# Exploring quantum geometry created by quantum matter

Abhay Ashtekar<EMAIL_ADDRESS>Physics Department & Institute for
Gravitation and the Cosmos
The Pennsylvania State University, University Park, PA 16802 U.S.A

###### Abstract

Exactly soluble models can serve as excellent tools to explore conceptual
issues in non-perturbative quantum gravity. In perturbative approaches, it is
only the two radiative modes of the linearized gravitational field that are
quantized. The goal of this investigation is to probe the ‘Coulombic’ aspects
of quantum geometry that are governed entirely by matter sources. Since there
are no gravitational waves in 3 dimensions, 3-d gravity coupled to matter
provides an ideal arena for this task. Our analysis will reveal novel aspects
of quantum gravity that bring out limitations of classical and semi-classical
theories in unforeseen regimes: non-linearities of general relativity can
magnify small quantum fluctuations in the matter sector to large effects in
the gravitational sector. Finally, this analysis leads to thought experiments
that bring out rather starkly why understanding of the nature of physical
reality depends sensitively on the theoretical lens with which it is probed.
As theories becomes richer, new scales emerge, triggering novel effects that
could not be imagined before. The model provides a concise realization of this
well-known chain.

##  I Introduction

This work has several different motivations, stemming from various aspects of
quantum gravity. A number of bold ideas have been put forward in 4-d quantum
gravity that appear to be plausible from one perspective but puzzling –and
even unsettling– from another. Exactly soluble models with local degrees of
freedom –sometimes called midi-superspace in the literature– can be used to
examine these issues _non-perturbatively_. Of course, one has to make
simplifying assumptions to arrive at these models. Therefore, can miss
important aspects of the full theory and one has to exercise due caution in
assessing the conclusions one arrives at. Nonetheless, these models can
provide pointers and bring to forefront new conceptual issues in a crisp
manner.

One such issue concerns the potential implications of ‘trans-Planckian modes’
associated with extremely high frequency fields that arise, e.g., in the
analysis of Hawking radiation. Over the years, there have been suggestions
that an adequate handling of these modes may lead to novel physics by, e.g.,
having to abandon local Lorentz invariance (see, e.g., sctj ). Exactly soluble
models provide a natural arena to analyze this issue. Models we will discuss,
for example, do have unforeseen effects induced by high frequency modes. It is
then natural to ask if they lead to a loss of local Lorentz invariance in the
final non-perturbative theory, or if this symmetry is preserved by the new
physics. Another example comes from the widely held view that the usual
counting of states in quantum field theory becomes inadequate at high
frequencies and the correct counting would lead to a holographic picture in
which physics in the bulk is fully captured by states residing on the boundary
(see, e.g., jb ; ls ; gth  for early works and hol-review  for a recent
review). Is this view supported in exactly soluble models?

A third example comes from another commonly held view –first introduced by
Wheeler jaw – that there is a ‘space-time foam’ at the microscopic level
because of the perpetual quantum fluctuations of the gravitational field. In
Wheeler’s paradigm, these fluctuations are to emerge as novel non-perturbative
effects associated with the quantum nature of geometry, rather than from the
ultraviolet behavior of graviton-mediated interactions in perturbative
approaches. This view then leads one to visualize the ‘non-perturbative ground
state’ of full quantum gravity as having very different micro-structure than
that suggested by the smooth, tame geometry of Minkowski space and the
associated perturbative vacuum. Is this idea borne out in exactly soluble
models? If so, the standard positive energy theorem of classical general
relativity may not extend to non-perturbative quantum gravity; the energy
density in the ground starts could well be Planck density with either sign. Is
this what happens?

Exactly soluble models can also be well-suited to probe an issue at the
opposite corner of the theory: semi-classical gravity. This is a
mathematically well-defined theory in which gravity/geometry is treated
classically, matter fields quantum mechanically, and the two are related by
semi-classical Einstein’s equations in which the right hand side is provided
by the expectation value of stress-energy of quantum matter. It is generally
assumed that the theory would provide a good approximation to the predictions
of quantum gravity so long as space-time curvature is low compared to the
Planck scale. Exactly soluble models can shed light on the validity of this
conjecture. Can one arrive at this theory from full quantum gravity in the
appropriate regime? Are there potential surprises about its domain of
validity?

Another motivation for this work comes from the ongoing debate on whether
quantum matter can inject genuinely quantum information into the gravitational
field, or, if it is only the radiative –or ‘true’– degrees of the
gravitational field that are quantum mechanical (see, e.g. boseetal ; cmvv ;
cbp ; dgw ). The soluble models we consider can be thought of as 3-dimensional
general relativity coupled to the Maxwell (or Klein Gordon) fields. The
3-dimensional viewpoint provides a conceptually clean arena to probe this
issue since the gravitational field is now completely determined by matter;
there are no (gravitational waves or) gravitons in 3 dimensions. Nonetheless,
we will find that quantum geometry does exhibit specific and quintessentially
quantum features. In fact the nonlinear coupling between matter and gravity
can magnify small quantum fluctuations in the matter sector to large quantum
effects in space-time geometry, leading to consequences that could not have
been foreseen classically.

Finally, these soluble models also suggest thought experiment that yield more
general conceptual insights. Our understanding of physical reality is deeply
influenced by the theoretical paradigm we choose to work in. Indeed,
possibilities we envisage and experiments we design to test subsequent
hypotheses are dictated this paradigm. For the specific models we will
discuss, two constants of Nature play a fundamental role: Planck’s constant
$\hbar$ that dictates the size of quantum effects and Newton’s constant $G$
that governs gravitational phenomena. We will find that ‘switching them on and
off’ results in drastic changes in our understanding of the underlying
physical reality.

The material is organized as follows. In section II we discuss the classical
‘midi-superspace models’ and in section III the unforeseen quantum gravity
effects they present us with. Section IV summarizes the main results and puts
them in a broader context. The key mathematical results presented here were
summarized in a Letter aa-prl  quite some time ago. This article provides
detailed derivations –that have been been circulated only in private
communications so far– and examines them from different angles, putting them
in a broader conceptual perspective. Since significant time has lapsed, I will
also discuss related results in other works that have appeared since (in
particular in schmidt ; rgjp ; mena ; beetle ; admt ; boseetal ; cmvv ; cbp ;
dgw ; abs1 ; abs2 ), as well as the thought experiment, mentioned above, that
has not appeared in the literature.

In the main discussion we will work with 2+1 dimensional general relativity
coupled with a scalar or a Maxwell field. Space-time metric $g_{ab}$ has
signature -,+,+ and its derivative operator is denoted by $\nabla$; its
Riemann tensor is defined by
$R_{abc}{}^{d}k_{d}=2\nabla_{[a}\nabla_{b]}k_{c}$, the Ricci tensor by
$R_{ac}=R_{abc}{}^{b}$, its scalar curvature by $R=g^{ac}\,R_{ac}$, and its
Einstein tensor by $G_{ab}$. We will set $c=1$ but display the dependence of
various quantities on $G$ and $\hbar$. For convenience of the reader we note
that, in 3-d, Newton’s constant has dimensions $[G]\,\sim\,M^{-1}$. Therefore,
unlike in 4-d general relativity, there is a natural mass scale in the
classical theory. But one cannot associate a given mass $M_{\circ}$ with a
length; there is no notion of ‘Schwarzschild radius’ of $M_{\circ}$. It is
also helpful to keep in mind the differences that arise in quantum gravity
from the more familiar 4-d situation. While there is a natural notion of
Planck length ($[\hbar G]\sim L$) and therefore of Planck frequency, there is
no notion of Planck mass that would be relevant to quantum gravity. In section
II we will briefly discuss how the 3-d models we consider arise from a
symmetry reduction of 4-d general relativity, a la Kaluza-Klein. There the
4-dimensional fields will explicitly carry a prefix $4$.

##  II Exactly soluble Models

In this section, we first recall the model we will primarily work with and
then briefly discuss other similar models that have been considered in the
literature. These are often referred to as ‘midi-superspace models’ because
they result from a symmetry reduction of 4-dimensional general relativity but
have an infinite number of true degrees of freedom (in contradistinction with
the ‘mini-superspace’ of homogeneous cosmological models which have only a
finite number of degrees of freedom). While these models and their structure
has been discussed in the literature, we review the salient features for
completeness because the paper is intended for a broad audience. This
discussion will also enable us to fix the notation used subsequently.

As mentioned above, in most of this paper we will work primarily in 3 space-
time dimensions and with signature (-,+,+). Let us denote this space-time by
$(M,g_{ab})$. $M$ will be assumed to be topologically $R^{3}$ and our metric
$g_{ab}$ will be asymptotically flat both at null and spatial infinity in the
sense of Refs. aamv ; abs1 ; abs2  respectively. Matter fields will be taken
to be either Maxwell fields $F_{ab}$ or Klein-Gordon fields $\Phi$. As is
well-known, in 3-dimensions there is a duality between the two. Maxwell’s
equations imply that $F_{ab}$ satisfies Maxwell’s equations if an only if its
the dual ${}^{\star}\\!F_{a}:=\frac{1}{2}\,\epsilon_{a}{}^{bc}F_{bc}$ is exact
–i.e., ${}^{\star}\\!F_{a}=:\nabla_{a}\Phi$– where $\Phi$ satisfies the Klein-
Gordon equation $\nabla^{a}\nabla_{b}\Phi=0$. (Here we have used the fact that
the topology of $M$ is trivial). Interestingly the dictionary
${}^{\star}\\!F_{a}:=\nabla_{a}\Phi$ translates the stress-energy tensor
$T_{ab}^{\rm Max}$ to the stress-energy $T_{ab}^{\rm KG}$:

$\displaystyle T_{ab}^{\rm{Max}}$ $\displaystyle\equiv$ $\displaystyle
F_{a}{}^{m}\,F_{bm}-\frac{1}{4}g_{ab}\,F_{mn}F^{mn}=\,{}^{\star}\\!F_{a}\,{}^{\star}\\!F_{b}-\frac{1}{2}{}^{\star}\\!F^{m}{}^{\star}\\!F_{m}\,g_{ab}$
(1) $\displaystyle=$
$\displaystyle\nabla_{a}\Phi\,\nabla_{b}\Phi-\frac{1}{2}\nabla^{m}\Phi\nabla_{m}\Phi\,g_{ab}\,\equiv\,T_{ab}^{\rm
KG}\,.$

Therefore, the pair $(g_{ab},\,\Phi)$ satisfies the coupled Klein-Gordon _and_
Einstein’s equations if and only if the pair $(g_{ab},\,F_{ab})$ satisfies the
Einstein-Maxwell equations. From notational simplicity, we will work with a
Klein-Gordon field $\Phi$ in calculations. But conceptually it will often be
more convenient to to regard the source as a Maxwell field $F_{ab}$ and its
quantum excitations as photons.

Thus, the Klein-Gordon field $\Phi$ and the space-time metric satisfy the
coupled set of equations:

$\Box\Phi=0\qquad{\rm and,}\qquad G_{ab}=8\pi
G\,T_{ab}\quad\Leftrightarrow\quad R_{ab}=8\pi G\,\nabla_{a}\nabla_{b}\Phi\,.$
(2)

Hamiltonian analysis of this system has also been carried out in some detail
mv . But because (2) are rather complicated partial differential equations, we
still do not have good control on detailed properties of their solutions.
However, it is well-known that equations simplify greatly if we restrict
ourselves to the axisymmetric sector of this system consisting of solutions in
which $g_{ab}$ admits a rotational Killing field $\partial/\partial\theta$,
with regular axis, and $\Phi$ is Lie-dragged by it. Then one can cast the
space-time metric in the form

$g_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}=e^{G\Gamma(R,T)}\,(\mathrm{d}T^{2}+\mathrm{d}R^{2})\,+\,R^{2}\mathrm{d}\theta^{2}\,$
(3)

where $T\in(-\infty,\infty)$ and $0\leq R<\infty$. The chart is unique up to
the transformation $T\to T+\rm{const}$, and by inspection $R$ is the norm of
the rotational Killing field. Note that if $\Gamma(R,T)=0$ the metric becomes
the Minkowski metric $g^{\circ}_{ab}$. The second consequence of the
assumption of axisymmetry is technically more powerful: it is easy to check
that a scalar field $\Phi$ satisfies the Klein-Gordon equation $\Box\Phi=0$
w.r.t. the metric $g_{ab}$ if and only if it satisfies $\mathring{\Box}\Phi=0$
w.r.t. the Minkowski metric $g^{\circ}_{ab}$. Therefore, the two equations in
(2) now decouple: we can just solve the wave equation $\mathring{\Box}\Phi=0$
in Minkowski space, obtain a solution $\Phi$ and then use it in $R_{ab}=8\pi
G\,\nabla_{a}\Phi\nabla_{b}\Phi$ to determine the only unknown metric
coefficient, $\Gamma(R,T)$. The final simplification is that this last
equation can be readily solved to obtain

$\Gamma(R,T)=\frac{1}{2}\,\int_{0}^{R}\mathrm{d}\bar{R}\,\bar{R}\,\,\,[\,(\partial_{T}\Phi)^{2}+(\partial_{\bar{R}}\Phi)^{2}\,](\bar{R},T)\,.$
(4)

Note that the right side has a simple interpretation: It is simply the energy
of the scalar field $\Phi$ _w.r.t. the Minkowski metric_ in a disc of radius
$R$ at time $T$. Therefore, in the axisymmetric case, there is a very large
class of asymptotically flat solutions on which we have excellent control abs1
; abs2 . _This is our midi-superspace._

To summarize, in the midi-superspace of interest, to solve the field equations
(2), one only needs an axisymmetric solution $\Phi(R,T)$ to the Klein-Gordon
equation in _Minkowski space-time_ $(M,g^{\circ}_{ab})$. Given $\Phi$, we can
simply write down the metric (3) such that the pair $(g_{ab},\Phi)$ solves the
coupled Einstein-Klein-Gordon equations. This is the _general_ solution in the
axisymmetric sector of 3-dimensional gravity coupled to a Klein-Gordon (or
Maxwell) field. Finally, as is well-known a general solution $\Phi$ of
$\mathring{\Box}\Phi=0$ can be readily constructed by going to the frequency
domain: It is given as an expansion in the basis
$f^{+}_{\omega}(R,T):=J_{0}(\omega R)e^{-i\omega T}$ of positive frequency
solutions, where $J_{0}$ is the zeroth order Bessel function of first kind:

$\Phi(R,T)=\int_{0}^{\infty}\mathrm{d}\omega\,\big{[}\phi(\omega)f^{+}_{\omega}(R,T)\,+\,\bar{\phi}(\omega)\bar{f^{+}}_{\omega}(R,T)\big{]}\,.$
(5)

This is the precise sense in which the midi-superspace under consideration is
an exactly soluble sector of 3-dimensional gravity (and, as we will see, also
of 4-d vacuum gravity). For each choice of a regular function $\phi(\omega)$
we obtain a solution $\Phi(R,T)$ to $\mathring{\Box}\Phi=0$ which then
determines the only unknown coefficient $\Gamma(R,T)$ in the metric (3) such
that $(\Phi,g_{ab})$ satisfy the coupled Einstein-Klein-Gordon system (2).
This is a midi-superspace because the system has one local degree of freedom
that is neatly coded in the function $\phi(\omega)$ in the above formulation.

To make the conceptual considerations sharper, will often focus on solutions
in which $\Phi(R,T)$ has initial data of compact support on a Cauchy slice
(although this restriction can be relaxed by allowing the much milder fall-off
specified in aamv ). It then follows that the $\Phi(R,T)$ vanishes outside
some radius $R=R_{\circ}(T)$, say, for each $T$. Denote by $M^{\prime}$ the
complement of the support of the given solution $\Phi$. In this region
$M^{\prime}$ we have:

$\displaystyle\Gamma(R,T)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\int_{0}^{\infty}\mathrm{d}R\,R\,\,[(\partial_{T}\Phi)^{2}+(\partial_{R}\Phi)^{2}](R,T)\,$
(6) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathrm{d}\omega\,\omega\,|\phi(\omega)|^{2}=:H_{\circ}(\phi)$

where, in the first step, we could extend the initial integral over the
interval $R\in(0,R_{\circ}(T))$ to $R\in(0,\infty)$ because $\Phi(R,T)$
vanishes for $R>R_{\circ}(T)$. Note that $H_{\circ}(\phi)$ is a constant
because it equals the total, conserved energy of the solution $\Phi$ w.r.t.
the Minkowski metric $g^{\circ}_{ab}$; it is independent of $T$. Therefore, on
$M^{\prime}$ the metric assumes the form

$g_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}\mid_{M^{\prime}}\,\,=\,e^{GH_{\circ}}\,(\mathrm{d}T^{2}+\mathrm{d}R^{2})\,+\,R^{2}\mathrm{d}\theta^{2}\,$
(7)

showing that it is flat! Indeed, this is also obvious from the fact that Ricci
tensor $R_{ab}$ vanishes in absence of sources and in 3-dimensions $R_{abcd}$
is determined entirely by $R_{ab}$. However, unless $H_{\circ}=0$, –i.e.,
unless $\Phi$ vanishes identically– the physical metric $g_{ab}$ is not
globally flat: Although it is everywhere smooth on all of $M$, it has a
‘conical structure with a deficit angle’ in $M^{\prime}$. Note also that
$g_{ab}$ admits, a time translation Killing vector $\partial/\partial_{T}$ on
$M^{\prime}$, in addition to the rotational Killing vector. The solution is
asymptotically flat both at spatial and null infinity aamv ; abs1 ; abs2 .

Hamiltonian analysis of this midi-superspace aamv ; aamp  reveals a striking
result: The Hamiltonian $H(g,\Phi)$ –i.e., the generator of the asymptotic
time translation for the full system– is _not_ the energy $H_{\circ}(\phi)$ of
the Klein Gordon field in Minkowski space, but is _non-polynomially_ related
to it:

$H(g,\Phi)=\frac{1}{4G}\,\big{(}1-e^{-4GH_{\circ}(\phi)}\big{)}$ (8)

This is the physical energy of the system in the Einstein-Klein-Gordon theory;
as with the ADM energy in 4-d, it includes all contributions, including those
from gravity.111While the Hamiltonian analysis refers to spatial infinity, $H$
is also the past limit of the Bondi energy at null infinity abs1 . $H$ is
manifestly positive and, in striking contrast to $H_{\circ}$, it is _bounded
above_. This property is _genuinely non-perturbative_ : if we were to expand
$H$ in powers of $G$, to the leading order $G^{\,0}$, we obtain $H_{\circ}$
and if we truncate the series to any finite order, the expression is again
unbounded above. It is only when one sums all the terms that one finds that
$H$ is bounded. For a flat conic space-time, this result was obtained by
Henneaux already in 1983 henneaux  and, for geometries sourced by $N$ point
particles, by Deser, Jackiw and ’t Hooft djt  in 1984. But in fact positivity
and boundedness of $H$ is a general result in 3-d general relativity, so long
as the matter satisfies the positive energy condition
$T_{ab}\,t^{a}_{1}\,t^{b}_{2}\geq 0$ for all time-like vectors
$t^{a}_{1},t^{a}_{2}$ aamv . Finally, note that the non-perturbative form (8)
is possible because the 3-d Newton’s constant has physical dimensions of
inverse mass.

In 4-d gravity the Arnowitt-Deser-Misner (ADM) or the Bondi energy is
expressible as a 2-surface ‘charge-integral’ at infinity that involves only
geometrical/gravitational fields. In the 3-d case the expression (8) of
energy, on the other hand, involves $H_{\circ}$ which is a bulk integral
involving just the matter fields. However, since it measures the deficit angle
of the metric $g_{ab}$ in $M^{\prime}$, it has a geometric interpretation (see
djt  for geometries sourced by point particles). Moreover, as we now explain,
it can also be expressed as a _line integral involving only geometry_. Since
the isometry group in the tangent space of any point of $(M,g_{ab})$ is the
3-d Lorentz group, there is a natural notion of $SU(1,1)$ spinors,
$\lambda_{A}$. By parallel transporting any $\lambda_{A}$ at a point $p\in M$
along a closed curve $C$ starting and ending at $p$ yields another spinor
$\lambda^{\prime}_{A}$ at $p$, related to the original $\lambda_{A}$ via
$\lambda^{\prime}_{A}=U^{(C)}{}_{A}{}^{B}\,(p)\,\lambda_{B}$, where
$U^{(C)}{}_{A}{}^{B}\,(p)$ is an $SU(1,1)$ rotation. Now, ${\rm Tr}\,U^{(C)}$
= $U^{(C)}{}_{A}{}^{A}(p)$ is independent of the choice of the point $p$ on
the curve. Furthermore, since the curvature of $g_{ab}$ vanishes in
$M^{\prime}$, ${\rm Tr}\,U^{(C)}$ is also independent of the choice of the
curve $C$ that lies entirely in $M^{\prime}$. Let us restrict ourselves to
such curves. Then, the trace is a function only of the total energy
$H(g,\Phi)$ of the system:

${\rm Tr}\,U^{(C)}(g)=2\cos\Big{[}\frac{\pi H(g,\phi)}{4G}\Big{]}\,.$ (9)

Consequently, there is a simple thought experiment that we can make in the
asymptotic region to determine $H$, and therefore also the metric coefficient
$g_{RR}$ (or $G_{TT}$) in the source-free region $M^{\prime}$: transport
particles with spin $\frac{1}{2}$ around any closed loop $C$ in $M^{\prime}$,
measure the $SU(1,1)$ rotation $U_{C}(p)$ they undergo, and take its trace.
Since $\Phi$ vanishes on $M^{\prime}$, this experiment can be carried out
_without reference to matter fields_. We will return to this experiment in
section IV.

Finally, there is another striking feature of 3-d gravity: the absence of
gravitational collapse leading to the formation of black holes. This is
related to a fact we noted above: in 3-d there is no analog of the
Schwarzschild radius associated with a given mass $M_{\circ}$. In our midi-
superspace, this feature is seen in detail: For regular initial data for
$\Phi$ (e.g. of compact support), no matter how densely we pack the scalar
field, we obtain a non-singular solution $g_{ab}$ on all of $M={R}^{3}$
without any horizons.

We conclude with a number of remarks addressed primarily to the general
relativity community.

_Remarks:_

(1) An important difference in the asymptotic structure of 4-d and 3-d gravity
is the following. In 4-d we can and do ask that all physical metrics $g_{ab}$
of interest approach a _fixed_ Minkowski metric $g^{\circ}_{ab}$
asymptotically and the information about the mass is encoded in the leading
deviation. In 3-d, as we just saw, $g_{ab}$ does not approach the fixed
Minkowski metric even to the zeroth order because of the factor involving
$H_{\circ}$ that varies from one space-time to another, depending on its
energy content. This fact complicates the discussion of the asymptotic
symmetry group in both spatial aamv  and null abs1  regimes. In particular,
the notion of ‘time translation’ is now more subtle. This issue of time is
discussed in, e.g., aamp ; mena .

(2) The model we presented is a symmetry reduction of the celebrated Einstein-
Rosen waves of 4-d, vacuum general relativity EinRos ; exact . These solutions
have cylindrical symmetry –i.e., two hypersurface orthogonal, commuting
Killing vector fields, a spatial translation (say along direction $z$
direction) and a rotation (around the $z$ axis). The Kaluza Klein reduction is
carried about for the translational Killing field with norm, say $e^{\psi}$
and the 3-d scalar field $\Phi$ is given by: $\Phi=\psi/4\pi G$ abs1 . The
rotational Killing field of ${}^{4}\\!g_{ab}$ descends from the 4-manifold
${}^{4}\\!M$ to the 3-manifold $M$ and becomes a Killing field of the 3-metric
$g_{ab}$. Einstein’s vacuum equations ${}^{4}\\!R_{ab}=0$ in 4-d are
equivalent to the 3-dimensional field equations (2). For details, especially
of the asymptotic analysis from a 4-d perspective, see abs2 . In the context
of Einstein-Rosen waves, $H_{\circ}$ is called the ‘c-energy’ and was
sometimes interpreted as the energy ‘per unit length along the $z$-axis’ in
these waves. Careful Hamiltonian analysis aamv ; aamp  shows that it is more
appropriate to assign this interpretation to $H$.

(3) There is an interesting variation schmidt  of the midi-superspace of 4-d
Einstein-Rosen waves in which the space-time topology is $R^{2}\times T^{2}$
rather than $R^{4}$ and, consequently, the corresponding $\mathcal{I}^{+}$ has
topology $T^{2}\times R$. In the Kaluza-Klein reduction, the 3-d space-time
has topology $R^{2}\times S^{1}$, rather than $R^{3}$. Locally, the structure
of equations is the same as the one we discussed but there are global
differences. In particular, there is a right and a left spatial infinity. This
model has been analyzed from a Hamiltonian perspective in detail and then used
as a point of departure for non-perturbative quantization in beetle . The
canonical analysis along the same lines as aamp  but with interesting
conceptual differences that arise from the difference in boundary conditions.
Another interesting exactly soluble model is provided by Gowdy space-times,
where the spatial topology is that of a 3-torus $T^{3}$. Here the midi-
superspace is significantly richer in that the 4-d Killing vector used in the
Kaluza-Klein reduction is not hypersurface orthogonal. Consequently, there is
an additional local degree of freedom in the 3-dimensional description. The
phase space and non-perturbative canonical quantization of this model is
discussed in mena  in the setting of 4-dimensional connection-dynamics.

##  III Non-perturbative quantum theory

This section is divided into three parts. In the first we introduce the
quantum framework, in the second we discuss the unforeseen results on the
nature of quantum geometry that it leads to, and in the third we discuss a
thought experiment that brings out the deep interplay between the notion of
physical reality and the theoretical paradigm used to frame it in.

###  III.1 The framework

Classically the model is exactly soluble because we can decouple the matter
field $\Phi$ and the dynamical space-time metric $g_{ab}$. This procedure led
us to encode the unconstrained degree of freedom of the total system in $\Phi$
satisfying $\mathring{\Box}\,\Phi=0$, or, equivalently, in the freely
specifiable function $\phi(\omega)$. Therefore, a natural strategy is to first
focus on $\Phi$ and quantize it and then investigate the nature of the quantum
geometry $\hat{g}_{ab}$ it determines. This procedure will make the underlying
mathematical structure simple since the first step involves just quantum
fields in 3-d Minkowski space-time $(M,g^{\circ}_{ab})$.

Let us first collect the equations we will need from the textbook quantization
of the Klein-Gordon field $\Phi(R,T)$ in the 3-d Minkowski space-time
$(M,g^{\circ}_{ab})$. The positive and negative frequency expansion (5) of the
solution immediately leads us to the ‘field operator’

$\hat{\Phi}(R,T)=\int_{0}^{\infty}\mathrm{d}\omega\,\big{[}\phi^{+}_{\omega}(R,T)\,\hat{A}(\omega)+\,\overline{\phi^{+}}_{\omega}(R,T)\,\hat{A}^{\dagger}(\omega)\big{]}\,$
(10)

where $\hat{A}(\omega)$ and $\hat{A}^{\dagger}(\omega)$ are the annihilation
and creation operators satisfying
$[\hat{A}(\omega)\,\hat{A}^{\dagger}(\omega^{\prime})]=\hbar\,\delta(\omega,\omega^{\prime})$.
They operate on the symmetric Fock space $\mathcal{F}$, where the underlying
1-particle Hilbert space $\mathcal{H}$ is spanned by $\phi(\omega)$ –or,
equivalently, classical solutions $\Phi(R,T)$– _which have a finite norm_ :

$||\phi||^{2}:=\frac{1}{\hbar}\,\int_{0}^{\infty}\\!\mathrm{d}\omega\,|\phi(\omega)|^{2}\,.$
(11)

Note that, while solutions $\Phi(R,T)$ and their ‘frequency components’
$\phi(\omega)$ are classical concepts that make no reference to $\hbar$,
presence of $\hbar$ is essential in the expression (11) since the norm has to
be dimensionless; (11) is a quintessentially quantum expression. The
Hamiltonian operator $\hat{H}_{\circ}$ has the standard expression

$\hat{H}_{\circ}=\int_{0}^{\infty}\mathrm{d}\omega\,\omega\hat{A}^{\dagger}(\omega)\,\hat{A}(\omega)\,.$
(12)

(There is no explicit factor of $\hbar$ because
$[\hat{A}(\omega),\,\hat{A}^{\dagger}({\omega}^{\prime})]=\hbar\,\delta(\omega,\omega^{\prime})$.)

As is well-known, in the free field theory coherent states play a key role in
the discussion of the relation the quantum theory to classical. Given a
classical solution $\mathring{\Phi}(R,T)$, or equivalently,
$\mathring{\phi}({\omega})$, we have a normalized coherent state
$|\Psi_{\mathring{\phi}}\rangle$:

$\displaystyle|\Psi_{\mathring{\phi}}\rangle$ $\displaystyle=$ $\displaystyle
N\,\,e^{\frac{1}{\hbar}\int\mathrm{d}{\omega}\,\mathring{\phi}({\omega})\hat{A}^{\dagger}(\omega)}\,|0\rangle$
$\displaystyle{\rm where}\quad N$ $\displaystyle=$ $\displaystyle
e^{\frac{1}{2\hbar}\,\int\mathrm{d}\omega\,|\mathring{\phi}({\omega})|^{2}}\quad{\hbox{\rm
is the normalization constant.}}$ (13)

Recall that the expectation values of $\hat{\Phi}(R,T)$ and $\hat{H}$ in this
state just yield the classical solution $\mathring{\Phi}(R,T)$ and its energy:

$\langle\Psi_{\mathring{\phi}}|\hat{\Phi}(R,T)|\Psi_{\mathring{\phi}}\rangle\,=\,\Phi_{\circ}(R,T)\qquad{\rm
and}\qquad\langle\Psi_{\mathring{\phi}}|\hat{H_{\circ}}(R,T)|\Psi_{\mathring{\phi}}\rangle\,=\,H_{\circ}(\mathring{\phi})\,.$
(14)

Furthermore, as is well-known, the state $|\Psi_{\mathring{\phi}}\rangle$ is
sharply peaked at the classical solution $\Phi_{\circ}(R,T)$ in the sense that
the uncertainty in the field and its momentum is saturated and equally
distributed in the appropriate sense at all times $T$. The uncertainty in the
Hamiltonian $H_{\circ}$ is given by:

$\big{(}\Delta\hat{H_{\circ}}\big{)}^{2}=\hbar\int_{0}^{\infty}\mathrm{d}\omega\,\omega^{2}\,|\mathring{\phi}({\omega})|^{2}\quad{\rm
and}\quad\frac{\big{(}\Delta\hat{H_{\circ}}\big{)}}{\big{(}\langle\hat{H}\rangle\big{)}}\,\approx\,\frac{1}{\big{(}\frac{1}{\hbar}\int_{0}^{\infty}\mathrm{d}\omega\,|\mathring{\phi}|^{2}({\omega})\big{)}^{\frac{1}{2}}}\,=\,\frac{1}{\langle\hat{\mathcal{N}}\rangle^{\frac{1}{2}}}\,.$
(15)

In the second equation we have considered a profile
$\mathring{\phi}({\omega})$ that is sharply peaked at some fixed frequency to
bring out the physical meaning of the right side, and denoted the number
operator on the Fock space $\mathcal{F}$ by $\hat{\mathcal{N}}$. Thus, the
relative uncertainty is proportional to the inverse of the square root of the
expected number of ‘photons’ in the coherent state under consideration. More
intense the classical beam $\mathring{\phi}({\omega})$ –i.e., greater the
value of $\int\mathrm{d}\omega\,|\mathring{\phi}|^{2}({\omega})$ relative to
$\hbar$– less the relative uncertainty in the quantum state, and more
trustworthy are the classical results vis a vis the correct quantum answer.
Note that the frequency at which $\mathring{\phi}({\omega})$ is peaked is
irrelevant in this consideration –we could shift the peak to arbitrarily high
frequencies and the classical approximation will continue to be excellent so
long as
$\int_{0}^{\infty}\mathrm{d}\omega\,|\mathring{\phi}|^{2}({\omega})\,\gg\hbar$.
This is a standard result in quantum optics.

But all these considerations referred only to the matter sector –either a
scalar field $\Phi$, or a Maxwell field $F_{ab}=\epsilon_{abc}\nabla^{c}\Phi$–
which, thanks to the integrability of the model, could be analyzed without
knowing the physical metric $g_{ab}$. Now, in the classical theory, Einstein’s
equations determine $g_{ab}$ completely for any given matter field
$\mathring{\Phi}(R,T)$. In the quantum theory, we have chosen to put the true
degree of freedom in $\Phi(R,T)$ and represented it as an operator (10) on
$\mathcal{F}$. Therefore, the metric coefficient $g_{RR}=-g_{TT}$ in the
asymptotic region –namely, $e^{GH_{\circ}}$ of (7)– also becomes an operator
on $\mathcal{F}$:

$\hat{g}_{RR}\mid_{\rm
asym}\,:=\,e^{G\hat{H_{\circ}}}\,=\,e^{G\int_{0}^{\infty}\mathrm{d}{\omega}\,{\omega}\,\hat{A}^{\dagger}(\omega)\,\hat{A}(\omega)}\,.$
(16)

Similarly, the non-perturbative Hamiltonian $H$ of the total system –matter
_and_ gravity– becomes the operator

$\hat{H}\,:=\,\frac{1}{4G}\,\big{(}1-e^{-4G\hat{H_{\circ}}}\big{)}\,$ (17)

on $\mathcal{F}$. These operators encode the non-perturbative information
contained in gravity-matter interactions.

Generally non-trivial models are exactly soluble because they can be mapped to
a trivial model – in our case, the Einstein-Klein-Gordon (or Einstein-Maxwell)
system $(g_{ab},\Phi)$ could be mapped to a free field $\Phi(R,T)$ in
Minkowski space. Non-triviality is then transferred to the map that _relates
the two models_. In our case, the relation is given by (7) and (8) in the
classical theory and (16) and (17) in the quantum theory. In section III.2 we
will find that these equations have certain unforeseen consequences that bring
out the non-triviality of the original Einstein-matter system.

_Remarks:_

1\. While defining various operators one encounters the issue of factor
ordering. For the free Hamiltonian $\hat{H_{\circ}}$ of the scalar field
$\hat{\Phi}$ in Minkowski space, there is an unambiguous answer: normal
ordering used in (12). In view of the relation (7) between the classical
$H_{\circ}$ and metric coefficients $g_{RR}=-g_{TT}$, and the relation (8)
between $H_{\circ}$ and the Hamiltonian $H$ of the full system then led us to
define quantum operators $\hat{g}_{RR}|_{\rm asym}$ and $\hat{H}$. However, it
is natural to ask whether one could also use other choices. For example, could
we not have normal ordered the operator _after_ exponentiation and used
$\hat{H}^{\prime}\,=\,\,:\\!\hat{H}\\!:$ instead? The operator
$\hat{H}^{\prime}$ shares several attractive properties with $\hat{H}$. In
particular, it also annihilates the vacuum state $|0\rangle\in\mathcal{F}$,
and its expectation value in coherent states $|\Psi_{\mathring{\phi}}\rangle$
also yields the value of the classical Hamiltonian $H(\mathring{\phi})$.
However, in sharp contrast to $\hat{H}$, the spectrum of $\hat{H}^{\prime}$ is
the _entire real line_. Given that the classical observable $H$ takes values
only in the finite interval $[0,\,\frac{1}{4G}]$, $\hat{H}^{\prime}$ is an
inadmissible choice for the corresponding quantum observable. The choice we
made in (16) and (17) is free of such drawbacks. In particular, the spectrum
of $\hat{H}$ is precisely $[0,\,\frac{1}{4G}]$.

2\. The ‘field operator’ $\hat{\Phi}(R,T)$ of (10) is in fact an operator-
valued distribution. Consequently, there are technical subtleties in giving
precise mathematical meaning to $\hat{g}_{RR}=-\hat{g}_{TT}$ in full space-
time (discussed, e.g., in aamp ). However, for conceptual issues that are at
the forefront of our analysis, we could bypass this issue by focusing only on
the asymptotic region, where the classical matter fields $\Phi_{\circ}(R,T)$
–at which the coherent states $|\Psi_{\mathring{\phi}}\rangle$ are peaked–
vanish. More precisely, the careful treatment yields the same results as those
we obtained by restricting the discussion to the asymptotic region from the
beginning.

3\. In our framework the true degree of freedom lies in the scalar Field
$\Phi(R,T)$. Therefore, it was natural to construct the Hilbert space of
states for this field and represent geometric observables by appropriate
operators thereon. However, one might be concerned that this procedure is
somewhat unnatural from the perspective of general relativity and _only_ the
physical metric $g_{ab}$ should be used in the quantization procedure. This is
indeed possible using a canonical quantization procedure which refers only to
the physical metric as was first discussed in the 4-d context of cylindrically
symmetric waves in a remarkably early work by Kuchař kuchar , and later in the
3-d context in allen  (without however realizing that the second is a Kaluza-
Klein reduction of the first). But these investigations overlooked subtle
issues related to boundary conditions and the distinction between
diffeomorphisms representing gauge and true dynamics they imply. A careful
handling of these issues is needed to put the Hamiltonian theory and the
canonical quantization procedure on a firm footing. When this is done, the
framework also brings out subtleties associated with the issue of time.
Finally, the Hilbert space ($\mathcal{F}$ in our discussion above) of quantum
states can also be selected without having to introduce the Minkowski metric
$g^{\circ}_{ab}$. For details on both these points, see aamp .

###  III.2 Unforeseen quantum gravity effects

The Hilbert space of quantum states is the Fock space $\mathcal{F}$,
constructed entirely from the matter sector. As we saw, if one uses
observables –such as $\hat{\Phi}$ and $\hat{H_{\circ}}$– that refer only to
the matter sector, then the classical theory is an excellent approximation if
the system is in a coherent state $|\Psi_{\mathring{\phi}}\rangle$. In these
states, the expectation values of quantum observables equal values of their
classical counterparts and their quantum fluctuations are negligible provided
the expected number of ‘photons’, $\langle\Psi_{\mathring{\phi}}|\hat{{\cal
N}}|\Psi_{\mathring{\phi}}\rangle$, is large, irrespective of the choice of
$\mathring{\phi}$. Now, observables –such as $\hat{g}_{RR}$ that encodes the
quantum metric and $\hat{H}$ that represents the Hamiltonian $\hat{H}$ of the
full system, including gravity– are also represented by operators on
$\mathcal{F}$. In the classical theory, values of these observables can be
computed using the matter sector alone. Therefore one’s first expectation
would be that the classical theory would again provide an excellent
approximation for these observables if the system is in a coherent state
$|\Psi_{\mathring{\phi}}\rangle$. However, as we will now show, this
expectation is not borne out. Even though geometry is completely determined by
matter and coherent states $|\Psi_{\mathring{\phi}}\rangle$ are sharply peaked
on classical configurations $\mathring{\phi}$, quantum properties of
gravitational observables in these states can be very different from their
classical analogs. For notational simplicity, from now on we will drop the
suffix ‘asymp’ in $g_{RR}\\!\mid_{\rm asym}$ since our discussion will refer
only to the asymptotic metric.

Let us begin with the action of the operator
$\hat{g}_{RR}=e^{G\hat{H_{\circ}}}$ on coherent states:

$\displaystyle e^{G\hat{H_{\circ}}}|\Psi_{\mathring{\phi}}\rangle\,$
$\displaystyle=$ $\displaystyle
N\,e^{G\hat{H_{\circ}}}\,e^{\frac{1}{\hbar}\int_{0}^{\infty}\,\mathrm{d}{\omega}\,\mathring{\phi}({\omega})\,\hat{A}^{\dagger}(\omega)}\,\,|0\rangle$
(18) $\displaystyle=$ $\displaystyle
N\,e^{G\hat{H_{\circ}}}\,\sum_{n=0}^{\infty}\,\frac{1}{\hbar^{n}\,n!}\,\int_{0}^{\infty}\,\mathrm{d}{\omega}_{1}\ldots\mathrm{d}{\omega}_{n}\,\,\mathring{\phi}({\omega}_{1})\ldots\phi({\omega}_{n})\,\,|{\omega}_{1},\ldots{\omega}_{n}\rangle$
$\displaystyle=$ $\displaystyle
N\,\sum_{n=0}^{\infty}\,\frac{1}{\hbar^{n}\,n!}\,\int_{0}^{\infty}\,\mathrm{d}{\omega}_{1}\ldots\mathrm{d}{\omega}_{n}\,\,\mathring{\phi}({\omega}_{1})\ldots\phi({\omega}_{n})\,\,e^{G\hbar({\omega}_{1}+\ldots+{\omega}_{n})}\,\,|{\omega}_{1},\ldots{\omega}_{n}\rangle$
$\displaystyle=$ $\displaystyle
N\,\sum_{n=0}^{\infty}\,\frac{1}{\hbar^{n}\,n!}\,\int_{0}^{\infty}\,\mathrm{d}{\omega}_{1}\ldots\mathrm{d}{\omega}_{n}\,\,\mathring{\bm{\varphi}}({\omega}_{1})\ldots\mathring{\bm{\varphi}}({\omega}_{n})\,\,\,\,|{\omega}_{1},\ldots{\omega}_{n}\rangle\,,$

where, as before,
$N=e^{\frac{1}{2\hbar}\,\int\mathrm{d}{\omega}\,|\mathring{\phi}({\omega})|^{2}}$
is the normalization constant for the coherent state, and in the last step we
have set
$\mathring{\bm{\varphi}}({\omega})=e^{G\hbar{\omega}}\,\mathring{\phi}({\omega})$.
Using the definition of the normalized coherent state
$|\Psi_{\mathring{\bm{\varphi}}}\rangle$, we can express the result as:

$e^{G\hat{H_{\circ}}}|\Psi_{\mathring{\phi}}\rangle\,=\,e^{\frac{1}{2\hbar}\int_{0}^{\infty}\mathrm{d}{\omega}\,|\mathring{\phi}({\omega})|^{2}(e^{2G\hbar\omega}-1)}\,|\Psi_{\mathring{\bm{\varphi}}}\rangle\,.$
(19)

Thus, the action _shifts_ the peak of the coherent state from
$\mathring{\phi}$ to $\mathring{\bm{\varphi}}$ and multiplies it by a
constant. Note that the shift rescales $\mathring{\phi}({\omega})$ by a factor
that is _exponential_ in $G\hbar\omega$. Finally, since the inner-product
between these two coherent states is given by

$\langle\Psi_{\mathring{\phi}}|\Psi_{\mathring{\bm{\varphi}}}\rangle\,=\,e^{-\frac{1}{2\hbar}\,\int_{0}^{\infty}\mathrm{d}{\omega}|\mathring{\phi}({\omega})|^{2}(1+e^{2G\hbar{\omega}}-2e^{G\hbar{\omega}})}$
(20)

the expectation value of $\hat{g}_{RR}|_{\rm asym}$ is given by

$\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}\,|\Psi_{\mathring{\phi}}\rangle\,=\,e^{\frac{1}{\hbar}\int_{0}^{\infty}\mathrm{d}{\omega}\,|\mathring{\phi}({\omega})|^{2}\,(e^{G\hbar\omega}-1)}\,,$
(21)

in contrast to the classical value

$g_{RR}=e^{G\int_{0}^{\infty}\mathrm{d}{\omega}\,{\omega}\,|\mathring{\phi}({\omega})|^{2}}\,.$
(22)

There are several notable differences between the expectation value (21) and
the classical value (22). First, even though the expectation value is
evaluated in a coherent state, it depends explicitly on $\hbar$. Thus, unlike
$\langle\Psi_{\mathring{\phi}}|\hat{\Phi}(R,T)|\Psi_{\mathring{\phi}}\rangle$
or
$\langle\Psi_{\mathring{\phi}}|\hat{H_{\circ}}|\Psi_{\mathring{\phi}}\rangle$,
in the matter sector, in the geometric sector the expectation value (21)
itself carries a signature of quantum effects. Second, since we only had
$\hbar$ at our disposal in the matter sector, we did not have a preferred
frequency scale. With the availability of both $G$ and $\hbar$, we do: The
Planck frequency is given by $\omega_{\rm Pl}=(G\hbar)^{-1}$. Therefore we can
examine the expectation value in various limits by assuming that
$\mathring{\phi}({\omega})$ is sharply peaked at various frequencies
$\mathring{\omega}$. Let us first consider the low frequency limit,
$G\hbar\mathring{\omega}\ll 1$. Then, the expectation value (21) can be
approximated as:

$\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}\,|\Psi_{\mathring{\phi}}\rangle\,\,\approx\,\,e^{G\int_{0}^{\infty}\mathrm{d}{\omega}\,{\omega}\,|\mathring{\phi}({\omega})|^{2}}\,\,\big{(}1\,+{\cal
N}\,(G\hbar\mathring{\omega})^{2}\big{)}\,.$ (23)

where as before ${\cal N}$ is the expected number of ‘photons’ in the state
$|\Psi_{\mathring{\phi}}\rangle$. Thus, we recover the classical value of
$g_{RR}$ _provided the frequency is so low that_ ${\cal
N}\,G\hbar\mathring{\omega}\ll 1$. Interestingly, while in the matter sector
the classical approximation becomes better as ${\cal N}$ increases, for
geometry, it becomes worse as ${\cal N}$ increases even when
$\mathring{\omega}$ is in the low frequency regime!222I thank Don Marolf for
pointing out that in hindsight this second condition can be understood by
first noting the relation $g_{RR}=e^{GH_{\circ}}$ and then using the form (15)
of fluctuations in $\hat{H_{\circ}}$. Although the result refers only to to
the expectation value of $\hat{g}_{RR}$ and not to fluctuations, the argument
makes the disagreement between classical expectations and exact quantum
results plausible. In the high frequency regime, $G\hbar\mathring{\omega}\gg
1$, there is _always a huge disagreement_ between the expectation value and
the classical expression:

$\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}\,|\Psi_{\mathring{\phi}}\rangle\,\,\approx\,\,e^{{\cal
N}(e^{G\hbar\mathring{\omega}})}\quad\hbox{\rm so
that}\quad\frac{\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}\,|\Psi_{\mathring{\phi}}\rangle}{g_{RR}}\,\approx\,e^{{\cal
N}\,(e^{G\hbar\mathring{\omega}}\,-\,G\hbar\mathring{\omega})}\,.$ (24)

In this regime, $\hbar$ does not disappear; in fact terms involving $\hbar$
swamp the classical term since in the numerator $G\hbar\mathring{\omega}$
appears _exponentially_ in exponent itself! And again, larger the expected
number of ‘photons’ in the source, larger is the disagreement between the
quantum and the classical results.

We can also calculate the quantum fluctuations. In the matter sector, these
are very small provided ${\cal N}\gg 1$, irrespective of the frequency at
which $\mathring{\phi}({\omega})$ is concentrated. The situation is again
quite different for quantum geometry. Since
$\hat{G}_{RR}^{2}=e^{2G\hat{H_{\circ}}}$ the same calculation we carried out
to evaluate
$\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}\,|\Psi_{\mathring{\phi}}\rangle$
yields:

$\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}^{2}\,|\Psi_{\mathring{\phi}}\rangle\,=\,e^{\frac{1}{\hbar}\int_{0}^{\infty}\mathrm{d}{\omega}\,|\mathring{\phi}({\omega})|^{2}\,(e^{2G\hbar\omega}-1)}\,,$
(25)

whence the relative uncertainty is given by

$\Big{(}\frac{\triangle\hat{g}_{RR}}{\langle\hat{g}_{RR}\rangle}\Big{)}^{2}\,\,=\,\,[e^{{\frac{1}{\hbar}}\int\mathrm{d}{\omega}\,|\mathring{\phi}(\omega)|^{2}(1-e^{G\hbar{\omega}})^{2}}\,\,\,-\,1]\,.$
(26)

We can again simplify this exact expression using $\mathring{\phi}({\omega})$
that is sharply peaked at some frequency $\mathring{\omega}$. Then, in the low
frequency regime with $G\hbar\mathring{\omega}\ll 1$ we obtain

$\Big{(}\frac{\triangle\hat{g}_{RR}}{\langle\hat{g}_{RR}\rangle}\Big{)}^{2}\,\,\approx\,\,e^{{\cal
N}(G\hbar\mathring{\omega})^{2}}\,\,-1\ ,$ (27)

Again, we find that the uncertainties in geometry grow with the number ${\cal
N}$ of ‘photons’. Thus, there is an interesting tension: To reduce quantum
fluctuations in the matter sector, we need a large number ${\cal N}$ of
‘photons’ in the coherent state $|\Psi_{\mathring{\phi}}\rangle$. However, for
a fixed $\mathring{\omega}$, larger the value of ${\cal N}$ –i.e., more
intense the beam– larger is its influence on gravity. In quantum theory, this
gives rise to larger relative uncertainties. The qualitative nature of this
effect is not surprising. Nonetheless, it is pleasing to see it appear in a
sharp, precise and quantitative form. This is possible because the model is
exactly soluble.

In the high frequency regime, as one would expect from our results on
expectation values, quantum fluctuations in geometry are huge

$\Big{(}\frac{\triangle\hat{g}_{RR}}{\langle\hat{g}_{RR}\rangle}\Big{)}^{2}\,\,\approx\,\,e^{{\cal
N}\,(e^{2G\hbar{\omega}_{o}})}\,$ (28)

since $G\hbar{\omega}\gg 1$ and appears _exponentially_ in the first exponent.
Again, these effects refer to the asymptotic form of the metric in a region in
which the classical profile $\mathring{\phi}$ can be vanishingly small. In the
classical theory the metric in this region is as tame as it can be since the
curvature vanishes. Yet quantum geometry exhibits very large, quintessentially
quantum mechanical effects even though it has no degrees of freedom of its own
and is completely determined by the matter sector!

Finally, for the total Hamiltonian $\hat{H}$ of the matter+gravity system, the
analysis can be carried out using the same techniques. For completeness, let
us list the final results. The expectation value of $\hat{H}$ is given by:

$\langle\Psi_{\mathring{\phi}}|\hat{H}|\Psi_{\mathring{\phi}}\rangle\,=\,{\frac{1}{4G}}\,\,\big{[}1-e^{{\frac{1}{\hbar}}\int_{0}^{\infty}\mathrm{d}{\omega}\,|\mathring{\phi}({\omega})|^{2}(e^{-4G\hbar{\omega}}-1)}\big{]}\,,$
(29)

while the value of the classical Hamiltonian $H(\mathring{\phi})$ is

$H(\mathring{\phi})=\frac{1}{4G}\,\big{(}1-e^{-4G\int_{0}^{\infty}\mathrm{d}{\omega}|\mathring{\phi}({\omega})|^{2}}\big{)}\,.$
(30)

In the low frequency regime, $G\hbar\mathring{\omega}\ll 1$ we have

$\langle\Psi_{\mathring{\phi}}|\hat{H}|\Psi_{\mathring{\phi}}\rangle\approx
H(\mathring{\phi})-\frac{4}{G}{\cal
N}(G\hbar\mathring{\omega})^{2}e^{-4GH_{o}(\mathring{\phi})}\,,$ (31)

while in the high frequency limit $G\hbar\mathring{\omega}\gg 1$ we have:

$\langle\Psi_{\mathring{\phi}}|\hat{H}|\Psi_{\mathring{\phi}}\rangle\approx\frac{1}{4G}\,\big{(}1-e^{-\frac{H_{\circ}(\mathring{\phi})}{\hbar\mathring{\omega}}}\big{)}$
(32)

which is quite different from $H_{\circ}(\mathring{\phi})$! The uncertainties
can also be calculated exactly. But the significance of quantum effects is is
now overshadowed by the fact that the spectrum of $\hat{H}$ is bounded in the
finite interval $[0,\,\frac{1}{4G}]$.

Let us summarize. One of our primary motivations is to analyze the quantum
effects on geometry induced by quantum matter. Therefore, we are led to ask:
Can matter induce interesting quantum effects on the geometrical/gravitational
sectors, even when it is itself in ‘tame’ quantum states? This question led us
to focus on standard coherent states in the matter sector because they provide
canonical ‘quantum representations of classical matter fields’. And we found
that the quantum effects induced on geometry/gravity can be very large even
when quantum fluctuations in the matter sector are small. This unforeseen
behavior arises because of the non-linear structure that is specific to
Einstein dynamics: These non-linearities can magnify small fluctuations in the
matter sector to huge quantum effects on geometry/gravity. Not surprisingly,
the effect is very pronounced when the classical configuration
$\mathring{\phi}({\omega})$ is peaked at a trans-Planckian frequency
$\mathring{\omega}\gg\omega_{\rm Pl}=\frac{1}{\hbar G}$. Then the expectation
value
$\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}\,|\Psi_{\mathring{\phi}}\rangle$
is wildly different from the classical $g_{RR}(\mathring{\phi})$, even in the
asymptotic region where $\mathring{\phi}$ vanishes. But there are interesting
effects also in the ‘tame’ sub-Planckian frequency regime
$G\hbar\mathring{\omega}\ll 1$, because what enters in the explicit
expressions of quantum effects in the gravitational sector is $\sqrt{{\cal
N}}\times G\hbar\mathring{\omega}$ –rather than just
$G\hbar\mathring{\omega}$– where ${\cal N}$ is the expected number of
‘photons’ in the coherent state $|\Psi_{\mathring{\phi}}\rangle$. In the
matter sector, classical theory provides better and better approximation to
the full quantum theory as $\sqrt{{\cal N}}$ increases. Therefore, it is also
natural to consider the limit in which $G\hbar\mathring{\omega}\ll 1$ but
$\sqrt{{\cal N}}\,(G\hbar\mathring{\omega})>1$. In this regime, the matter
sector is extremely well approximated by the classical theory. But (23)
implies that
$\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}\,|\Psi_{\mathring{\phi}}\rangle$
will not be well approximated by the $g_{RR}$ of the classical solution.
Furthermore, (27) implies that in this regime quantum fluctuations will be
small also in the geometrical sector. Thus, the state
$|\Psi_{\mathring{\phi}}\rangle$ will be peaked on a classical geometry but
different from the one determined by classical source
$\mathring{\phi}({\omega})\sim\mathring{\Phi}(R,T)$!

Finally, since $\hat{g}_{RR}$ is a (positive definite) self-adjoint operator
on $\mathcal{F}$, we can use its spectral decomposition to construct states in
$\mathcal{F}$ that are sharply peaked at a classical value
$g_{RR}(\mathring{\phi})$. What happens to the surprising quantum effects in
these states? This interesting issue is discussed in rgjp . It turns out that
the large quantum effects are then transferred to the matter sector. We chose
to focus on $|\Psi_{\mathring{\phi}}\rangle$ because they are better suited to
analyze quantum effects on geometry _induced by matter_ : States in which
matter observables have tame behavior are best suited to bring out this
induced effect.

_Remark:_ In our analysis, we began with the standard description of general
relativity in terms of the metric. Now, in the asymptotically flat context,
there exists in the literature an alternate formulation of Einstein’s
equations where the basic variable is not a metric $g_{ab}$ but a ‘cut-
function’ $Z$ that, at the end of the analysis, determines $g_{ab}$. In the
final picture, $Z$ specifies intersections of light cones, emanating from
space-time points, with $\mathcal{I}^{+}$ NSF1 . The underlying idea was to
pass to quantum general relativity via an appropriate quantization of $Z$
–rather than $g_{ab}$– with the hope that in the final theory space-time
points themselves would become fuzzy NSF2 ; NSF3 . In the 2+1 theory under
consideration, the metric $g_{ab}(R,T)$ is replaced by a single function
$Z(\xi;R,T)$ where $(u,\xi)$ label points of $\mathcal{I}^{+}$ and in the
final picture the 3-parameter family of ‘cuts’ defined by space-time points is
given by $u=Z$. In the 3+1 theory, it has not been possible to implement this
idea beyond linearized approximation. However, it could be carried out non-
perturbatively in the 2+1 theory, using the framework outlined in this section
as point of departure admt . Interestingly, it was found that, even though
there are large quantum fluctuations in the metric, fluctuations in $\hat{Z}$
are strongly damped. In this framework there are dynamical variables that
correspond to space-time points and their fluctuations are similar to those of
$\hat{g}_{RR}$ we found. In this sense, the idea of ‘fuzzy points’ is realized
in an interesting manner: space-time points can lave large fluctuations but
fluctuations of the intersections of their light cones with $\mathcal{I}^{+}$
are highly suppressed.

###  III.3 Lessons from a thought experiment

As explained in section I, exactly soluble models are often well suited to
sharpen conceptual issues. One such issue is: how does one’s description of
physical reality change when one passes from a less accurate theory to a more
accurate one? General relativity, for example, opened entirely new classes of
phenomena and possibilities that could not be envisaged in Newtonian gravity.
These arise because, with both $G$ and the velocity of light $c$ at one’s
disposal, a new scale arises. One can now associate a length with a mass –the
Schwarzschild radius $R_{\rm Sch}=2GM/c^{2}$ that is not available in the
Newtonian theory since it knows only about $G$– and this scale then unleashes
an entirely new class of phenomena. Similarly, in the quantum theory Planck’s
constant $\hbar$ provides new scales, dramatically changing our understanding
of the atomic world and leading to a plethora of unforeseen phenomena that
have shaped physics of the micro-world. In quantum gravity one has access to
all three of these fundamental constants and there have been speculations,
dating back to Planck himself, on the nature of new physics that would arise
at Planck length, Planck frequency, and Planck density. Exactly soluble models
provide a clean-cut platform to discuss the nature of this new physics. In
this subsection we will leave $c$ unchanged but switch on and off $\hbar$ and
$G$ and examine how the nature of physical reality changes.

Consider then a trivial thought experiment in a 3-dimensional space-time:
Switch on a beam of laser light and then switch it off after some time.333The
laser beam would be naturally axisymmetric. We wish to focus just on the light
beam and not details of the source that produced it. Therefore, we will
consider as our system the retarded minus advanced solution, that satisfies
the source-free Maxwell’s equation. Our task is to describe what ‘really’
happens in the space-time around us when this is done. Interestingly, we
obtain four _quite different_ answers to this question: (i) A description of a
_classical physicist_ who ignores both general relativity (GR) and quantum
field theory (QFT) and sees only a Maxwell field propagating in flat space-
time, setting $\hbar=G=0$; (ii) that of a _quantum field theorist_ who knows
of $\hbar$ and describes the laser beam as a coherent state of photons but
sets $G=0$; (iii) that of a _general relativist_ who ignores $\hbar$ but knows
that the beam curves space-time with surprising consequences in the non-
perturbative regime; and, finally, (iv) that of a _quantum gravity theorist_
who knows about both $G$ and $\hbar$. As our discussion of the last two
subsection shows, the fourth description of physical reality has both subtle
and truly novel elements.

Let us begin with the first description. The physical system of interest for
the classical physicist is the Maxwell field
$\mathring{F}_{ab}=\epsilon_{abc}\nabla^{c}\mathring{\Phi}$ that propagates in
a 3-d Minkowski $(M,\,g^{\circ}_{ab})$. It has initial data of compact support
on every Cauchy surface and therefore vanishes in a finite neighborhood of
$i^{\circ}$. The total energy in the system is entirely from the Maxwell field
and equals $H_{\circ}(\mathring{F})$, which can also be read-off at
$\mathcal{I}$ (for detailed expressions, see, abs2 ). It is unbounded above
and increases linearly with the frequency and intensity (i.e.,
$|\mathring{\phi}(\omega)|^{2}$) of the beam. With only the velocity of light
$c$ at one’s disposal, there is no frequency (nor energy) scale in the theory.
Therefore one cannot speak of low or high frequency (nor of low or high
energy) fields and the general description holds for any frequency or
intensity of the laser beam. The second description is that of a quantum field
theorist. With availability of $\hbar$, they can regard the beam as consisting
of photons. They will point out that what the classical physicist referred to
as a classical Maxwell field $\mathring{F}_{ab}\sim\mathring{\Phi}$ is in fact
a coherent state $|\Psi_{\mathring{\phi}}\rangle$ and calculate the expected
number ${\cal
N}=\frac{1}{\hbar}\,\int_{0}^{\infty}\mathrm{d}{\omega}\,|\mathring{\phi}|^{2}$
of photons in it. They now point out that although the expectation value
$\langle\Psi_{\mathring{\phi}}|\,\hat{F}_{ab}(R,T)\,|\Psi_{\mathring{\phi}}\rangle$
of the Maxwell field operator agrees with what the classical physicist’s
$\mathring{F}_{ab}(R,T)$, there are quantum fluctuations that the classical
physicist missed and these would become measurable if ${\cal N}$ were small,
decaying as $1/\sqrt{{\cal N}}$. But in this description, since $G=0$, the
physical space-time is again Minkowski space $(M,g^{\circ}_{ab})$.

The third description is that of general relativist, who has access to $G$ and
Einstein’s equations, but not to $\hbar$. Therefore, they continue to describe
the light beam using the classical Maxwell field. But their description of
physical reality is quite different from that of the classical physicist. Now
the physical metric is $g_{ab}$ rather than $g^{\circ}_{ab}$. It is dynamical
within the support of the beam. More interestingly, although it is flat
outside the support, if one parallel transport a test spinning particle around
a closed curve $C$ lying _entirely_ in this region, one finds that its state
changes when it returns to the starting point. The change is encoded in the
non-trivial holonomy $U^{(C)}$ of (9). Thus, by performing experiments with
probes that never interact with the light beam, they can determine $H_{\circ}$
–whence also the non-trivial metric components $g_{RR}$ ($=-g_{TT})$– and know
that space-time is _not_ described by Minkowski metric $g^{\circ}_{ab}$ even
in the asymptotic region. Perhaps the most startling discovery they would make
is that total energy $H$ of the system –including the gravitational
contributions– is _bounded from above, irrespective of high high the frequency
of the intensity of the laser beam is!_ This qualitatively new phenomenon
occurs because $G^{-1}$ has dimensions of mass that sets the scale for the
upper limit. Furthermore, though a theoretical investigation, the general
relativist would discover that this interesting fact about Nature cannot be
captured using a perturbative expansion in $G$, no matter how high an order
$n$ in $(GM)^{n}$ you truncate the theory.

Finally, let us bring-in the quantum gravity expert. With both $G$ and $\hbar$
available, they can introduce a new notion of Planck frequency $\omega_{\rm
Pl}=\frac{1}{G\hbar}$. With this new scale available, not only can they design
new experiments to probe aspects of physical reality that had remained
unexplored, but also solve some mysteries that the other three were left with.
They can design experiments to probe the quantum nature of geometry in the
asymptotic region where there are no photons at all. Since, furthermore, there
are no gravitons –or radiative degrees of freedom– in the gravitational field,
_these are ‘clean’ tests of the nature of quantum information in geometry
induced entirely by the quantum properties of matter_. If they could create
laser beams peaked at a trans-Planckian frequency
$\mathring{\omega}>1/G\hbar$, they would discover that even in the asymptotic
regions,
$\langle\Psi_{\mathring{\phi}}|\,\hat{g}_{RR}\,|\Psi_{\mathring{\phi}}\rangle$
is _very_ different from $g_{RR}$ predicted by the classical theory. Much to
the surprise of the general relativist, this can occur even when the amplitude
of $\mathring{\phi}({\omega})$ is chosen to be sufficiently small for the
space-time curvature computed using classical Einstein’s equations to be quite
small everywhere. A second regime would surprise the quantum field theorist.
Consider the case when the intensity of the beam is high so that ${\cal N}\gg
1$. Then the quantum fluctuations in the matter sector are small and the
quantum field theorist would expect the induced quantum fluctuations in the
geometry would be at least as small, if not much smaller. We found that this
is not the case; as ${\cal N}$ increases, the relative quantum uncertainties
in $\hat{g}_{RR}$ grow both in the low and high frequency regimes (see (27)
and (28) respectively). Finally, consider the regime
$G\hbar\mathring{\omega}\ll 1$ but ${\cal N}\times G\hbar\mathring{\omega}>1$,
discussed at the end of section III.2. Neither the quantum field theorist nor
the general relativist would have a reason to suspect their descriptions
because they would see nothing unusual in the matter sector. The quantum field
theorist would only see a coherent state with a large number of photons, that
is therefore well-approximated by a smooth, classical Maxwell field
$\mathring{F}_{ab}$, and the general relativist would see the gravitational
field $\mathring{F}_{ab}$ sources. Yet, both descriptions are inadequate. Had
general relativist carried out a careful measure of $g_{RR}$ they would have
found that their theoretical prediction is significantly different from the
observed value. This would have been regarded as an anomaly by the community,
analogous to that in the perihelion shift of mercury before the discovery of
general relativity. Had the quantum field theorist carried out a careful
measurement of the $g_{RR}$ and constructed a statistical distribution, they
would have been puzzled that an irreducible minimum persists in the relative
uncertainty, no matter how accurately they measure this (to them) ‘classical
quantity’; somewhat like the puzzle experienced by Penzias and Wilson before
they knew about the cosmic microwave background. Thus, the quantum gravity
expert would be able to resolve the uncomfortable puzzles that the general
relativity and quantum field theory communities had encountered by confronting
their detailed theoretical calculations with careful measurements in specific
regimes.

##  IV Discussion

In the last two sections we analyzed an exactly soluble sector of non-
perturbative quantum gravity in 2+1 dimensions. Let us now explore the lessons
it offers. In 3+1 quantum gravity, we have a number of paradigms and
conjectures. Are they realized in our soluble sector? Do our results offer new
hints or bring out unforeseen subtleties?

Let us begin with the list of illustrative examples of proposals from section
I. The first proposal pertains the possibility of violation of the local
Lorentz invariance due to trans-Planckian quantum effects. Thanks to the
underlying simplicity of the sector, we could carry out explicit calculations.
They brought to forefront the surprising role of trans-Planckian frequencies.
If the matter profile $\mathring{\phi}({\omega})$ at which the coherent state
$|\Psi_{\mathring{\phi}}\rangle$ is peaked has even a small ‘blip’ at a
frequency $\mathring{\omega}>\omega_{\rm Pl}$, the _induced_ quantum effects
on the space-time geometry become large: even in the asymptotic region away
from the support of the matter source, the classical value $g_{RR}$ is a poor
approximation to the expectation value
$\langle\Psi_{\mathring{\phi}}|\,\hat{G}_{RR}\,|\Psi_{\mathring{\phi}}\rangle$
and the relative quantum fluctuations in $\hat{g}_{RR}$ are large.444Recall
that, starting from section III we have dropped the suffix ‘asym’ in
$\hat{g}_{RR}|_{\rm asym}$. Thus, the intuition that trans-Planckian
frequencies may change the paradigm suggested by classical GR and standard QFT
is indeed borne out in a precise sense. Do these effects, then, lead to the
violation of local Lorentz invariance? The frequency $\mathring{\omega}$ does
refer to the asymptotic time-translation Killing field, $\partial/\partial T$.
However, the _dynamics_ does not break _local_ Lorentz invariance. The field
equation satisfied by $\hat{\Phi}(R,T)$, for example, is fully covariant. In
particular, there is no quantum gravity induced vector field in dynamical
equations that can be the source of such a breakdown. Next, let us consider
the status of ‘holography’ in this model. As in 3+1 dimensions, neither
general relativity nor quantum field theory has a built-in length scale to
define an ultraviolet regime. Quantum gravity does, and we found that this
theory does depart radically from classical GR and QFT in the ultraviolet
regime, $\omega>\omega_{\rm Pl}=\frac{1}{G\hbar}$. Do these effects then force
us to adopt a holographic view in the non-perturbative quantum description? In
the classical theory, the true degrees of freedom are encoded in the matter
field that resides in the 3-manifold $M$. We found that the situation is the
same in non-perturbative quantum theory; the quantum scalar field continues to
reside in the bulk as in any standard quantum field theory. Bulk degrees of
freedom are not replaced by ‘surface degrees of freedom’. Thus, even though
the model exhibits novel features in the Planck regime, they are realized
within the standard bulk degrees of freedom of quantum field theory. I should
add that one _can_ construct the quantum Hilbert space $\mathcal{F}$ using
just null infinity $\mathcal{I}^{+}$ abs1 ; aamp . But this is not what is
meant by ‘holography’. Indeed, one can also construct the standard Fock space
of photons in 3+1 dimensions using just $\mathcal{I}^{+}$.

Next, recall that quantum geometry also has novel features already in the
_low_ frequency regime. In particular, we found that many ‘seemingly tame’
classical solutions are physically ‘spurious’. Consider, for simplicity,
solutions $(\mathring{\Phi},\,g_{ab})$ for which $\mathring{\phi}$ is sharply
peaked at a frequency $\mathring{\omega}\ll\omega_{\rm Pl}$. It would seem
natural to assume that such solutions are ‘tame’ and should arise in the
classical limit of the full quantum theory. That is, one would have expected
the quantum theory to admit states $|\Psi_{\mathring{\phi}}\rangle$ that are
sharply peaked at such solutions. However, we saw that for this to occur,
_two_ inequalities have to be satisfied: ${\cal N}\gg 1$ and ${\cal
N}(G\hbar\mathring{\omega})^{2}\ll 1$. The first is required to ensure that
$|\Psi_{\mathring{\phi}}\rangle$ is sharply peaked at $\mathring{\Phi}$ but
the second inequality is surprising in that it requires that ${\cal N}$ cannot
be ‘too large’! Why is it needed? Suppose ${\cal N}$ is large as per first
inequality and ${\cal N}(G\hbar\mathring{\omega})^{2}=K$. Then
$\langle\Psi_{\mathring{\phi}}|\hat{g}_{RR}|\Psi_{\mathring{\phi}}\rangle\approx
e^{\frac{K}{2}}\,g_{RR}$. Thus, the classical answer would not be a good
approximation even at the expectation value level unless $K\ll 1$.
Consequently, a large class of apparently ‘tame’ classical solutions are
simply not realized as approximations to the quantum theory; from the more
complete perspective of quantum gravity, they are spurious. This surprising
limitation seems to re-enforce Wheeler’s ‘space-time foam’ idea that the
quantum vacuum state would also be quite different from the perturbative
vacuum that is peaked at Minkowski space. Is this expectation borne out?
Recall that the perturbative vacuum $|0\rangle$ is the coherent state
$|\Psi_{\mathring{\phi}}\rangle$ with $\mathring{\phi}=0$, and is annihilated
by the matter Hamiltonian $\hat{H_{\circ}}$. Now, the full Hamiltonian
$\hat{H}$ and the asymptotic metric operator $\hat{g}_{RR}$ are both self-
adjoint and well-defined functions of $\hat{H_{\circ}}$. Therefore,
$|0\rangle$ is an eigenstate of both, with eigenvalue $0$ for $\hat{H}$ and
$1$ for $\hat{g}_{RR}$. Thus the perturbative vacuum $|0\rangle$ is in fact an
eigenstate of the quantum metric operator with Minkowski metric
$g^{\circ}_{ab}$ as its eigenvalue. Contrary to one’s first expectation, there
is nothing unruly about the ground state! Finally, as we noted in section II,
there is a positive energy theorem in classical 2+1 dimensional general
relativity aamv . It goes over to the quantum theory since $\hat{H}$ is a non-
negative self-adjoint operator and the vacuum $|0\rangle$ is the unique state
that is annihilated by it. Thus, again we find that while several aspects of
proposals for novel physics in quantum gravity are realized, closely related
aspects that were assumed to follow are not.

These considerations have implications also on semi-classical gravity, an
approximation in which gravity is treated classically but matter quantum
mechanically and the effect of quantum matter on the classical metric is
computed using semi-classical Einstein’s equations $G_{ab}=8\pi
G\,\langle\,\hat{T}_{ab}\,\rangle$, in which the left side refers to a
classical metric tensor $g_{ab}$ and the right to the expectation value of the
(renormalized) stress-energy tensor of quantum matter that propagates on the
classical geometry defined by $g_{ab}$. It is often assumed that this theory
would serve as an excellent approximation to full quantum gravity away from
the Planck regime. In 3+1 dimensions, exact solutions to semi-classical
gravity are difficult to come by and, more importantly, it has not been
possible to check if they are good approximations because we do not have full
quantum gravity. Let us examine this issue through the lens of our model where
we can work out predictions of full quantum gravity and compare them with the
semi-classical theory. A solution to semi-classical gravity consists in
specifying a classical metric $g_{ab}$ and a quantum state of matter
$|\Psi\rangle$ such that the quantum field $\hat{\Phi}$ satisfies its field
equation with respect to $g_{ab}$ and the pair $(g_{ab},\,\,|\Psi\rangle)$
satisfies the semi-classical Einstein’s equation. Choose a classical solution
$(g_{ab},\Phi)$ in our model. Then it follows that in the asymptotic region
the pair $(g_{ab},\,\,|\Psi_{\mathring{\phi}}\rangle)$ solves semi-classical
equations exactly. Thus, now we have an infinite class of solutions to semi-
classical gravity. However, as we just saw, there are regimes –i.e. choices of
$\mathring{\phi}({\omega})$– for which $g_{ab}$ is a poor approximation to
full quantum geometry. And this occurs even in the asymptotic region where
curvature vanishes identically! This suggests that the domain of validity of
the semi-classical approximation can be quite subtle; it may not be dictated
just by space-time curvature.

Interaction between quantum gravity and quantum information communities has
increased over the last 5 years or so, giving rise to a healthy exchange of
ideas. As mentioned in section I, this dialog has sparked interest in
manifestation of quantum gravity effects that do not involve the ‘radiative
degrees of freedom’ of gravity but are instead induced by the quantum nature
of matter sources (see, e.g. boseetal ; cmvv ; cbp ; dgw ). Thus, they involve
the ‘Coulombic’ part of the gravitational field. As we discussed in section
II, 2+1 gravity is especially well suited to analyze these quantum effects
because the gravitational field does not carry any radiative degrees of
freedom. It is determined entirely by matter: while there is an infinite
number of degrees of freedom, in our analysis they all reside in the matter
sector. Therefore _all_ quantum properties of geometry/gravity are induced by
the quantum properties of matter. We found that the metric has a number of
quintessentially quantum properties. If the quantum state of matter
$|\Psi_{\mathring{\phi}}\rangle$ is sharply peaked on a classical field
$\mathring{\Phi}(R,T)$, we are in a regime that is tame from the perspective
of quantum matter. Yet, the expectation value of $\hat{g}_{RR}$ can differ
significantly from that of the classical solution determined by
$\mathring{\Phi}(R,T)$. The difference is entirely due to non-perturbative
quantum effects, and of course goes to zero in the $\hbar\to 0$ limit. There
is also a trans-Planckian regime in which matter continues to behaves in a
tame manner –its quantum state $|\Psi_{\mathring{\phi}}\rangle$ is sharply
peaked at a classical field
$\mathring{\Phi}(R,T)\sim\mathring{\phi}({\omega})$ with very small
fluctuations– but the quantum metric has experiences very large quantum
fluctuations. In both these regimes, one can carry out thought experiments.
One can use spinning test particles in the asymptotic region where $\Phi(R,T)$
vanishes: there is no direct interaction between quantum matter that sources
the gravitational field and test particles. One can parallel propagate the
spinning particles around a closed loop $C$ and measure the change in its
spin. This results in a measurement of the holonomy $U^{(C)}$ of the
gravitational spin connection. Although the curvature of the physical metric
vanishes everywhere along $C$, the holonomy is non-trivial –this is the
gravitational analog of the well-known Aharonov-Bohm effect in electro-
magnetism. Measurement of this holonomy determines the non-trivial metric
component $g_{RR}=-g_{TT}$ as well as the Hamiltonian/energy of the _total_
system, matter+gravity. One can carry out repeated measurements and determine
the expectation values and quantum uncertainties in these observables. Since
these experiments are carried out entirely in the gravitational sector, they
provide unambiguous test of the quantum nature of geometry/gravity, that is
induced entirely by matter. As we saw, the specific non-linear coupling
between matter and geometry of general relativity can enhance the small
quantum fluctuations in the matter sector to produce large quantum effects in
geometry. These effects are _non-perturbative_ and not considered in current
discussions, where gravity is treated perturbatively and, furthermore, matter
is generally modeled by non-relativistic particles, rather than quantum
fields. Our framework is free of both these limitations. It provides a proof
of principle that quantum information can be transferred to the gravitational
sector from the matter sector through Coulombic interaction without any
reference to gravitons or radiative modes.

It is natural to ask if the effects that were uncovered by our analysis are
specific to gravity or if they would also arise in electromagnetism.555I thank
Norbert Straumann for raising this question. The calculation I summarize below
resulted from that inquiry. Are there interesting quantum effects on the non-
radiative, Coulombic aspects of the Maxwell field by charged quantum fields
that sources them? Now, as we saw, in 2+1 dimensions, Maxwell fields are dual
to scalar fields and even with the axis-symmetric restriction, they carry
radiative degrees of freedom. Therefore, in the 2+1 dimensional
electromagnetic analog of our model the Maxwell field would not be determined
entirely by charged sources; it would have its own radiative degree of
freedom. A better candidate is provided by the spherically symmetric sector of
3+1 dimensional system consisting of Maxwell and charged Klein Gordon fields.
In this sector, there are no photons, just as in our 2+1 dimensional model
there are no gravitons. Therefore, if we use a charged Klein-Gordon quantum
field $\hat{\Phi}$ as the source, the quantum Maxwell field $\hat{F}_{ab}$ is
represented as an operator on the Hilbert space of $\hat{\Phi}$. Thus, any
quantum features we find in the Maxwell field $\hat{F}_{ab}$ are induced by
the quantum Klein-Gordon field via Coulomb interaction. It turns out that this
model is also exactly soluble in Minkowski space-time. Spherical symmetry
singles out a time-translation Killing field $t^{a}$ which one can use to
decompose $F_{ab}$ into its electric and magnetic parts and introduce a
canonical notion of frequency for the Klein-Gordon field. In the asymptotic
region, we can express the only non-trivial component ${E}^{r}$ of the
electric field in terms of the total charge ${Q}$ of the Klein-Gordon field,
just as we could express $g_{RR}=-g_{TT}$ in terms of the total (Minkowski)
energy $H_{\circ}$ of matter sources in the gravitational case. We can again
construct coherent states $|\Psi_{\mathring{\phi}}\rangle$ in the charged
Klein-Gordon sector where (for simplicity) $\mathring{\Phi}$ has compact
support on every Cauchy slice and calculate expectation values and
fluctuations of $\hat{E}^{r}$ in the asymptotic region. They are non-zero.
However, whereas the sophisticated non-linearities of Einstein dynamics in the
2+1 theory imply the relation $g_{RR}=e^{GH_{\circ}}$, the 3+1 Maxwell theory
implies that the relation between $E^{r}$ and $Q$ is linear,
$E^{r}=\frac{Q}{r^{2}}$. Therefore, while quantum fluctuations in the Klein-
Gordon sector do induce quantum effects in the Maxwell sector even in absence
of photons, they are not exponentially magnified as in the gravitational case.
Indeed, the expressions one obtains in the Maxwell case resemble those in the
weak field approximation of our 2+1 gravitational analysis.666In general
relativity, the gravitational interaction ‘dresses’ the bare mass to yield
surprising results also in 3+1 dimensions. These are genuinely non-
perturbative effects that are lost if one expands the result in powers of
Newton’s constant. For a discussion of this phenomenon, see, e.g., Chapter 1
of aa-book .

These concrete insights on a number of conceptual issues could be arrived at
because the model is exactly soluble. However, the very reasons that make it
exactly soluble also limit its reach. As explained in section II, from a 4-d
perspective, the model represents Einstein-Rosen waves which have cylindrical
symmetry. These symmetries make it totally unsuitable for the laboratory
experiments that are being considered. Nonetheless, the analysis brought out a
number of unforeseen effects and conceptual subtleties in 3-d quantum gravity
coupled to matter. To what extent can one take them over to 4 dimensions? An
obvious strategy would be to consider general relativity coupled to scalar
fields in 4-d, but restrict oneself to the spherically symmetric sector. Then
again, we would have an infinite dimensional midi-superspace where there are
again no ‘radiative degrees of freedom’ in the gravitational field. The metric
would be determined entirely by matter and one can investigate quantum gravity
effects that are induced by quantum matter, just as in the present case. But
now, the analysis becomes significantly more complicated mathematically
because one cannot decouple the scalar field dynamics from that of the metric.
From a physical perspective, there is now the possibility of a gravitational
collapse leading to a black hole. While this is a rich sector, it has its own
deep puzzles associated with the Hawking evaporation, and therefore not
directly useful extract clean insights of the type that our 2+1 dimensional
model provided. However, one can focus just on ‘sub-critical’ initial data
sets so that the scalar waves come in from infinity and scatter off to
infinity. This restriction would prevent us from addressing interesting issues
such as the Bekenstein bound jb  and related conjectures ls ; gth  that led to
the idea of holography. But one could still probe the status of other issues
discussed in this section. That analysis will have to take in to account
another key difference between the 2+1 and 3+1 dimensional general relativity.
In 2+1 dimensions, the metric is flat outside sources while in the 3+1 context
it only approaches the flat metric as $1/r$ as one recedes from the matter
sources. Therefore, in 4-dimensional asymptotically flat situations, effects
we found –such as those due to the presence of trans-Planckian frequencies–
will decay as one moves away from sources and the fluctuations will be non-
negligible only near the sources. However, this limitation is present also in
the experiments that are currently contemplated. Thus, while the most striking
predictions of the 3-d model will not carry over to 4-d situations, the
subtleties brought out by the analysis may provide useful guidance. At a
conceptual level, its detailed analysis unearthed directions in which one can
look for novel effects, and at a mathematical level it suggests strategies to
represent matter sources by quantum fields, rather than non-relativistic
particles, and go beyond perturbation theory in the gravitational sector.

## Acknowledgements:

This work was supported by the NSF grant PHY-1806356 and the Eberly Chair
funds of Penn State. I would like to thank Fernando Barbero, Chris Beetle,
Alex Corichi, Rodolfo Gambini, Badri Krishnan, Guillermo Mena, Monica Pierri,
Jorge Pullin, Norbert Straumann, Thomas Thiemann and Madhavan Varadarajan for
numerous discussions. Many of the results were presented at the Erwin
Schrödinger Institute, Vienna; Schmidt-fest at the Albert Einstein Institute,
Potsdam; Gravitational Waves Workshop at IIT Gandhinagar; and seminars
elsewhere. I would like to thank the participants for their comments and
questions that added clarity to the presentation of this work.

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