Also at ]Department of Philosophy, University of Toronto, Toronto, Ontario,
Canada M5R 2M8

# Taming Fluctuations for Gaussian States in Loop Quantum Cosmology

Patrick Fraser  [<EMAIL_ADDRESS>Department of Physics, University
of Toronto, Toronto, Ontario, Canada M5S 1A7.

###### Abstract

We do not observe quantum effects on cosmological scales. Thus, if loop
quantum cosmology (LQC) is to provide an accurate depiction of the real world,
it must allow for quantum states of spacetime geometry which are semi-
classical in two respects: they must be sharply peaked around a single,
classical geometry, and they must have small quantum fluctuations. It is
generally assumed that Gaussian states exhibit both of these properties. After
all, they do in ordinary quantum mechanics. In this paper, we derive exact
closed-form expressions for the fluctuations of Gaussian states in LQC and
their lower bound given by the Robertson-Schrödinger inequality. We
demonstrate that, contrary to ordinary quantum mechanics, fluctuations for
Gaussian states in spatially flat, homogeneous and isotropic LQC diverge as
the state variance increases (as well as in related cosmological models with
the same kinematic Hilbert space and canonical observables). However, when the
holonomy length is made to scale with a volume regularization parameter, these
fluctuations may be arbitrarily suppressed by taking the fiducial volume to be
large, providing analytic control over their divergence. Finally, we show
that, despite this, Gaussian states in LQC generally do not minimize
uncertainty. Moreover, it is conjectured that no such minimal-uncertainty
states exist. Throughout this work, it becomes clear how important the often-
assumed condition of holonomy length volume-scaling is; we show that when this
condition is violated, the resulting theory exhibits operator closure
pathologies and other exotic algebraic features.

Loop Quantum Cosmology, Fluctuations, Gaussian States, Semi-Classicality

##  I Introduction

In loop quantum cosmology (LQC), spacetime exists in a quantum state which may
be regarded as a superposition of geometries. However, the observable universe
appears to be sharp; one does not observe coherent quantum effects on
cosmological scales. Thus, the actual universe, as represented by LQC, must be
described by a state which is sharply peaked around a single geometry.
Moreover, such quantum geometries exhibit the same sort of quantum
fluctuations as are present in ordinary quantum mechanics. Classical general
relativity does not exhibit these fluctuations so the semi-classical states of
LQC are expected to be states for which these fluctuations are very small,
ideally vanishingly so.

These two properties – sharpness and minimal fluctuation – may be taken to be
constitutive features of semi-classical states (see, for instance, [1, 2, 3]).
In ordinary quantum mechanics, such states are easy to find: Gaussian states –
and coherent states more generally – satisfy both of these conditions.
However, the Hilbert space structure of ordinary quantum mechanics is much
different from that of LQC. Therefore, we cannot hastily assume that the same
intuitions which hold for such ‘nice’ states in ordinary quantum mechanics
also hold for the analogous states in LQC.

In Section II we introduce the basic theory of LQC. We then discuss general
features of uncertainty relations in Section III. In Section IV, we define
Gaussian states in the volume representation of LQC and explicitly compute the
fluctuations of the canonical LQC observables for these Gaussian states in
Section V. We show that there are two important cases to consider: that in
which the shift operator is closed on the superselection sector (whence the
holonomy length scales with a fiducial volume $V_{0}$) and otherwise. We show
that the fluctuations of Gaussian states may diverge as their variance
increases in either case for a fixed $V_{0}$, but that this divergence may be
suppressed by taking $V_{0}\to\infty$ in the case where holonomy length is
scales with $V_{0}$. In Section VI, we show that Gaussian states do not
saturate uncertainty, and so this measure of semi-classicality fails for
Gaussian states in general (though it is asymptotically satisfied when $V_{0}$
scaling is introduced). Finally, in Section VII we sketch a further argument
that there are in fact no physical squeezed states which minimize uncertainty.

##  II Loop Quantum Cosmology

LQC describes superpositions of classical spacetime geometries resulting in a
theory of cosmology with several nice features [4, 5], such as a bouncing
evolution with no spacetime singularities [6, 7, 8, 9]. As a simplifying
assumption, we shall restrict ourselves to FLRW-spacetimes. That is, we assume
spacetime is a superposition of geometries which each have a line element of
the form

$ds^{2}=-dt^{2}+a(t)^{2}[dx^{2}+dy^{2}+dz^{2}].$

Such classical spacetimes may be foliated into time-like slices which each
carry a fiducial 3-metric $\mathring{q}_{ab}$ (whose determinant is denoted
$\mathring{q}$). For classical FLRW-spacetimes, we may encode this fiducial
metric using Ashtekar variables given by

$A_{a}^{i}=\dot{a}(t)(dx^{i})_{a},\qquad
E_{i}^{a}=a(t)^{2}\sqrt{\mathring{q}}(\partial_{i})^{a}.$

Defining $c:=\dot{a}(t)$ and $p:=a(t)^{2}$, these become

$A_{a}^{i}=c(dx^{i})_{a},\qquad
E_{i}^{a}=p\sqrt{\mathring{q}}(\partial_{i})^{a}.$

The action which describes the dynamics of such spacetimes under the Einstein
Field Equations is the Holst action, which has a symplectic term [10, 11] of
the form

$\frac{1}{8\pi G\gamma}\int\dot{A}^{i}_{a}E^{a}_{i}d^{3}x=\frac{1}{8\pi
G\gamma}\int\dot{c}p\sqrt{\mathring{q}}d^{3}x.$

Unless the spacetime in question is compact, this term is divergent and must
be regularized. Thus, one considers a spacetime region $\mathcal{V}$ with a
finite fiducial volume $V_{0}=\int_{\mathcal{V}}\sqrt{\mathring{q}}$. Since
FLRW-spacetimes are homogeneous, the location of this region is arbitrary. It
is common (in both the classical and quantum theories) to scale the canonical
coordinates with $V_{0}$ such that its dependency is transformed away [5, 7,
12]. However, when we switch to the quantum theory, it has been shown that
$V_{0}$ is not a gauge quantity [2] (indeed, even after transforming it away
from the canonical variables, the symplectic form still depends on it [5]),
and so we leave it for now. It shall play an interesting role in the analysis
to come.

With this regularization, the Poisson bracket of $c$ and $p$ is given by

$\\{c,p\\}=\frac{8\pi G\gamma}{V_{0}}$

where $\gamma$ is the Barbero-Immirzi parameter. While this bracket is
invariant under re-scaling, it still dependents on $V_{0}$ [2]. It is common
now to make a canonical coordinate transformation, taking $V=p^{3/2}=a(t)^{3}$
to be the spatial volume of a time-like slice of the foliation at $t$, and
taking $\beta=c/\sqrt{p}=\dot{a}(t)/a(t)$ to be the Hubble parameter (which is
the conjugate momentum to the volume coordinate). The canonical variables $V$
and $\beta$ may then be readily use to study the cosmological behaviour of
FLRW-spacetimes under classical Hamiltonian general relativity. These are the
variables which are quantized to yield a quantum theory of cosmology.

In Wheeler-de Witt theory, one quantizes $V$ and $\beta$ using the usual Dirac
quantization procedure. In LQC, we instead quantize the canonical variables
using the quantization procedure of loop quantum gravity (LQG) – which makes
use of an alternative representation of the Weyl algebra – to obtain a
different quantum cosmological model. In this sense, LQC is essentially a
symmetry reduced model of LQG with only a single degree of freedom (the scale
factor $a(t)$ on each time slice).

Taking volume to be the coordinate observable of the resulting quantum theory
of spatially flat, homogeneous and isotropic cosmology, the kinematic volume
Hilbert space (in the so-called polymer representation [13]) is the non-
separable Hilbert space containing all countable complex linear combinations
of the basis $\\{|V\rangle|V\in\mathbb{R}\\}$ (whose coefficients $c_{i}$
satisfy $\sum_{i}|c_{i}|^{2}<\infty$). The inner-product is defined by
$\langle V_{x}|V_{y}\rangle=\delta_{xy}$, the Kronecker-$\delta$. The sign of
$V$ indicates manifold orientation. Thus, for any positive volume $V$, the
value $-V$ here indicates the same volume with the opposite orientation, not a
negative volume as such; there is no new physics introduced with negative
values of $V$. (That said, it has been proposed by Christodoulou et al. [14]
that fermionic phases could be used to detect this orientation.)

There are two natural of operators to consider on this space; the volume
operator $\hat{V}$, and the holonomy operator $\hat{h}_{\lambda}$ (also called
the shift), which generates translations in the volume coordinate. Following
the loop quantization procedure, these two operators are defined on the
polymer Hilbert space in the following way:

$\hat{V}|V\rangle=V|V\rangle,\qquad\hat{h}_{\lambda}|V\rangle=\left|V+\frac{\alpha\lambda}{V_{0}}\right\rangle$
(1)

where $\alpha=4\pi G\gamma\hbar$ is a constant and $\lambda$ is the holonomy
length. We see that $\hat{h}_{\lambda}$ depends on $V_{0}$. This will be
important in what follows, for it ensures that the $\hat{h}_{\lambda}$
operator is never closed on superselection sectors unless $\lambda$ is made to
scale with $V_{0}$.

While a coordinate translation arises from exponentiating the conjugate
momentum in ordinary quantum mechanics, complications arise when one tries to
extract a conjugate momentum observable $\hat{\beta}$ from $\hat{h}_{\lambda}$
(i.e. a holonomy flux operator), where periodicity and other issues become
apparent (see, for instance, [12]). Specifically, on the polymer Hilbert
space, $\hat{h}_{\lambda}$ is not strongly continuous in $\lambda$, and
therefore the Stone-von Neumann theorem which usually ensures a unique
representation of quantum mechanics fails. Thus, there does not exist a unique
operator $\hat{\beta}$ for which
$\hat{h}_{\lambda}=\exp\\{-i\lambda\hat{\beta}\\}$. A different choice of
conjugate momentum may therefore be made, and it is here that the
representation of LQG is different from that of ordinary quantum mechanics.

The canonical observables of LQC are $\hat{V}$ and
$\frac{1}{\lambda}\hat{S}_{\lambda}$ where
$\hat{S}_{\lambda}:=\frac{i}{2}(\hat{h}_{\lambda}-\hat{h}^{*}_{\lambda})$. We
may also define
$\hat{C}_{\lambda}:=\frac{1}{2}(\hat{h}_{\lambda}+\hat{h}^{*}_{\lambda})$.These
operators are both self-adjoint and if there did exist a unique $\hat{\beta}$
such that $\hat{h}_{\lambda}$ were its exponentiation, then
$\hat{S}_{\lambda}$ and $\hat{C}_{\lambda}$ would look like its sine and
cosine and in small $\lambda$, $\frac{1}{\lambda}\hat{S}_{\lambda}$ would
approximate $\hat{\beta}$, whence it resembles the conjugate momentum of
$\hat{V}$.

These are the observables which enable one to make use of the Hamiltonian
formalism to study the dynamics of quantum models of a spatially flat,
homogeneous and isotropic spacetime. However, the analysis to come occurs
solely in the kinematic Hilbert space in which one defines quantum states for
spatial geometries in terms of their volume. That is, the dynamics are
irrelevant for our considerations. As such, the following discussion is in
principle applicable to a broader class of quantum cosmological models.
Spatial homogeneity is needed for volume regularization to be well-motivated.
Otherwise, any spacetime geometries whose quantization yields the above
kinematic Hilbert space and which takes $\hat{V}$ and
$\frac{1}{\lambda}\hat{S}_{\lambda}$ to be the canonical observables may be
subjected to this analysis.

The most comprehensive summary of LQC beyond the spatially flat, homogeneous
and isotropic case is found in [5]. Here, we see that, while the Hamiltonian
constraint which generates the dynamics of LQC is different for different
models, the kinematic Hilbert space and canonical observables are unchanged in
the positive-curvature ($k=+1$) FLRW-spacetimes as well as those models with a
non-zero cosmological constant, so much of our analysis survives in these
settings as well. The $\Lambda>0$ case yields challenges in ensuring that the
phase space may be extended over the entire dynamics, but this does not impact
the kinematics at a fixed time. Negative-curvature models ($k=-1$) require
different operators and so demonstrating analogous results in that setting is
non-trivial.

There is an important caveat when considering different cosmological models:
the arbitrary scaling of the fiducial volume $V_{0}$ which plays an important
role in gaining analytic control over divergent fluctuations is only possible
in spatially non-compact spacetimes. In compact spacetimes (such as $k=+1$),
one may only take $V_{0}$ to be the actual finite volume of spacetime and no
larger. In such cases, the ‘taming’ procedure described is of limited use.

With these basic notions from LQC established, we now discuss the fluctuations
of these observables.

##  III Uncertainty

Let $\hat{A}$ and $\hat{B}$ be two symmetric operators on some Hilbert space
$\mathcal{H}$. Then the Robertson-Schrödinger uncertainty relation is given by

$\Delta_{A}^{2}\Delta_{B}^{2}\geq\left|\frac{1}{2}\langle\\{\hat{A},\hat{B}\\}\rangle-\langle\hat{A}\rangle\langle\hat{B}\rangle\right|^{2}+\left|\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle\right|^{2}$
(2)

where $\\{\cdot,\cdot\\}$ is the anti-commutator [15, 16]. The Robertson-
Schrödinger relation is simply a more strict bound on the usual Heisenberg
uncertainty relation. Indeed, the Heisenberg relation is obtained from
Equation (2) by truncating the first term (which is always non-negative)
yielding

$\Delta_{A}^{2}\Delta_{B}^{2}\geq\left|\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle\right|^{2}.$
(3)

It is not a priori obvious that there exist states which saturate the either
(2) or (3) for a given pair of observables, but saturation of the latter
implies saturation of the former. The inequality (2) is well-defined provided
it is evaluated for states $|\psi\rangle$ for which $\hat{A}|\psi\rangle$ is
in the domain of $\hat{B}$ and vice-versa [17, 18]. We see that $\hat{V}$ and
$\hat{h}_{\lambda}$ satisfy this condition for all LQC states.

Rovelli and Wilson-Ewing [10] provided a detailed analysis of the influence of
fluctuations on cosmological effects by describing the large-scale
phenomenology of LQC under the assumption that the lower bound of the
Heisenberg relation is saturated. However, these results offer limited clarity
without having on hand any states which saturate (3). The typical choice for
such states from ordinary quantum mechanics are Gaussian states. We shall see,
however, that these states do not minimize fluctuation generically in the LQC
setting.

Let us now calculate the Robertson-Schrödinger fluctuations for these
observables in the polymer representation. The commutator and anti-commutator
of $\hat{V}$ with $\hat{h}_{\lambda}$ and $\hat{h}^{*}_{\lambda}$ are:

$\displaystyle[\hat{V},\hat{h}_{\lambda}]=\frac{\alpha\lambda}{V_{0}}\hat{h}_{\lambda},\qquad\\{\hat{V},\hat{h}_{\lambda}\\}=\hat{h}_{\lambda}\left(2\hat{V}+\frac{\alpha\lambda}{V_{0}}\hat{I}\right)$
(4)
$\displaystyle[\hat{V},\hat{h}_{\lambda}^{*}]=-\frac{\alpha\lambda}{V_{0}}\hat{h}^{*}_{\lambda},\qquad\\{\hat{V},\hat{h}_{\lambda}^{*}\\}=\hat{h}^{*}_{\lambda}\left(2\hat{V}-\frac{\alpha\lambda}{V_{0}}\hat{I}\right)$
(5)

Where $\hat{I}$ is the identity operator. Substituting (4) and (5) into (2)
yields

$\Delta_{V}^{2}\Delta_{\frac{1}{\lambda}S_{\lambda}}^{2}\geq\left|\frac{i}{4\lambda}\left\langle\frac{\alpha\lambda}{V_{0}}(\hat{h}_{\lambda}-\hat{h}^{*}_{\lambda})\right\rangle+\frac{i}{2\lambda}\langle(\hat{h}_{\lambda}-\hat{h}^{*}_{\lambda})\hat{V}\rangle-\langle\hat{V}\rangle\left\langle\frac{i}{2\lambda}(\hat{h}_{\lambda}-\hat{h}^{*}_{\lambda})\right\rangle\right|^{2}+\left|\frac{\alpha}{4V_{0}}\langle\hat{h}_{\lambda}+\hat{h}^{*}_{\lambda}\rangle\right|^{2}.$
(6)

But we may readily compute that
$\langle(\hat{h}_{\lambda}-\hat{h}^{*}_{\lambda})\hat{V}\rangle=\langle\hat{V}\rangle\langle\hat{h}_{\lambda}-\hat{h}^{*}_{\lambda}\rangle$.
Thus, we have

$\Delta_{V}^{2}\Delta_{\frac{1}{\lambda}S_{\lambda}}^{2}\geq\left|\frac{i\alpha}{4V_{0}}\langle(\hat{h}_{\lambda}-\hat{h}^{*}_{\lambda})\rangle\right|^{2}+\left|\frac{\alpha}{4V_{0}}\langle\hat{h}_{\lambda}+\hat{h}^{*}_{\lambda}\rangle\right|^{2}.$
(7)

This may be re-written as

$\Delta_{V}^{2}\Delta_{\frac{1}{\lambda}S_{\lambda}}^{2}\geq\left(\frac{\alpha}{2V_{0}}\right)^{2}\left(|\langle\hat{S}_{\lambda}\rangle|^{2}+|\langle\hat{C}_{\lambda}\rangle|^{2}\right).$
(8)

When does this expression reduce to the Heisenberg uncertainty relation (3),
i.e. when does $\langle\hat{S}_{\lambda}\rangle=0$? Without determining a
solution in general, we note that any state $|\psi\rangle$ satisfying
$\psi(V)=\psi(-V)$ will do this. Since spacetime is symmetric under a parity
transformation (i.e. a change of orientation taking $V\mapsto-V$), this holds
in general for physically meaningful LQC states.

Rovelli and Wilson-Ewing [10] discuss the behaviour of LQC states which are
sharply peaked by analyzing the lower bound of the Heisenberg relation (3) and
supposing it is saturated by some such state. However, they do not explicitly
construct any such states to show that this bound is saturated. It is this gap
which we now fill in. It will be shown that, contrary to popular intuition,
Gaussian state do not saturate this lower bound. Nevertheless, under the right
conditions, their fluctuations do approach the lower bound asymptotically in
the limit considered by Rovelli and Wison-Ewing.

##  IV Gaussian States

In ordinary quantum mechanics, Gaussian states play a special role, for they
are sharply peaked. Their probability amplitudes are centered on a particular
point and decay exponentially away from that point at a rate over which we may
have analytic control (by manipulating the variance $\sigma$). Additionally,
they minimize the Heisenberg uncertainty relation (3) between position and
momentum observables. These two facts offer Gaussian states as a natural
choice for semi-classical states: classical systems have no quantum
fluctuations, and have definite locations; Gaussian states approximate both of
these features to maximal precision. Sharpness is an obvious feature of
Gaussian states. However, the fact that they minimize fluctuations is not
trivial.

It has been argued [1] that the constancy of fluctuations of Gaussian states
is also a crucial ingredient their semi-classicality. Time independence
ensures that their fluctuations do not spread out under dynamical evolution,
and thus remain minimal. While this is certainly important, the discussion to
follow shall show that even the condition of instantaneous minimization is
often too much to ask in the LQC setting (i.e. on a single time slice of the
foliation). We therefore leave the study of the time evolution of these
fluctuations open for future investigation.

In the volume representation of LQC, following Willis [19, Eq. 2.4.11],
Gaussian states centered at $V=0$ take the form

$|\psi\rangle=c\sum_{n\in\mathbb{Z}}e^{-(nl)^{2}/2\sigma^{2}}|nl\rangle$ (9)

where $l$ is a chosen lattice spacing and $\sigma>0$ is the Gaussian variance.
The value $c$ is a normalization constant. An analogous definition for more
general coherent states may be found in [12, p. 257]. In LQC, we choose $l$ to
be minimal by taking it to be the Planck length $l_{P}$. The normalization is
given by

$|c|^{2}=\frac{1}{\sum_{n\in\mathbb{Z}}e^{-(nl)^{2}/\sigma^{2}}}.$ (10)

(We here normalize to 1.) If we wish to consider Gaussian states not centered
at $V=0$, we must make a small modification. Spacetime is thought to be have a
parity symmetry under changes in manifold orientation. While field theories
within spacetimes may violate parity symmetries (e.g. the weak interaction in
the Standard Model [20]), the underlying spacetime itself does not. Hence, we
require that quantum states exhibit this parity symmetry as well, and thus
satisfy $\psi(V)=\psi(-V)$ [5, 21]. However, Gaussian states as defined in (9)
violate this symmetry if one simply shifts the given state by a certain number
of lattice sites $\mu\in\mathbb{Z}$. To account for this, if we wish to
consider Gaussian state in the present framework with a non-zero ‘mean,’ we
must include symmetric positive and negative orientation modes. (This value
$\mu$ is not the mean of $\hat{V}$ but rather of $|\hat{V}|$, see Appendix A
for details) Therefore, a generic Gaussian state ‘centered’ around a point
$\mu\in\mathbb{Z}$ will be of the form

$|\psi\rangle=\frac{c}{\sqrt{2}}\sum_{n\in\mathbb{Z}}\left[e^{-(nl-\mu
l)^{2}/2\sigma^{2}}+e^{-(nl+\mu l)^{2}/2\sigma^{2}}\right]|nl\rangle.$ (11)

This may equivalently be written as

$|\psi\rangle=\frac{c}{\sqrt{2}}\sum_{n\in\mathbb{Z}}e^{-(nl)^{2}/2\sigma^{2}}\left(|nl+\mu
l\rangle+|nl-\mu l\rangle\right).$ (12)

The normalization in this case is then given by

$|c|^{2}=\frac{1}{\sum_{n\in\mathbb{Z}}\left[e^{-(nl-\mu
l)^{2}/2\sigma^{2}}+e^{-(nl+\mu l)^{2}/2\sigma^{2}}\right]^{2}}$ (13)

which reduces to (10) when $\mu=0$. Since a choice of units is irrelevant to
the physics of the theory, we shall henceforth set $l=l_{P}=1$. Expanding in
$u:=\exp\\{-1/\sigma^{2}\\}$, one sees that

$|c|^{2}=\frac{1}{2\left(1+e^{-(\mu/\sigma)^{2}}\right)\vartheta_{3}(u)}$ (14)

where $\vartheta_{3}(u)=\vartheta_{3}(0,u)$ is the third Jacobi theta function
defined by

$\vartheta_{3}(z,q):=\sum_{n\in\mathbb{Z}}q^{n^{2}}e^{2niz}.$

This function is well-defined for all $u\in[0,1]$ and so for all values of
$\sigma\neq 0$. One may readily evaluate the following limits:

$\lim_{\sigma\to 0}|c|^{2}=0,\qquad\lim_{\sigma\to\infty}|c|^{2}=\frac{1}{2}.$
(15)

Monotonicity ensures that $0<|c|^{2}<1/2$ for all values of $\sigma>0$ and
$\mu$. We could just as well define Gaussian states in the dual
$\beta$-representation (i.e. holonomy-flux coordinates in
$L^{2}(\mathbb{R}_{\text{Bohr}},d\mu)$) as done in [2, 22, 23], however, the
volume representation is easier to calculate in.

##  V Exact Gaussian Fluctuations

We now compute the exact solutions for the fluctuations of $\hat{V}$ and
$\frac{1}{\lambda}\hat{S}_{\lambda}$ for Gaussian states in the volume
representation. For a given $l$, the super-selection sector of the non-
separable polymer Hilbert space is the subspace generated by the lattice of
volume eigenstates of the form $|nl\rangle$ for $n\in\mathbb{Z}$. Fixing $l=1$
as we have done makes this space unique. Such a sector is isomorphic to the
usual separable Hilbert space $\ell^{2}$, and so if all of the relevant
operators of the theory are closed on such a sector, we may simply discuss
that particular sector, whence the theory reduces to a quantum theory on a
separable Hilbert space.

If one transforms away the the factors of $V_{0}$ in the canonical variables
of the theory before quantization (e.g. [12]), there is no such factor in the
holonomy operator, and so one may choose the value of $\lambda$ to be a scaled
integer multiple of $l$, reducing LQC to a separable Hilbert space. However,
if one keeps the original coordinates, as we have done, the holonomy operator
is not closed on the superselection sector unless $\lambda$ is made
$V_{0}$-dependent. This detail will be important in the following analysis.

It is easy to show, due to parity symmetry, that $\langle\hat{V}\rangle=0$ for
Gaussian states. Following Willis [19], we may compute
$\langle\hat{V}^{2}\rangle$ for Gaussian states using the Poisson summation
formula. Given a function $g(y)$, this formula states that

$\sum_{n\in\mathbb{Z}}g(x+n)=\sum_{n\in\mathbb{Z}}e^{i2\pi
xn}\int_{-\infty}^{\infty}g(y)e^{-i2\pi yn}dy.$

For Gaussian states, we then have

$\displaystyle\langle\hat{V}^{2}\rangle=\sum_{n\in\mathbb{Z}}g(0+n)$
$\displaystyle
g(y):=\frac{|c|^{2}}{2}y^{2}\left(e^{-(y-\mu)^{2}/\sigma^{2}}+e^{-(y+\mu)^{2}/\sigma^{2}}+2e^{-(y^{2}+\mu^{2})/\sigma^{2}}\right)$

We may evaluate this expression (noting that $\mu\in\mathbb{Z}$, whence the
complex phase which arises from integration vanishes) to obtain

$\begin{split}\langle\hat{V}^{2}\rangle=&\sum_{n\in\mathbb{Z}}\int_{-\infty}^{\infty}y^{2}\left(e^{-(y-\mu)^{2}/\sigma^{2}}+e^{-(y+\mu)^{2}/\sigma^{2}}+2e^{-(y^{2}+\mu^{2})/\sigma^{2}}\right)e^{-i2\pi
yn}dy\\\
=&\frac{|c|^{2}\sqrt{\pi}}{2}\sigma\sum_{n\in\mathbb{Z}}e^{-(\pi\sigma
n)^{2}}\left[2\mu^{2}+\sigma^{2}(1+e^{-\mu^{2}/\sigma^{2}})(1-2\pi^{2}n^{2})\right]\\\
=&\frac{|c|^{2}\sqrt{\pi}}{2}\sigma\left\\{\left[2\mu^{2}+\sigma^{2}\left(1+e^{-\mu^{2}/\sigma^{2}}\right)\right]\vartheta_{3}(v)-2\pi^{2}\sigma^{4}e^{-\pi^{2}\sigma^{2}}\left(1+e^{-\mu^{2}/\sigma^{2}}\right)\left(\frac{d}{dv}\vartheta_{3}(v)\right)\right\\}\\\
=&\frac{\sqrt{-\ln v}}{4\sqrt{\pi}\left(1+e^{\mu^{2}\pi^{2}/\ln
v}\right)\vartheta_{3}(v^{1/\pi^{2}\sigma^{4}})}\left\\{\left[2\mu^{2}-\frac{\ln
v}{\pi^{2}}\left(1+e^{\mu^{2}\pi^{2}/\ln
v}\right)\right]\vartheta_{3}(v)-\left(1+e^{\mu^{2}\pi^{2}/\ln
v}\right)\left(\frac{2v(\ln
v)^{2}}{\pi^{2}}\right)\left(\frac{d}{dv}\vartheta_{3}(v)\right)\right\\}\end{split}$
(16)

where $v:=\exp\\{-\pi^{2}\sigma^{2}\\}=u^{\pi^{2}\sigma^{4}}$. Note that this
is independent of $\lambda$ and $V_{0}$. Since $\langle\hat{V}\rangle=0$, we
see that $\Delta^{2}_{\hat{V}}=\langle\hat{V}^{2}\rangle$. These volume
fluctuations are computed numerically and plotted against $\sigma$ in Figure
1.

Figure 1: The magnitude of the fluctuations $\langle\hat{V}^{2}\rangle$ for
Gaussian states with variance $\sigma$ about $\alpha\lambda/V_{0}=1$ with
$\mu=0,1,2$ (with minima increasing with $\mu$).

We may now look at the fluctuations in the conjugate operator
$\frac{1}{\lambda}\hat{S}_{\lambda}$. First, it is known that parity
transformations take $\hat{S}_{\lambda}\mapsto-\hat{S}_{\lambda}$ (cf. the
Appendix in [24]). Thus, for the states in question, we readily see that
$\langle\frac{1}{\lambda}\hat{S}_{\lambda}\rangle=0$. Finally, we have

$\begin{split}\left\langle\left(\frac{1}{\lambda}\hat{S}_{\lambda}\right)^{2}\right\rangle=&\left\langle\left(-\frac{\hat{h}_{\lambda}-\hat{h}^{*}_{\lambda}}{2i\lambda}\right)^{2}\right\rangle\\\
=&\frac{1}{4\lambda^{2}}\left\langle
2\hat{I}-\hat{h}^{2}_{\lambda}-\hat{h}^{*2}_{\lambda}\right\rangle\\\
=&\frac{1}{4\lambda^{2}}\left\langle
2\hat{I}-\hat{h}_{2\lambda}-\hat{h}_{-2\lambda}\right\rangle.\end{split}$ (17)

For the Gaussian state about $\mu$, we have

$\begin{split}\langle\hat{h}_{\pm
2\lambda}\rangle=&\frac{|c|^{2}}{2}\sum_{m,n\in\mathbb{Z}}e^{-m^{2}/2\sigma^{2}}e^{-n^{2}/2\sigma^{2}}\left(\langle
m+\mu|+\langle
m-\mu|\right)\left(\bigg{|}n+\mu\pm\frac{2\alpha\lambda}{V_{0}}\bigg{\rangle}+\bigg{|}n-\mu\pm\frac{2\alpha\lambda}{V_{0}}\bigg{\rangle}\right)\\\
=&\frac{|c|^{2}}{2}\sum_{m,n\in\mathbb{Z}}e^{-m^{2}/2\sigma^{2}}e^{-n^{2}/2\sigma^{2}}\bigg{[}2\delta\left(m-n\mp\frac{2\alpha\lambda}{V_{0}}\right)+\delta\left(m-n+2\mu\mp\frac{2\alpha\lambda}{V_{0}}\right)+\delta\left(m-n-2\mu\mp\frac{2\alpha\lambda}{V_{0}}\right)\bigg{]}\\\
=&\begin{cases}\frac{|c|^{2}}{2}\bigg{[}2u^{2(\pm\alpha\lambda/V_{0})^{2}}\left(\sum_{n\in\mathbb{Z}}u^{n^{2}}u^{\pm
2n\alpha\lambda/V_{0}}\right)+u^{2(\pm\alpha\lambda/V_{0}-\mu)^{2}}\left(\sum_{n\in\mathbb{Z}}u^{n^{2}}u^{2n(\pm\alpha\lambda/V_{0}-\mu)}\right),&\frac{2\alpha\lambda}{V_{0}}\in\mathbb{Z}\\\
\qquad+u^{2(\pm\alpha\lambda/V_{0}+\mu)^{2}}\left(\sum_{n\in\mathbb{Z}}u^{n^{2}}u^{2n(\pm\alpha\lambda/V_{0}+\mu)}\right)\bigg{]}&\\\
0&\text{otherwise}\end{cases}\end{split}$ (18)

where $u=\exp\\{-1/\sigma^{2}\\}$. The last line is obtained by noting from
Cauchy’s criterion that the series converges for all values of the relevant
parameters. Thus, we may expand the sum and evaluate each Kronecker-$\delta$
term independently. We see the dichotomy in cases here because, when
$\frac{2\alpha\lambda}{V_{0}}\notin\mathbb{Z}$, the action of
$\hat{h}_{2\lambda}$ shifts every term of the Gaussian state off of the
permitted lattice sites; it is not closed on a superselection sector. In terms
of operator-algebraic considerations, this dichotomy is related to rotation
algebras, see Appendix B. The inner-product is unforgiving here; since none of
the offset terms line up with any of the other points on the lattice, the
state overlap vanishes everywhere. We may prevent this from happening by
choosing $\lambda$ to be an integer multiple of $V_{0}/2\alpha$. Let us
proceed by considering both cases.

###  V.1 The non-integer case

Suppose that $\frac{2\alpha\lambda}{V_{0}}\notin\mathbb{Z}$. Then we may
readily compute from (17) that

$\left\langle\left(\frac{1}{\lambda}\hat{S}_{\lambda}\right)^{2}\right\rangle=\frac{1}{2\lambda^{2}}.$

Thus, since
$\langle\hat{V}\rangle=\langle\frac{1}{\lambda}\hat{S}_{\lambda}\rangle=0$, we
see that

$\begin{split}\Delta_{V}^{2}\Delta_{\frac{1}{\lambda}S_{\lambda}}^{2}=&\left\langle\hat{V}^{2}\right\rangle\left\langle\left(\frac{1}{\lambda}\hat{S}_{\lambda}\right)^{2}\right\rangle\\\
=&\frac{\sqrt{-\ln v}}{8\lambda^{2}\sqrt{\pi}\left(1+e^{\mu^{2}\pi^{2}/\ln
v}\right)\vartheta_{3}(v^{1/\pi^{2}\sigma^{4}})}\left\\{\left[2\mu^{2}-\frac{\ln
v}{\pi^{2}}\left(1+e^{\mu^{2}\pi^{2}/\ln
v}\right)\right]\vartheta_{3}(v)-\left(1+e^{\mu^{2}\pi^{2}/\ln
v}\right)\left(\frac{2v(\ln
v)^{2}}{\pi^{2}}\right)\left(\frac{d}{dv}\vartheta_{3}(v)\right)\right\\}\end{split}$
(19)

with $v=\exp\\{-\pi^{2}\sigma^{2}\\}$. Overall, the fluctuations here behave
just like $\langle\hat{V}^{2}\rangle$ in Figure 1 with an overall suppression
by a factor $2\lambda^{2}$. If we allow $\lambda$ to be an arbitrary
independent parameter of the theory, this case is generic. Let us now consider
the integer case.

###  V.2 The integer case

Suppose $\frac{2\alpha\lambda}{V_{0}}\in\mathbb{Z}$. From (18), taking
$q=u=\exp\\{-1/\sigma^{2}\\}$ and making a perspicuous choice of $z$ for each
term as the arguments for the Jacobi $\vartheta$-function
$\vartheta_{3}(z,q)$, we obtain

$\begin{split}\langle\hat{h}_{\pm
2\lambda}\rangle=&\frac{|c|^{2}}{2}\bigg{[}2u^{2(\pm\alpha\lambda/V_{0})^{2}}\vartheta_{3}\left(\frac{\mp\alpha\lambda/V_{0}}{i\sigma^{2}},u\right)+u^{2(\pm\alpha\lambda/V_{0}-\mu)^{2}}\vartheta_{3}\left(\frac{\mp\alpha\lambda/V_{0}+\mu}{i\sigma^{2}},u\right)+u^{2(\pm\alpha\lambda/V_{0}+\mu)^{2}}\vartheta_{3}\left(\frac{\mp\alpha\lambda/V_{0}-\mu}{i\sigma^{2}},u\right)\bigg{]}.\end{split}$
(20)

This calculation may be substituted into (17) to obtain an expression for the
expected variance in $\frac{1}{\lambda}\hat{S}_{\lambda}$. This expectation
value is plotted against $\sigma$ in Figure 2. One can see readily that, for
large $\sigma$, these fluctuations become essentially constant at $1/4$.

We may also check the behaviour of these fluctuations is $\mu$ varies. This is
illustrated in Figure 3. Note that this quantity rapidly converges as $\mu$
gets large for any fixed value of $\sigma$ with an asymptotic minimum at $1/4$
as $\sigma\to\infty$ and $\mu\to\infty$.

In the integer case, the sine fluctuations also vary as $\alpha\lambda/V_{0}$
changes. We plot these fluctuations as a function of $\sigma$ for several
integer values of $2\alpha\lambda/V_{0}$ in Figure 4. We see that the rapid
convergence to 1/4 persists, but larger integer values of
$\alpha\lambda/V_{0}$ result in a slower initial rate of convergence.

We may now analyze the overall fluctuations
$\Delta_{V}^{2}\Delta_{\frac{1}{\lambda}S_{\lambda}}^{2}$. By supposing
$\frac{2\alpha\lambda}{V_{0}}\in\mathbb{Z}$, we assume that there exists some
integer $k$ such that $\lambda=kV_{0}/2\alpha$. Thus, substituting this value,
from (16) and (17), the overall expression for the fluctuations in the integer
case is:

$\begin{split}\Delta_{V}^{2}\Delta_{\frac{1}{\lambda}S_{\lambda}}^{2}=&\left\langle\hat{V}^{2}\right\rangle\left\langle\left(\frac{1}{\lambda}\hat{S}_{\lambda}\right)^{2}\right\rangle\\\
=&\frac{\alpha^{2}\sqrt{\pi}\sigma}{4k^{2}V_{0}^{2}\left(1+e^{-\mu^{2}/\sigma^{2}}\right)\vartheta_{3}(e^{-1/\sigma^{2}})}\left\\{\left[2\mu^{2}+\sigma^{2}\left(1+e^{-\mu^{2}/\sigma^{2}}\right)\right]\vartheta_{3}(e^{-\pi^{2}\sigma^{2}})-2\pi^{2}\sigma^{4}e^{-\pi^{2}\sigma^{2}}\left(1+e^{-\mu^{2}/\sigma^{2}}\right)\left(\frac{d}{dv}\vartheta_{3}(v)\right)\right\\}\\\
&\times\bigg{(}2-\frac{1}{4(1+e^{-\mu^{2}/\sigma^{2}})\vartheta_{3}\left(e^{-1/\sigma^{2}}\right)}\bigg{\\{}2e^{-2(\alpha\lambda/V_{0})^{2}/\sigma^{2}}\bigg{[}\vartheta_{3}\left(\frac{-\alpha\lambda/V_{0}}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)+\vartheta_{3}\left(\frac{\alpha\lambda/V_{0}}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)\bigg{]}\\\
&+e^{-2(\alpha\lambda/V_{0}-\mu)^{2}/\sigma^{2}}\bigg{[}\vartheta_{3}\left(\frac{-\alpha\lambda/V_{0}+\mu}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)+\vartheta_{3}\left(\frac{\alpha\lambda/V_{0}-\mu}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)\bigg{]}\\\
&+e^{-2(\alpha\lambda/V_{0}+\mu)^{2}/\sigma^{2}}\bigg{[}\vartheta_{3}\left(\frac{-\alpha\lambda/V_{0}-\mu}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)+\vartheta_{3}\left(\frac{\alpha\lambda/V_{0}+\mu}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)\bigg{]}\bigg{\\}}\bigg{)}\end{split}$
(21)

with $v=\exp\\{-\pi^{2}\sigma^{2}\\}$ in the $\frac{d}{dv}\vartheta_{3}(v)$
term for brevity (where all other $v$ and $u$ terms have been expanded in
$\sigma$).

We may now state the two main results of this section. First, if we analyze
the numerically generated plots for the integer case in Figures 1, 2, 3, and
4, we see that these fluctuations diverge as $\sigma$ gets large, effectively
scaling as $\langle\hat{V}^{2}\rangle/4$. This is very different from what is
observed in ordinary quantum mechanics where the fluctuations of Gaussian
states in their canonical coordinates are not only bounded, but constant for
all values of $\sigma$.

The other important realization from this discussion is that from (21), we
observe

$\lim_{V_{0}\to\infty}\Delta_{V}^{2}\Delta_{\frac{1}{\lambda}S_{\lambda}}^{2}=0.$
(22)

This was the claim posited by Rovelli and Wilson-Ewing in [10]. This result
was, for them, somewhat unexpected since $V_{0}$ is just a regularization
parameter and thus should not be physically significant. However, their
original analysis was non-constructive; they examined the lower bound of (3)
but did not construct explicit semi-classical states for which the
fluctuations saturate this bound in the appropriate limits. Here, we have
provided such explicit states and derived this result exactly. The benefit to
this is that it allows us to see where this surprising result comes from.

This scaling of fluctuations with $V_{0}$ arises only when one introduces a
$V_{0}$-dependence to $\lambda$. One cannot take such a transformation to just
be a scaling of the theory variables because, if $\lambda$ is fixed, the shift
operator $\hat{h}_{\lambda}$ will discontinuously jump between being closed on
the superselection sector, and not being closed. Hence, this dependency is
connected to the basic Hilbert space structure of the theory, and so it is
unsurprising that it should impact the resulting phenomenology. Interestingly,
however, while scaling $V_{0}$ allows one to reduce the fluctuations of these
Gaussian states, it does not ensure that fluctuations are minimal for any
fixed value of $V_{0}$. We shall show this in the next section by comparing
the results presented here with the lower bound of the Robertson-Schrödinger
inequality.

It should be noted that if one generalizes this analysis to the context of
spatially compact spacetimes with volume $V_{\text{Max}}$, then the
regularization parameter $V_{0}$ only makes sense when $V_{0}\leq
V_{\text{Max}}$. Indeed, it is most natural to simply fix
$V_{0}=V_{\text{Max}}$. As such, the $V_{0}\to\infty$ limit is no longer
possible in this context; one may only scale the fiducial volume so far. This
means that there is a limit to how far one may suppress the fluctuations of
Gaussian states using this procedure, and so there will still generally be
many Gaussian states with extremely large fluctuations.

Figure 2: The magnitude of the fluctuations
$\langle\left(\frac{1}{\lambda}\hat{S}_{\lambda}\right)^{2}\rangle$ for
Gaussian states with variance $\sigma$ and volume mean $\mu=0$. Figure 3: The
magnitude of the fluctuations
$\langle\left(\frac{1}{\lambda}\hat{S}_{\lambda}\right)^{2}\rangle$ for
Gaussian states with variance $\sigma$ and mean $\mu$. Figure 4: The magnitude
of the fluctuations
$\langle\left(\frac{1}{\lambda}\hat{S}_{\lambda}\right)^{2}\rangle$ for
Gaussian states with variance $\sigma$ at different fixed integer values of
$\alpha\lambda/V_{0}$ (with $\mu=0$).

##  VI Saturating Uncertainty

We now examine the lower bound of the uncertainty inequality (8) (which
reduces to (3)) for the fluctuations of Gaussian states. Generically, the
lower bound is given by

$\left(\frac{\alpha}{2V_{0}}\right)^{2}|\langle\hat{C}_{\lambda}\rangle|^{2}=\left(\frac{\alpha}{4V_{0}}\right)^{2}|\langle\hat{h}_{\lambda}+h_{-\lambda}\rangle|^{2}.$

Using an identical sector closure argument as above,
$\langle\hat{h}_{\pm\lambda}\rangle=0$ if
$\alpha\lambda/V_{0}\not\in\mathbb{Z}$, whence the lower bound is zero. That
is, we now have a split between the cases when $k=2\alpha\lambda/V_{0}$ is a
is an even integer or not. In case where $k$ is not an even integer, this
lower bound is zero. Otherwise, substituting this value for $\lambda$, the
lower bound may be computed to be

$\begin{split}\left(\frac{\alpha}{2V_{0}}\right)^{2}|\langle\hat{C}_{\lambda}\rangle|^{2}=&\left(\frac{\alpha}{8V_{0}\left(1+e^{-(\mu/\sigma)^{2}}\right)\vartheta_{3}(e^{-1/\sigma^{2}})}\right)^{2}\\\
&\times\bigg{|}2e^{-k^{2}/8\sigma^{2}}\vartheta_{3}\left(\frac{-k}{i4\sigma^{2}},e^{-1/\sigma^{2}}\right)+e^{-2(k/4-\mu)^{2}/\sigma^{2}}\vartheta_{3}\left(\frac{-k/4+\mu}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)\\\
&+e^{-2(k/4+\mu)^{2}/\sigma^{2}}\vartheta_{3}\left(\frac{-k/4-\mu}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)+2e^{-k^{2}/8\sigma^{2}}\vartheta_{3}\left(\frac{k}{i4\sigma^{2}},e^{-1/\sigma^{2}}\right)\\\
&+e^{-2(-k/4-\mu)^{2}/\sigma^{2}}\vartheta_{3}\left(\frac{k/4+\mu}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)+e^{-2(-k/4+\mu)^{2}/\sigma^{2}}\vartheta_{3}\left(\frac{k/4-\mu}{i\sigma^{2}},e^{-1/\sigma^{2}}\right)\bigg{|}^{2}.\end{split}$
(23)

The degree to which this bound is saturated by the actual fluctuations is
given by the difference between (21) and (23). There are three cases to
consider; the non-integer case where $\alpha\lambda/V_{0}\not\in\mathbb{Z}$
whence both sides of the inequality are simplified, the odd-$k$ case where
$2\alpha\lambda/V_{0}\in\mathbb{Z}$ but
$\alpha\lambda/V_{0}\not\in\mathbb{Z}$, whence the exact value is complicated
but the lower bound vanishes, and the even-$k$ case where
$\alpha\lambda/V_{0}\in\mathbb{Z}$, whence both sides have a complicated form.

###  VI.1 Non-integer case

When $2\alpha\lambda/V_{0}\not\in\mathbb{Z}$, the Roberson-Schrödinger-
inequality becomes

$\begin{split}0\leq&\frac{\sqrt{\pi}\sigma}{8\lambda^{2}\left(1+e^{-\mu^{2}/\sigma^{2}}\right)\vartheta_{3}(e^{-1/\sigma^{2}})}\left\\{\left[2\mu^{2}+\sigma^{2}\left(1+e^{-\mu^{2}/\sigma^{2}}\right)\right]\vartheta_{3}(v)-2\pi^{2}\sigma^{4}e^{-\pi^{2}\sigma^{2}}\left(1+e^{-\mu^{2}/\sigma^{2}}\right)\left(\frac{d}{dv}\vartheta_{3}(v)\right)\right\\}.\end{split}$
(24)

Any roots of this expression correspond to all of the instances in which the
Robertson-Schrödinger inequality is saturated. However, by inspecting the
behaviour of $\langle\hat{V}^{2}\rangle$ in Figure 1, we see that it has no
roots. The next problem, then, is to determine under what circumstances these
fluctuations are minimal, if non-zero. But the right-hand side of the
inequality is monotonically increasing so we may conclude that it always
decreases as $\sigma\to 0$ and is thus never minimal. Every Gaussian state in
this case always violates the Robertson-Schrödinger inequality, and is only
made optimal by taking $\sigma$ to be as small as possible.

###  VI.2 Odd-$k$ case

When $k=2\alpha\lambda/V_{0}$ is an odd integer, the exact fluctuations are
given by (21), while the lower bound is zero. Thus, the analysis is identical
to the above so we exclude it for brevity, noting that the uncertainty
inequality is never saturated, and the fluctuations asymptotically diverge
from the lower bound.

###  VI.3 Even-$k$ case

Now suppose that $\alpha\lambda/V_{0}\in\mathbb{Z}$ so that $k$ is an even
integer. In this case, the uncertainty relation is given by (21) being greater
than or equal to (23). One can readily check that, since $e^{-1/\sigma^{2}}<1$
and $e^{-\pi^{2}\sigma^{2}}<1$ for all values of $\sigma>0$, all of the
$\vartheta$-terms in (21) and (23) are suppressed by their exponential
prefactors exponentially quickly as $k$ gets large. Thus, in the large $k$
regime, the fluctuations in the even-$k$ case reduce to the non-integer case.
We therefore see that, for large holonomy lengths $\lambda$, that is the
generic case. We plot the difference between the exact fluctuations and the
lower bound for several values of $k$ in Figure 5. Again, these fluctuations
do not saturate the uncertainty relation.

Figure 5: The difference between the exact fluctuations
$\Delta_{\hat{V}}^{2}\Delta_{\frac{1}{\lambda}\hat{S}_{\lambda}}^{2}$ and the
lower bound given by the Robertson-Schrödinger inequality for a Gaussian state
with variance $\sigma$ and mean $\mu=0$ in the integer case
$\alpha\lambda/V_{0}\in\mathbb{Z}$ (where $\lambda=\frac{kV_{0}}{2\lambda}$
for $k\in\mathbb{Z}$).

The important fact from this section is that, contrary to the case in ordinary
quantum mechanics, Gaussian states do not saturate uncertainty relations in
LQC.

##  VII Squeezed States

We have seen that Gaussian states do not minimize the uncertainty relation
exactly, though may do so asymptotically in the $V_{0}\to\infty$ limit. This
naturally raises the question: are there any well-defined states on the
superselection sector which minimize uncertainty? In this section, I sketch an
argument that, barring the trivial zero-volume state $|0\rangle$, the answer
is likely no. This argument is not a formal proof but rather of a rough
motivation.

For any symmetric operators $\hat{A}$ and $\hat{B}$, it is known [18, p. 244]
that (3) is saturated by a pure state $|\psi\rangle$ if and only if
$|\psi\rangle$ is an eigenstate of $\hat{A}$ or $\hat{B}$, or if
$|\psi\rangle$ is an eigenstate of $\hat{A}+i\xi\hat{B}$ for some
$\xi\in\mathbb{R}$. The last case are the so-called $\xi$-squeezed coherent
states of $\hat{A}$ and $\hat{B}$.

As a simplifying assumption, let us suppose that $\alpha\lambda/V_{0}=1$ so
that $\hat{h}_{\lambda}|V\rangle=|V+1\rangle$. Taking more general integer
values will in principle yield a similar result, but the necessary analysis
becomes too unwieldy to be presented here. Let us suppose for reductio that
$|\psi\rangle$ is a state on the superselection lattice which is an eigenstate
of $\hat{V}+\frac{i\xi}{\lambda}\hat{S}_{\lambda}$ and thus minimizes the
uncertainty relation for the operators $\hat{V}$ and
$\frac{1}{\lambda}\hat{S}_{\lambda}$. Then $|\psi\rangle$ may be written as

$|\psi\rangle=\sum_{n\in\mathbb{Z}}\psi(n)|n\rangle$ (25)

where $\psi(n):\mathbb{Z}\to\mathbb{C}$ obeys $\sum_{n}|\psi(n)|^{2}<\infty$.
Then we have for some eigenvalue $A$:

$\begin{split}A|\psi\rangle=\left(\hat{V}+\frac{i\xi}{\lambda}\hat{S}_{\lambda}\right)|\psi\rangle=&\sum_{n\in\mathbb{Z}}n\psi(n)|n\rangle-\psi(n)\frac{\xi}{2\lambda}\left(|n-1\rangle-|n+1\rangle\right)\\\
=&\sum_{n\in\mathbb{Z}}\left(n\psi(n)-\frac{\xi}{2\lambda}\psi(n+1)+\frac{\xi}{2\lambda}\psi(n-1)\right)|n\rangle=\sum_{n\in\mathbb{Z}}A\psi(n)|n\rangle.\end{split}$
(26)

The coefficients $\psi(n)$ only satisfy this condition if they obey the
difference equation

$2(A-n)\psi(n)=\frac{\xi}{\lambda}[\psi(n-1)-\psi(n+1)]$ (27)

for all $n\in\mathbb{Z}$. This difference equation may be solved to yield:

$\psi(n)=c_{1}I_{n-A}\left(\frac{-\xi}{\lambda}\right)+c_{2}K_{n-A}\left(\frac{\xi}{\lambda}\right)$
(28)

for arbitrary constants $c_{1}$ and $c_{2}$ where $I$ and $K$ are the modified
Bessel functions of the first and second kind, respectively. In order for a
state with coefficients $\psi(n)$ to be an element of the Hilbert space, we
require that

$\sum_{n\in\mathbb{Z}}\left|c_{1}I_{n-A}\left(\frac{\xi}{\lambda}\right)+c_{2}K_{n-A}\left(\frac{-\xi}{\lambda}\right)\right|^{2}<\infty.$
(29)

However, for any fixed $\xi$, as $n\to+\infty$, $K_{n-A}(\frac{\xi}{\lambda})$
diverges, yet $I_{n-A}(\frac{-\xi}{\lambda})$ converges to zero (and so cannot
be scaled to counter-act this divergence) so for this condition to be
satisfied, we must set $c_{2}=0$. Thus, the only viable states are those of
the form

$\psi(n)=c_{1}I_{n-\lambda}\left(\frac{-\xi}{\lambda}\right)$ (30)

However, as $n\to-\infty$, one may check that
$|I_{n-\lambda}(\frac{-\xi}{\lambda})|^{2}$ likewise diverges. Thus, the
boundedness condition further requires that $c_{1}=0$ as well. Hence, there
are no non-zero vectors in this Hilbert space which are $\xi$-squeezed states
for any value of $\xi$. Thus, there are no $\xi$-squeeze states.

The only alternatives, then, for minimizing uncertainty are eigenstates of
$\hat{V}$ or $\frac{1}{\lambda}\hat{S}_{\lambda}$. The only eigenstate of
$\hat{V}$ which respects the requisite parity symmetry $\psi(V)=\psi(-V)$ is
the zero-volume state $|\psi\rangle=|0\rangle$. Thus, this is the trivial
unique volume eigenstate which minimizes uncertainty. It is non-trivial to
compute eigenstates of $\frac{1}{\lambda}\hat{S}_{\lambda}$, but it is
conjectured that any such states will likewise be unphysical, or at least fail
to be sharply peaked, and so fail to be semi-classical.

##  VIII Conclusion

Volume-regularized loop quantum cosmology is not a single theory, but rather a
large class of different theories differentiated from one another by their
associated choice of fiducial volume $V_{0}$ and their chosen holonomy length
$\lambda$. We have here constructed generic families of Gaussian states on
superselection sectors in these theories and computed their fluctuations with
respect to the canonical observables of these theories, namely, volume, and
the sine of holonomy. These fluctuations were then compared with their
fundamental lower bound given by the Robertson-Schrödinger inequality, a
generalization of the uncertainty principle. Three salient results were shown.

  1. 1.

For a Gaussian state with a fixed width $\sigma$ and a fixed parity-symmetric
mean $\mu$, one can always choose a theory of LQC with a sufficiently large
fiducial volume $V_{0}$ such that the fluctuations of this state become
negligible. This fails to be true for spatially compact spacetimes.

  2. 2.

Within a fixed theory of LQC (with a given finite $V_{0}$), one can always
find a Gaussian state with sufficiently large $\sigma$ such that its
fluctuations become arbitrarily large.

  3. 3.

The relation between $\lambda$ and $V_{0}$ play an important role in
determining the phenomenology of a theory of LQC, namely, they determine
whether or not the holonomy operator $\hat{h}_{\lambda}$ is closed on the
relevant superselection sector.

These results indicate that Gaussian states are not universally semi-classical
in theories of LQC and that the semi-classical sector of a theory of LQC
depends sensitively upon its basic constitutive parameters.

##  Appendix A The $|\hat{V}|$ Operator

The ‘physical’ volume of spacetime is independent of its orientation. Thus,
the operator which corresponds to the ‘physical’ volume of a quantum spacetime
is not $\hat{V}$, but rather $|\hat{V}|$, defined on the polymer Hilbert space
by

$|\hat{V}||V\rangle=\begin{cases}V|V\rangle,&V\geq 0\\\
-V|V\rangle,&V<0\end{cases}.$ (31)

In this view, the definition of a Gaussian state given in (11) more closely
resembles the traditional notion of a Gaussian state, for in this case, we see
that $\langle|\hat{V}|\rangle\propto\mu$, and so $\mu$ is properly the mean.
To see this, we may compute

$|\hat{V}||\psi\rangle=\frac{c}{\sqrt{2}}\sum_{n>0}n\left(e^{-(n-\mu)^{2}/2\sigma^{2}}+e^{-(n+\mu)^{2}/2\sigma^{2}}\right)(|n\rangle+|-n\rangle)$
(32)

From which it follows that

$\langle|\hat{V}|\rangle=2|c|^{2}\sum_{n>0}n\left(e^{-(n-\mu)^{2}/\sigma^{2}}+e^{-(n+\mu)^{2}/\sigma^{2}}\right)^{2}$
(33)

This is hard to solve analytically, however, large $n$ terms get exponentially
suppressed and so we may approximate this series for small $\mu$ and non-small
$\sigma$ with a cutoff. Plotting the first 50 terms against $\mu$ with
$\sigma=1,2,3$ in Figure 6, we see that this is nicely fit by $\mu/\sqrt{2}$
(the extra factor arises from the state normalization). We may readily note
that $\hat{V}^{2}=|\hat{V}|^{2}$, and so the rest of the above analysis of
oriented volume fluctuations persists in when orientation is dispenses with.

Figure 6: The first 50 terms of the volume mean $\langle|\hat{V}|\rangle$ for
Gaussian states with mean $\mu$ at values $\sigma=1,2,3$.

##  Appendix B $\hat{h}_{\lambda}$ Closure and Rotation Algebras

In $C^{*}$-algebra theory, the rotation algebra $\mathcal{A}_{\theta}$ is
characterized by the universal property of containing two unitary elements
$U_{1}$ and $U_{2}$ which satisfy

$U_{1}U_{2}=e^{i2\pi\theta}U_{2}U_{1}.$ (34)

There are three cases of this algebra to consider: (i) the trivial commutative
case where $\theta\in\mathbb{Z}$, (ii) the case where $\theta\in\mathbb{Q}$
(called a rational rotation algebra), and (iii) the case where
$\theta\notin\mathbb{Q}$ (called an irrational rotation algebra). The three
cases are radically different, and the spectral theory of the later two
(especially irrational rotation algebras) is notoriously rich.

In the present context, fixing $\lambda$, we see that the $C^{*}$-algebra
generated by $\hat{V}$ and $\hat{h}_{\lambda}$ carries a representation of
$\mathcal{A}_{\theta}$ for $\theta=\frac{2\alpha\lambda}{V_{0}}$.
Specifically, defining a the unitary operator $\hat{U}=e^{i2\pi\hat{V}}$
generated by a series expansion in $\hat{V}$, we readily see that

$\hat{U}\hat{h}_{\lambda}=e^{i2\pi\theta}\hat{h}_{\lambda}\hat{U}.$ (35)

In the instance when $\frac{2\alpha\lambda}{V_{0}}\in\mathbb{Z}$ (whence
$\hat{h}_{\lambda}$ is closed on the superselection sector), we see that
$\mathcal{A}_{\theta}$ is commutative. In the case where $\hat{h}_{\lambda}$
is not closed on the relevant superselection sector, one finds that
$\mathcal{A}_{\theta}$ becomes a non-trivial rotation algebra and has much
richer features.

###### Acknowledgements.

I wish to thank Francesca Vidotto, Edward Wilson-Ewing, and Carlo Rovelli for
valuable discussions and manuscript feedback. I acknowledge the support of the
Natural Sciences and Engineering Research Council of Canada (NSERC), funding
reference number USRA-554658-2020. This work is based on research supported by
the John Templeton Foundation under grant #61048.

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