# The Classical Equations of Motion of Quantized Gauge Theories, Part I:
General Relativity

David E. Kaplan  Department of Physics & Astronomy, The Johns Hopkins
University, Baltimore, MD 21218, USA Tom Melia  Kavli IPMU (WPI), UTIAS, The
University of Tokyo, Kashiwa, Chiba 277-8583, Japan  Surjeet Rajendran
Department of Physics & Astronomy, The Johns Hopkins University, Baltimore, MD
21218, USA

###### Abstract

In this and a companion paper, we show that quantum field theories with gauge
symmetries permit a broader class of classical dynamics than typically
assumed. In this article, we show that the dynamics extracted from the path
integral or Hamiltonian formulation of general relativity allows for classical
states that do not satisfy the full set of Einstein’s equations. This amounts
to loosening the Hamiltonian and momentum constraints that are imposed on the
initial state. Nevertheless, the quantum theory permits gauge invariant time
evolution of these states. The time evolution of these states is such that at
the classical level the full set of Einstein’s equations would appear to hold,
with the physical effects of these states being attributable to an auxiliary,
covariantly conserved energy-momentum tensor with no internal degrees of
freedom. We derive the generalized Einstein equations for these states and
show that a homogeneous and isotropic initial background state contributes to
expansion identical to cold dark matter. The inhomogeneous components of this
state could source curvature perturbations that grow linearly at linear order.
This auxiliary contribution to Einstein’s equations could have either sign and
thus provide a trivial way to violate the null energy condition, enabling
novel gravitational dynamics such as cosmic bounces and wormholes.

###### Contents

  1. 1 Introduction
  2. 2 The Classical Limit
  3. 3 Minisuperspace
    1. 3.1 Canonical Quantization
    2. 3.2 Coherent States and a Path Integral Description
  4. 4 General Relativity
    1. 4.1 Cosmological Implications
  5. 5 Discussion
  6. A Quantum Field Theory on a Manifold
  7. B Coherent States and the Path Integral
    1. B.1 Coherent State Basis Operators

##  1 Introduction

Why do we observe nature obeying certain classical equations of motion? At the
level of classical mechanics, these equations are axioms and our belief in
them is the result of experimental validation. But, the fundamental laws of
nature are quantum mechanical and classical physics is merely a limit of the
quantum theory. In this context, the classical equations of motion are not
axiomatic. Instead, they are identities that relate the expectation values of
various quantum mechanical operators. These identities are automatically
enforced whenever the quantum state evolves per the sole axiomatic equation of
quantum mechanics - Schrödinger’s equation. This equation governs the time
evolution of all quantum states, including that of macroscopic, classical
systems. In particle mechanics, these identities are proven by Ehrenfest’s
theorem. For field theories, similar identities can be obtained through the
Schwinger-Dyson procedure. The purpose of this paper (and its companion) is to
highlight an important subtlety that arises when this procedure is applied to
quantum field theories that possess gauge symmetries and explore the
consequences of this subtlety.

The equations of motion of a classical field theory are obtained by taking a
classical action, varying it and setting the variation to zero. In field
theories with a gauge symmetry, some of the degrees of freedom are not
physical - they can be gauged away. As a result, there are fewer second order
dynamical equations of motion than the number of fields used to represent the
theory in the classical action - these equations govern the evolution of the
physical degrees of freedom of the theory. Naive variations of the action
along the gauge degrees of freedom yield additional equations - but, since
there are no associated physical degrees of freedom corresponding to these,
the resultant equations of motion take the form of constraint equations on the
physical degrees of freedom. There are thus more equations than available
degrees of freedom and one might worry that this system of equations cannot be
consistently solved. However, it can be shown that if the dynamical equations
are given initial conditions that obey the constraints, the evolution
equations, in concert with the conservation equations associated with the
gauge symmetry, will ensure that the time evolution continues to automatically
obey the constraints.

What if the evolution equations were given initial conditions that did not
obey the constraints? The second order equations for the physical degrees of
freedom can still be solved and one would obtain unambiguous time evolution
for these states. In classical mechanics we would reject such states simply
because of their failure to obey the classical constraints. But, classical
mechanics is a limit of quantum mechanics and thus the correct question to ask
is if the classical equations of motion emerge as identities of the quantum
dynamics described by Schrödinger’s equation. Since Schrödinger’s equation is
a dynamical equation, it can enforce the dynamical equations of the classical
theory but not the constraint equations. Do the constraint equations follow
from the quantum theory? In conventional quantization schemes, these
constraint equations are imposed as additional restrictions on the quantum
states of the theory. These restrictions are motivated by a desire to
reproduce the observed classical laws of nature and justified by the belief
that these restrictions are necessary to obtain gauge invariant physics.

But, the only logical restriction that needs to be imposed on the quantum
states of the theory are that they lead to well defined, gauge invariant
physics. The quantum theory, being the deeper underlying theory, can deviate
away from classical physics. With this relaxed but logically consistent
criterion, we show that there are a broader class of quantum states that can
be consistently evolved in gauge theories than conventionally assumed. These
states do not obey the constraint equations, but they are nevertheless well
defined and give rise to gauge invariant physics. The time evolution of these
states will violate the constraint equations - but, as we will see, the gauge
symmetry and the associated conservation equations restrict these variations
to a simple class of deviations. These deviations have observable consequences
and it is thus a matter of experiment to see if the quantum state of our
universe belongs to the set of quantum states that are restricted to obey
these constraint equations or if the state belongs to this broader class that
violates these constraints.

In this paper, we focus on general relativity while the companion paper
focuses on electromagnetism. The rest of this paper is organized as follows.
In Section 2, we review the derivation of the classical equations of motion
using the Schwinger-Dyson procedure and highlight the subtleties that arise in
applying this procedure to gauge theories. Then, proceeding with the
quantization of general relativity, in Section 3, we apply this mathematical
framework to a toy model of gravity, namely minisuperspace, which describes a
homogeneous and isotropic cosmology. With the understanding from this section,
we derive the quantum dynamics of gravity in Section 4. In the final section,
we conclude and discuss a broader set of phenomenological questions for such
theories. We include two appendices for the sake of completeness. In Appendix
A, we describe the mathematical framework necessary to describe general
relativity as a quantum field theory on a manifold and show how the theory can
be described in a gauge invariant manner. In Appendix B, we describe the
coherent state basis in minisuperspace. Throughout we use Greek letters for
four-vector indices, Latin letters for three-vector (spatial) indicies, and
overdots for coordinate time derivatives. We use standard notation for normal
partial derivatives ${}_{,\mu}\equiv\partial/\partial x^{\mu}$.

##  2 The Classical Limit

In this section, we review the Schwinger-Dyson procedure that shows how the
classical equations of motion of a field theory are automatically obeyed by
expectation values of various quantum field operators whenever the time
evolution of the state obeys Schrödinger’s equation. Consider, for example,
the theory of a scalar field $\phi$. In the quantum theory, the field and its
conjugate momentum are promoted to operators $\hat{\phi}$ and $\hat{\pi}$
respectively. A Hamiltonian $\hat{H}$ is constructed from these operators.
Quantum states, constructed for example as a Fock space using the operators
$\hat{\phi}$ and $\hat{\pi}$, evolve as per Schrödinger’s equation. It is then
straightforward to see that the following equations hold: 111We are loose with
notation here. The final terms should be interpreted as functional derivatives
of the classical Hamiltonian with respect to fields $\phi$ and $\pi$ and then
replacing all fields with operators, keeping ordering fixed.

$\displaystyle\partial_{t}\langle\hat{\phi}\rangle=i\left\langle\left[\hat{H},\hat{\phi}\right]\right\rangle=\left\langle\frac{\partial\hat{H}}{\partial\hat{\pi}}\right\rangle$
(1)
$\displaystyle\partial_{t}\langle\hat{\pi}\rangle=i\left\langle\left[\hat{H},\hat{\pi}\right]\right\rangle=-\left\langle\frac{\partial\hat{H}}{\partial\hat{\phi}}\right\rangle$
(2)

These are Hamilton’s equations of motion in expectation value. This can also
be shown using the path integral approach. In the path integral approach, the
time evolution operator $T\left(t_{2};t_{1}\right)$ that describes the time
evolution of a quantum state from time $t_{1}$ to $t_{2}$ is defined in terms
of its transition matrix elements on field basis states $|\phi\rangle$. This
transition matrix element is given by:

$\langle\phi_{f}|T\left(t_{2};t_{1}\right)|\phi_{i}\rangle=\int_{\phi\left(t_{1}\right)=\phi_{i}}^{\phi\left(t_{2}\right)=\phi_{f}}D\phi\,e^{iS[\phi]}$
(3)

where $S=\int{\cal L}d^{4}x$ is the classical action. Redefine the field
variables in the integral by a shift $\phi\rightarrow\phi+\delta\phi$ where
$\delta\phi$ is taken to vanish at the boundaries of integration. The variable
shift should not change the physics of the quantum states. By setting this
variation to zero, we get the following identity:

$\int{\cal D}\phi e^{iS\left[\phi\right]}=\int{\cal D}\phi
e^{iS\left[\phi\right]+i\delta S\left[\phi+\delta\phi\right]}\implies\int{\cal
D}\phi\left(1+i\frac{\delta
S}{\delta\phi}\delta\phi\right)e^{iS\left[\phi\right]}=0$ (4)

Thus, for an arbitrary (but infinitesimal) variation $\delta\phi$, we have

$\left\langle\frac{\delta
S}{\delta\phi}\right\rangle=\left\langle\frac{\delta{\cal
L}}{\delta\phi}-\partial_{\mu}\frac{\delta{\cal
L}}{\delta\left(\partial_{\mu}\phi\right)}\right\rangle=0$ (5)

In other words, the classical equations of motion, derived by demanding that
the classical action is stationary with respect to variations of any field in
the action, is satisfied at the expectation value level.

Observe the following subtlety in applying this procedure to theories with
gauge symmetries. In these, the number of physical degrees of freedom are
smaller than the number of fields used to represent the theory in the
classical action. In the quantum theory, these superfluous or gauge degrees of
freedom are identified by the fact that these degrees of freedom do not posses
corresponding conjugate momenta. Since these degrees of freedom are not
physical, they need to be fixed through some procedure in order to define a
Hamiltonian. But, once the degree of freedom is fixed, what does it mean to
vary the Hamiltonian with respect to this fixed degree of freedom as per (1)?
If the quantum theory is defined using the path integral, the extraction of
the transition matrix elements (3) requires the gauge degrees of freedom to be
factorized. This factorization effectively enforces a gauge choice on the path
integral. Thus the action that appears in the path integral that defines the
transition matrix elements (3) is not the original classical action $S$ but
instead a suitably gauge fixed action $S_{gf}$. Since $S_{gf}\neq S$, the
identities implied by changing the variables of integration in the path
integral are different from the conventional classical equations obtained by
varying $S$ (see discussion in [1]).

Note the key distinction between the above quantum method of deriving the
equations of motion and the classical approach. In the classical theory, the
classical action is first varied and the equation of motion is obtained by
setting these variations to zero. A gauge choice is subsequently made to solve
these equations of motion. In the quantum theory, the definition of the
Hamiltonian or the path integral requires a gauge choice – without such a
choice, either the Hamiltonian has ill-defined operators or the gauge
degeneracies have not been factorized from the path integral. This changes the
derived classical equations. In the following, we describe how these issues
impact the quantization of general relativity and the extraction of the
classical limit of this quantum theory.

##  3 Minisuperspace

We start the discussion with a toy (but illustrative) model of cosmology known
as minisuperspace. It is general relativity in four dimensions and it
describes homogeneous and isotropic space-times. Namely, it is generated from
a metric

$g_{\mu\nu}\rightarrow
ds^{2}=-N\left(t\right)^{2}dt^{2}+a\left(t\right)^{2}\left(dx^{2}+dy^{2}+dz^{2}\right)$
(6)

The spatial degrees of freedom are frozen and the theory describes the time
evolution of the scale factor $a$ in concert with other homogeneuous and
isotropic forms of matter. The action is the standard Einstein-Hilbert term
plus matter

$S_{ms}=\int dt\sqrt{-g}\left({M_{pl}^{2}}R+{\cal L}_{matter}\right)$ (7)

where $M_{pl}=(8\pi G_{N})^{-1/2}$ is the reduced Planck mass, and $R$ and $g$
are the Ricci scalar and the metric determinant respectively. The matter
fields are position independent and the volume integral (an overall volume
factor) has been removed.

Finding the paths that extremize the action leads to the standard Friedmann
equations. For example, for a scalar field $\phi$ with matter action
$\mathcal{L}_{matter}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)$,
and the metric (6), the full action is222We have tacitly integrated by parts
to remove terms with $\ddot{a}$. This is equivalent to including a Gibbons-
Hawking-York term needed to produce the correct equations of motion for
manifolds with boundaries [2, 3].

$S_{ms}=\int
dt\left(-6M_{pl}^{2}\frac{a(t)\dot{a}(t)^{2}}{N(t)}+\frac{a(t)^{3}\dot{\phi}(t)^{2}}{2N(t)}-N(t)a(t)^{3}V(\phi)\right)$
(8)

leading to classical equations of motion (dropping the $t$ arguments of $N$
and $a$ for clarity)

$\displaystyle\frac{\delta S_{ms}}{\delta N}$ $\displaystyle=$ $\displaystyle
a^{3}\left(6M_{pl}^{2}\frac{\dot{a}^{2}}{N^{2}a^{2}}-\frac{\dot{\phi}^{2}}{2N^{2}}-V(\phi)\right)=0$
(9) $\displaystyle\frac{\delta S_{ms}}{\delta a}$ $\displaystyle=$
$\displaystyle
3Na^{2}\left(4M_{pl}^{2}\frac{\ddot{a}}{N^{2}a}+2M_{pl}^{2}\frac{\dot{a}^{2}}{N^{2}a^{2}}-4M_{pl}^{2}\frac{\dot{a}\dot{N}}{N^{3}a}+\frac{\dot{\phi}^{2}}{2N^{2}}-V(\phi)\right)=0$
(10) $\displaystyle\frac{\delta S_{ms}}{\delta\phi}$ $\displaystyle=$
$\displaystyle-\frac{a^{3}}{N}\left(\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}-\frac{\dot{N}\dot{\phi}}{N}+N^{2}\frac{\partial
V(\phi)}{\partial\phi}\right)=0$ (11)

Upon fixing a time coordinate by choosing, for example, $N(t)=1$, the above
reduce to Friedmann’s first and second equations and the equation of motion
for the scalar.

While the above is the standard procedure, we have been a bit cavalier. If we
take the action $S_{ms}$ and redefine the path for this integral
$N(t)\rightarrow N(t)+\delta N(t)$ (with vanishing $\delta N$ at the
boundaries), we would require the shift in the action
$S_{ms}\left[N,a,\phi\right]\rightarrow S_{ms}\left[N+\delta N,a,\phi\right]$
to be trivial. However, it is already true that $S_{ms}\left[N+\delta
N,a,\phi\right]=S_{ms}\left[N,a,\phi\right]$ for any path. This is because a
change in path of $N$ can be compensated for by defining a new time coordinate
integration variable $dt^{\prime}=(1+\frac{\delta N}{N})dt$. This suggests
that in the quantum theory, the path integral does not guarantee this first
equation in the classical limit. We will see how this bears out below.

###  3.1 Canonical Quantization

Now we want a quantum version of this theory. We see that $\dot{N}(t)$ does
not appear in the action and thus is not dynamical. It also plays a central
role in the reparameterization invariance of the time coordinate, (namely,
$dt\rightarrow f(t)dt$ and $N(t)\rightarrow N(t)/f(t)$), and thus is something
like a gauge degree of freedom. To canonically quantize, let’s first look at
the classical Hamiltonian. On the metric side, only $a(t)$ has a conjugate
momentum, $\pi=\delta{\cal L}/\delta\dot{a}=-12M_{pl}^{2}a\dot{a}/N$, since
$\delta{\cal L}/\delta\dot{N}=0$. The scalar’s conjugate variable is
$\pi_{\phi}=\delta{\cal L}/\delta\dot{\phi}=(a^{3}/N)\dot{\phi}$. The
resulting Hamiltonian

$H=\left[\pi\dot{a}+\pi_{\phi}\dot{\phi}-{\cal
L}\right]_{\dot{a}=\cdots,\dot{\phi}=\cdots}=-\frac{N}{24M_{pl}^{2}a}\pi^{2}+\frac{N}{2a^{3}}\pi_{\phi}^{2}+Na^{3}V(\phi)$
(12)

is interestingly $-N(\delta S_{ms}/\delta N)\equiv N\tilde{H}$, or $N$ times
the function that vanishes by the first Friedmann equation (9) when replacing
$\dot{a}$ and $\dot{\phi}$ by functions of their conjugate momenta.

In the quantum theory, there is now the issue of what to do with $N$, which
appears as an auxiliary field that naively commutes with all operators. The
natural thing to do is fix a gauge, noting that the Schrödinger equation with
the above Hamiltonian would be of the form:

$i\partial_{t}|\psi\rangle=N(t)\hat{\tilde{H}}|\psi\rangle$ (13)

and is covariant with respect to time reparameterizations. Thus, to define the
quantum theory, a sensible choice is to pick a time coordinate by way of
fixing $N(t)$.

Now, once we fix $N$, we can write a quantum Hamiltonian operator by
interpreting $a,\pi,\phi,\pi_{\phi}$ as operators with standard commutation
relations. There are remaining issues, such as defining inverse field
operators and operator ordering, which we will address below when we derive
the path integral.

Our main point is that once $N$ is fixed, we only get the following equations
in the gravitational sector:

$\displaystyle\partial_{t}\langle\hat{a}\rangle=-i\langle\left[\hat{\tilde{H}},\hat{a}\right]\rangle$
(14)
$\displaystyle\partial_{t}\langle\hat{\pi}\rangle=-i\langle\left[\hat{\tilde{H}},\hat{\pi}\right]\rangle$
(15)

When appropriately defining the commutators, inverse operators, and operator
ordering, the first equation reproduces the definition of the conjugate field
$\pi$, while the second equation reproduces the second Friedmann equation. The
first Friedmann equation would require $\langle\hat{\tilde{H}}\rangle=0$, but
is not necessarily a consequence of this quantum theory. In the classical
theory, this equation is produced by varying $N$ \- but in the quantum theory,
to write down the Schrödinger equation we had to pick a time parameterization
and thus fix $N$. Once $N$ has been fixed, there aren’t additional equations
that can be derived from varying it (see discussion in section 2). Of course,
the theory can be defined with any choice of $N$ \- but this simply changes
the definition of time and does not restrict the physical states of the
theory.

While not guaranteed by the above theory, what is guaranteed is

$\partial_{t}\langle\hat{\tilde{H}}\rangle=i\langle\left[\hat{\tilde{H}},\hat{\tilde{H}}\right]\rangle=0$
(16)

which means if the initial state is chosen to satisfy the first Friedmann
equation (in expectation value), then it satisfies it at all times. This is
equivalent to the classical equations, where the time derivative of the first
Friedmann equation is linearly dependent of the second Friedmann equation and
matter equations of motion. Thus, one can impose
$\langle\hat{\tilde{H}}\rangle=0$ on the initial state and it will be
satisfied at all times. Note, classically this is just fixing the initial
boundary condition for $\dot{a}(t)$.

What is instead often imposed in the literature is a second order constraint –
namely that the Hilbert space is restricted to the states $|\psi\rangle$ such
that $\hat{{H}}|\psi\rangle=0$. This is known as the Wheeler-DeWitt equation
[4], and while at the equation level seems to reproduce classical physics, it
in fact generates a theory without time evolution – the Schrödinger equation
becomes trivial and the quantum state is static. It has been suggested that
time could emerge from this “timeless” quantum state by the entanglement of
one degree of freedom with the other degrees of freedom so that the relative
evolution between these states would appear as time evolution. This suggestion
has never been actually implemented and it has largely been ruled out in
general terms [5]. Moreover, while the idea of time being connected to the
relative evolution of quantum states has merit, the restriction imposed by the
Wheeler-DeWitt equation that the quantum state remain static implies that the
only permitted states of the theory are those where the quantum state is a
macroscopic superposition over the entire history of the cosmos [6].

###  3.2 Coherent States and a Path Integral Description

In quantum mechanics (or quantum field theory), a path integral represents the
time evolution operator, typically in the general coordinate (or field) basis.
It can be derived from a Hamiltonian in a fixed gauge, or can be gauge-fixed
in situ. Below, we will instead derive the path integral in the coherent state
basis, which makes the classical-like evolution of these states even more
manifest.

For minisuperspace, we can start by taking the quantum version of the
Hamiltonian (12) with $N=1$, and derive the path integral. At this point we
are required to define the operators that comprise the kinetic terms. To do
so, we will define a basis of states which are a set of coherent states:
$|\pi_{cl}a_{cl}\rangle$. One can define these states using standard tricks
from the quantum harmonic oscillator. First, define a ‘ground state’
$|0\rangle$ as a state annihilated by the operator
$\hat{b}\equiv(\hat{a}+i\hat{\pi})/\sqrt{2}$, which obeys
$\left[\hat{b},\hat{b}^{\dagger}\right]=1$. One can now build coherent states
in the standard way by finding eigenstates of $\hat{b}$:

$\displaystyle|\beta\rangle=e^{-|\beta|^{2}/2}e^{\beta\hat{b}^{\dagger}}|0\rangle$
$\displaystyle\hat{b}|\beta\rangle=\beta|\beta\rangle$

where $\beta$ is a c-number. One can easily show that
$\langle\beta|\hat{a}|\beta\rangle=(\beta+\beta^{*})/\sqrt{2}$ and
$\langle\beta|\hat{\pi}|\beta\rangle=-i(\beta-\beta^{*})/\sqrt{2}$. Thus, we
define the real and imaginary parts of $\beta$ as $\beta_{r}\equiv
a_{cl}/\sqrt{2}$ and $\beta_{i}\equiv\pi_{cl}/\sqrt{2}$, respectively.

We now define a Hamiltonian that will generate on average classical time
evolution by defining operators in terms of coherent states, which will ensure
such states evolve dominantly in a classical way. It is known that the
coherent state basis is over-complete, and that an operator can be fully
defined in this basis by its diagonal elements (a brief proof is in Appendix
B.1). Thus, taking the gauge-fixed version of the classical Hamiltonian (12),
an initial proposal for the quantum Hamiltonian of mini-superspace is

$\hat{H}=-\frac{1}{24M_{pl}^{2}}\widehat{\left[\frac{\pi^{2}}{a}\right]}+\frac{1}{2}\hat{\pi}_{\phi}^{2}\widehat{\left[\frac{1}{a^{3}}\right]}+\widehat{\left[a^{3}\right]}V(\hat{\phi})$
(17)

where the operators in square brackets are defined by their diagonal matrix
elements in the $|\pi_{cl}a_{cl}\rangle$ basis, e.g.,

$\langle\pi_{cl}a_{cl}|\widehat{\left[\frac{\pi^{2}}{a}\right]}|\pi_{cl}a_{cl}\rangle=\frac{\pi_{cl}^{2}}{a_{cl}}$
(18)

and they commute with matter operators $\hat{\phi}$ and $\hat{\pi}_{\phi}$.

It turns out that for simply defined coherent states (which in the field
basis, do not vanish at $a=0$ and have exponentially small support in the
$a<0$ region), the matter kinetic term operator is ill-defined. There are two
ways to address this. The first is to use a more sophisticated coherent state
basis, one whose states vanish as you approach $a\rightarrow 0$. Such coherent
states have been build for the coherent states of the hydrogen atom, namely
addressing the radial coordinate [7, 8], and one could build similar states
here. A perturbative approach is to shift the scale factor operator by a fixed
value $\hat{a}\rightarrow a_{0}+\hat{\Delta a}$. This amounts to a change of
basis of coherent states, as the operator $\hat{\delta a}$ still has the
standard commutator with $\hat{\pi}$. Now, operators can be Taylor expanded,
e.g.,

$\widehat{\left[\frac{\pi^{2}}{a_{0}+\Delta
a}\right]}\equiv\frac{1}{a_{0}}\sum_{n}(-1)^{n}\frac{\widehat{\left[\pi^{2}\Delta
a^{n}\right]}}{a_{0}^{n}}$ (19)

where each term in the sum is again defined by its diagonal matrix elements.
These operators are now perturbatively well-defined (both on and off diagonal
matrix elements). Once expanded in fact, all terms in the power series have
positive powers of $\hat{\pi}$ and $\Delta\hat{a}$. Thus, these bracketed
operators are equivalent to the normal ordered operator, e.g.,
$\widehat{\left[\pi^{2}\Delta
a^{n}\right]}\equiv\;:\\!\hat{\pi}^{2}\Delta\hat{a}^{n}\\!:$ . While these
coherent states are not eigenstates of the Hamiltonian, they can be shown to
evolve classically up to small wave-function spreading if the initial spread
is small (which is true if $a_{0}\gg\Delta a_{cl}\gg 1$ with the above
normalization). This Hamiltonian and coherent state basis is useful when the
initial expectation value
$\langle\Delta\hat{a}\rangle=a_{cl}-a_{0}\equiv\Delta a_{cl}\ll a_{0}$. As the
expectation value evolves outside the domain of validity of the Taylor
expansion, one can match the state onto a new basis expanded around a
different value of $a_{0}$.

Now we are poised to build the path integral using the coherent-state basis.
Using the completeness relation for coherent states (for example, for the
metric fields):

$\frac{1}{\pi}\int\left[d\pi\right]\left[da\right]|\pi a\rangle\langle\pi
a|=1$ (20)

one can break a transition matrix element into infinitesimal time steps
$\delta t$ by inserting a complete set of coherent states, as we do in
Appendix B. The resulting path integral is

$\langle\pi_{f}a_{f}\pi_{\phi_{f}}\phi_{f}|e^{-i\hat{H}t}|\pi_{i}a_{i}\pi_{\phi_{i}}\phi_{i}\rangle=\int_{\pi_{i},a_{i},\pi_{\phi_{i}},\phi_{i}}^{\pi_{f},a_{f},\pi_{\phi_{f}},\phi_{f}}\,{\cal
D}\pi\,{\cal D}a\,{\cal D}\pi_{\phi}\,{\cal D}\phi\,e^{i\int
dt\left[\frac{1}{2}(\dot{a}\pi-\dot{\pi}a+\dot{\phi}\pi_{\phi}-\dot{\pi}_{\phi}\phi)-H\left[\pi,a,\pi_{\phi},\phi\right]\right]}$
(21)

where the boundaries of the path integral are labeled by the expectation
values of $\hat{\pi}$ and $\hat{a}$ in the initial and final states. For fixed
initial values, $a_{i}$ and $\pi_{i}$, this path integral can be taken to be a
wave function of $a_{f}$ and $\pi_{f}$ in coherent state space. It is easy to
show that for a given initial state, this wave function is sharply peaked
around the classical predictions for $a_{f}$ and $\pi_{f}$ dictated by the
second Friedmann equation. If one wants the system to satisfy the first
Friedman equation, one can choose an initial value of $\pi_{i}$ to satisfy
this constraint, after which the $\pi_{f}$ peak value will also satisfy this
constraint.

We can now write the resulting classical equations of motion for this theory,
first by solving the equations for $\pi$ and $\pi_{\phi}$. What is new is that
the first Friedmann equation is not required to be satisfied (but its time
derivative is). We will define the amount of violation by the constant
$\mathbb{H}_{0}$, allowing us to write the equations:

$\displaystyle
6M_{pl}^{2}a\dot{a}^{2}=a^{3}(\frac{1}{2}\dot{\phi}^{2}+V(\phi))+\mathbb{H}_{0}$
(22)
$\displaystyle-2M_{pl}^{2}(2a\ddot{a}+\dot{a}^{2})=a^{2}(\frac{1}{2}\dot{\phi}^{2}-V(\phi))$
(23) $\displaystyle
a^{3}(\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}+\partial_{\phi}V(\phi))=0$ (24)

What we see (after scaling out by the appropriate powers of $a$) is that the
non-compliance with the first Friedmann equation or Hamiltonian constraint
(22) leads to a more general initial condition for $\dot{a}$, which could be
interpreted as the existence of a new but non-substantive source of energy
density that has zero pressure. In other words, it contributes to the
equations as a component of dust or a dark matter would. Interestingly, the
violation of (22), parameterized by $\mathbb{H}_{0}$ could take either sign,
and thus could contribute as a substance with negative energy density. There
are of course no additional fields here, but an quantum state of field
configurations of the non-dynamical gravitational fields. Note, this and
similar effects has also been seen looking at constrained gravitational
instantons [9], in Horava-Lifschitz gravity [10, 11], and in Einstein-aether
theories [12, 13].

##  4 General Relativity

We will now take what we learned from the minisuperspace example and look at
the classical limits from a quantized field theory with a symmetric (metric)
tensor, $g_{\mu\nu}$, coupled to itself and matter in a general coordinate
invariant way.

The classical action that generates general relativity is of course the
Einstein-Hilbert action

$S=\int d^{4}x\sqrt{-g}\left({M_{pl}^{2}}R+{\cal L}_{matter}\right)+S_{GHY}$
(25)

where the matter Lagrangian can contain independent fields as well as the
metric, its inverse, and covariant derivatives associated with the metric. We
have also included the final Gibbons-Hawking-York term explicitly to remove
boundary terms. Minimization of this action with respect to the path of the
metric tensor produces Einstein’s equations:

$\frac{\delta S}{\delta
g_{\mu\nu}}=\sqrt{-g}\left(M_{pl}^{2}G^{\mu\nu}-T^{\mu\nu}\right)=0$ (26)

where from the second term in (25), $S_{m}=\int\sqrt{-g}{\cal L}_{matter}$:

$T^{\mu\nu}=-\frac{1}{\sqrt{-g}}\frac{\delta S_{m}}{\delta g_{\mu\nu}}$ (27)

These plus the matter equations of motion determine the classical dynamics as
a function of initial conditions. As in the classical version of
minisuperspace, some of these equations are in fact constraints on the initial
conditions.

To canonically quantize, one must find a suitable Hamiltonian operator. Like
the previous examples of minisuperspace and QED, there are redundancies in
this theory in the form of reparameterization invariance. The coordinates can
be redefined in terms of four independent functions,
$x^{\mu}\rightarrow\xi^{\mu}(x)$. This imposed invariance also results in some
components of $g_{\mu\nu}$ being non-dynamical or lacking standard kinetic
terms. Thus we have a problem when attempting to define conjugate momenta for
components of $g_{\mu\nu}$, namely that some vanish. Define
$L_{grav}\equiv\int d^{3}x(\sqrt{-g}M_{pl}^{2}R+L_{boundary})$, where the last
term is to remove higher derive/boundary terms. As the matter part of the
action does not contain time derivatives of metrics, the conjugate momenta for
the metric components are naturally defined as:

$\displaystyle\pi^{ij}$ $\displaystyle\equiv\frac{\delta L_{grav}}{\delta
g_{ij,0}}$ $\displaystyle\pi^{i}$ $\displaystyle\equiv\frac{\delta
L_{grav}}{\delta g_{0i,0}}=0$ $\displaystyle\pi$
$\displaystyle\equiv\frac{\delta L_{grav}}{\delta g_{00,0}}=0$

There is now a clear choice to make – effectively a gauge choice before we
define the Hamiltonian operator. By choosing $g_{0\mu}=-\delta^{0}_{\mu}$, we
both remove the fields (traditionally the ‘lapse function’ and ‘shift vector’)
without conjugates and get the benefit of choosing a time coordinate and time
slices on which to identify initial states.

The gravitational part of the Lagrangian density in this gauge (now setting
$M_{pl}=1$) can be shown to be

${\cal
L}=\sqrt{\gamma}\left(\frac{1}{4}(\gamma^{ik}\gamma^{j\ell}-\gamma^{ij}\gamma^{k\ell})\gamma_{ij,0}\gamma_{k\ell,0}\,+\\!\\!\phantom{!}^{{}^{(3)}}\\!\\!R\right)$
(28)

where $\\!\\!\phantom{!}{}^{{}^{(3)}}\\!\\!R$ is the three-dimensional Ricci
scalar associated with the spatial metric $g_{ij}\equiv\gamma_{ij}$, $\gamma$
is the determinant of the spatial metric, and spatial indices are raised and
lowered with this metric.

In this gauge we see that from the Lagrangian standpoint, the equations of
motion are reduced to the spatial part of Einstein’s equations, and thus these
will be the only ones guaranteed in expectation value in the quantum theory.

We can now build a Hamiltonian operator for the quantum theory in terms of the
conjugate momentum, which classically is

$\pi^{ij}=\frac{1}{2}\sqrt{\gamma}(\gamma^{ki}\gamma^{\ell
j}-\gamma^{ij}\gamma^{k\ell})\gamma_{k\ell,0}$ (29)

Solving for the time derivative of the metric

$\gamma_{ij,0}=\frac{2}{\sqrt{\gamma}}(\gamma_{ia}\gamma_{jb}-\frac{1}{2}\gamma_{ij}\gamma_{ab})\pi^{ab}$
(30)

allows us to construct the classical Hamiltonian density

${\cal
H}=\frac{1}{\sqrt{\gamma}}(\pi_{ij}\pi^{ij}-\frac{1}{2}\pi^{2})-\sqrt{\gamma}\phantom{!}^{{}^{(3)}}\\!\\!R$
(31)

where $\pi\equiv\gamma_{ij}\pi^{ij}$. Now we would like to use the classical
intuition to produce a Hamiltonian operator in a quantum theory. Again, the
kinetic terms are non-trivial, and thus a way to guarantee the classical limit
is approached is, as we did with minisuperspace, to define the metric-
dependent terms in terms of their diagonal matrix elements in a coherent-state
basis, $|\pi_{cl}\gamma_{cl}\rangle$. We see no impediment to doing this by
replacing the terms in the classical Hamiltonian by operators as we did in the
homnogeneous case (17) of the previous section.

Thus, in the coherent state basis, the path integral for quantum gravity
should read:

$\langle\pi_{f}\gamma_{f}\cdots|e^{-i\hat{H}t}|\pi_{i}\gamma_{i}\cdots\rangle=\int_{\pi_{i},\gamma_{i}}^{\pi_{f},\gamma_{f}}\,{\cal
D}\pi\,{\cal D}\gamma\,\cdots\,e^{i\int
d^{4}x\left[\frac{1}{2}(\dot{\gamma}_{ij}\pi^{ij}-\dot{\pi}^{ij}\gamma_{ij})-{\cal
H}\left[\pi,\gamma\right]\right]+\cdots}$ (32)

where the various elipses represent the state quantum numbers on the left side
of the equation, and the measure parts and action parts for any and all matter
fields on the right side. This path integral can be though of as a functional
of the final values $\pi_{f}$ and $\gamma_{f}$, and this functional is sharply
peaked around the classical expectation values of these functions. The key
point is that the six degrees of freedom and their conjugates only produce (in
expectation value) six pairs of Hamilton equations of motion, which correspond
to the spatial parts of Einstein’s equations.

Minimizing the variation of the classical action with respect to the metric
led to the equations $\sqrt{-g}(G^{\mu\nu}-8\pi G_{N}T^{\mu\nu})=0$. In
synchronous gauge, we have found only the spatial equations emerge from the
quantum field theory in the classical limit. Let’s package the loosening of
restriction as we did for minisuperspace:

$\displaystyle G^{00}$ $\displaystyle=$
$\displaystyle{8\pi}G_{N}T^{00}+{8\pi}G_{N}\frac{\mathbb{H}}{\sqrt{-g}}$
$\displaystyle G^{0i}$ $\displaystyle=$
$\displaystyle{8\pi}G_{N}T^{0i}+{8\pi}G_{N}\frac{{\mathbb{P}}^{i}}{\sqrt{-g}}$
$\displaystyle G^{ij}$ $\displaystyle=$ $\displaystyle{8\pi}G_{N}T^{ij}$

for, as of yet, arbitrary functions $\mathbb{H}$ and $\mathbb{P}^{i}$. In this
language, we define an auxiliary energy-momentum tensor

$T_{\rm
aux}^{\mu\nu}=\frac{1}{\sqrt{-g}}\begin{pmatrix}\mathbb{H}&\mathbb{P}^{1}&\mathbb{P}^{2}&\mathbb{P}^{3}\\\
\mathbb{P}^{1}&0&0&0\\\ \mathbb{P}^{2}&0&0&0\\\
\mathbb{P}^{3}&0&0&0\end{pmatrix}$ (33)

Now from these constructed classical equations, we can find restrictions on
$T_{\rm aux}$ by noting that the tensor $T$ is covariantly conserved and $G$
satisfies the Bianchi identity. Thus, since $\nabla_{\mu}(G^{\mu\nu}-8\pi
G_{N}T^{\mu\nu})=0$, then:

$0=\nabla_{\mu}T_{\rm aux}^{\mu\nu}=\partial_{\mu}T_{\rm
aux}^{\mu\nu}+\Gamma_{\mu\lambda}^{\mu}T_{\rm
aux}^{\lambda\nu}+\Gamma_{\mu\lambda}^{\nu}T_{\rm aux}^{\mu\lambda}$ (34)

These four equations simplify in this gauge, as
$\Gamma_{00}^{0}=\Gamma_{0i}^{0}=\Gamma_{00}^{i}=0$. We can use this condition
to constrain the functions $\left[\mathbb{H},\mathbb{P}\right]$. Using the
identity $\Gamma_{\mu\nu}^{\mu}=-\sqrt{-g}\partial_{\nu}(1/\sqrt{-g})$ and
defining $t_{\rm aux}^{\mu\nu}\equiv\sqrt{-g}T_{\rm aux}^{\mu\nu}$, we can
write the $\nu=0$ and $\nu=i$ equations respectively as

$\displaystyle\partial_{0}t_{\rm aux}^{00}+\partial_{i}t_{\rm aux}^{i0}=0$
$\displaystyle\partial_{0}t_{\rm aux}^{0i}+2\Gamma_{j0}^{i}t_{\rm aux}^{j0}=0$

which (noting that $\gamma_{ij}\equiv g_{ij}$, $\gamma=-g$, and
$\gamma^{ij}\gamma_{jk}=\delta^{i}_{k}$ in this gauge) simplifies further to

$\displaystyle\partial_{0}\mathbb{H}=-\partial_{i}\mathbb{P}^{i}$
$\displaystyle\partial_{0}\left(\gamma_{ij}\mathbb{P}^{j}\right)=0$

We thus see that this auxiliary shadow matter is made up of three time-
independent functions and one whose time dependence is fixed by the other
three (and the metric).

###  4.1 Cosmological Implications

We will discuss some preliminary observations about the cosmological
implications of this additional source term and leave a fuller analysis of the
cosmology to a companion paper [14] and of the effects on non-linear general
relativity to future work.

To analyze the cosmological effects, we expand the metric around a homogeneous
background to linear order in fields as in cosmological perturbation theory:

$ds^{2}=-dt^{2}+a(t)^{2}(\delta_{ij}+h_{ij})dx^{i}dx^{j}$ (35)

where we further break up $h_{ij}$ into irreducible components of the three-
dimensional Euclidean group: two scalars $h$ and $\eta$, one divergenceless
vector $w_{i}$, and one transverse-traceless tensor $s_{ij}$

$h_{ij}=h\delta_{ij}+D_{ij}\eta+(\partial_{i}w_{j}+\partial_{j}w_{i})+s_{ij})$
(36)

where the differential operator
$D_{ij}=\partial_{i}\partial_{j}/\nabla^{2}-(1/3)\delta_{ij}$ (and
$\nabla^{2}$ is the standard spatial Laplacian). The inverse derivatives will
be defined such that for the Fourier transforms of the fields (e.g.,
$\tilde{\eta}(k)$ is the transform of $\eta(x)$), this differential operator
becomes $\tilde{D}_{ij}=\hat{k}_{i}\hat{k}_{j}-\delta_{ij}/3$.

Because the different representations decouple at linear order, we can focus
on the scalar perturbations. Following [15] (but switching from conformal time
to proper time), we can write the spatial part of the source-free Einstein
equations, $G^{i}_{j}=0$, in terms of the scalar modes. In Fourier space, we
define

$h_{ij}(\vec{x},t)=\int
d^{3}ke^{i\vec{k}\cdot\vec{x}}\left(\hat{k}_{i}\hat{k}_{j}\tilde{h}(\vec{k},t)+(\hat{k}_{i}\hat{k}_{j}-\frac{1}{3}\delta_{ij})6\tilde{\eta}(\vec{k},t)\right)$
(37)

The two scalar equations are the trace part and the longitudinal traceless
part of the space-space equations:

$\displaystyle\ddot{\tilde{h}}+3\frac{\dot{a}}{a}\dot{\tilde{h}}-2\frac{k^{2}}{a^{2}}\tilde{\eta}=0$
(38)
$\displaystyle\ddot{\tilde{h}}+3\frac{\dot{a}}{a}\dot{\tilde{h}}+6\left(\ddot{\tilde{\eta}}+3\frac{\dot{a}}{a}\dot{\tilde{\eta}}\right)-2\frac{k^{2}}{a^{2}}\tilde{\eta}=0$
(39)

The first thing to note is that there are no wave-like solutions to these
equations. Subtracting (38) from (39) produces an equation for $\tilde{\eta}$
which has two solutions: a constant and a solution in which
$\dot{\tilde{\eta}}\sim 1/a^{3}$. The constant solution can be inserted in
(38) and we see that during matter domination, $\tilde{h}\propto a$ is a
solution. Thus, whatever sources the expansion and the initial perturbations
(whether real or shadow matter), the metric perturbations grow linearly during
’matter’ domination.

We can also think of the source terms
$\mathbb{H}\equiv\mathbb{H}_{0}+\delta\mathbb{H}$ and $\mathbb{P}^{i}$ in
perturbation theory, where we assume $\mathbb{H}_{0}$ as homogeneous and
constant, and $\delta\mathbb{H}$ and $\mathbb{P}^{i}$ as small inhomogeneous
perturbations. While $\mathbb{H}_{0}$ could be the dominant homogeneous
contribution to the expansion rate, we could ask about the evolution of the
perturbative sources. Assuming $\mathbb{H}_{0}$ is the only homogeneous
contribution, the time-time equation at zeroth order is

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi}{3}G_{N}\frac{\mathbb{H}_{0}}{a^{3}}$
(40)

whereas the time-time and the longitudinal part of the time-space equations of
motion at linear order are

$\displaystyle\frac{\dot{a}}{a}\dot{\tilde{h}}-2\frac{k^{2}}{a^{2}}\tilde{\eta}$
$\displaystyle=$ $\displaystyle 8\pi
G_{N}\frac{1}{a^{3}}\left[-\frac{1}{2}\tilde{h}\mathbb{H}_{0}+\delta\mathbb{H}\right]$
(41) $\displaystyle-2i\frac{k}{a^{2}}\dot{\tilde{\eta}}$ $\displaystyle=$
$\displaystyle 8\pi G_{N}\frac{\mathbb{P}_{\parallel}}{a^{3}}\,.$ (42)

where $\delta\mathbb{H}$ and
$\mathbb{P}_{\parallel}\equiv\hat{k}_{i}\mathbb{P}^{i}$ are taken to be in
Fourier space. The leading growth of $\tilde{h}$ is cancelled in the first of
these equations. The contant part of $\delta\mathbb{H}$ falls off as $1/a$
relative to $\tilde{h}$, i.e., effectively redshifts with the background
density. The perturbation $\mathbb{P}_{\parallel}$ also only contributes to
the dying mode of $\dot{\tilde{\eta}}$ and $\mathbb{P}_{\parallel}\sim
1/a^{2}$. This is consistent with the constraint equations

$\displaystyle\partial_{0}\delta\mathbb{H}=-ik\mathbb{P}_{\parallel}$
$\displaystyle\partial_{0}\left(a^{2}\mathbb{P}_{\parallel}\right)=0$

##  5 Discussion

By all accounts, quantum field theory is the correct underlying theory of
nature and classical physics is a limit of this quantum theory. As classical
observers, we can readily observe and obtain the classical equations of
motion. We are then faced with the task of obtaining the correct quantum
mechanical description from the known classical equations. But, since
classical physics is a limit of quantum mechanics, this is a tricky inverse
problem.

The conventional approach to this inverse problem has been to take a classical
theory and replace the Poisson brackets that generate its equations of motion
by various quantum commutators and use these relations to reconstruct the
underlying quantum theory. While this approach works for particle mechanics
and scalar field theories, we recognize that it fails for a broader class of
quantum field theories. As is well known, in theories with fermions, the
commutators need to be replaced with anti-commutators. In gauge theories,
given gauge redundancies, some gauge degrees of freedom need to be suitably
fixed in order for the theory to yield sensible results. When these “rules”
are directly applied to general relativity, the procedure yields a trivial
Hamiltonian that is manifestly incorrect. There are thus no rigid and sacred
set of rules that “derives” a quantum theory from the known classical dynamics
of the theory. Instead, the correct prescription is to see if a given quantum
field theory (i.e. either a Hamiltonian or path integral description) is
logically consistent and yields a classical limit that is consistent with
observation.

With this point of view, there arises an interesting possibility - since
classical physics is a limit of quantum mechanics, could the quantum theory
permit more freedom than inferred by the classical observer? In this paper, we
have seen an example of such freedom where we see that certain Hamiltonian
constraints that are imposed on the theory by the classical equations of
motion do not exist at the quantum level. The principal reason for this
relaxation is that the entire quantum evolution is described by just one
equation - the Schrödinger Equation. This is a first order differential
equation and thus given any initial state, it is able to describe its time
evolution. This is unlike the classical theory where additional constraint
equations are imposed on the system on top of the dynamical equations of
motion.

In this paper, we have shown that this relaxation permits a broader class of
initial states in general relativity that result in gauge invariant time
evolution. In a companion paper, we explore the consequences of such
relaxation in electromagnetism. These states do not satisfy the Hamiltonian
constraint of general relativity. But, their quantum mechanical evolution is
such that in the classical limit of the theory, these states would behave as
though they were a peculiar covariantly conserved classical stress tensor.
But, there are no new degrees of freedom associated with this stress tensor -
it is simply a quantum state of the metric.

In a cosmological setting, the homogeneous piece of this classical stress
tensor mimics the evolution of a homogeneous dark matter field. Interestingly,
this component could have either sign. Irrespective of the sign, the
associated perturbative degrees of freedom are gravitons which have positive
kinetic energy terms. If this component had a negative sign, it would violate
the null energy condition without any of the disastrous ghost-like
instabilities often associated with such sources. Consequently, these
gravitational states provide a trivial way to violate the null energy
condition and yield interesting gravitational phenomena such as cosmic
bounces.

The inhomogeneous pieces of these components are phenomenologically
interesting. First, in the linear regime, they grow just like dark matter and
may have cosmological significance. In this paper, we have not analyzed the
behavior of these inhomogeneous components in the nonlinear regime. This is
however an important task since it could lead to cosmologically relevant
signatures. Further, since the inhomogeneous components can also locally
violate the null energy condition, they may be able to support exotic
phenomena such as wormholes.

An important feature of these initial states is that their effects can be
dynamically obliterated from our observable universe by a period of inflation.
Conventional classical general relativity is thus an attractor solution if
there is a period of inflation. This is unsurprising since inflation erases
sensitivity to initial conditions and the nontrivial quantum effects
identified by us in this paper are tied to the initial quantum state of the
universe. It is thus of great interest to develop the full phenomenology
(gravitational and electromagnetic) of these quantum states since those may
give rise to observational effects. If such effects are observed, they would
rule out an inflationary period in the early universe, forcing us to re-
explore the origins of our universe.

## Acknowledgements

We thank Peter Graham, Vivian Poulin, Tristan Smith, Raman Sundrum and Jed
Thompson for fruitful discussions. This work was supported by the U.S.
Department of Energy (DOE), Office of Science, National Quantum Information
Science Research Centers, Superconducting Quantum Materials and Systems Center
(SQMS) under Contract No. DE-AC02-07CH11359. D.E.K. and S.R. are supported in
part by the U.S. National Science Foundation (NSF) under Grant No.
PHY-1818899. S.R. is also supported by the DOE under a QuantISED grant for
MAGIS. The work of S.R. was also supported by the Simons Investigator Award
No. 827042. T.M. is supported by the World Premier International Research
Center Initiative (WPI) MEXT, Japan, and by JSPS KAKENHI grants JP19H05810,
JP20H01896, JP20H00153, and JP22K18712.

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##  Appendix A  Quantum Field Theory on a Manifold

Let’s clarify the mathematical structure we are assuming for the quantum
theory of gravity. We want a quantum field theory of a metric tensor coupled
to other quantum fields and living on a fixed manifold. The manifold is taken
to be a four-dimensional manifold with topology $\mathbb{R}\times\Sigma$. A
conventional choice for $\Sigma$ is $\mathcal{R}^{3}$ but other choices are
possible. There is one question the field theory should be able to answer:
given an initial quantum state on $\Sigma$, how does this state evolve in
“time”?

The subtlety in answering this question is that the manifold does not have a
preferred time coordinate, or potentially even slicing, and physical questions
should have answers which are time reparameterization invariant. In addition,
of the ten fields in the metric tensor, only two are dynamical, whereas four
are pure gauge degrees of freedom, and the other four are fixed by classical
constraints. All of this extra degeneracy should be taken care of in the
quantum theory without generating additional degrees of freedom while
maintaining the reparameterization invariance of the manifold. We will
describe the theory using the path integral as it offers an easier path to
handle these issues.

To define the path integral, we begin by taking the manifold and picking some
arbitrary foliation $f\left(t\right)$ where the function $f\left(t\right)$ is
a one to one function that assigns to the parameter $t$ a set of points
$\Sigma\left(t\right)$ whose topology is $\Sigma$. These foliations are the
“time” slices that we have picked for this manifold. On each
$\Sigma\left(t\right)$, pick some spatial coordinates
$\left(x^{1},x^{2},x^{3}\right)$. The above assignments are performed at the
level of the manifold and these do not require specification of the metric. On
this foliation, the quantum field that is the metric tensor is expressed in
the form:

$g_{\mu\nu}\rightarrow ds^{2}=-N^{2}dt^{2}+N_{i}dtdx^{i}+s_{ij}dx^{i}dx^{j}$
(43)

Similarly, the non-gravitational quantum fields in the theory are described as
various vector, scalar or spinor fields on this foliation. Given these, we can
define the usual Einstein-Hilbert Lagrangian $\mathcal{L}_{GR}$ and matter
Lagrangian for the theory. Note that the definition of the Lagrangian requires
a foliation since the Lagrangian is implicitly a function of space-time and
contains space-time derivatives.

Now comes the physics via the path integral. For simplicity, we will focus on
the pure gravitational part of the theory. From the Lagrangian, we know that
the fields $N$ and $N_{i}$ do not contain conjugate momenta. These degrees of
freedom are not physical and they are undetermined. Thus, one can arbitrarily
set, for example, $N=1$ and $N_{i}=0$ and assert that the matrix elements of
the time evolution operator of the degrees of freedom $s_{ij}$ from time
$t_{1}$ to $t_{2}$ are given by the path integral:

$\langle
s^{f}_{ij}|T\left(t_{2};t_{1}\right)|s^{i}_{ij}\rangle=\int_{s_{ij}\left(t_{1}\right)=s^{i}_{ij}}^{s_{ij}\left(t_{2}\right)=s^{f}_{ij}}Ds_{ij}\,e^{iS}$
(44)

where the states $|s^{i,f}_{ij}\rangle$ are field basis states333A similar
construction can also be employed for coherent states. We will in fact adopt
this construction from Section 3 onwards. and
$S=\int_{t_{1}}^{t_{2}}d^{4}x\,\mathcal{L}_{GR}$ is the Einstein-Hilbert
action. Notice that this path integral does not involve integration over the
functions $N$ and $N_{i}$ which have been fixed. The integral is over the six
degrees of freedom $s_{ij}$. The above path integral is well defined and it
can be calculated.

The next task is to show that this construction where we took an arbitrary
foliation and made choices of the functions $N,N_{i}$ gives rise to physics
that is invariant under these choices. To see this, we first fix the initial
and final slices $\Sigma\left(t_{1}\right)$ and $\Sigma\left(t_{2}\right)$,
respectively. Now pick any arbitrary foliation in the middle. This amounts to
keeping the parameters $t_{1}$ and $t_{2}$ fixed on the manifold while
reparameterizing $t$ in between $t_{1}$ and $t_{2}$. The transformation from
the initial foliation to this new foliation is a coordinate transformation
under which all the fields have well defined transformations. These
transformations will in general mix the elements of the metric to yield a new
metric $\tilde{g}_{\mu\nu}=\frac{\partial x^{\alpha}}{\partial
x^{{}^{\prime}\mu}}\frac{\partial x^{\beta}}{\partial
x^{{}^{\prime}\nu}}g_{\alpha\beta}$. Thus, the metric components
$\tilde{g}_{00},\tilde{g}_{0i}$ will be different from the initial values
($-1$ and $0$, respectively) that we had arbitrarily chosen. We thus recognize
that coordinate transformations allow us to change between different values of
$N,N_{i}$. Fixing $N,N_{i}$ is thus a gauge fixing procedure where we pick the
coordinates on which the path integral is performed - without fixing the
coordinates, one cannot evaluate the path integral since the integral is a
function known in terms of coordinates.

The change in time slices also means that fields that were defined to be at
the “same time” in one foliation will be mapped to unequal points in “time” in
other foliations. To keep track of these changes, one can discretize the
spatial manifold $\Sigma$ and track the fields values at discrete spatial
points. Once this is performed, the path integral is now an integral over
functions defined at these discrete spatial points with a different
parameterization of time when the foliation is changed. The metric that
appears in the path integral (44) will contain more non-trivial elements.
However, the number of physical functions that are allowed to vary in the path
integral will still be six. The measure of the path integral does not change
under the reparameterization since we are still integrating over the same
field values at these spatial points - all that has been done is a relabelling
of the time parameter. Since the action of the theory is invariant under this
reparameterization, the transition matrix elements $\langle
s^{f}_{ij}|T\left(t_{2};t_{1}\right)|s^{i}_{ij}\rangle$ are invariant.

Second, let us return to the original foliation that we had picked. Retaining
$N_{i}=0$, pick $N$ to be any arbitrary function of $t$. Due to the
reparameterization invariance of the theory, one can show that $\langle
s^{f}_{ij}|T\left(t_{2};t_{1}\right)|s^{i}_{ij}\rangle=\langle
s^{f}_{ij}|T\left(\tilde{t}_{2};\tilde{t}_{1}\right)|s^{i}_{ij}\rangle$ where
the time labels $\tilde{t}$ are obtained by solving the equation
$d\tilde{t}=N\left(t\right)dt$. We thus understand the meaning of $N$ \- it
simply reflects a choice of time parameterization. Even though (44) is
explicitly dependent on the choice of time parameterization, the physics that
it describes, namely, the relative phase between various quantum states as
described by the transition matrix elements, is invariant as long as we
recognize that the time evolution operator (44) is suitably changed when the
time parameter is changed by a new foliation. The physics described by a
parameterization $t$ and a function $N\left(t\right)$ is identical to the
physics described by the parameterization $\tilde{t}$ and the function
$\tilde{N}\left(\tilde{t}\right)$ where
$N\left(t\right)dt=\tilde{N}\left(\tilde{t}\right)d\tilde{t}$. Neither $N$ or
$t$ are individually physical - but this differential combination is physical.
Note that all other changes to the foliations and the functions $N,N_{i}$ in
the middle of these two time slices can be constructed as a composition of
these two cases and thus lead to gauge invariant physics.

Finally, let us consider a foliation which also changes the initial and final
slices
$\Sigma\left(t_{1,2}\right)\rightarrow\Sigma\left(\tilde{t}_{1,2}\right)$.
Since the initial and final slices now correspond to different points on the
manifold, there is no canonical identification that permits us to show that
the physics is invariant. This is a particularly important aspect of quantum
mechanics in comparison to classical physics. In classical field theory, one
could evaluate the values of various functions at the same point on the
manifold and potentially compare those values using well defined coordinate
transformation properties. However, a quantum state is inherently non-local
since it naturally includes entanglement with modes of the field that could be
localized at different points on the slices $\Sigma$ and thus there is no
simple local comparison that one can do. Formally, the Hilbert space is
different on different time slicing. However, it is clear that the physics is
invariant under such a foliation change since one may regard this kind of
foliation change as being naturally embedded in the “middle” of a larger set
of foliations describing the evolution of a quantum state from
$\Sigma\left(t^{-}_{1}\right)$ to $\Sigma\left(t^{+}_{2}\right)$ where
$t^{-}_{1}\leq t_{1}\leq t\leq t_{2}\leq t^{+}_{2}$. This foliation change is
then already described by the situations above.

An extremely important aspect of these coordinate transformations is that
these transformations are simply functions that map one set of coordinates on
the manifold to another. They are thus independent of the basis state whose
time evolution is being described by the path integral. This implies that
certain metric redefinitions, which one might describe as “gauge” choices,
performed in classical general relativity have to be properly interpreted in
the quantum context. See the discussion of this point in [1].

The above construction treats $\Sigma\left(t\right)$ as a time slice in
performing the evolution of the system. In general relativity, given a set of
time slices, for some initial data, the time evolution could result in the
time slices becoming spatial. This is a fact even in classical general
relativity and it cannot be avoided in the quantum theory. This implies that
in certain situations the computation of the path integral for certain
foliations can become ill defined since functions like the Lagrangian will
contain divergent or ill defined terms as the time slice evolves into a
spatial slice. But, the solution for these issues is similar to the approach
taken in classical general relativity. In the case where these issues arise
due to coordinate singularities, one may choose a different time slice and
perform the evaluation and see if these coordinate singularities can be
avoided.

##  Appendix B Coherent States and the Path Integral

First we construct a coherent state basis based on an arbitrary harmonic
oscillator. Independent of the width of the harmonic oscillator, we know this
is an over-complete basis. For the field operator $\hat{a}$ and its conjugate
$\hat{\pi}$ with standard commutators, we define coherent states
$|\pi_{cl}a_{cl}\rangle$ such that

$\displaystyle\langle\pi_{cl}a_{cl}|\hat{a}|\pi_{cl}a_{cl}\rangle$
$\displaystyle=$ $\displaystyle a_{cl}$
$\displaystyle\langle\pi_{cl}a_{cl}|\hat{\pi}|\pi_{cl}a_{cl}\rangle$
$\displaystyle=$ $\displaystyle\pi_{cl}$ (45)

Now define $\hat{A}\equiv(\sigma\hat{a}+(1/\sigma)\hat{\pi})/\sqrt{2}$ and a
state $|0\rangle_{\sigma}$ such that $\hat{A}|0\rangle_{\sigma}=0$, where
$\sigma$ is an arbitrary real parameter. We know that any state

$|\alpha\rangle\equiv
e^{-|\alpha|^{2}/2}e^{\alpha\hat{A}^{\dagger}}|0\rangle_{\sigma}$ (46)

(with $\alpha$ a complex number) satisfies
$\hat{A}|\alpha\rangle=\alpha|\alpha\rangle$ and $\langle\alpha\rangle=1$.
Using $\left[\hat{A},\hat{A}^{\dagger}\right]=1$, and (45), that
$|\pi_{cl}a_{cl}\rangle=|\alpha\rangle$ with the real and imaginary parts of
$\alpha$ defined as $\alpha^{R}=\sigma a_{cl}/\sqrt{2}$ and
$\alpha^{I}=\pi_{cl}/\sqrt{2}\sigma$, respectively.

Now define the Hamiltonian:

$\hat{H}=-\frac{1}{24M_{pl}^{2}}\widehat{\left[\frac{\pi^{2}}{a}\right]}+\frac{1}{2}\hat{\pi}_{\phi}^{2}\widehat{\left[\frac{1}{a^{3}}\right]}+\widehat{\left[a^{3}\right]}V(\hat{\phi})$
(47)

where the operators in square brackets are defined by their diagonal matrix
elements in the $|\pi_{cl}a_{cl}\rangle$ basis, e.g.,

$\langle\pi_{cl}a_{cl}|\widehat{\left[\frac{\pi^{2}}{a}\right]}|\pi_{cl}a_{cl}\rangle=\frac{\pi_{cl}^{2}}{a_{cl}}$
(48)

The matter coupling requires a different basis as for the coherent states
defined around this origin, this operator is ill-defined. For simplicity, we
will stick to the pure gravity case.

Now we are poised to build the path integral using the coherent-state basis.
Using the completeness relation for coherent states:

$\frac{1}{\pi}\int\left[d\pi\right]\left[da\right]|\pi a\rangle\langle\pi
a|=1$ (49)

one can break a transition matrix element into infinitesimal time steps
$\delta t$ by inserting a complete set of coherent states:

$\langle\pi^{\prime}a^{\prime}|e^{-i\hat{H}t}|\pi
a\rangle=\prod_{k=0}^{N}\int\left[\frac{d\pi_{k}}{\pi}\right]\left[da_{k}\right]\langle\pi_{k}a_{k}|\hat{1}-i\hat{H}\delta
t|\pi_{k-1}a_{k-1}\rangle\delta(\pi_{N}-\pi^{\prime})\delta(a_{N}-a^{\prime})\delta(\pi_{0}-\pi)\delta(a_{0}-a)$
(50)

Let’s compute one of the infinitesimal matrix elements in (50),
$\langle\pi_{k}a_{k}|\hat{1}-i\delta t\hat{H}|\pi_{k-1}a_{k-1}\rangle$. The
identity matrix element can be approximated:

$\langle\pi_{k}a_{k}|\pi_{k-1}a_{k-1}\rangle\simeq\langle\pi_{k}a_{k}|\pi_{k}a_{k}\rangle-\langle\pi_{k}a_{k}|\frac{|\pi_{k}a_{k}\rangle-|\pi_{k-1}a_{k-1}\rangle}{\delta
t}\delta t$ (51)

Approximating the last term as a total time derivative in the vanishing step-
size limit, $\sim\frac{d}{dt}\langle\pi a|\pi a\rangle(t)\delta t$, we can
rewrite the full term as

$\langle\pi_{k}a_{k}|\hat{1}-i\delta t\hat{H}|\pi_{k-1}a_{k-1}\rangle\\\
\simeq
1-\left[\dot{\pi}_{k}\frac{\partial}{\partial\pi_{k-1}}\langle\pi_{k}a_{k}|\pi_{k-1}a_{k-1}\rangle+\dot{a}_{k}\frac{\partial}{\partial
a_{k-1}}\langle\pi_{k}a_{k}|\pi_{k-1}a_{k-1}\rangle\right]_{k-1\rightarrow
k}\delta t-i\langle\pi_{k}a_{k}|\hat{H}|\pi_{k}a_{k}\rangle\delta t$ (52)

we can evaluate the field derivatives using the explicit form of the coherent
states (46), e.g.,

$\displaystyle\frac{\partial}{\partial
a_{cl}}|\pi_{cl}a_{cl}\rangle=\frac{\sigma}{\sqrt{2}}\frac{\partial}{\partial\alpha^{R}}|\pi_{cl}a_{cl}\rangle=\frac{\sigma}{\sqrt{2}}(-\alpha^{R}+\hat{A}^{\dagger})|\pi_{cl}a_{cl}\rangle$
(53)
$\displaystyle\frac{\partial}{\partial\tilde{a}_{cl}}\langle\pi_{cl}a_{cl}|{\pi}_{cl}\tilde{a}_{cl}\rangle\bigg{|}_{\tilde{a}\rightarrow
a}=\frac{\sigma}{\sqrt{2}}(-\alpha^{R}+(\alpha^{R}-i\alpha^{I})=-i\frac{\sigma}{\sqrt{2}}\alpha^{I}=-\frac{i}{2}\pi_{cl}$
(54)

and similarly $\partial\langle\pi
a|\tilde{\pi}a\rangle/\partial\tilde{\pi}\rightarrow ia/2$. Thus, for an
infinitesimal time step, we have (exponentiating):

$\langle\pi_{k}a_{k}|\hat{1}-i\delta t\hat{H}|\pi_{k-1}a_{k-1}\rangle\simeq
e^{i\left(\frac{1}{2}(\dot{a}_{k}\pi_{k}-\dot{\pi}_{k}a_{k})-H(\pi_{k},a_{k})\right)\delta
t}$ (55)

from which we can derive the path integral (21).

###  B.1 Coherent State Basis Operators

Here we show how the relationship between the number-operator basis (the
eigenstates of a harmonic oscillator) and the coherent state basis of
operators. We will also show that an operator in the coherent state basis is
completely defined by its diagonal elements. As we will see, one can derive
all off-diagonal matrix elements from diagonal ones. We will follow the nice
notes of [16], but see also [17]. For generalization to fields, see, for
example, the classic paper by Sudarshan [18]

Coherent states, states that are eigenvectors of the lowering operator
$\hat{A}$, can be written in the number-operator basis:

$|\alpha\rangle=e^{-|\alpha|^{2}/2}\sum_{n}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle$
(56)

Define an operator $\hat{O}$ in the number-operator basis as
$O_{mn}\equiv\langle m|\hat{O}|n\rangle$. And define

${\cal O}_{\bar{\alpha}\beta}\equiv
e^{+(|\alpha|^{2}+|\beta|^{2})/2}\langle\alpha|\hat{O}|\beta\rangle=\sum_{mn}O_{mn}\frac{\bar{\alpha}^{m}\beta^{n}}{\sqrt{m!n!}}$
(57)

writing $\alpha^{*}\rightarrow\bar{\alpha}$ for convenience. One can
immediately see that all of the matrix elements of $\hat{O}$ in the number-
operator basis can be derived from the diagonal matrix elements in the
coherent state basis:

$O_{mn}=\frac{1}{\sqrt{m!n!}}\frac{\partial^{m}}{\partial\bar{\alpha}^{m}}\frac{\partial^{n}}{\partial\alpha^{n}}{\cal
O}_{\bar{\alpha}\alpha}\bigg{|}_{\bar{\alpha}=\alpha=0}$ (58)

One can also derive the off-diagonal matrix elements in the coherent state
basis as well by plugging (58) into (57).