$ has to be finite as far as $\ell_{P}$ is finite. The
technical reason is the following: The $O(\ell_{P}^{2})$ correction in
$\mathscr{G}_{n}$ may become non-negligible, when the spacing becomes
comparable to the Planck length, i.e. $a_{n}\sim\ell_{P}$ Giesel and Thiemann
(2007b); Zhang  _et al._ (2022). Fixing $V_{\rm tot}$, $L$ has to be bounded
by $L_{0}=V_{\rm tot}^{1/3}/(N_{1}\ell_{P})$ in order that the semiclassical
approximation can be applied to all $\mathscr{G}_{n}$ in the sum. Therefore,
the refinement limit $L\to\infty$ can only be taken together with the
semiclassical limit $\ell_{P}\to 0$. Further investigation is needed to go
beyond the semiclassical approximation when computing $\mathscr{G}_{n}$. Then
one may consider $L\to\infty$ without the semiclassical limit.

Our result shares the similarity with the random LFT Christ  _et al._ (1982);
Pang and Ren (1986); Perantonis and Wheater (1988); Griffin and Kieu (1993) by
the average over lattices. But a key difference is that we sum over lattices
with different numbers of vertices relating to the lattice refinement, while
the random lattice often fixes the number of vertices. It is interesting to
further explore the relation to LFT, in order to see the impact from quantum
geometry to high energy particle physics. Our computation should extend to
n-point functions and further probe the UV behavior. It is also important to
study the chiral anomaly on quantum geometry, which is a research undergoing.

###### Acknowledgements.

M.H. acknowledges Chen-Hung Hsiao for discussions at early stage of this work.
M.H. receives support from the National Science Foundation through grants
PHY-1912278 and PHY-2207763. M.H. also acknowledges funding provided by the
Alexander von Humboldt Foundation for his visit at the Friedrich-Alexander-
Universität Erlangen-Nürnberg. C. Z. acknowledges the support by the Polish
Narodowe Centrum Nauki, Grant No. 2018/30/Q/ST2/00811, and NSFC with Grants
No. 11961131013 and No. 11875006.

##  Appendix A A. Hamiltonian opertors

In LQG, to promote the classical Hamiltonian of gravity $H^{G}\equiv
E-(1+\beta^{2})L$ to the operator, we need to regularize $H^{G}$ based on a
graph $\gamma$, and express $H^{G}$ in terms of the basic variables $h_{e}$
and $p_{e}^{a}$. Consider $\gamma$ to be a cubic lattice, and denote by $e(I)$
the edge along the $I$-direction. The quantization of $H^{G}\equiv
E-(1+\beta^{2})L$ gives the graph-preserving operator
$\hat{H}^{G}_{\gamma}\equiv\hat{E}_{\gamma}-(1+\beta^{2})\hat{L}_{\gamma}$,
where

$\displaystyle\hat{E}_{\gamma}$
$\displaystyle=\frac{-1}{4i\kappa\beta\ell_{P}^{2}}\sum_{v\in
V(\gamma)}\sum_{e(I),e(J),e(K)\text{ at
}v}\epsilon^{IJK}\tr(h_{\alpha_{IJ}}h_{e(K)}^{-1}[h_{e(K)},\hat{V}_{v}]).$
(11) $\displaystyle\hat{L}_{\gamma}$
$\displaystyle=-\frac{4\kappa}{i\beta^{7}\ell_{P}^{10}}\sum_{v\in
V(\gamma)}\sum_{e(I),e(J),e(K)\text{ at
}v}\varepsilon^{IJK}\tr([h_{e(I)},[\hat{V}_{\gamma},\hat{E}_{\gamma}]]h_{e(I)}^{-1}[h_{e(J)},[\hat{V}_{\gamma},\hat{E}_{\gamma}]]h_{e_{J}}^{-1}[h_{e(K)},\hat{V}_{v}]h_{e(K)}^{-1})$

$\alpha_{IJ}$ is a minimal loop in $\gamma$ containing edges $e(I)$ and
$e(J)$. $\hat{V}_{v}$ is the volume operator associated to the vertex $v$, and
$\hat{V}_{\gamma}=\sum_{v\in V(\gamma)}\hat{V}_{v}$ is the total volume
operator.

The fermion Hamiltonian operator is given by
$\hat{H}_{\gamma}^{F}=\sum_{\langle
v,v^{\prime}\rangle}\hat{\zeta}_{v}^{\dagger}M_{v,v^{\prime}}(\hat{h},\hat{p})\hat{\zeta}_{v^{\prime}}$,
where $M_{v,v^{\prime}}(\hat{h},\hat{p})$ reads

$\displaystyle
M_{v,v^{\prime}}(\hat{h},\hat{p})=\delta_{v,v^{\prime}}\hat{V}_{v}^{-\frac{1}{2}}\left(-\frac{(1+3\beta^{2})}{4\beta}\sum_{e\text{
at
}v}\hat{p}_{e}^{a}{\bm{\sigma}}_{a}-\frac{\kappa}{i\ell_{P}^{2}\beta}\left[\hat{E}_{\gamma},\hat{V}_{v}\right]\mathbf{1}_{2\times
2}\right)\hat{V}_{v}^{-\frac{1}{2}}$ (12)
$\displaystyle-\frac{\beta-i}{4}\left(\sum_{e\text{ at
}v}\delta_{v^{\prime},t_{e}}\hat{V}_{v}^{-\frac{1}{2}}{\bm{\sigma}}_{a}\hat{h}_{e}\hat{p}_{e}^{a}\hat{V}_{v}^{-\frac{1}{2}}\right)-\frac{\beta+i}{4}\left(\sum_{e^{\prime}\text{
at
}v^{\prime}}\delta_{v,t_{e^{\prime}}}\hat{V}_{v}^{-\frac{1}{2}}\hat{p}_{e^{\prime}}^{a}h_{e^{\prime}}^{-1}{\bm{\sigma}}_{a}\hat{V}_{v}^{-\frac{1}{2}}\right)$

where $t_{e}$ denotes the target of the edge $e$. Here
$\hat{V}_{v}^{-\frac{1}{2}}$, densely defined on $\mathcal{H}$, is not the
inverse of $\sqrt{\hat{V}_{v}}$. Indeed, $\sqrt{\hat{V}_{v}}$ takes $0$ as its
eigenvalue, so its inverse is not densely defined.
$\hat{V}_{v}^{-\frac{1}{2}}$ is obtained by regularizing $\det(e)/\det(q)$ in
a neighorhood at $v$ ($\det(e)$ is the determinant of cotriad) Giesel and
Thiemann (2006); Yang and Ma (2016). The expression of
$\hat{V}_{v}^{-\frac{1}{2}}$ reads

$\displaystyle\hat{V}_{v}^{-\frac{1}{2}}=-\frac{16}{3}\left(\frac{1}{i\ell_{P}^{2}\beta}\right)^{3}\sum_{e(I),e(J),e(K)\text{
at
}v}\epsilon^{IJK}\tr\left([\hat{h}_{e(I)},\hat{V}_{v}^{1/2}]\hat{h}_{e(I)}^{-1}[\hat{h}_{e(J)},\hat{V}_{v}^{1/2}]\hat{h}_{e(J)}^{-1}[\hat{h}_{e(K)},\hat{V}_{v}^{1/2}]h_{e(K)}^{-1}\right).$
(13)

Semiclassically $\hat{V}_{v}^{-\frac{1}{2}}$ is not always positive, and the
sign relates to the orientation $\mathrm{sgn}(\det(e))$. However, the sign is
canceled in the classical limit of $M_{v,v^{\prime}}$, since every term
contains a pair of $\hat{V}_{v}^{-\frac{1}{2}}$.

##  Appendix B B. Initial state

Given the cubic graph $\gamma_{n}$, Thiemann’s coherent state
$|\psi_{g(\gamma_{n})}\rangle$ is peaked at the phase space point
$g(\gamma_{n})\equiv\\{h_{e},p_{e}^{a}\\}_{e\subset\gamma_{n}}$ Thiemann and
Winkler (2001a, b). $h_{e}$ is the SU(2) holonomy of Ashtekar-Barbero
connection $\Gamma^{a}_{I}+\beta K^{a}_{I}$ ($\Gamma$ is the spin connection
and $K$ is the extrinsic curvature), and $p^{a}_{e}$ smears the densitized
triad $\sqrt{\det(q)}\bm{e}^{I}_{a}$ on the 2-face dual to $e$. The
expectation values
$\langle\psi_{g(\gamma_{n})}|\hat{h}_{e}|\psi_{g(\gamma_{n})}\rangle=h_{e}+O(\ell_{P}^{2})$
and
$\langle\psi_{g(\gamma_{n})}|\hat{p}^{a}_{e}|\psi_{g(\gamma_{n})}\rangle=p_{e}^{a}+O(\ell_{P}^{2})$
endow $\gamma_{n}$ with semiclassical internal and external geometries.
$|\Omega\rangle\in\mathcal{H}$ relates to
$|\psi_{\gamma_{n}}\rangle\equiv|\psi_{g_{\rm flat}(\gamma_{n})}\rangle$ (up
to normalization) peeked at the flat geometry $g_{\rm
flat}(\gamma_{n})=\\{h_{e(I)}=\mathbf{1}_{2\times
2},\,p_{e(I)}^{a}=p_{1}\delta^{a}_{I}\\}_{e(I)}$ where $e(I)$ denotes the edge
along $I$-direction ($I=1,2,3$). $|\psi_{\gamma_{n}}\rangle$ endow
$\gamma_{n}$ with the vanishing extrinsic curvature and the flat lattice
geometry with constant lattice spacing $a_{n}=\sqrt{p_{n}}$. The effective
fermion Hamiltonian $\hat{H}_{\gamma_{n}}^{F}(\psi_{\gamma_{n}})$ equals the
standard fermion Hamiltonian $\hat{\bm{H}}_{\gamma_{n}}^{F}$ on the flat
lattice plus $O(\ell_{P}^{2})$ corrections, because
$\langle\psi_{\gamma_{n}}|M_{v,v^{\prime}}(\hat{h},\hat{p})|\psi_{\gamma_{n}}\rangle\|\psi_{\gamma_{n}}\|^{-2}$
equals to $M_{v,v^{\prime}}({h},{p})$ evaluated at
${h}_{e}=\mathbf{1}_{2\times 2},\ {p}_{e(I)}^{a}=p_{1}\delta^{a}_{I}$ up to
$O(\ell_{P}^{2})$ Giesel and Thiemann (2007b). Namely,

$\displaystyle\hat{H}_{\gamma_{n}}^{F}(\psi_{\gamma_{n}})=\hat{\bm{H}}_{\gamma_{n}}^{F}\left[1+O(\ell_{P}^{2})\right],\quad\hat{\bm{H}}_{\gamma_{n}}^{F}=\frac{i\hbar}{2a_{n}}\sum_{v\in\gamma_{n}}\sum_{I=1}^{3}\left(\hat{\zeta}_{v}^{\dagger}{\bm{\sigma}}^{I}\hat{\zeta}_{v+\delta_{I}}-\hat{\zeta}_{v+\delta_{I}}^{\dagger}{\bm{\sigma}}^{I}\hat{\zeta}_{v}\right).$
(14)

It is standard to diagonalize $\hat{\bm{H}}_{\gamma_{n}}^{F}$ by Fourier
transformation
$\hat{\zeta}_{\vec{k}}=\sqrt{\frac{1}{N_{n}^{3}}}\sum_{v\in\gamma_{n}}\hat{\zeta}_{v}e^{-ia_{n}\vec{k}\cdot\vec{v}}$
($\vec{v}\in\mathbb{Z}^{3}$ and $\vec{v}=\vec{v}+N_{n}$):

$\displaystyle\hat{\bm{H}}_{\gamma_{n}}^{F}=\frac{-\hbar}{a_{n}}\sum_{k\in\mathrm{FBZ}(\gamma_{n})}\hat{\zeta}_{k}^{\dagger}\left[\sum_{I=1}^{3}\sin(a_{n}k_{I})\bm{\sigma}^{I}\right]\hat{\zeta}_{k}=\frac{-\hbar}{a_{n}}\sum_{k\in\mathrm{FBZ}(\gamma_{n})}\hat{\zeta}_{k}^{\dagger}\Theta(k)^{\dagger}\left(\begin{array}[]{cc}-\mathfrak{s}({k})&0\\\
0&\mathfrak{s}({k})\end{array}\right)\Theta({k})\hat{\zeta}_{k}$ (17)

where $\mathfrak{s}({k})=\sqrt{\sum_{I=1}^{3}\sin^{2}(a_{n}k_{I})}$, and
$\Theta(k)$ is the unitary matrix diagonalizing
$\sum_{I=1}^{3}\sin(a_{n}k_{I})\bm{\sigma}^{I}$. The ground state
$|\omega_{\gamma_{n}}\rangle$ associates to the negative zero-point energy
$-\mathcal{E}_{0}\equiv-\frac{\hbar}{a_{n}}\sum_{k\in\mathrm{FBZ}(\gamma_{n})}\mathfrak{s}({k})[1+O(\ell_{P}^{2})]$.

The expectation value of physical Hamiltonian $\hat{\bf H}$ at
$|\psi_{\gamma_{n}}\rangle\otimes|\omega_{\gamma_{n}}\rangle$ is negative and
equals $-\mathcal{E}_{0}$. The existence of zero-point energy is because the
fermion Hamiltonian $H_{\gamma}^{F}$ in $\hat{\bf H}$ is not normal ordered.
It does not make sense to normal order $H_{\gamma}^{F}$ given that $\hat{\bf
H}$ is background independent. Interestingly, we have the semiclassical
relation ${\bf H}+E_{\rm dust}=0$ where $E_{\rm dust}$ is the energy of
Gaussian dust Giesel and Thiemann (2015); Kuchar and Torre (1991). This
relation is one of the starting point of the reduced phase space quantization.
Viewing ${\bf H}$ as the expectation value determines $E_{\rm
dust}=\mathcal{E}_{0}$. Then $\langle\psi_{\gamma_{n}}|\hat{\bf H}+E_{\rm
dust}|\psi_{\gamma_{n}}\rangle=\hat{H}_{\gamma_{n}}^{F}(\psi_{\gamma_{n}})+E_{\rm
dust}$ cancels the fermion zero-point energy and gives the normal ordered
Hamiltonian on the flat spacetime up to $O(\ell_{P}^{2})$.

##  Appendix C C. Semiclassical analysis

Figure 3: A $N$-side polygon path starting from $T_{i}$, passing $\tau_{2}$,
arriving at $\tau_{1}$ and returning $T_{i}$. $|Z_{0}(\gamma_{n})\rangle$ and
$|Z_{N}(\gamma_{n})\rangle$ take the inner products with
$|\Omega_{\gamma_{n}}\rangle$ and $|\Omega^{\prime}_{\gamma_{n}}\rangle$ in
(20) and (Fermions on Quantum Geometry and Resolution of Doubling Problem).

Let us consider the gauge invariant observable
$\mathscr{G}_{h}(\tau_{1},\tau_{2},v_{1},v_{2})$. Similar to (5), we have

$\displaystyle\mathscr{G}_{h}(\tau_{1},\tau_{2};v_{1},v_{2})$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{L}|w_{n}|^{2}\mathscr{G}_{h,n}(\tau_{1},\tau_{2};v_{1},v_{2}),$
(18) $\displaystyle\mathscr{G}_{h,n}(\tau_{1},\tau_{2};v_{1},v_{2})$
$\displaystyle=$
$\displaystyle\langle\Omega_{\gamma_{n}}|(\hat{h}_{c})_{BA}\mathcal{T}[\hat{\Psi}^{A}(\tau_{1},v_{1})\hat{\Psi}^{B}{}^{\dagger}(\tau_{2},v_{2})]|\Omega_{\gamma_{n}}\rangle.$
(19)

We can obtain the path integral formula of
$\mathscr{G}_{h,n}(\tau_{1},\tau_{2};v_{1},v_{2})$ on $\gamma_{n}$ as

$\displaystyle\mathscr{G}_{h,n}(\tau_{1},v_{1},\tau_{2},v_{2})$
$\displaystyle=\int\limits_{\mathcal{G}}\differential
u\int\prod_{j=0}^{N}\mathrm{d}g_{j}(\gamma_{n})e^{S^{G}_{\gamma_{n}}(g,u)}((\hat{h}_{c})_{BA})_{N}(V^{-\frac{1}{2}}_{v_{1}})_{j_{1}}(V^{-\frac{1}{2}}_{v_{2}})_{j_{2}}\mathcal{I}^{AB}_{n}(g,u)$
(20) $\displaystyle\mathcal{I}^{AB}_{n}(g,u)$
$\displaystyle=\int\prod_{j=0}^{N}\differential\zeta_{j}(\gamma_{n})\,e^{S^{F}_{\gamma_{n}}(g,\zeta)}(\zeta_{v_{1}}^{A})_{j_{1}}(\zeta_{v_{2}}^{B*})_{j_{2}}\langle\omega_{\gamma_{n}}|\phi_{\zeta_{N}}\rangle\langle\phi_{\zeta_{0}}|u|\omega_{\gamma_{n}}\rangle$

where the integral with respect $u=\\{u_{v}\\}_{v}\in\mathcal{G}$ comes from
the projection $\hat{\mathbf{P}}_{\mathcal{G}}$ introduced in defining
$|\Omega_{\gamma_{n}}\rangle$. A pair of $\langle\Omega_{\gamma_{n}}$ and
$|\Omega_{\gamma_{n}}\rangle$ only give one $u$-integral because the operator
is gauge invariant and
$\hat{\mathbf{P}}_{\mathcal{G}}^{2}=\hat{\mathbf{P}}_{\mathcal{G}}$.

In the derivation of the path integral, we insert the Heisenberg operator in
$\mathscr{G}_{h}$, and have the following expression in terms of the time-
evolution operators

$\displaystyle\mathscr{G}_{h,n}(\tau_{1},v_{1},\tau_{2},v_{2})=\langle\Omega_{\gamma_{n}}|(\hat{h}_{c})_{BA}e^{\frac{i}{\hbar}\hat{\bf
H}_{\gamma_{n}}(\tau_{1}-T_{i})}\hat{\Psi}^{A}(v)e^{-\frac{i}{\hbar}\hat{\bf
H}_{\gamma_{n}}(\tau_{1}-\tau_{2})}\hat{\Psi}^{B\dagger}(v)e^{-\frac{i}{\hbar}\hat{\bf
H}_{\gamma_{n}}(\tau_{2}-T_{i})}|\Omega_{\gamma_{n}}\rangle,$ (21)

when $\tau_{1}>\tau_{2}$. It leads to that in the path integral (20), the
paths $Z_{j}(\gamma_{n})\equiv(g_{j}(\gamma_{n}),\zeta_{j}(\gamma_{n}))$ start
from $T_{i}$, pass $\tau_{2}$, arrive at $\tau_{1}$, and finally return to
$T_{i}$. As in the standard derivation, we begin with considering the $N$-side
polygon paths as shown in Fig. 3 and, then, let $N$ approaches to $\infty$. In
(20) and (7), $S_{\gamma_{n}}^{G}$ reads

$\displaystyle S_{\gamma_{n}}^{G}=$
$\displaystyle\sum_{j=-1}^{N}\sum_{e\subset\gamma_{n}}\frac{2\eta_{j+1,j}(e)^{2}-\eta_{j+1,j+1}(e)^{2}-\eta_{j,j}(e)^{2}}{2\ell_{P}^{2}}-\frac{i}{\hbar}\sum_{\begin{subarray}{c}j=0\end{subarray}}^{N-1}\rho_{j}\delta\tau\left(\frac{\langle\psi_{g_{j+1}(\gamma_{n})}|\hat{\mathbf{P}}_{\gamma_{n}}\hat{H}_{\gamma_{n}}^{G}\hat{\mathbf{P}}_{\gamma_{n}}|\psi_{g_{j}(\gamma_{n})}\rangle}{\langle\psi_{g_{j+1}(\gamma_{n})}|\psi_{g_{j}(\gamma_{n})}\rangle})\right),$

where $\rho_{j}=1$ or $-1$ at instances before or after $\tau_{1}$. $j=-1$ and
$j=N+1$ correspond to the initial and final state. The coherent label
$g_{j}(\gamma_{n})=\\{g_{e}\\}_{e\subset\gamma_{n}}$ relates to
$h_{e},p_{e}^{a}$ by $g_{e}=e^{-p_{e}^{a}\bm{\sigma}^{a}/2}h_{e}$.
$\eta_{i,j}(e)$ satisfies
$\tr(g_{i}^{\dagger}(e)g_{j}(e))=2\cosh(\eta_{i,j}(e))$. At $j=-1$,
$g_{-1}(\gamma_{n})=u\cdot g_{\rm flat}(\gamma_{n})$ is the gauge
transformation of $g_{\rm flat}(\gamma_{n})$.

$\mathcal{I}^{AB}_{n}$ is independent of $\hbar$. The integral of
$\differential u$ and $\differential g_{j}(\gamma_{n})$ is studied with the
stationary phase approximation to obtain a semiclassical expansion in
$\ell_{P}^{2}$. With the chosen initial and final states, the solution of
equations of motion gives $u=\mathring{u}=\mathbf{1}$ for all vertices $v$ and
$g=\mathring{g}(\gamma_{n})=(\mathring{p},\mathring{h})$ corresponding to the
flat spacetime: $\mathring{h}_{e,j}=\mathbf{1}_{2\times 2}$ and
$\mathring{p}_{e(I),j}^{a}=a_{n}^{2}\delta^{a}_{I}$. Thus as far as the
leading order in the semiclassical expansion is concerned, we can just freeze
$u$ in $\mathcal{I}_{n}^{AB}$ to $\mathbf{1}$, and freeze
$((\hat{h}_{c})_{BA})_{N}(V^{-\frac{1}{2}}_{v_{1}})_{j_{1}}(V^{-\frac{1}{2}}_{v_{2}})_{j_{2}}$
to their corresponding classical value at the flat geometry. Therefore,
Eq.(20) can be approximated by $\delta_{BA}\mathscr{G}^{AB}_{n}$
semiclassically, where $\mathscr{G}^{AB}_{n}$ is given by

$\mathscr{G}^{AB}_{n}=\frac{1}{V_{\rm
tot}}\mathcal{I}^{AB}_{n}(\mathring{g},\mathring{u})\int\limits_{\mathcal{G}}\mathrm{d}u\int\prod_{j=0}^{N}\mathrm{d}g_{j}(\gamma_{n})\,e^{S^{G}_{\gamma_{n}}(g,u)}\left[1+O(\ell_{P}^{2})\right]$

The integral
$\int\mathrm{d}u\int\prod_{j=0}^{N}\mathrm{d}g_{j}(\gamma_{n})\,e^{S^{G}_{\gamma_{n}}(g,u)}$
is the transition amplitude between a pair of
$|\tilde{\psi}_{\gamma_{n}}\rangle\equiv\hat{\mathbf{P}}_{\mathcal{G}}\hat{\mathbf{P}}_{\gamma_{n}}|\psi_{\gamma_{n}}\rangle$,
the quantum-geometry sectors of the initial state, with respect to the pure
gravity Hamiltonian
$\hat{\mathfrak{h}}_{\gamma_{n}}^{G}\equiv\hat{\mathbf{P}}_{\gamma_{n}}\hat{H}_{\gamma_{n}}^{G}\hat{\mathbf{P}}_{\gamma_{n}}$.
Recall that the paths are the ones as in FIG.3. We have

$\displaystyle\int\limits_{\mathcal{G}}\mathrm{d}u\int\prod_{j=0}^{N}\mathrm{d}g_{j}(\gamma_{n})\,e^{S^{G}_{\gamma_{n}}(g,u)}=\langle\tilde{\psi}_{\gamma_{n}}|e^{\frac{i}{\hbar}\hat{\mathfrak{h}}^{G}_{\gamma_{n}}(\tau_{1}-T_{i})}e^{-\frac{i}{\hbar}\hat{\mathfrak{h}}^{G}_{\gamma_{n}}(\tau_{1}-\tau_{2})}e^{-\frac{i}{\hbar}\hat{\mathfrak{h}}^{G}_{\gamma_{n}}(\tau_{2}-T_{i})}|\tilde{\psi}_{\gamma_{n}}\rangle=\langle\tilde{\psi}_{\gamma_{n}}|\tilde{\psi}_{\gamma_{n}}\rangle=1+O(\ell_{P}^{2}).$

where in the last step, we use
$1=\langle\Omega_{\gamma_{n}}|\Omega_{\gamma_{n}}\rangle=\int_{\mathcal{G}}\mathrm{d}u\langle\psi_{\gamma_{n}}|\mathbf{P}_{\gamma_{n}}u\mathbf{P}_{\gamma_{n}}|\psi_{\gamma_{n}}\rangle\langle\omega_{\gamma}|u|\omega_{\gamma}\rangle=\int\mathrm{d}u\langle\psi_{\gamma_{n}}|\mathbf{P}_{\gamma_{n}}u\mathbf{P}_{\gamma_{n}}|\psi_{\gamma_{n}}\rangle[1+O(\ell_{P}^{2})]$,
because $\|\omega_{\gamma_{n}}\|=1$ and
$\langle\psi_{\gamma_{n}}|\mathbf{P}_{\gamma_{n}}u\mathbf{P}_{\gamma_{n}}|\psi_{\gamma_{n}}\rangle$
is a Gaussian-like function peaked at $u=\mathbf{1}$. Therefore

$\mathscr{G}^{AB}_{n}=\frac{1}{V_{\rm
tot}}\mathcal{I}^{AB}_{n}(\mathring{g},\mathring{u})[1+O(\ell_{P}^{2})].$

Substituting the flat-geometry data $\mathring{p}$ and $\mathring{h}$ into
$S^{F}_{\gamma_{n}}$, and performing the Fourier transformation to
$\zeta_{i,v}$ along spatial directions, we get $S_{\gamma_{n}}^{F}$ in terms
of the Fourier coefficients $\zeta_{j,\vec{k}}$ as
$S_{N}^{F}=\sum_{j,\vec{k},j^{\prime},\vec{k}^{\prime}}\zeta_{j,\vec{k}}^{\dagger}\mathscr{H}_{j,\vec{k},j^{\prime},\vec{k}^{\prime}}\zeta_{j^{\prime},\vec{k}^{\prime}}$,
with
$\mathscr{H}_{j,\vec{k},j^{\prime},\vec{k}^{\prime}}=-\delta_{j,j^{\prime}}\mathbf{1}_{2\times
2}+\delta_{j,j^{\prime}+1}\left(\mathbf{1}_{2\times
2}+\frac{i\delta\tau}{a_{n}}\sum_{I=1}^{3}\sin(a_{n}k_{I}){\bm{\sigma}}^{I}\right)$.
With these results, $\mathcal{I}^{AB}_{n}(\mathring{g},\mathring{u})$ can be
computed by using the standard Fermionic Gaussian integral. The result is
obtained by taking the limit $N\to\infty$:

$\displaystyle\mathcal{I}^{AB}_{n}(\mathring{g},\mathring{u})=\sum_{\vec{k}\in\text{FBZ}(\gamma_{n})}\int_{\mathbb{R}}\frac{\mathrm{d}\omega}{2\pi
i}\,e^{-i\omega(\tau_{1}-\tau_{2})+ia_{n}\vec{k}\cdot\left(\vec{v}_{1}-\vec{v}_{2}\right)}\,G_{\omega,\vec{k}}(\gamma_{n},a_{n}).$
(22)

## References

  * Page and Geilker (1981) D. N. Page and C. D. Geilker, Phys. Rev. Lett. 47, 979 (1981).
  * Deser and van Nieuwenhuizen (1974) S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 10, 411 (1974).
  * ’t Hooft and Veltman (1974) G. ’t Hooft and M. J. G. Veltman, Ann. Inst. H. Poincare Phys. Theor. A 20, 69 (1974).
  * Goroff and Sagnotti (1986) M. H. Goroff and A. Sagnotti, Nucl. Phys. B 266, 709 (1986).
  * Ashtekar  _et al._ (2021) A. Ashtekar, T. De Lorenzo, and M. Schneider, Adv. Theor. Math. Phys. 25, 7 (2021), arXiv:2107.08506 [gr-qc] .
  * Hawking (1975) S. W. Hawking, Commun. Math. Phys. 43, 199 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * Donà  _et al._ (2014) P. Donà, A. Eichhorn, and R. Percacci, Phys. Rev. D 89, 084035 (2014), arXiv:1311.2898 [hep-th] .
  * Perez and Sudarsky (2019) A. Perez and D. Sudarsky, Phys. Rev. Lett. 122, 221302 (2019), arXiv:1711.05183 [gr-qc] .
  * Ashtekar  _et al._ (1989) A. Ashtekar, J. D. Romano, and R. S. Tate, Phys. Rev. D 40, 2572 (1989).
  * Thiemann (1998a) T. Thiemann, Class. Quant. Grav. 15, 1281 (1998a), arXiv:gr-qc/9705019 [gr-qc] .
  * Sahlmann and Thiemann (2006) H. Sahlmann and T. Thiemann, Class. Quant. Grav. 23, 867 (2006), arXiv:gr-qc/0207030 [gr-qc] .
  * Lewandowski and Zhang (2021) J. Lewandowski and C. Zhang, (2021), arXiv:2112.08865 [gr-qc] .
  * Bodendorfer  _et al._ (2013) N. Bodendorfer, T. Thiemann, and A. Thurn, Class. Quant. Grav. 30, 045004 (2013), arXiv:1105.3706 [gr-qc] .
  * Bojowald and Das (2008) M. Bojowald and R. Das, Phys. Rev. D 78, 064009 (2008), arXiv:0710.5722 [gr-qc] .
  * Domagala  _et al._ (2010) M. Domagala, K. Giesel, W. Kaminski, and J. Lewandowski, Phys. Rev. D 82, 104038 (2010), arXiv:1009.2445 [gr-qc] .
  * Kisielowski and Lewandowski (2019) M. Kisielowski and J. Lewandowski, Class. Quant. Grav. 36, 075006 (2019), arXiv:1807.06098 [gr-qc] .
  * Bianchi  _et al._ (2013) E. Bianchi, M. Han, C. Rovelli, W. Wieland, E. Magliaro, and C. Perini, Class. Quant. Grav. 30, 235023 (2013), arXiv:1012.4719 [gr-qc] .
  * Oriti and Ryan (2006) D. Oriti and J. Ryan, Class. Quant. Grav. 23, 6543 (2006), arXiv:gr-qc/0602010 .
  * Mansuroglu and Sahlmann (2021) R. Mansuroglu and H. Sahlmann, Phys. Rev. D 103, 066016 (2021), arXiv:2011.00233 [gr-qc] .
  * Zhang and Ma (2011) X. Zhang and Y. Ma, Phys. Rev. D 84, 104045 (2011), arXiv:1107.5157 [gr-qc] .
  * Kamiński  _et al._ (2006) W. Kamiński, J. Lewandowski, and M. Bobieński, Classical and Quantum Gravity 23, 2761 (2006).
  * Montvay and Münster (1994) I. Montvay and G. Münster,  _Quantum Fields on a Lattice_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1994).
  * Nielsen and Ninomiya (1981) H. B. Nielsen and M. Ninomiya, Phys. Lett. B 105, 219 (1981).
  * Barnett and Smolin (2015) J. Barnett and L. Smolin, (2015), arXiv:1507.01232 [gr-qc] .
  * Gambini and Pullin (2015) R. Gambini and J. Pullin, Phys. Lett. B 749, 374 (2015), arXiv:1506.08794 [gr-qc] .
  * Han and Rovelli (2013) M. Han and C. Rovelli, Class. Quant. Grav. 30, 075007 (2013), arXiv:1101.3264 [gr-qc] .
  * ’t Hooft (2016) G. ’t Hooft, Journal of Physics: Conference Series 701, 012014 (2016).
  * Wheeler (1955) J. A. Wheeler, Physical Review 97, 511 (1955).
  * Hawking (1978) S. W. Hawking, Nucl. Phys. B 144, 349 (1978).
  * Kogut and Susskind (1975) J. B. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975).
  * Creutz  _et al._ (2002) M. Creutz, I. Horvath, and H. Neuberger, Nucl. Phys. B Proc. Suppl. 106, 760 (2002), arXiv:hep-lat/0110009 .
  * Lewandowski  _et al._ (2006) J. Lewandowski, A. Okolow, H. Sahlmann, and T. Thiemann, Commun. Math. Phys. 267, 703 (2006), arXiv:gr-qc/0504147 [gr-qc] .
  * Ashtekar and Lewandowski (1994) A. Ashtekar and J. Lewandowski, in  _Knots and Quantum Gravity_ (1994) p. 21.
  * Thiemann (2020) T. Thiemann, Front. in Phys. 8, 548232 (2020), arXiv:2003.13622 [gr-qc] .
  * Giesel and Thiemann (2015) K. Giesel and T. Thiemann, Class. Quant. Grav. 32, 135015 (2015), arXiv:1206.3807 [gr-qc] .
  * Giesel and Thiemann (2010) K. Giesel and T. Thiemann, Classical and Quantum Gravity 27, 175009 (2010).
  * Giesel and Thiemann (2007a) K. Giesel and T. Thiemann, Class. Quant. Grav. 24, 2465 (2007a), arXiv:gr-qc/0607099 [gr-qc] .
  * Giesel and Thiemann (2007b) K. Giesel and T. Thiemann, Class. Quant. Grav. 24, 2565 (2007b), arXiv:gr-qc/0607101 .
  * Kuchar and Torre (1991) K. V. Kuchar and C. G. Torre, Phys. Rev. D43, 419 (1991).
  * Han and Liu (2020a) M. Han and H. Liu, Phys. Rev. D 102, 024083 (2020a), arXiv:2005.00988 [gr-qc] .
  * Dittrich (2006) B. Dittrich, Class. Quant. Grav. 23, 6155 (2006), arXiv:gr-qc/0507106 .
  * Thiemann (2006) T. Thiemann, Class. Quant. Grav. 23, 1163 (2006), arXiv:gr-qc/0411031 .
  * Rovelli (2002) C. Rovelli, Phys. Rev. D65, 124013 (2002), arXiv:gr-qc/0110035 [gr-qc] .
  * Rovelli (1991) C. Rovelli, Class. Quant. Grav. 8, 297 (1991).
  * Thiemann and Winkler (2001a) T. Thiemann and O. Winkler, Classical and Quantum Gravity 18, 2561 (2001a).
  * Thiemann and Winkler (2001b) T. Thiemann and O. Winkler, Classical and Quantum Gravity 18, 4629 (2001b).
  * Flori and Thiemann (2008) C. Flori and T. Thiemann, (2008), arXiv:0812.1537 [gr-qc] .
  * Thiemann (1998b) T. Thiemann, Class. Quant. Grav. 15, 1487 (1998b), arXiv:gr-qc/9705021 .
  * Giesel and Thiemann (2006) K. Giesel and T. Thiemann, Class. Quant. Grav. 23, 5667 (2006), arXiv:gr-qc/0507036 .
  * Yang and Ma (2016) J. Yang and Y. Ma, Phys. Rev. D 94, 044003 (2016).
  * Note (1) In $\mathscr{G}_{h}$, $\langle\Omega|$ can be replaced by $\langle\Omega^{\prime}|$ for free, because the operator is gauge invariant and $\mathaccentV{hat}05E{\mathbf{P}}_{\mathcal{G}}^{2}=\mathaccentV{hat}05E{\mathbf{P}}_{\mathcal{G}}$.
  * Han and Liu (2020b) M. Han and H. Liu, Phys. Rev. D 101, 046003 (2020b), arXiv:1910.03763 [gr-qc] .
  * Finocchiaro and Oriti (2021) M. Finocchiaro and D. Oriti, Front. in Phys. 8, 552354 (2021), arXiv:2004.07361 [hep-th] .
  * Zhang  _et al._ (2022) C. Zhang, S. Song, and M. Han, Phys. Rev. D 105, 064008 (2022), arXiv:2102.03591 [gr-qc] .
  * Christ  _et al._ (1982) N. H. Christ, R. Friedberg, and T. D. Lee, Nucl. Phys. B 202, 89 (1982).
  * Pang and Ren (1986) Y. Pang and H.-c. Ren, Phys. Lett. B 172, 392 (1986).
  * Perantonis and Wheater (1988) S. J. Perantonis and J. F. Wheater, Nucl. Phys. B 295, 443 (1988).
  * Griffin and Kieu (1993) C. J. Griffin and T. D. Kieu, Phys. Rev. Lett. 70, 3844 (1993), arXiv:hep-lat/9210005 .


[a]Derek Leinweber

# Impact of Dynamical Fermions on Centre Vortex Structure

James Biddle  Waseem Kamleh

###### Abstract

This presentation examines the centre vortex structure of Monte-Carlo
generated gauge-field configurations using modern visualisation techniques.
This time, the manner in which light dynamical fermion degrees of freedom
impact the centre-vortex structure is explored. Focusing on the thin vortices
identified by plaquettes having a non-trivial centre phase, the vortex
structure is illustrated through 3D renderings of oriented plaquettes. The
impact of light dynamical fermions is not subtle, changing both the density of
vortices and the complexity of the vortex structures observed. The role of
vortex branching points in full QCD is highlighted in the survey of results
presented.

##  1 Introduction

The salient emergent features of the QCD ground-state vacuum fields are: the
dynamical generation of mass through chiral symmetry breaking, and the
confinement of quarks. Understanding the most fundamental aspects of QCD-
vacuum field structure giving rise to these phenomena is a contemporary
problem of interest.

The most fundamental mechanism proposed to give rise to these phenomena are
centre vortices [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] within the ground-
state vacuum fields. Here, the eight degrees of freedom contained within an
$SU(3)$ gauge-field link are replaced by one of the three cube roots of unity.
It is this extreme level of simplification that makes the centre-vortex
proposal so compelling as the most fundamental essence of QCD vacuum
structure.

Within the pure-gauge sector of $SU(3)$-colour, there is encouraging evidence
that centre vortices underpin both confinement and dynamical chiral-symmetry
breaking.

  * •

Removal of $SU(3)$ centre vortices removes confinement, while consideration of
the vortices alone provides confinement [13, 14, 15].

  * •

The planar vortex density of centre-vortex degrees of freedom scales with the
lattice spacing providing an well defined continuum limit [13].

  * •

Removal of vortices suppresses the infrared enhancement of the gluon
propagator. Again the vortices alone contain the long distance structure of
the gluon fields responsible for the well-known infrared enhancement [16].

  * •

A connection between centre vortices and instantons was established through
gauge-field smoothing [15]. An understanding of the phenomena linking these
degrees of freedom was illustrated in Ref. [17].

  * •

Centre vortices have been shown to give rise to mass splitting in the low-
lying hadron spectrum [18, 19, 20].

  * •

Through studies of the nonperturbative quark propagator of the overlap-Dirac
fermion operator, evidence that centre vortices underpin dynamical chiral
symmetry breaking in $SU(3)$ gauge theory was reported in Ref. [19].

  * •

The removal of centre vortex degrees of freedom from the gluon fields restores
chiral symmetry [18].

But how do these results hold up in QCD where the virtual transition of gluons
into dynamical quark-antiquark pairs can significantly alter the vacuum
structure [21]. In this presentation we investigate how these dynamical
fermions change the centre-vortex structure of the ground-state vacuum fields.

##  2 Centre Vortex Identification

Centre vortices are identified through a gauge fixing procedure designed to
bring the lattice link variables as close as possible to the identity, up to a
phase. Here, we consider the original Monte-Carlo generated configurations and
gauge transform them directly to Maximal Centre Gauge [22, 23, 13], avoiding
any preconditioning [14]. In doing so, the lattice link variables $U_{\mu}(x)$
are brought close to the centre elements of $SU(3)$,

$Z=\exp\left(2\pi i\,\frac{m}{3}\right)\,\mathbf{I},\textrm{ with }m=-1,0,1.$
(1)

This is implemented through gauge transformations $\Omega$ such that,

$\sum_{x,\mu}\,\left|\mathrm{tr}\,U_{\mu}^{\Omega}(x)\,\right|^{2}\stackrel{{\scriptstyle\Omega}}{{\to}}\mathrm{max}\,.$
(2)

One then projects the link variables to the centre

$U_{\mu}(x)\to Z_{\mu}(x)\textrm{ where }Z_{\mu}(x)=\exp\left(2\pi
i\,\frac{m_{\mu}(x)}{3}\right)\mathbf{I}\,,$ (3)

where $m_{\mu}(x)=-1,0,1.$

The centre-line of an extended vortex in three dimensions is identified by the
presence of nontrivial centre charge, $z$, found in the product of centre-
projected links around a plaquette,

$z=\prod_{\Box}Z_{\mu}(x)=\exp\left(2\pi i\,\frac{m}{3}\right)\,.$ (4)

A right-handed ordering of the dimensions is selected in calculating the
centre charge. If $z=1$, no vortex pierces the plaquette. If $z\neq 1$ a
vortex with charge $z$ pierces the plaquette. We refer to the centre charge of
a vortex via the value of $m=\pm 1$.

##  3 Centre Vortex Visualisation

The centre lines of extended vortices are illustrated on the dual lattice by a
jet piercing the plaquette producing a nontrivial centre charge. The
orientation of the jet follows a right-handed coordinate system. For example,
with reference to Eq. (4), an $m=+1$ vortex in the $x$-$y$ plane is plotted in
the $+\hat{z}$ direction as a blue jet. Similarly, an $m=-1$ vortex in the
$x$-$y$ plane is plotted in the $-\hat{z}$ direction. Figure 1 provides an
illustration of this assignment.

$Z_{y}(n+\hat{x})\ $${}\ \
Z_{x}^{*}(n+\hat{y})$$Z_{y}^{*}(n)$$Z_{x}(n)$$x$$y$$z$

$Z_{y}(n+\hat{x})\ $$Z_{x}^{*}(n+\hat{y})$$Z_{y}^{*}(n)$$Z_{x}(n)$$x$$y$$z$

Figure 1: Rendering the centre charge of Eq. (4) associated with a plaquette
in the $x$-$y$ plane at lattice site $n$. (left) An $m=+1$ vortex with centre
charge $z=\exp(2\pi i/3)$ is rendered as a jet pointing in the $+\hat{z}$
direction. (right) An $m=-1$ vortex with centre charge $z=\exp(-2\pi i/3)$ is
rendered as a jet in the $-\hat{z}$ direction.

As the centre charge transforms to its complex conjugate under permutation of
the two dimensions describing the plaquette, the centre charge can be thought
of as a directed flow of charge $z=\exp(2\pi i/3)$.

##  4 Centre Vortex Structure

The projected centre vortices (P-vortices), identified on the lattice as
described above, form surfaces in four dimensional space-time, analogous to
the centre line of a vortex in fluid dynamics that maps out a surface as it
moves through time. As the surface cuts through the three-dimensional volume
of our visualisation, a P-vortex line mapping the flow of centre charge is
rendered.

Our point of focus in this presentation is to discover the impact of
dynamical-fermion degrees of freedom on the vortex structure of a gauge field.
Here we the PACS-CS $(2+1)$-flavour full-QCD ensembles [24], made available
through the ILDG [25]. These ensembles use a $32^{3}\times 64$ lattice, and
employ a renormalisation-group improved Iwasaki gauge action with $\beta=1.90$
and non-perturbatively ${\cal O}(a)$-improved Wilson quarks, with $C_{\rm
SW}=1.715$. We consider their lightest $u$\- and $d$-quark-mass ensemble
identified by a pion mass of 156 MeV [24], and set the scale using the Sommer
parameter with $r_{0}=0.4921$ fm providing a lattice spacing of $a=0.0933$ fm
[24].

For comparison, we have created a matched $32^{3}\times 64$ pure-gauge
ensemble using the same improved Iwasaki gauge action with $\beta=2.58$
providing a Sommer-scale spacing of $a=0.100$ fm, facilitating comparisons
with all the PACS-CS ensembles.

Figures 3 and 3 illustrate the centre-vortex structure of pure-gauge and
dynamical-fermion ground-state vacuum fields respectively. Inspection of the
vortices reveals the flow of centre charge, intersection points and a
prevalence of branching points resembling monopole or anti-monopole
contributions, where three jets emerge from or converge to a point. With the
introduction of dynamical fermions the structure becomes more complicated,
both in the abundance of nontrivial centre charge and in the proliferation of
branching points.

Figures 3 and 3 are interactive illustrations which can be activated in Adobe
Reader111Open this pdf document in Adobe Reader 9 or later. Linux users can
install Adobe acroread version 9.4.1, the last edition to have full 3D
support. From the “Edit” menu, select “Preferences…” and ensure “3D &
Multimedia” is enabled and “Enable double-sided rendering” is selected. by
clicking on the image. Once activated, click and drag to rotate, Ctrl-click to
translate, Shift-click or mouse wheel to zoom, and right click to access the
“Views” menu. Several views have been created to facilitate and inspection of
the centre-vortex structure. The presence of a percolating vortex cluster is a
characteristic feature of the confining phase [26]. These illustrations are
representative of the ensemble in that the vortex vacuum is typically
dominated by a single large percolating cluster. Moreover, dynamical fermions
significantly increase the number of vortices observed.

Considering an ensemble of 200 configurations with $32$ three-dimensional
slices each, the average number of vortices composing the primary cluster in
these $32^{2}\times 64$ spatial slices is

  * •

$3,277\,\pm 156$ vortices in the Pure Gauge theory, versus

  * •

$5,924\,\pm 239$ vortices in Full QCD.

Noting that there are $32^{2}\times 64\times 3=196,608$ spatial plaquettes on
these lattices, one sees that the presence of a vortex is still a relatively
rare occurrences.

[ noplaybutton, 3Dtoolbar, 3Dmenu, 3Dviews=U3D/Primary-Secondary-PG-28.vws,
3Dcoo = 16 16 32, 3Dc2c=0.245114266872406 0.8673180341720581
0.43321868777275085, 3Droo = 110.0, 3Droll =-98.1, 3Dlights=Default,
width=1.04]U3D/Primary-Secondary-PG-28.u3d

Figure 2:  The centre-vortex structure of a ground-state vacuum field
configuration in pure SU(3) gauge theory. (Click to activate.) The flow of
$+1$ centre charge through a gauge field is illustrated by the jets. Blue jets
illustrate the single percolating vortex structure, while other colours
illustrate smaller structures.

[ noplaybutton, 3Dtoolbar, 3Dmenu, 3Dviews=U3D/Primary-Secondary-DF-18.vws,
3Dcoo = 16 16 32, 3Dc2c=0.245114266872406 0.8673180341720581
0.43321868777275085, 3Droo = 110.0, 3Droll =-98.1, 3Dlights=Default,
width=]U3D/Primary-Secondary-DF-18.u3d

Figure 3:  The centre-vortex structure of a ground-state vacuum field
configuration in dynamical 2+1 flavour QCD. (Click to activate.) The flow of
$+1$ centre charge through a gauge field is illustrated by the jets. Symbols
are as described in Fig. 3.

Figures 5 and 5 illustrate the secondary loop structure left behind as one
removes the single large percolating structure. The introduction of dynamical
fermions increases both the number of loops observed, and the complexity of
their structure by a proliferation of branching points (or monopoles [27]).
Figure 5 contains many views to facilitate the observation of this complexity.

[ noplaybutton, 3Dtoolbar, 3Dmenu, 3Dviews=U3D/Secondary-PG-28.vws, 3Dcoo = 16
16 32, 3Dc2c=0.245114266872406 0.8673180341720581 0.43321868777275085, 3Droo =
110.0, 3Droll =-98.1, 3Dlights=Default, width=1.04]U3D/Secondary-PG-28.u3d

Figure 4:  The centre-vortex structure of secondary loops in a ground-state
vacuum field configuration of pure SU(3) gauge theory. (Click to activate.)
The flow of $+1$ centre charge in the secondary loops – left behind as the
single percolating structure is removed – is illustrated.

[ noplaybutton, 3Dtoolbar, 3Dmenu, 3Dviews=U3D/Secondary-DF-18.vws, 3Dcoo = 16
16 32, 3Dc2c=0.245114266872406 0.8673180341720581 0.43321868777275085, 3Droo =
110.0, 3Droll =-98.1, 3Dlights=Default, width=1.05]U3D/Secondary-DF-18.u3d

Figure 5:  The centre-vortex structure of secondary loops in a ground-state
vacuum field configuration of dynamical 2+1 flavour QCD. (Click to activate.)
The flow of $+1$ centre charge in the secondary loops is illustrated.

##  5 Outlook

In summary,

  * •

Centre-vortex structure is complex.

  * •

We observe a proliferation of branching points in $SU(3)$ gauge theory with
further enhancement as light dynamical fermion degrees of freedom are
introduced in simulating QCD.

  * •

Each ground-state configuration is dominated by a long-distance percolating
centre-vortex structure.

  * •

We have observed an approximate doubling in the size of the percolating vortex
structure in in going from pure-gauge theory to full QCD.

  * •

We have also observed an enhancement in the number of small vortex paths upon
the introduction of dynamical fermions.

  * •

Increased complexity in the vortex paths is also observed as the number of
monopole-antimonopole pairs is significantly increased with the introduction
of dynamical fermions.

In short, dynamical-fermion degrees of freedom radically alter the centre-
vortex structure of the ground-state vacuum fields.

Having gained insight into the impact of dynamical fermions on ground-state
vacuum field structure, future work will aim to quantify these effects. Here
the distribution of path lengths is of particular interest as it may be
possible to quantify the change in the branching probability of vortex paths
in $SU(3)$ gauge theory as dynamical fermions are introduced.

## Acknowledgements

We thank the PACS-CS Collaboration for making their $2+1$ flavour
configurations available and the ongoing support of the International Lattice
Data Grid (ILDG). This research is supported with supercomputing resources
provided by the Phoenix HPC service at the University of Adelaide and the
National Computational Infrastructure (NCI) supported by the Australian
Government. This research is supported by the Australian Research Council
through Grants No. LE190100021, DP190102215, and DP210103706. WK is supported
by the Pawsey Supercomputing Centre through the Pawsey Centre for Extreme
Scale Readiness (PaCER) program.

## References

  * [1] G. ’t Hooft, “On the Phase Transition Towards Permanent Quark Confinement,” Nucl. Phys. B 138 (1978) 1–25.
  * [2] G. ’t Hooft, “A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories,” Nucl. Phys. B 153 (1979) 141–160.
  * [3] L. Del Debbio, M. Faber, J. Greensite, and S. Olejnik, “Center dominance and Z(2) vortices in SU(2) lattice gauge theory,” Phys. Rev. D 55 (1997) 2298–2306, arXiv:hep-lat/9610005.
  * [4] M. Faber, J. Greensite, and S. Olejnik, “Casimir scaling from center vortices: Towards an understanding of the adjoint string tension,” Phys. Rev. D 57 (1998) 2603–2609, arXiv:hep-lat/9710039.
  * [5] L. Del Debbio, M. Faber, J. Giedt, J. Greensite, and S. Olejnik, “Detection of center vortices in the lattice Yang-Mills vacuum,” Phys. Rev. D 58 (1998) 094501, arXiv:hep-lat/9801027.
  * [6] R. Bertle, M. Faber, J. Greensite, and S. Olejnik, “The Structure of projected center vortices in lattice gauge theory,” JHEP 03 (1999) 019, arXiv:hep-lat/9903023.
  * [7] M. Faber, J. Greensite, S. Olejnik, and D. Yamada, “The Vortex finding property of maximal center (and other) gauges,” JHEP 12 (1999) 012, arXiv:hep-lat/9910033.
  * [8] M. Engelhardt and H. Reinhardt, “Center projection vortices in continuum Yang-Mills theory,” Nucl. Phys. B 567 (2000) 249, arXiv:hep-th/9907139.
  * [9] R. Bertle, M. Faber, J. Greensite, and S. Olejnik, “P vortices, gauge copies, and lattice size,” JHEP 10 (2000) 007, arXiv:hep-lat/0007043.
  * [10] J. Greensite, “The Confinement problem in lattice gauge theory,” Prog. Part. Nucl. Phys. 51 (2003) 1, arXiv:hep-lat/0301023.
  * [11] M. Engelhardt, M. Quandt, and H. Reinhardt, “Center vortex model for the infrared sector of SU(3) Yang-Mills theory: Confinement and deconfinement,” Nucl. Phys. B 685 (2004) 227–248, arXiv:hep-lat/0311029.
  * [12] J. Greensite, “Confinement from Center Vortices: A review of old and new results,” EPJ Web Conf. 137 (2017) 01009, arXiv:1610.06221 [hep-lat].
  * [13] K. Langfeld, “Vortex structures in pure SU(3) lattice gauge theory,” Phys. Rev. D69 (2004) 014503, arXiv:hep-lat/0307030.
  * [14] A. O’Cais, W. Kamleh, K. Langfeld, B. Lasscock, D. Leinweber, et al., “Preconditioning Maximal Center Gauge with Stout Link Smearing in SU(3),” Phys.Rev. D82 (2010) 114512, arXiv:0807.0264 [hep-lat].
  * [15] D. Trewartha, W. Kamleh, and D. Leinweber, “Connection between center vortices and instantons through gauge-field smoothing,” Phys. Rev. D92 no. 7, (2015) 074507, arXiv:1509.05518 [hep-lat].
  * [16] J. C. Biddle, W. Kamleh, and D. B. Leinweber, “Gluon propagator on a center-vortex background,” Phys. Rev. D98 no. 9, (2018) 094504, arXiv:1806.04305 [hep-lat].
  * [17] J. C. Biddle, W. Kamleh, and D. B. Leinweber, “Visualization of center vortex structure,” Phys. Rev. D 102 no. 3, (2020) 034504, arXiv:1912.09531 [hep-lat].
  * [18] D. Trewartha, W. Kamleh, and D. Leinweber, “Centre vortex removal restores chiral symmetry,” J. Phys. G44 no. 12, (2017) 125002, arXiv:1708.06789 [hep-lat].
  * [19] D. Trewartha, W. Kamleh, and D. Leinweber, “Evidence that centre vortices underpin dynamical chiral symmetry breaking in SU(3) gauge theory,” Phys. Lett. B747 (2015) 373–377, arXiv:1502.06753 [hep-lat].
  * [20] E.-A. O’Malley, W. Kamleh, D. Leinweber, and P. Moran, “SU(3) centre vortices underpin confinement and dynamical chiral symmetry breaking,” Phys. Rev. D 86 (2012) 054503, arXiv:1112.2490 [hep-lat].
  * [21] P. J. Moran and D. B. Leinweber, “Impact of Dynamical Fermions on QCD Vacuum Structure,” Phys. Rev. D 78 (2008) 054506, arXiv:0801.2016 [hep-lat].
  * [22] L. Del Debbio, M. Faber, J. Greensite, and S. Olejnik, “Center dominance and Z(2) vortices in SU(2) lattice gauge theory,” Phys. Rev. D55 (1997) 2298–2306, hep-lat/9610005.
  * [23] K. Langfeld, H. Reinhardt, and O. Tennert, “Confinement and scaling of the vortex vacuum of SU(2) lattice gauge theory,” Phys. Lett. B419 (1998) 317–321, arXiv:hep-lat/9710068.
  * [24] PACS-CS Collaboration, S. Aoki et al., “2+1 Flavor Lattice QCD toward the Physical Point,” Phys. Rev. D 79 (2009) 034503, arXiv:0807.1661 [hep-lat].
  * [25] M. G. Beckett, B. Joo, C. M. Maynard, D. Pleiter, O. Tatebe, and T. Yoshie, “Building the International Lattice Data Grid,” Comput. Phys. Commun. 182 (2011) 1208–1214, arXiv:0910.1692 [hep-lat].
  * [26] M. Engelhardt, K. Langfeld, H. Reinhardt, and O. Tennert, “Deconfinement in SU(2) Yang-Mills theory as a center vortex percolation transition,” Phys. Rev. D 61 (2000) 054504, arXiv:hep-lat/9904004.
  * [27] F. Spengler, M. Quandt, and H. Reinhardt, “Branching of Center Vortices in SU(3) Lattice Gauge Theory,” Phys. Rev. D98 no. 9, (2018) 094508, arXiv:1810.04072 [hep-th].