Quantum Gravity Effective Action Provides Entropy of The Universe

Ken-ji Hamada

Institute of Particle and Nuclear Studies, KEK, Tsukuba 305-0801, Japan
and Graduate Institute for Advanced Studies, SOKENDAI, Tsukuba 305-0801, Japan

###### Abstract

The effective action in renormalizable quantum theory of gravity provides
entropy because the total Hamiltonian vanishes. Since it is a renormalization
group invariant that is constant in the process of cosmic evolution, we can
show conservation of entropy, that is an ansatz in the standard cosmology.
Here we study renormalizable quantum gravity that exhibits conformal dominance
at high energy beyond the Planck scale. The current entropy of the universe is
derived by calculating the effective action under the scenario of quantum
gravity inflation caused by its dynamics. We then argue that ghost modes must
be unphysical, but necessary for the Hamiltonian to vanish and for entropy to
exist in gravitational systems.

##  1 Introduction

The origin of a huge amount of cosmic entropy is a big mystery, most of which
is currently carried by the cosmic microwave background radiation (CMB).
Moreover, the standard cosmology assumes that it is conserved during evolution
[1]. The conservation law has been explained thermodynamically, but is that
really sufficient? The observed CMB anisotropy spectra [2, 3] show that the
whole universe is entangled in every corner. It suggests that there was a
moment in the past when they were correlated with each other. It is natural to
think that the origin of entropy and its conservation law were ruled by the
laws of physics at that time.

In general, the effective action in renormalizable quantum field theory is
finite and renormalization group (RG) invariant [4]. If the theory is
diffeomorphism invariant, the energy-momentum tensor is also finite, i.e.
normal product, and RG invariant [5, 6, 7, 8, 9]. Furthermore, when gravity is
quantized, the energy-momentum tensor vanishes as an identity, as shown by the
Schwinger-Dyson equation for the gravitational field [10]. Since the
Hamiltonian vanishes, the effective action of quantum gravity gives entropy
itself, and thus

$\displaystyle\Gamma_{\rm QG}=-S_{\rm Univ}.$ (1)

The $S_{\rm Univ}$ is entropy of the universe, conserved as an RG invariant.
This is a general conclusion derived from diffeomorphism invariance and
renormalizability.

On the other hand, as if to deny this idea, it is often emphasized that
conventional renormalization methods are not applicable to quantum theory of
gravity. In fact, Einstein’s theory of gravity is unrenormalizable. To escape
this problem, researchers usually introduce a finite ultraviolet (UV) cutoff
in the Planck scale and often think of it as an entity of spacetime
quantization. However, this idea breaks diffeomorphism invariance. To preserve
diffeomorphism invariance, renormalizability is necessary, then spacetime
exists continuously while fluctuating greatly even beyond the Planck scale.

The problem with renormalization is relevant to the existence of
singularities. Since the Einstein-Hilbert action does not contain the Riemann
tensor $R_{\mu\nu\lambda\sigma}$ that controls curvature, singularities cannot
be removed. If we introduce a positive-definite fourth-derivative action
containing $R_{\mu\nu\lambda\sigma}^{2}$ to make gravity renormalizable [11,
12, 13, 14], then singularities are statistically forbidden as objects for
which the action diverges. But, if we formulate it simply in the usual
graviton picture defined by a perturbation expansion around flat spacetime,
another difficulty arises called the ghost problem.

However, the existence of ghost modes itself is not the problem. The local
particle world has to be positive definite, but the universe as a whole is not
so. They are rather essential elements for the total Hamiltonian to vanish,
i.e., to preserve diffeomorphism invariance. As a matter of fact, Einstein’s
theory of gravity has a ghost mode due to the indefinitness of the Einstein-
Hilbert action, which allows non-trivial solutions such as the Friedmann
solution to exist. If all modes were positive definite, the only state in
which the Hamiltonian vanishes would be the trivial vacuum, and thus no
entropy. Ghost modes cause problems only when they appears as physical
objects.

In order to solve these problems, renormalizable quantum gravity that exhibits
background freedom asymptotically has been proposed [15, 16, 17, 18, 19]. A
distinctive feature of the theory is that it implements a novel perturbation
technique expanding around spacetime where the Weyl tensor
$C_{\mu\nu\lambda\sigma}$ vanishes, instead of the conventional perturbation
expansion around flat spacetime. In the UV limit of this perturbation, the
conformal mode of the gravitational field determining distance is not subject
to any restrictions, and thus fluctuates largely in a nonperturbative manner.
Hence, scalar fluctuations dominate in the early universe. The correctness of
this approach is supported by the idea of inflation [20, 21, 22, 23, 24, 25,
26].

This theory has a new mechanism for constraining ghosts, called the BRST
conformal invariance [27, 28, 29, 30, 31, 32, 33], which arises as part of
diffeomorphism invariance in the UV limit [15, 16, 17, 18, 19]. It represents
the background freedom that all different conformally-flat spacetimes are
gauge equivalent. The BRST conformal invariance condition shows that all ghost
modes involved in the gravitational field are unphysical and can never be
seen, while there are an infinite number of physical states, which are only
scalar-type (primary scalars), no tensor-type states [31, 32, 33].

In this paper, we show that the effective action of this quantum gravity
indeed provides entropy of the present universe.

##  2 Renormalizable Quantum Gravity

The action of renormalizable quantum gravity that indicates conformal
dominance asymptotically is given by [15, 16, 17, 18, 19]

$\displaystyle I=\int
d^{4}x\hbox{$\sqrt{-g}$}\biggl{\\{}-\frac{1}{t^{2}}C_{\mu\nu\lambda\sigma}^{2}-bG_{4}+\frac{1}{\hbar}\left(\frac{1}{16\pi
G}R+{\cal L}_{\rm M}\right)\biggr{\\}}.$

The first is the Weyl action and the second is the Euler term, where
$G_{4}=R^{2}_{\mu\nu\lambda\sigma}-4R^{2}_{\mu\nu}+R^{2}$, both of which are
conformally invariant. The third is the Einstein-Hilbert action and the fourth
denotes matter field actions that are conformally invariant in the UV limit.

The Planck constant $\hbar$ appears only before the lower derivative terms,
which is due to the fact that the gravitational field is completely
dimensionless. This implies that all of the fourth-derivative gravitational
actions here and below describe purely quantum dynamics and produce entropy of
spacetime. In the following, $\hbar=1$.

Using the conformal mode $\phi$ and the traceless tensor mode
$h^{\mu}_{~{}\nu}$, the gravitational field is decomposed as
$g_{\mu\nu}=e^{2\phi}{\bar{g}}_{\mu\nu}$ with ${\bar{g}}_{\mu\nu}=(\eta
e^{h})_{\mu\nu}=\eta_{\mu\lambda}(\delta^{\lambda}_{~{}\nu}+h^{\lambda}_{~{}\nu}+h^{\lambda}_{~{}\sigma}h^{\sigma}_{~{}\nu}/2+\cdots)$.111Applying
this expansion method leaves diffeomorphism
$\delta_{\zeta}\phi=\zeta^{\lambda}\partial_{\lambda}\phi+\partial_{\lambda}\zeta^{\lambda}/4$
and
$\delta_{\zeta}h_{\mu\nu}=\zeta^{\lambda}\partial_{\lambda}h_{\mu\nu}+(\partial_{\mu}\zeta_{\lambda}-\partial_{\lambda}\zeta_{\mu})h^{\lambda}_{~{}\nu}/2+(\partial_{\nu}\zeta_{\lambda}-\partial_{\lambda}\zeta_{\nu})h^{\lambda}_{~{}\mu}/2$
with a gauge parameter $\zeta^{\lambda}$ satisfying conformal Killing
equations at the UV limit of $t=0$. This is the BRST conformal
transformations. The conformal factor $e^{2\phi}$ is handled
nonperturbatively, while $h^{\mu}_{~{}\nu}$ is expanded as being small at the
UV limit. The coupling constant for the expansion is $t$ introduced in front
of the Weyl action, while $b$ is not an independent coupling because the Euler
term does not contain a kinetic term. The flat metric
$\eta_{\mu\nu}=(-1,1,1,1)$ defines the comoving frame with coordinates
$x^{\mu}=(\eta,{\bf x})$.

The key to quantization is to rewrite the theory into a quantum field theory
defined on the familiar flat spacetime. The partition function is then
expressed as [15, 16, 17, 18, 19, 31, 32, 33]

$\displaystyle\int[dg]_{g}\,e^{iI(g)}=\int[d\phi
dh]_{\eta}\,e^{iS(\phi,{\bar{g}})+iI(g)},$

where $S$ is the Wess-Zumino action [34] for the conformal anomaly [35, 36,
37, 38], which is necessary to preserve diffeomorphism invariance when
rewriting the path integral measure to the practical measure on flat
spacetime.

The Wess-Zumino action $S$ is responsible for fourth-derivative dynamics of
the conformal mode $\phi$. The Riegert action associated with the Euler-type
conformal anomaly [27] that remains even in the zeroth order of $t$ is
particularly important, which is

$\displaystyle S_{\rm R}=\int
d^{4}x\biggl{\\{}-\frac{b_{c}}{(4\pi)^{2}}B\biggl{[}2\phi{\bar{\Delta}}_{4}\phi+\biggl{(}\bar{G}_{4}-\frac{2}{3}{\bar{\nabla}}^{2}{\bar{R}}\biggr{)}\,\phi\biggr{]}\biggr{\\}},$

providing a kinetic term of $\phi$. The quantities with the bar denote those
defined by ${\bar{g}}_{\mu\nu}$ and $\sqrt{-g}\Delta_{4}$ is a fourth-order
differential operator that is conformally invariant for scalars, defined by
$\Delta_{4}=\nabla^{4}+2R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-2R\nabla^{2}/3+\nabla^{\mu}R\nabla_{\mu}/3$.
The lowest coefficient is given by $b_{c}=(N_{\rm X}+11N_{\rm W}/2+62N_{\rm
A})/360+769/180$ [13, 15, 30, 35, 36, 37, 38], where $N_{\rm X}$, $N_{\rm F}$,
and $N_{\rm A}$ are the number of scalar fields, Weyl fermions, and gauge
fields in the matter sector, respectively, and $b_{c}=7.0$ for the Standard
Model. A correction by $t$ is denoted as $B=1-\gamma_{1}t^{2}/4\pi+o(t^{4})$.

The beta function is negative as $\mu dt/d\mu=-\beta_{0}\,t^{3}+o(t^{5})$ with
$\beta_{0}=[(N_{\rm X}+3N_{\rm F}+12N_{\rm A})/240+197/60]/(4\pi)^{2}$ [13,
15, 30, 35, 36, 37, 38], where $\mu$ is an arbitrary mass scale introduced
upon quantization. The dynamics of the tensor mode causes running of the
coupling constant expressed by
$\bar{t}^{2}(Q)=[\beta_{0}\log(Q^{2}/\Lambda_{\rm QG}^{2})]^{-1}$, where
$Q^{2}=q^{2}/e^{2\phi}$ is physical momentum squared, $q^{2}$ is comoving one,
and the $\phi$-dependence comes from the Wess-Zumino actions [16, 24, 39]. The
new dynamical energy scale $\Lambda_{\rm QG}\,(=\mu\,e^{-1/2\beta_{0}t^{2}})$
is a physical constant, i.e., an RG invariant satisfying $d\Lambda_{\rm
QG}/d\mu=0$, as is the Planck mass $M_{\rm P}$ [40]. The correlation length is
given by $\xi_{\Lambda}=1/\Lambda_{\rm QG}$, so that spacetime will be
dynamically discretized in this length [41].

The inflationary solution exists only for $M_{\rm P}>\Lambda_{\rm QG}$ [23,
24, 25, 26]. This relation also serves as a unitarity condition to ensure that
ghost modes do not appear as particles in the world after the spacetime phase
transition. In quantum spacetime, all ghost modes are constrained by the BRST
conformal invariance, but this cannot be applied after the transition.
Therefore, the transition must occur below the Planck scale so that ghost
gravitons with mass about $M_{\rm P}$ do not appear.

##  3 Entropy of The Universe

Most of entropy in a quantum gravity state is carried by the conformal mode
rather than tensor or matter modes. Its dynamics cause inflation, and entropy
in that period will give entropy of the universe. The homogeneous component of
the conformal mode is then responsible for them significantly, which is
denoted as ${\hat{\phi}}$.

The effective action for this mode is given from the Riegert and Einstein-
Hilbert sectors as

$\displaystyle\Gamma_{\rm QG}=V_{3}\int
d\eta\biggl{\\{}-\frac{b_{c}}{8\pi^{2}}B{\hat{\phi}}\partial_{\eta}^{4}{\hat{\phi}}+3M_{\rm
P}^{2}e^{2{\hat{\phi}}}\bigl{(}\partial_{\eta}^{2}{\hat{\phi}}+\partial_{\eta}{\hat{\phi}}\partial_{\eta}{\hat{\phi}}\bigr{)}\biggr{\\}},$
(2)

where $V_{3}=\int d^{3}{\bf x}$. The Weyl and matter sectors are disregarded
here due to conformal invariance, while the running of the coupling constant
$t$ is taken into account below.

The equation of motion for the homogeneous mode is given by

$\displaystyle-\frac{b_{c}}{4\pi^{2}}B\partial_{\eta}^{4}{\hat{\phi}}+6M_{\rm
P}^{2}e^{2{\hat{\phi}}}\bigl{(}\partial_{\eta}^{2}{\hat{\phi}}+\partial_{\eta}{\hat{\phi}}\partial_{\eta}{\hat{\phi}}\bigr{)}=0.$
(3)

This equation has a stable inflationary de Sitter solution at $t\to 0\,(B\to
1)$ [23]. Introducing the proper time defined by $d\tau=ad\eta$ with the scale
factor $a=e^{\hat{\phi}}$ and using the Hubble variable
$H=\partial_{\tau}a/a$, the solution is expressed as

$\displaystyle H=H_{\bf D},\qquad H_{\bf
D}=\sqrt{\frac{8\pi^{2}}{b_{c}}}M_{\rm P}.$ (4)

This shows that the scale factor increases exponentially as $a=e^{H_{\bf
D}\tau}$, where let it be unity at $\tau=0$ representing infinite energy so
that $V_{3}$ is the spatial volume far before inflation begins.

The inflationary expansion starts at the Planck scale $H_{\bf D}$ and
terminates at the dynamical scale $\Lambda_{\rm QG}$ where a spacetime phase
transition occurs [23, 24, 25, 26]. The running coupling constant
$\bar{t}^{2}$ indicates that spacetime is initially conformally invariant,
while when energy drops to the vicinity of $\Lambda_{\rm QG}$, $\bar{t}^{2}$
increases rapidly and deviates from such a spacetime. The phase transition is
expressed as a process in which the conformal gravity dynamics disappears as
$\bar{t}^{2}$ grows and shifts to the present universe dominated by the
Einstein-Hilbert action. At this time, since energy of the whole system is
preserved to be zero due to diffeomorphism invariance, the disappeared energy
of quantum gravity is transferred to that of matters, causing the big bang.
Entropy of the universe (1) is also inherited as it is in matters as a
conserved quantity.

To describe such dynamics, we have proposed a model that approximates the
running coupling constant by a time-dependent mean field
$\bar{t}^{2}(\tau)=[\beta_{0}\log(\tau_{\Lambda}^{2}/\tau^{2})]^{-1}$ that
diverges at the dynamical time $\tau_{\Lambda}=1/\Lambda_{\rm QG}$ [24, 25,
26]. The running behavior is incorporated by replacing $t^{2}$ with
$\bar{t}^{2}(\tau)$, and the replaced $B$ is denoted as $\bar{B}(\tau)$. The
disappearance of conformal gravity dynamics is expressed by assuming its form
as $\bar{B}(\tau)=[1+\gamma_{1}{\bar{t}}^{2}(\tau)/4\pi]^{-1}$, where
$\beta_{0}$ and $\gamma_{1}$ are treated as positive phenomenological
parameters. However, to be able to carry out calculations analytically, here
we mainly consider a radically simplified model so that $\bar{t}^{2}$ remains
almost zero and diverges sharply at $\tau_{\Lambda}$, so that $\bar{B}$ is
almost $1$ and abruptly goes to zero at $\tau_{\Lambda}$ as a step function.
In the meanwhile, we refer to the original unsimplified dynamical model as
appropriate.

The number of e-foldings ${\cal N}_{e}=\log[a(\tau_{\Lambda})/a(\tau_{\rm
P})]$ from the Planck time $\tau_{\rm P}=1/H_{\bf D}$ to the dynamical time
$\tau_{\Lambda}$ is then given by the ratio of two energy scales as

$\displaystyle N=\frac{H_{\bf D}}{\Lambda_{\rm QG}}.$

If we solve (3) with the dynamical factor $\bar{B}(\tau)$ rather than the step
function, $H$ becomes larger than $H_{\bf D}$ near the phase transition and
the number of e-foldings becomes slightly larger than $N$ (see Fig.1).

Figure 1: The inflationary solution presented by renormalizable and
asymptotically background-free quantum gravity. The left is the time evolution
of the scale factor $a$ and the right is that of the Hubble variable $H$,
where $H_{\bf D}$ is normalized to unity and $t\,(=H_{\bf D}\tau)$ is a
normalized proper time. The solid lines (violet and blue) are the solution of
the dynamical model approximating the running coupling constant with the time-
dependent mean field, and the dashed lines (green and red) are the radically
simplified case. The vertical dotted line represents the dynamical time
$\tau_{\Lambda}$, set to $60$, at which the spacetime phase transition occurs.
After time bigger than $\tau_{\Lambda}$, the variables are calculated using
the low-energy effective gravity theory defined by derivative expansions
centered on the Einstein-Hilbert action (see [24] in detail), while the dashed
lines display the Friedmann solution.

If $N$ is fixed, the number of e-foldings is determined, and the energy scale
$\Lambda_{\rm QG}$ is also determined. Scalar spacetime fluctuations, which
account for most of initial fluctuations, have been shown to reduce in
amplitude during inflation [24, 25, 26]. The magnitude at the phase transition
can be roughly estimated as $\delta R/R\sim\Lambda_{\rm QG}^{2}/12H_{\bf
D}^{2}=1/12N^{2}$, where the denominator is the de Sitter curvature. It should
be the order of the CMB anisotropy, $10^{-4}\sim 10^{-5}$ [2, 3]. Furthermore,
the comoving dynamical scale $\lambda=\Lambda_{\rm QG}/a_{0}$, where $a_{0}$
is the current scale factor, will be approximately given by the current Hubble
constant $H_{0}\simeq 0.00023\,{\rm Mpc}^{-1}$ to explain the sharp falloff in
low multipole components of the CMB anisotropy spectrum [23]. From these
conditions, $N$ is determined to be about $60$, and $\Lambda_{\rm QG}$ is
given on the order of $10^{17}$GeV, two orders of magnitude lower than the
Planck energy [23, 24, 25, 26]. The magnitude of $a_{0}$ that expresses how
much the universe has expanded since before inflation is determined to be
about $10^{59}$ (see Fig.2). From the previous studies, we have found that
$\Lambda_{\rm QG}$ is a tight parameter whose value cannot be changed very
much.

Figure 2: The evolution scenario of quantum gravity inflation. This shows that
most of the universe we see today, represented by the Hubble distance, was
originally the size of a quantum gravity excitation given by the correlation
length $\xi_{\Lambda}\,(=1/\Lambda_{\rm QG})$. That is, $1/H_{0}\simeq
10^{59}\xi_{\Lambda}$ or $H_{0}\simeq\lambda$. This suggests that the universe
is an entangled quantum system.

Rewriting the effective action (2) with the equation of motion (3) yields

$\displaystyle\Gamma_{\rm QG}$ $\displaystyle=$ $\displaystyle-3M_{\rm
P}^{2}V_{3}\int
d\eta\bigl{(}{\hat{\phi}}-1\bigr{)}e^{2{\hat{\phi}}}\bigl{(}\partial_{\eta}^{2}{\hat{\phi}}+\partial_{\eta}{\hat{\phi}}\partial_{\eta}{\hat{\phi}}\bigr{)}.$
(5)

If the inflationary period given by (4) continues until it terminates at
$\tau_{\Lambda}$ in the proper time, entropy of quantum gravity is given from
the formula (1) as

$\displaystyle S_{\rm Univ}$ $\displaystyle=$ $\displaystyle 6M_{\rm
P}^{2}H_{\bf D}^{2}V_{3}\int^{\tau_{\Lambda}}_{0}d\tau e^{3H_{\bf
D}\tau}\bigl{(}H_{\bf D}\tau-1\bigr{)}$ $\displaystyle=$ $\displaystyle
2M_{\rm P}^{2}H_{\bf D}V_{3}\biggl{[}e^{3H_{\bf
D}\tau_{\Lambda}}\biggl{(}H_{\bf
D}\tau_{\Lambda}-\frac{4}{3}\biggr{)}+\frac{4}{3}\biggr{]}.$

Omitting the negligible last term and rewriting with the scale ratio $N$, we
finally get

$\displaystyle S_{\rm Univ}=2\sqrt{\frac{8\pi^{2}}{b_{c}}}M_{\rm
P}^{3}e^{3N}\biggl{(}N-\frac{4}{3}\biggr{)}V_{3}.$ (6)

Matters are created when the quantum gravity entropy (6) is released at the
phase transition. The entropy is an RG invariant quantity, and thus it should
match the current entropy, given in density by [1],

$\displaystyle s=\frac{S_{\rm Univ}}{V_{3}^{\rm
today}}=\frac{2\pi^{2}}{45}g_{*S}T^{3}\simeq 2.91\times 10^{3}\,{\rm
cm}^{-3},$ (7)

where $g_{*S}=3.91$ and $T=2.73\,{\rm K}=11.9{\rm cm}^{-1}$. When comparing
the two expressions, note that $V_{3}$ in (6) is the initial three-dimensional
volume before inflation. The universe has undergone inflation and the
subsequent Friedmann eras, increasing the current scale factor to $a_{0}$.
Therefore, from $V_{3}^{\rm today}=a_{0}^{3}V_{3}$, the current entropy
density can be expressed as

$\displaystyle s=2\sqrt{\frac{8\pi^{2}}{b_{c}}}M_{\rm
P}^{3}e^{3N}\biggl{(}N-\frac{4}{3}\biggr{)}a_{0}^{-3}.$

Here, fixing the parameters to $b_{c}=7$ and $a_{0}=10^{59}$ as derived before
and finding the value of $N$ so that this agrees with (7), we get $N=62.2$,
where $M_{\rm P}=2.436\times 10^{18}\,{\rm GeV}=1.235\times 10^{32}\,{\rm
cm}^{-1}$ is used. Thus, we find that the current entropy of the universe is
consistent with the scenario of quantum gravity inflation.

The ratio between $2.73$K and $\Lambda_{\rm QG}$ suggests that the universe
expands about $10^{29}$ times after settling into the Friedmann spacetime.
Therefore, the universe has to expand about $10^{30}$ times during the
inflationary era. This corresponds to ${\cal N}_{e}\simeq 70$, which is
greater than $60$. This contradiction can be resolved by considering a more
realistic dynamical model described by the time-dependent running coupling
constant above. Fig.1 shows an inflationary solution calculated by choosing
the phenomenological parameters to be $\beta_{0}=0.171$ and $\gamma_{1}=0.1$
with $N=60$ so that entropy obtained by evaluating (5) with this solution
numerically agrees with (7). In any case, quantum gravity can generate
sufficient entropy.

##  4 Conclusion

The current entropy of the universe is derived from the effective action of
renormalizable quantum gravity with asymptotic background freedom. The result
suggests that entropy of the universe is originated from quantum gravity. The
conservation of entropy can be shown from diffeomorphism invariance and
renormalizability. Hence, these conditions should be taken as guiding
principles even in the trans-Planckian world. The ghost modes in gravity as
unphysical beings are necessary to provide entropy while preserving the total
Hamiltonian to be zero, and the zero-point energy then disappears due to them
[10]. The cosmic time, which is a change in central value of the conformal-
mode fluctuation given by the inflation and Friedmann solutions, is also one
of things dynamically generated due to the existence of the ghost modes.
Instead of denying the existence, we should consider what it brings to a
quantum system of the entire universe. For gravitational systems, since the
Hamiltonian vanishes identically, we cannot directly apply no-go theorems
proving that energy spectrum is unbound from below such as the Ostrogradsky
theorem [42].

Reconciling quantum mechanics and gravity requires a deep understanding of
what it means for the Hamiltonian to vanish. The ghost mode reminds us of the
Bohm’s “hidden variables” in the sense that it is “invisible, but being”.
However, note that there is no such variable in special relativity, and the
ghosts are necessary only when gravity has a substantial contribution. The
weak-field (graviton) approximation is nothing more than reducing the
gravitational system to a system in special relativity, in which case ghost
modes should not exist essentially.

##  Appendix A Notes on Unitarity or Reality

Quantum spacetime where quantum gravity is activated is described in terms of
conformal field theory so that the particle picture no longer holds true and
the scattering matrix is not defined.

The issue of unitarity is then easier to understand if we consider quantum
field theory in Euclidian space by performing the Wick rotation. If the
Euclidized action $I$ is real and bounded below, the weight $e^{-I}$ becomes
positive and finite, thus the path integral is correctly defined. In this
case, the reality of the field (positivity of two-point function) is never
lost, and unitarity holds. This also states that a singular configuration
where $I$ is positively divergent can be removed as a unphysical state that
does not exist stochastically because the weight vanishes, as mentioned in
Introduction.

On the other hand, if $I$ is an indefinite action that can be negative
infinity, the path integral will diverge, the field reality will be sacrificed
to regularize the divergence. As can be seen from this fact, the path integral
cannot be defined correctly by the Einstein-Hilbert action alone, thus the
square of the curvature is necessary to ensure positive definiteness.

Why does the ghost problem occur? In the end, it turns out that the problem
lies in not employing an appropriate perturbation method. Usually, it is
carried out choosing a picture where gravitons propagate in flat spacetime. In
Einstein’s theory of gravity, this choice is actually better at eliminating
ghosts locally, thus is an adequate setting. In fourth-derivative quantum
gravity, however, this poses the problems that the positive- and negative-
metric modes behave asymptotically as independent physical degrees of freedom,
thus a correlation function for the negative-metric mode has physical meaning,
but it is not correctly defined. On the contrary, if we consider physical
quantities using the gravitational field as a “fundamental field” without
separating into positive- and negative-metric modes, then the correlation
function is correctly defined because the field action itself is positive-
definite, and thus the reality of the field is never violated.222Note the
difference that, in Einstein’s theory of gravity, $\phi$ itself becomes a
ghost mode, whereas in fourth-derivative quantum gravity, ghost modes arise
only when the fundamental fields $\phi$ and $h_{\mu\nu}$ are further expanded
into modes.

Hence, in order to define fourth-derivative quantum gravity, it is clear that
when dealing with physical quantities, there must be some kind of constraint
that does not allow the separation of the field into positive and negative-
metric modes, so that the Hamiltonian is bounded. The BRST conformal
invariance plays exactly that role.

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## Appendix A: Experimental Scenario Variables

This appendix contains visualizations of the tasks and variables (backgrounds,
obstacle types, and obstacle colors) that define the scenarios of the
simulation experiments.

### Tasks

Figures 23–26 illustrate the setups of the tasks in Gazebo. Each figure
depicts the start and end of each task, in addition to a baseline end-effector
trajectory that does not consider obstacles.

(a) Start

(b) End

Figure 23: Simulation Task 1 setup. A baseline trajectory is illustrated in
green.

(a) Start

(b) End

Figure 24: Simulation Task 2 setup. The obstacle’s trajectory and a baseline
end-effector trajectory are illustrated in red and green.

(a) Start

(b) End

Figure 25: Simulation Task 3 setup. A baseline trajectory is illustrated in
green.

(a) Start

(b) End

Figure 26: Simulation Task 4 setup. The obstacle’s trajectory and a baseline
end-effector trajectory are illustrated in red and green.

### Backgrounds, Object Types and Object Colors/Textures

Figures 27 and 28 depict the different values of the background, obstacle
type, and obstacle color variables presented in Table 1.

(a) Empty

(b) Office

(c) Store (high illumination)

(d) Kitchen (low illumination)

Figure 27: Different backgrounds (Gazebo worlds) used in task scenarios listed
in Table 1.

(a) Box:
Brick Pattern

(b) Spiky Sphere:
Y-B Pattern

(c) Rock:
White

(d) Buckyball:
Red

Figure 28: Different obstacle types used in task scenarios, listed in Table 1.
Each obstacle also illustrates one of the possible obstacle colors.

## Appendix B: List of Experiment Scenarios

This appendix contains the lists of task scenarios that were used in the
simulated and real experiments in Tables 7 and 8.

Table 7: List of tuning, validation, and testing scenarios. Scenario ID |  Set |  Scenario Specification
---|---|---
0 |  Pre-Tuning |  Task 1, Empty, Cracker box
1 |  Tuning |  Task 1, Store, Red Box
2 |  Tuning |  Task 4, Store, Brick, Rock, Medium Speed
3 |  Tuning |  Task 3, Empty, Brick
4 |  Validation |  Task 1, Office, Red, Buckyball
5 |  Validation |  Task 1, Office, Yellow-Black, Rock
6 |  Validation |  Task 1, Empty, Brick, Spiky Sphere
7 |  Validation |  Task 4, Empty, Yellow-Black, Box, Medium Speed
8 |  Validation |  Task 4, Office, Yellow-Black, Buckyball, High Speed
9 |  Validation |  Task 4, Store, White, Spiky Sphere, Low Speed
10 |  Validation |  Task 3, Office, White
11 |  Validation |  Task 3, Store, Yellow-Black
12 |  Testing |  Task 1, Empty, Yellow-Black, Spiky Sphere
13 |  Testing |  Task 1, Store, Brick, Buckyball
14 |  Testing |  Task 1, Kitchen, White, Buckyball
15 |  Testing |  Task 1, Empty, Yellow-Black, Box
16 |  Testing |  Task 1, Store, Yellow-Black, Rock
17 |  Testing |  Task 2, Office, Red, Buckyball, High Speed
18 |  Testing |  Task 2, Office, Red, Rock, Medium Speed
19 |  Testing |  Task 2, Store, Brick, Box, Medium Speed
20 |  Testing |  Task 2, Kitchen, Yellow-black, Box, High Speed
21 |  Testing |  Task 2, Empty, Red, Spiky Sphere, Medium Speed
22 |  Testing |  Task 3, Empty, Red
23 |  Testing |  Task 3, Empty, Yellow-Black
24 |  Testing |  Task 3, Kitchen, White
25 |  Testing |  Task 3, Kitchen, Red
26 |  Testing |  Task 3, Store, Red
27 |  Testing |  Task 4, Store, Yellow-Black, Buckyball, Low Speed
28 |  Testing |  Task 4, Store, Red, Buckyball, High Speed
29 |  Testing |  Task 4, Empty, Yellow-Black, Spiky Sphere, Medium Speed
30 |  Testing |  Task 4, Empty, Yellow-Black, Box, Low Speed
31 |  Testing |  Task 4, Empty, Brick, Buckyball, Low Speed
Table 8: List of real robot experiment task scenarios. Scenario ID |  Set |  Scenario Specification
---|---|---
R1 |  Testing (Real) |  Task 1, Lab Background 1, Wooden Block
R2 |  Testing (Real) |  Task 2, Lab Background 2, Hand
R3 |  Testing (Real) |  Task 2, Lab Background 2, Metal Bar
R4 |  Testing (Real) |  Task 2, Lab Background 2, Wooden Block

## Appendix C: Evaluation Metrics and Criteria Details

This appendix contains more detailed descriptions of the quantitative metrics
and qualitative evaluation criteria presented in section 5.1.3.

### Quantitative Metrics

Table 9 lists the quantitative metrics we use to analyze the performances of
parameter sets and to evaluate our approach.

Table 9: Quantitative performance metrics. Metric |  Symbol |  Description
---|---|---
Execution Time |  $T$ |  Time required to complete the task
Trajectory Length |  $l_{\mathbf{Y}}$ |  Distance traveled by the end-effector
Number of Collisions |  $N_{collisions}$ |  No. of instances in which the EE intersects an obstacle (in simulation)
Distance to Goal |  $d_{G}$ |  Euclidean distance to the goal at the end of execution
Success |  S |  True iff no collisions and distance to goal $<\epsilon$
Velocities |  $\mathbf{\dot{y}}$ |  Magnitudes of instantaneous end-effector velocities
Accelerations |  $\mathbf{\ddot{y}}$ |  Magnitudes of instantaneous end-effector accelerations

Execution time, $T$, refers to how long it takes to execute the task. Relative
to baseline trajectories, it indicates how much more time is expended in
obstacle maneuvers.

The trajectory length estimates the distance traveled by the end-effector and
is similarly used to assess how far obstacle avoidance trajectory adaptations
tend to deviate from the baseline. It can be estimated by summing the
euclidean distances between consecutive end-effector positions:

$l_{\mathbf{Y}}=\sum_{i=1}^{N_{pos}}||\mathbf{y}_{i}-\mathbf{y}_{i-1}||_{2}$
(25)

$N_{pos}$ refers to the number of recorded positions while executing
trajectory $\mathbf{Y}$. It is generally desirable to minimize both $T$ and
$l_{\mathbf{Y}}$ as a secondary objective to successful obstacle avoidance and
reaching the goal.

In simulation, the number of collisions, $N_{collisions}$, represents a count
of the instances in which the end-effector intersects an obstacle (given that
physical collision effects are disabled). Intuitively, this provides an
indication of how often the end-effector collides and the degree to which
collisions are sustained or eventually adapted for.

The distance-to-goal metric measures how well a given execution fulfills the
goal reaching aspect of the task and is obtained from the euclidean distance
from the final position, $\mathbf{y}_{T}$, to the commanded goal position,
$g$:

$d_{G}=||\mathbf{y}_{T}-\mathbf{g}||_{2}$ (26)

Success is captured in binary metric $S$, which incorporates $N_{collisions}$
and $d_{G}$: for a success, a given trial must i) involve no collisions and
ii) end with the end-effector close to the goal, according to the goal-
reaching tolerance $\delta_{\mathbf{g}}$:

$S=\begin{cases}1,&\text{if }N_{collisions}=0\text{ and
}d_{G}<\delta_{\mathbf{g}}\\\ 0,&\text{otherwise}\\\ \end{cases}$ (27)

We also measure the magnitudes of end-effector velocities and accelerations,
which provide insights on motion properties particularly with regards to
qualitative criteria.

### Qualitative Metrics

Table 2 lists our qualitative evaluation criteria, which are described along
with their formalisms in the following.

Reliability can be defined as the degree to which a method produces
consistently positive results in successive trials under reasonably consistent
conditions. We expect a reliable system to repeatedly achieve similarly
positive results, as opposed to significant variances and/or failure rates,
provided that testing conditions do not substantially differ from the system’s
intended operating environment. One way to estimate the reliability is through
the ratio of successful trials to the total number of trials in imminent
collision situations, which are provided in the task scenarios defined in
section 5.1.1. A high value indicates that obstacle avoidance was consistently
successful across trials, while a lower value indicates an inconsistency that
shows unreliability, even if some success was achieved.

Predictability describes to what extent behaviour can be known or predicted in
advance, for which we use angular velocities that describe changes in
direction. The results of a controller that produces fewer directional changes
are easier to intuit or predict from historical observations (within or across
executions). We can quantify the local degree of direction change at every
end-effector position from the angle between the vectors formed by connecting
that positional point to the previous and the next point, as is done in
Salarpour and Khotanlou (2019). Given trajectory positions $\mathbf{y}_{k}$,
$\mathbf{y}_{k-1}$, and $\mathbf{y}_{k+1}$, and resultant vectors
$\mathbf{v}_{1}=(\mathbf{y}_{k}-\mathbf{y}_{k-1})$ and
$\mathbf{v}_{2}=(\mathbf{y}_{k+1}-\mathbf{y}_{k})$, the direction change,
$\zeta$, at $\mathbf{y}_{k}$ can be calculated from the dot product:

$\zeta=\arccos{\left(\frac{v_{1}\cdot v_{2}}{|v_{1}||v_{2}|}\right)}$ (28)

The magnitudes of $\zeta$ provide an indication of path curvature, and higher
values do not necessarily indicate unpredictable motions. On the other hand,
the magnitudes of $\dot{\zeta}$, which could be expressed as an angular
velocity, are related to how often and how drastically the end-effector
changes the direction in which it is moving.

Safety refers to the degree to which a system is unlikely to cause danger,
risk, or injury. In the case of the manipulation tasks addressed here, we can
examine the magnitudes of applied velocities and accelerations of the end-
effector as a measure of safety. Intuitively, higher velocities tend to
increase the risk of collisions with other objects or people in the
environment, and the magnitude of potential damage.

## Appendix D: Implementation Details

The components described in section 4 were implemented as ROS nodes, using the
ROS Noetic Ninjemys distribution and Python 3.8 for development. This section
summarizes details of our the ROS stack and the augmented Gazebo simulation of
the Kinova Gen3.

### ROS Components

Figure 29 shows a graph of the ROS nodes and the main communication channels
between them (ROS topics and services). When running experiments on the robot,
the Gazebo node is substituted by others, but the obstacle avoidance pipeline
is unchanged. In the following, we describe each of the nodes (depicted in
blue).

Figure 29: A graph of the main ROS nodes and the topics/services they
communicate on. Services are depicted with dotted arrows and italics; bold
arrows represent topics. The multiple nodes associated with the Kinova arm
drivers (from the ros_kortex package) have been reduced to a single “Kinova
Gen3 Nodes” block.

Our event_camera_emulation Python package provides functionalities for
generating, streaming, and visualizing event data derived from RGB images from
a camera or ROS publishers. An

    event_image_streamer} ROS node handles capturing RGB image frames, creating event images, and publishing the event data.
    %\paragraph

The

    snn_simulator} node receives event data and runs it through a simulated C-SNN to produce the neural activation maps required by the obstacle avoidance component.
    We use the open-source \textit

BindsNET package for running SNNs (presented in Hazan et al. (2018)). In order
to use the FST output representation described in section 4.4, we extended the
implementation to incorporate the computation and recording of first spike
times. Upon receiving event images, the node pre-processes the inputs by:

  1. 1.

Optionally applying an event filtering strategy

  2. 2.

Scaling the image to match the input layer size

  3. 3.

Assigning $r_{ON}$ and $r_{OFF}$ rate values for Poisson encoding (see section
4.3)

  4. 4.

Encoding the data into Poisson spike trains

The spike data is run through the C-SNN and network monitors are inspected to
extract statistics such as membrane potentials, spike counts, average firing
rate, and FSTs, which are published in a custom ROS message. The

    motion_controller} handles configuring the DMP, executing velocity commands from the PID controller to follow the trajectory plan, and adapting the plan by receiving SNN output data, decoding it, and adapting the DMP trajectory.
    We use the \textit

pydmps package to run the DMP.m, $\phi$ (see section 4.4). The ROS framework
allows for the DMP execution and the processing of SNN outputs to run
simultaneously, ensuring that the motion is adapted with the latest possible
information on sensed obstacles. A custom `KinovaArmController} class
interfaces with the Kinova’s \textit`kortex_driver (within ros_kortex),
sending twist commands computed by the PID controller (through ROS services)
while the driver executes the low-level velocity commands. This is the only
component of the node and the pipeline that is robot-specific. Therefore, the
implementation can be adapted to other platforms or applications by simply
integrating the associated low-level controller and adjusting parameters where
necessary. The

    gazebo_object_animator} node executes dynamic obstacle motions by moving a given object through a specified trajectory.
    During experiments, the \mintinline

Pythonmotion_controller publishes a trigger command (ROS boolean message) as
execution starts to induce a controlled, relative object motion. This is
useful when statistically evaluating performance, as it eliminates variance
that may arise due to slightly varying testing conditions.

### Simulation

We use an existing, open-source Gazebo simulation of the Kinova Gen3 arm, and
extend it to include an onboard RGB camera. The simulation integrates with the
same drivers and components that run on the real robot, which are accessible
through the ros_kortex ROS package. We extended the 7-DOF Gen3 arm simulation
with an Intel RealSense camera (from the realsense-ros package mounted at the
top of the end-effector and a plugin that enables configuring data capture and
accessing the camera data through ROS topics.

## Appendix E: Pipeline Component Parameters

Table 10 lists and describes the main parameters of our neuromorphic pipeline,
divided by component.

Table 10: The parameters of each pipeline component. A (*) indicates a
parameter which was not tuned in our experiments

.  Component Symbol Description Event Camera Emulator $\theta$ Event emission
threshold $s_{BE}$ Binary erosion filter structure size Convolutional SNN
$w_{c}$ SNN weight initialization constant $T_{sim}$ SNN simulation time
period $v_{thresh}$ SNN potential spiking threshold $T_{refrac}$ SNN
refractory spiking period $\tau_{v}$ SNN potential decay constant $v_{rest}$
(*) SNN resting potential $v_{reset}$ (*) SNN reset potential Obstacle
Avoidance Component $\phi_{max}$ Upper limit on $\bm{\mathbf{\phi}}$
$n_{\phi}$ $\bm{\mathbf{\phi}}$ history horizon $\eta$ Constant gain (PF;
Park) $C_{\delta}$ Gradient constant factor (PF; Park) $p_{0}$ (*) Obstacle
radius of influence (PF; Park) $t_{act}$ (*) FST activation threshold factor
Motion Planning and Control $\delta_{\mathbf{y}}$ Position reaching tolerance
$\delta_{obs}$ Obstacle avoidance distance tolerance $\delta_{safety,x}$
Safety strategy distance tolerance 1414endnote: 14We use two safety strategy
for the different task types (see section 6.1.2). The three safety strategy
parameters have distinct values for each, which are indexed by $x$
$\gamma_{v,x}$ Safety strategy velocity reduction factor $\gamma_{a,x}$ Safety
strategy acceleration reduction factor $\delta^{+}_{pos,i}$ Upper positional
limit (dimension $i$) $\delta^{-}_{pos,i}$ Lower positional limit (dimension
$i$) $K_{j}$ (*) Controller gain ($j$ $\in$ $\\{P,I,D\\}$)
$\delta_{\mathbf{g}}$ (*) Goal reaching tolerance

## Appendix F: Event Emulation Comparison Visualizations

This appendix contains figures that supplement the event emulation strategy
comparison presented in section 7.1. The event images produced by all five
methods for the three sample RGB inputs and the resulting SNN responses are
shown in Figures 30 and 31, respectively. In the latter, each sub-plot depicts
the recorded spike trains (left) and the total count of spikes (right) across
$T_{sim}$ at each output neuron. Note that the shared y-axes are indexed by a
row-major re-ordering of the neuron indices, which are originally arranged in
a two-dimensional grid, where the first and last indices are in the top-left
and right-bottom positions, respectively.

Figure 30: Event images produced by each emulation strategy described in
section 7.1. Although the event distributions are fairly similar, a few
notable differences include M2 (with blurring) producing less noisy events
than M1, the log-based methods M3 and M4 producing more events in darker
regions, and M1, M2, and M4 reacting more strongly to reflective surfaces.
Figure 31: SNN responses to input events from each emulation strategy
described in section 7.1 for each sample. The plots show the spike trains
(left) and spike counts (right) for each output neuron. The responses largely
match the input events shown in Figure 30 (e.g. more spikes in the first few
neurons for M1-M4, which correspond to the top regions of the event images).
Despite some differences in the input event data, the SNN responses are
similar (except for M5), indicating an inherent robustness to variations in
events.

## Appendix G: Qualitative Evaluation Visualizations

This appendix contains supplementary figures for the predictability
qualitative evaluations.

### Predictability: Angular Velocity Analysis

Figures 32 and 33 show examples of the estimated densities of angular
velocities, $\dot{\zeta}$, that are used to evaluate predictability from
simulated and real trials, respectively. Refer to the relevant discussions in
sections 6.1.4 and 6.2.

(a) Scenario 12 (Task 1)

(b) Scenario 17 (Task 2)

(c) Scenario 24 (Task 3)

Figure 32: Histograms and kernel density estimate plots depicting the
distributions of estimated angular end-effector velocities during testing
trials of three sample scenarios. We observe similar distributions from the
two cases and small values ([-2, 2] deg/s), which indicates that the
trajectories are generally predictable, as described in section 6.1.4. Figure
33: A histogram and kernel density estimate plot depicting the distributions
of estimated angular end-effector velocities measured in real experiments
(scenario R2). Similar to the simulation case, we observe low values that
indicate no sudden, large changes in the end-effector’s motion direction,
which corresponds to generally predictable robot motions.

## Appendix H: Results Metrics and Trajectory Plots

This appendix contains quantitative metric results and trajectory plots that
are referred to in discussions of the tuning, validation, and testing phases
of the simulation experiments in sections 6.1.2, 6.1.3, and 6.1.4:

  * •

Tuning phase trajectories executed in scenario 1 (parameter sets 1-12): Figure
34

  * •

Real experiment trajectories executed in scenarios R1-R4: Figure 35

  * •

Validation phase metrics in scenarios 4-11 (selected parameter sets): Figure
36

  * •

Testing phase metrics in scenarios 12-31 (selected parameter set): Figure 39

(a) Parameter Set 1

(b) Parameter Set 2

(c) Parameter Set 3

(d) Parameter Set 4

(e) Parameter Set 5

(f) Parameter Set 6

(g) Parameter Set 7

(h) Parameter Set 8

(i) Parameter Set 9

(j) Parameter Set 10

(k) Parameter Set 11

(l) Parameter Set 12

Figure 34: Trajectories executed in tuning scenario 1: parameter sets 1-12
(discussed in section 6.1.2).

(a) Scenario R1

(b) Scenario R2

(c) Scenario R3

(d) Scenario R4

Figure 35: Trajectories executed in real robot experiments with SNN feedback
in scenarios R1-R4 (discussed in section 6.2).

(a) Scenario 4

(b) Scenario 5

(c) Scenario 6

(d) Scenario 7

(e) Scenario 8

(f) Scenario 9

(g) Scenario 10

(h) Scenario 11

Figure 36: Quantitative metric results for validation scenarios 4-11: without
SNN feedback and with selected parameter sets.

(a) Scenario 12: {T1, Empty, Yellow, Spiky Sphere}

(b) Scenario 13: {T1, Store, Brick, Buckyball}

(c) Scenario 14: {T1, Kitchen, White, Buckyball}

(d) Scenario 15: {T1, Empty, Yellow-Black, Box}

Figure 37: Quantitative metric results without vs. with SNN feedback (best
parameter set): testing scenarios 12-15.

(a) Scenario 16: {T1, Store, Y-B, Rock}

(b) Scenario 17: {T2, Office, Red, Buckyball, High}

(c) Scenario 18: {T2, Office, Red, Rock, Med.}

(d) Scenario 19: {T2, Store, Brick, Box, Med.}

(e) Scenario 20: {T2, Kitchen, Y-B, Box, High}

(f) Scenario 21: {T2, Empty, Red, Spiky Sphere, Med.}

(g) Scenario 22: {T3, Empty, Red}

(h) Scenario 23: {T3, Empty, Y-B}

(i) Scenario 24: {T3, Kitchen, White}

(j) Scenario 25: {T3, Kitchen, Red}

(k) Scenario 26: {T3, Store, Red}

(l) Scenario 27: {T4, Store, Y-B, Buckyball, Low}

Figure 38: Quantitative metric results without vs. with SNN feedback (best
parameter set): testing scenarios 16-27. (cont.)

(a) Scenario 28: {T4, Store, Red, Buckyball, High}

(b) Scenario 29: {T4, Empty, Y-B, Spiky Sphere, Med.}

(c) Scenario 30: {T4, Empty, Y-B, Box, Low}

(d) Scenario 31: {T4, Empty, Brick, Buckyball, Low}

Figure 39: Quantitative metric results without vs. with SNN feedback (best
parameter set): testing scenarios 28-31. (cont.)