# Beyond PID Controllers: PPO with Neuralized PID Policy for Proton Beam
Intensity Control in Mu2e

C. Xu Equal contribution  J.YC. Hu∗ J. Jiang  S. Memik  R. Shi  A.M. Shuping
M. Thieme  H. Liu
Northwestern University Performed at Northwestern with support from the
Departments of Computer Science and Electrical and Computer Engineering
Evanston  IL USA
M.R. Austin  J.M. Arnold  J.R. Berlioz  P. Hanlet  K.J. Hazelwood  M.A.
Ibrahim  J.St. John  J. Mitrevski  V.P. Nagaslaev  D.J. Nicklaus  G. Pradhan
A.L. Saewert  B.A. Schupbach  K. Seiya  R.M. Thurman-Keup  N.V. Tran  A.
Narayanan
Fermi National Accelerator Laboratory Operated by Fermi Research Alliance, LLC
under Contract No.De-AC02-07CH11359 with the United States Department of
Energy. Additional funding provided by Grant Award No. LAB 20-2261 (Department
of Energy, Office of Science, 2020) Batavia  IL USA

###### Abstract

We introduce a novel Proximal Policy Optimization (PPO) algorithm aimed at
addressing the challenge of maintaining a uniform proton beam intensity
delivery in the Muon to Electron Conversion Experiment (Mu2e) at Fermi
National Accelerator Laboratory (Fermilab). Our primary objective is to
regulate the spill process to ensure a consistent intensity profile, with the
ultimate goal of creating an automated controller capable of providing real-
time feedback and calibration of the Spill Regulation System (SRS) parameters
on a millisecond timescale. We treat the Mu2e accelerator system as a Markov
Decision Process suitable for Reinforcement Learning (RL), utilizing PPO to
reduce bias and enhance training stability. A key innovation in our approach
is the integration of neuralized Proportional-Integral-Derivative (PID)
controller into the policy function, resulting in a significant improvement in
the Spill Duty Factor (SDF) by 13.6%, surpassing the performance of the
current PID controller baseline by an additional 1.6%. This paper presents the
preliminary offline results based on a differentiable simulator of the Mu2e
accelerator. It paves the ground works for real-time implementations and
applications, representing a crucial step towards automated proton beam
intensity control for the Mu2e experiment.

##  1 Introduction

We propose a novel RL-enhanced spill regularization system that incorporates a
neuralized PID policy function to tackle the beam regularization challenge in
the Mu2e experiments (Bartoszek et al., 2015) at Fermilab. Our objective is to
create an automated controller that ensures consistent spill (proton beams)
intensity during experiments meeting real-time control requirements (Narayanan
et al., 2021b). This objective falls under the broader scope of the
Accelerator Real-time Edge AI for Distributed Systems (READS) project
(Mitrevski, 2023; Seiya et al., 2021b; Hazelwood et al., 2021a). To achieve
this, we model the Mu2e accelerator system as a Markov Decision Process and
employ the Proximal Policy Optimization (PPO) algorithm (Schulman et al.,
2017) to cast the spill regulations as sequential decision-making problems.
Our main contribution is the integration of a neuralized PID policy function
(Zribi et al., 2018) for our RL framework, encompassing the inductive bias of
the standard PID controller (i.e. the proportional, integral, and derivative
information) to better capture states at different stages. Our experiments on
the Mu2e simulator show that we observed an average improvement of 13.6% in
the Spill Duty Factor (SDF). Additionally, our method outperforms the previous
PID controller approach (Narayanan et al., 2021b).

Figure 1: (a): The extraction (or ‘spill’) of protons from the Delivery Ring
is noisy (deviates from 1) without any regulation. (b): A snapshot of the beam
in physical space at the extraction location. As the horizontal beam size
increases, a slice of circulating beam (that is past the position of the
electrostatic septum) is extracted. (c): To create the muons, proton pulses
are made to hit a production target and muons are obtained from the
secondaries. The proton pulses with the required time structure are created by
extracting them from an accelerator ring called Delivery Ring at Fermilab and
sending it to the Mu2e production target.

The goal of Mu2e experiment (Bartoszek et al., 2015; Bernstein, 2019;
Narayanan et al., 2021a, 2022) at Fermilab is to search for new physics by
studying the decay of muons into electrons. This intricate experiment places
stringent demands on the quality of the proton beam directed at the muon
production target, see Figure 1. These requirements are essential to minimize
background particle physics processes that could obscure the discovery signal.
One of the key prerequisites for the Mu2e experiment includes achieving a
highly uniform extracted beam intensity during each spill of protons with 8
GeV kinetic energy. In order to achieve this, the Spill Regulation System
(SRS) (Thieme, 2022; Narayanan et al., 2021b; Ibrahim et al., 2019) is being
developed to govern the extraction process of the beam and mitigate various
sources of fluctuations in the spill profile.

The SRS adjusts the power supply currents of three dedicated fast quadrupoles
(resulting in varying its magnetic fields) and this variation results in
controlling the variations in the spill intensity (Figure 1 (c)). Ideally, the
SRS aims for the spill intensity to be perfectly uniform.

Our approach involves utilizing a Proximal Policy Optimization model with a
differentiable Mu2e simulator (Narayanan et al., 2021b) to regulate random
generated spills. In each episode (or spill), the simulator generates a series
of random spills, each with varying intensities (shown in Figure 1 (a)).
Subsequently, the PPO model intervenes in each individual time step to correct
these generated spills by adjusting the control signal of the Mu2e simulator.
The primary goal of our model is to bring spill rate of all spills as close to
a value of 1 as possible. To optimize this objective function, while
mitigating the influence of excessively high or low intensity spills, we
implement an exponential moving average (EMA) to measure the deviation of the
sequence (spill rate) from the desired value of 1. Additionally, we
incorporate neuralized PID controller, encompassing proportional, integral,
and derivative components in the state representation. By employing different
random seeds, our simulator can generate a diverse profile of noisy spills,
reflecting the real-world scenarios.

We demonstrate superior performance by numerically benchmarking our methods
with PID controller. Specifically, our experiments compare SDF (defined in
(2.1)) performance on different seeds. Our results show that our methods
consistently improve the SDF by 13.6% for 9 random generated spills and
achieve 1.6% improvements compared to the PID controller.

The rest of this paper commences by a detailed description of our proposed
method. Subsequently, we present numerical results to showcase the performance
of our approach. Finally, we conclude this paper with a discussion of
potential future directions and avenues for further research.

##  2 Methodology

In this section, we first introduce the machine learning’s role in optimizing
Fermilab’s accelerator parameters. Then, we provide a detailed design of our
RL-enhanced regularization system.

###  2.1 Problem Setup

Large fluctuations in the proton spill rate result in large fluctuations in
the intensity of produced muons. And this in turn causes background effects
that hinder the signals of new physics. To combat these, we approach the beam
intensity regulation challenge, specifically controlling the spill regulation
system, as a tracking control problem. The objective is to keep certain
signals (spill intensity) close to specific reference values ($\simeq 1$) by
controlling the quadrupole currents (power supply of 3 dedicated fast-ramping
quadrupoles) in the regulation system. By doing so, we adjust the magnetic
field, which subsequently adjusts the beam intensity throughout the delivery
ring, as depicted in Figure 1 (c).

Objective. To handle the constraint challenges posed by Mu2e experiment, we
regulate the uniformity of the extracted spill by increasing its Spill Duty
Factor (SDF),

$\displaystyle\text{SDF}\coloneqq
1/\left({1+\sigma_{\text{spill}}^{2}}\right),$ (2.1)

by regulating the extraction process (where $\sigma_{\text{spill}}$ is the
standard deviation in the spill rate). The ultimate goal for SRS in the Mu2e
experiment is to achieve a SDF of 0.6 or higher, with an ideal spill having a
constant spill rate value of 1 and an SDF of 1.

Mu2e Simulator. We employ a differentiable simulator proposed in (Narayanan et
al., 2021b) to replicate the beam physics process in Mu2e experiments
realistically. This physics simulator generates spill intensity and the
associated data. It subsequently conveys this data to the RL agent, which aids
in training the RL model. Once trained, the RL model transmits control signals
back to the simulator, allowing it to regulate the deviation in the spill rate
based on these signals and provide the modified data to the RL agent. This
process is shown in Figure 2 (a).

###  2.2 Proximal Policy Optimization (PPO) Controller for Spill Regulation
System

Figure 2: (a): The Mu2e simulator initially generates the noised spill data.
The agent proceeds to adjust the spill and the code employs this adjusted
spill to compute relevant information such as the state and reward. These
pieces of information are instrumental in training the RL agent, which in turn
offers new actions for the subsequent time step to refine the spill. (b): The
simulator refines (corrects) the spill derived from noisy data. It conveys the
state and reward, calculated using the corrected spill, to update the value
network responsible for evaluating the quality of the correction.
Subsequently, the state and loss generated by the value network contribute to
the adaptation of the policy network. The policy network, in response,
generates new actions for spill regulation.

Reward Function. Let $x_{t}$ be the observation of one single spill signal at
time step $t$ and $\sigma=1$ be the corresponding target reference value. The
reward function at $t$ is defined as the exponential moving average

$\displaystyle r_{t}=-{\rm
EMA}(t,\alpha),\alpha\in\left[0,1\right],\;\text{where}\;\;{\rm
EMA}(t,\alpha)=\alpha|x_{t}-\sigma|+(1-\alpha){\rm EMA}(t-1,\alpha).$ (2.2)

EMA gives more weight to recent spills and less weight to older spills. This
helps in reducing the impact of short-term fluctuations in the spills, making
it easier to identify underlying trends.

PID Controller. A PID controller in discrete-time operation captures past
details regarding tracking errors, their integrals, and derivatives within a
linear control strategy. We denote the time series of spill signal at time $t$
as $o_{t}=(x_{0},x_{1},\cdots,x_{t})$. In formal terms, the discrete-time PID
controller’s policy, characterized by its parameters $K_{P}$, $K_{I}$, and
$K_{D}$, is expressed as:

$\displaystyle\pi^{\text{PID}}(o_{t})=K_{P}\left(x_{t}-\sigma\right)+K_{I}\sum_{\tau=0}^{t}\left(x_{\tau}-\sigma\right)+K_{D}\frac{\left(x_{t}-\sigma\right)-\left(x_{t-1}-\sigma\right)}{\Delta
t},\vspace{-0.2truein}$ (2.3)

where $K_{P},K_{I},K_{D}$ are tuneable scalar coefficients and $\Delta t$ is
the discrete time interval.

Model: PPO with Neuralized PID Policy. Our model leverages the inductive bias
of PID (Proportional-Integral-Derivative) controller, and propose a neuralized
PID policy function. Specifically, we incorporate tracking errors, integrals,
and derivatives as components of the state vector $s_{t}$. Furthermore, we
employ a linear network to parametrize the standard PID controller (2.3), and
use it as a part of our policy function. Therefore, the policy function of our
model not only enables the extraction of external information (based on
previous actions) but also includes the PID control signals $K_{P}$, $K_{I}$,
and $K_{D}$ as its learnable parameters. Remarkably, when learned effectively,
our policy network outperforms the standard PID controller (2.3), making it a
highly adaptable solution.

Our approach incorporates three key components: the PPO algorithm, the EMA
reward function, and the neuralized PID controller. The PPO algorithm
(Schulman et al., 2017), a reinforcement learning technique, plays a central
role in optimizing policy functions to enhance decision-making in sequential
tasks and we specifically chose PPO to refine our reward function. Rather than
relying on a single spill for reward computation, we employ EMA within the
reward function. This approach allows us to both capture trends across a
sequence of spills and mitigate fluctuations. Additionally, we integrate the
neuralized PID bias into our policy network. In this setup, our policy network
consists of a trainable PID controller and a linear projection of past
actions. Let’s denote the state at time step $t$ as $s_{t}$, the action as
$a_{t}$. We combine the PID policy $\pi^{\text{PID}}$ and action policy
$\pi^{\text{action}}$ to formulate RL policies as $\pi$. The variable $a_{t}$
represents the control signal used for regulating the spill. The state $s_{t}$
encompasses both the previous action $a_{t-1}$ and the time series of the
spill signal $o_{t}$. As a result, the action at time $t$, $a_{t}$, can be
represented as follows:

$\displaystyle
a_{t}=\pi(s_{t})=\pi^{\text{PID}}(o_{t})+\pi^{\text{action}}(a_{t-1}).$ (2.4)

The learning process is shown in Figure 2 (b).

##  3 Experimental Studies

We validate our method by using the Mu2e differentialble simulator (Narayanan
et al., 2021b) as the RL environment. Our experiments involve the utilization
of both the Mu2e simulator and our RL-enhanced spill regularization system.

Figure 3: Spill intensity and SDF comparison in different seeds. (LHS):
Comparison of Spill Intensity: The spill intensity corrected by RL is closer
to 1 when compared to the PID-corrected spill. (RHS): Comparison of SDF: After
600 training iterations, the SDF achieved by RL outperforms or nears the SDF
obtained through PID.

Settings. We configure various hyperparameters for both the simulator and the
RL agent in our experiment. Specifically, we configure the simulator to
generate 430 spills per iteration, equivalent to 10 data points per
millisecond within a 43 ms spill duration, aligning closely with realistic
settings as detailed in (Narayanan et al., 2021b). In terms of reward
calculation, we choose a value of $\alpha=0.5$ for the EMA component.
Regarding the RL agent, we employ the stable-
baselines3111https://github.com/DLR-RM/stable-baselines3 PPO model (Raffin et
al., 2021). The actor network is designed as a single linear layer, while the
critic network took the form of a two-hidden layer ($64\times 64$) MLP
network. The learning rate is set at $1\crossproduct 10^{-4}$. We change the
random seed every 1000 (spills) epochs.

Results. In Figure 3, we examine the spill intensity in scenarios involving
unregularized, PID-regularized, and RL-regularized setups. Figure 3 (a)
demonstrates that the spill corrected by RL brings the spill rate closer to
the ideal value of 1 when compared to the PID-corrected spill, on the first
random seed configuration. Furthermore, we provide a visual representation of
the evolution of the SDF during the training process. Figure 3 (b) illustrates
that the SDF achieved through RL uniformly surpasses (or approaches) that of
PID regulation after 600 episodes.

Ablation Study. We systematically vary parameters and model architectural
configurations to differentiate their impact in these four aspects: 1), we
compare the usage of PID controller and a neural network in Policy network’s;
2), we experiment with various values of the smoothing factor ($\alpha$) in
the Exponential Moving Average (EMA); 3), we explore the effect of employing
different reward functions; 4), we select alternative states from the Mu2e
(spills) environment to replace the usage of inductive biases from the PID;
5), we compare two RL algorthims: Soft-Actor Critic (SAC) (Haarnoja et al.,
2018) and PPO. For all the experiments, we use the same random seed series. We
report the average improvements of SDF of our proposed methods, compared to
PID regularized SDF and unregularized SDF in Table 1.

Table 1: Evaluation of the impact of various parameters. “v.s. PID” represents
the SDF improvement by comparing our methods to PID method. “v.s. Noise” is
the SDF improvements of our methods compared to unregularized SDF. NN: neural
network policy; PID: neuralized PID policy; EMA: exponential moving average of
errors; $\alpha$: EMA’s smooth parameter; SUM: sum of errors; PPO: Proximal
Policy Optimization; SAC: Soft-Actor Critic; P: current error; I: sum of
errors; D: error difference; Act: previous action; CD: corrected spill
difference; Over-1: number of noisy spill intensity $\geq$ 1\.

v.s. PID | v.s. Noise | Policy Network | Reward Func. | RL Algorthim | State
---|---|---|---|---|---
-4.11 | 7.90 | PID | -EMA ($\alpha$=0.1) | PPO | P, I, D, Act
-11.12 | 0.89 | NN | -EMA ($\alpha$=0.5) | PPO | P, I, D, Act
1.42 | 13.55 | PID | -EMA ($\alpha$=0.9) | PPO | P, I, D, Act
1.95 | -10.06 | PID | -SUM | PPO | P, I, D, Act
1.46 | 13.47 | PID | -EMA ($\alpha$=0.5) | PPO | P, I, D
5.76 | -6.25 | PID | -EMA ($\alpha$=0.5) | PPO | CD, Over-1, P, Act
-12.60 | -24.61 | PID | -EMA ($\alpha$=0.5) | SAC | P, I, D, Act
1.65 | 13.67 | PID | -EMA ($\alpha$=0.5) | PPO | P, I, D, Act

##  4 Related Works in Real-time Edge AI for Distributed Systems (READS)

The READS project (Seiya et al., 2021a) includes two primary sub-projects: 1)
Beam Loss Deblending for the Main Injector and Recycler, and 2) Mu2e Spill
Regulation, which is the subject of this work.

Beam Loss Deblending for Main Injector and Recycler. In this sub-project, the
challenge is distinguishing between beam losses from two adjacent accelerators
(Recycler Ring and Main Injector) that share the same monitoring system. We
summarize some related works in resolving this challenge bellow. The
Deblending model (DBLN) (Hazelwood et al., 2021b) uses a dual-network MLP
setup to classify beam losses and predict their probabilities for each
accelerator and beam loss monitor (BLM). The Many Models (Hazelwood, 2023)
employs individual MLP models for each BLM and aggregates their outputs to
enhance local pattern recognition for beam losses. The semantic regression
model (Thieme et al., 2022) uses the U-Net architecture to capture each
accelerator’s localized and extensive beam loss patterns. As for hardware, a
series of work on Field Programmable Gate Array (FPGA)-based edge-AI systems
(Berlioz et al., 2022; Ibrahim et al., 2023; Shi et al., 2023; Arnold et al.,
2023) are developed and deployed for real-time identification of beam loss
sources in accelerator complex, enhancing operational accuracy and efficiency.

Mu2e Spill Regulation. There are some related works in solving challenges (as
described in Section 2.1) in Mu2e Spill Regulation project. The PID method
(Narayanan et al., 2021b) employs a hybrid machine learning approach that
integrates a neural network (NN) to optimize PID controller gains within an
end-to-end machine learning (ML) differentiable simulator simulator, thereby
enhancing the spill quality of a slow extraction system for proton delivery.
The GRU network (Narayanan et al., 2022) ingests a history of spill intensity
observations as input and makes predictions for the quadrupole adjustments
based on these inputs.

##  5 Conclusion

We present an innovative RL-enhanced spill control system, utilizing a neural
PID controller as the policy function, to tackle beam regulation issues in
Mu2e experiments. To simulate real-world spill control scenarios, we utilize a
differentiable Mu2e simulator to create spills, improve spill adjustments, and
establish reward signals. Furthermore, we harness an RL-based controller for
fine-tuning control signals during the regularization process. Our approach
outperforms the PID-based regularization model, achieving a 1.6% higher SDF
performance, with a 13.6% improvement in the SDF of unregularized spills,
confirmed across nine different settings with random seeds.

##  Appendix A Future Directions

We outline two potential directions for future investigations toward a model-
free RL controller for the Mu2e experiments.

#### Pretrained Transformers.

A main motivation for our method is the sequential modeling of noisy spill
rate hard to converge for naive RL algorithms. This challenge persists even
with the simplified simulator environment (Narayanan et al., 2021b). While our
approach incorporates model-based inductive bias (such as PID), another
popular method in the literature involves using transformers (Vaswani et al.,
2017) for generative trajectory modeling in RL (Paischer et al., 2022; Janner
et al., 2021; Chen et al., 2021). This approach is beneficial for our problem:
it scales up easily using existing transformer-based language and vision
models like BERT (Zhou et al., 2023; Ji et al., 2021; Devlin et al., 2018) and
GPT (Wei et al., 2021), enables stable training (Zhai et al., 2023), and
addresses the challenges of short-sighted behavior and long-term credit
assignment in RL (Chen et al., 2021).

However, a major issue with training transformers solely on observed samples
from the environment is their tendency to overfit, especially given the
typically limited data generated by the current policy (Chen et al., 2021). To
circumvent this, we propose using a Transformer pre-trained on extensive
observations/samples without any fine-tuning or weight adjustments (Paischer
et al., 2022). Modern Hopfield networks (Hu et al., 2023; Paischer et al.,
2022; Ramsauer et al., 2020) provide a natural approach for utilizing frozen
transformers via their associative memory interpretation. We notice that there
are works in (i) employing “frozen” modern Hopfield networks with pre-trained
transformers for RL (Paischer et al., 2022), and (ii) addressing noisy
sequence modeling using sparse modern Hopfield models (Hu et al., 2023). We
would like to explore the integration of (i) & (ii) in the future.

#### Feature Programming.

A key innovation in our approach is the integration of model-based inductive
bias from PID controllers into our policy network’s architecture. This
involves manually crafting features (PID parameters) from sequences of spills.
However, this method risks missing other crucial features and may introduce
instability in the training process, as evidenced in Table 1. To more
effectively identify and leverage informative features for RL model training,
we propose employing Feature Programming (Reneau et al., 2023), an automated
feature engineering framework for noisy multivariate time series. This
framework will be used to (i) generate informative features from noisy spill
sequences and (ii) incorporate model-based inductive bias from PID
controllers. In future work, we aim to combine both (i) and (ii) to develop a
model-free RL controller for the Mu2e experiments.

## Acknowledgments

JH and CW would like to thank Jiayi Wang for facilitating experimental
deployments. The authors would like to thank the anonymous reviewers and
program chairs for constructive comments.

AN was previously affiliated with Northern Illinois University for the earlier
part of the READS collaboration, DeKalb, IL, USA. Besides the aforementioned
support, JH is partially supported by the Walter P. Murphy Fellowship. HL is
partially supported by NIH R01LM1372201, NSF CAREER1841569, DOE DE-
AC02-07CH11359 and a NSF TRIPODS1740735. This research was supported in part
through the computational resources and staff contributions provided for the
Quest high performance computing facility at Northwestern University which is
jointly supported by the Office of the Provost, the Office for Research, and
Northwestern University Information Technology. The content is solely the
responsibility of the authors and does not necessarily represent the official
views of the funding agencies.

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terms found in Appendix A where in addition $x^{\mu}(\sigma)\rightarrow\langle
t(\sigma),r(\sigma),\theta(\sigma),\phi(\sigma)\rangle$. This notation for our
vector components signifies

$x^{a}=t(\sigma)\bigg{(}\frac{\partial}{\partial
t}\bigg{)}^{a}+r(\sigma)\bigg{(}\frac{\partial}{\partial
r}\bigg{)}^{a}+\theta(\sigma)\bigg{(}\frac{\partial}{\partial\theta}\bigg{)}^{a}+\phi(\sigma)\bigg{(}\frac{\partial}{\partial\phi}\bigg{)}^{a}$

where $(\partial/\partial t)^{a}$, $(\partial/\partial r)^{a}$,
$(\partial/\partial\theta)^{a}$, and $(\partial/\partial\phi)^{a}$ are
linearly independent tangent vectors which span the tangent spaces at each
point on the manifold. For example, we take our equation of motion for
$t(\sigma)$:

$\displaystyle
0=\frac{d^{2}t}{d\sigma^{2}}+\sum_{\mu,\rho}\Gamma^{t}_{\mu\rho}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\rho}}{d\sigma}$
$\displaystyle=\frac{d^{2}t}{d\sigma^{2}}+\Gamma^{t}_{tr}\frac{dt}{d\sigma}\frac{dr}{d\sigma}+\Gamma^{t}_{rt}\frac{dr}{d\sigma}\frac{dt}{d\sigma}+\Gamma^{t}_{\theta\theta}\frac{d\theta}{d\sigma}\frac{d\theta}{d\sigma}+\Gamma^{t}_{\phi\phi}\frac{d\phi}{d\sigma}\frac{d\phi}{d\sigma}$

However, we know from our work in Appendix A that
$\Gamma^{t}_{tr}=\Gamma^{t}_{rt}$. Consequently, we find after plugging in for
each affine connection term

$0=\frac{d^{2}t}{d\sigma^{2}}+\frac{2}{r}\frac{dt}{d\sigma}\frac{dr}{d\sigma}+\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$

Applying similar logic for $\nu=r$, $\theta$, and $\phi$:

$\displaystyle 0=\frac{d^{2}r}{d\sigma^{2}}+\Lambda
r\bigg{(}\frac{dt}{d\sigma}\bigg{)}^{2}-r\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
$\displaystyle
0=\frac{d^{2}\theta}{d\sigma^{2}}+2\frac{d\theta}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\frac{1}{r}\frac{dr}{d\sigma}\bigg{]}-\sin{\theta}\cos{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}$
$\displaystyle
0=\frac{d^{2}\phi}{d\sigma^{2}}+2\frac{d\phi}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\frac{1}{r}\frac{dr}{d\sigma}+\cot{\theta}\frac{d\theta}{d\sigma}\bigg{]}$

As briefly mentioned above, if one evaluates the norm of the ‘proper
velocity’, he/she will find:

$U^{a}U_{a}=\sum_{\mu,\nu}g_{\mu\nu}U^{\mu}U^{\nu}=\sum_{\mu,\nu}g_{\mu\nu}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\nu}}{d\sigma}=\left\\{\begin{array}[]{l
l}0&\quad\textrm{null geodesics}\\\ -1&\quad\textrm{time-like geodesics}\\\
\end{array}\right.$

In a relatively simple way, one can see this from our line element where
$-d\chi^{2}=\sum_{\mu,\nu}g_{\mu\nu}dx^{\mu}dx^{\nu}$. Massless particles
travel along null geodesics (i.e. our norm vanishes) whereas massive particles
travel along time-like geodesics. For comparison with special relativity and
general relativity, we express the component form of our time-like geodesics
where $\sigma=\chi$ in terms of the physically observable elapsed time as
measured by a clock carried along the given curve in a particular inertial
system, $\tau$:

$0=\frac{d^{2}\tau}{d\chi^{2}}\frac{dx^{\nu}}{d\tau}+\bigg{(}\frac{d\tau}{d\chi}\bigg{)}^{2}\cdot\bigg{[}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\sum_{\mu,\rho}\Gamma^{\nu}_{\mu\rho}\frac{dx^{\mu}}{d\tau}\frac{dx^{\rho}}{d\tau}\bigg{]}$

One immediately notices that the term in brackets represents the component
form of the geodesic equation for special and general relativity and would be
set equal to zero in both of these theories. However, since in the theory of
inertial centers $d^{2}\tau/{d\chi^{2}}\neq 0$ as
$d\chi/d\tau=\sqrt{\Lambda}\cdot r(\tau)$, the term in brackets is not
necessarily zero for our theory, and thus the observed inertial motion of
massive objects in our model characterized by the equation above is in fact
very different from inertial motion as seen in special and general relativity.

##  Appendix C Killing vector fields

As in our previous appendices, we provide a summary of [24] with regard to the
more general statements below (see Appendix C and Chapter 2 of [24]). In order
to understand the relevance of Killing vector fields with respect to inherent
symmetries associated with our manifold, we must begin with a brief
introduction to isometries and Lie derivatives. For two manifolds $M$ and $N$,
let $\phi$ be a smooth map from $M$ to $N$ ($\phi:M\rightarrow N$) and $f$ be
a function from $N$ to the reals ($f:N\rightarrow\mathbb{R}$). Then the
composition of $f$ with $\phi$, $f\circ\phi$, produces a function from
$M\rightarrow\mathbb{R}$ and $\phi$ is said to “pull back” $f$. In addition
$\phi$ “carries along” tangent vectors at a particular point $p\in M$ to
tangent vectors at $\phi(p)\in N$, and therefore defines a map
$\phi^{\star}:V_{p}\rightarrow V_{\phi(p)}$ in the following manner:

$(\phi^{\star}v)(f)=v(f\circ\phi)$

where $v\in V_{p}$, $\phi^{\star}v\in V_{\phi(p)}$, and $V_{p}$ denotes the
tangent vector space at $p$. One can also use $\phi$ to “pull back” dual
vectors at $\phi(p)$ to dual vectors at $p$ by defining a map
$\phi_{\star}:V^{\star}_{\phi(p)}\rightarrow V^{\star}_{p}$ requiring for all
$v^{a}\in V_{p}$

$(\phi_{\star}\mu)_{a}v^{a}=\mu_{a}(\phi^{\star}v)^{a}$

where $V^{\star}_{p}$ denotes the dual vector space at $p$. If
$\phi:M\rightarrow N$ is a diffeomorphism (i.e. a smooth function that is one-
to-one, onto, and its inverse is also smooth), then for an arbitrary tensor
$T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots a_{l}}$ of type $(k,l)$ at $p$ (type
$(k,l)$ refers to the number of dual vector “slots” and vector “slots”,
respectively), the tensor $(\phi^{\star}T)^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}$ at $\phi(p)$ is defined by

$(\phi^{\star}T)^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}(\mu_{1})_{b_{1}}\cdots(\mu_{k})_{b_{k}}(t_{1})^{a_{1}}\cdots(t_{l})^{a_{l}}=T^{b_{1}\ldots
b_{k}}{}_{a_{1}\ldots
a_{l}}(\phi_{\star}\mu_{1})_{b_{1}}\cdots([\phi^{-1}]^{\star}t_{l})^{a_{l}}$

as $(\phi^{-1})^{\star}:V_{\phi(p)}\rightarrow V_{p}$. If $\phi:M\rightarrow
M$ is a diffeomorphism and $T$ is a tensor field on $M$, then we refer to
$\phi$ as a symmetry transformation for the tensor field $T$ if
$\phi^{\star}T=T$. In addition, if

$(\phi^{\star}g)_{ab}=g_{ab}$

we refer to $\phi$ as an isometry.

To introduce the notion of Lie derivatives, we come back to diffeomorphisms
and define a one-parameter group of diffeomorphisms $\phi_{t}$ as a smooth map
from $\mathbb{R}\times M\rightarrow M$ such that for fixed $t\in\mathbb{R}$,
$\phi_{t}:M\rightarrow M$ is a diffeomorphism. As well, for all
$t,s\in\mathbb{R}$, $\phi_{t}\circ\phi_{s}=\phi_{t+s}$. In particular, this
requires $\phi_{t=0}$ to be the identity map. A vector field $v^{a}$ can be
thought of as the infinitessimal generator of a one-parameter group of finite
transformations of $M$ in the following manner. For fixed $p\in M$, we refer
to the curve $\phi_{t}(p):\mathbb{R}\rightarrow M$ as an orbit of $\phi_{t}$
which passes through $p$ at $t=0$. $v|_{p}$ is defined to be the tangent to
this curve at $t=0$. We also define the Lie derivative with respect to $v^{a}$
by

$\mathfrak{L}_{v}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}=\lim_{t\rightarrow 0}\bigg{\\{}\frac{\phi^{\star}_{-t}T^{b_{1}\ldots
b_{k}}{}_{a_{1}\ldots a_{l}}-T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}}{t}\bigg{\\}}$

where all tensors above are evaluated at a point $p$. $\mathfrak{L}_{v}$ is
then a linear map from smooth tensor fields of type $(k,l)$ to smooth tensor
fields of type $(k,l)$ and satisfies the Leibniz rule on outer products of
tensors. Since $v^{a}$ is tangent to the integral curves of $\phi_{t}$, for
functions $f:M\rightarrow\mathbb{R}$

$\mathfrak{L}_{v}(f)=v(f)$

In addition, if $\phi_{t}$ is a symmetry transformation for $T$, we have
$\mathfrak{L}_{v}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots a_{l}}=0$. Furthermore,
it is found that the Lie derivative with respect to $v^{a}$ of a vector field
$w^{a}$ is given by the commutator:

$\mathfrak{L}_{v}w^{a}=[v,w]^{a}$

where

$[v,w]^{a}=v^{b}\nabla_{b}w^{a}-w^{b}\nabla_{b}v^{a}$

and for a dual vector,

$\mathfrak{L}_{v}\mu_{a}=v^{b}\nabla_{b}\mu_{a}+\mu_{b}\nabla_{a}v^{b}$

The more general action of a Lie derivative with respect to $v^{a}$ on a
general tensor field $T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots a_{l}}$ is given by

$\mathfrak{L}_{v}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}=v^{c}\nabla_{c}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}-\sum_{i=1}^{k}T^{b_{1}\ldots c\ldots b_{k}}{}_{a_{1}\ldots
a_{l}}\nabla_{c}v^{b_{i}}+\sum_{j=1}^{l}T^{b_{1}\ldots b_{k}}{}_{a_{1}\ldots
c\ldots a_{l}}\nabla_{a_{j}}v^{c}$

where $\nabla_{a}$ is our derivative operator compatible with the metric
$g_{ab}$ (i.e. $\nabla_{c}g_{ab}=0$). Then a Killing vector field $\xi^{a}$ is
defined to be the vector field which generates a one-parameter group of
isometries $\phi_{t}:M\rightarrow M$ of the metric,
$(\phi^{\star}_{t}g)_{ab}=g_{ab}$. As remarked earlier, the necessary
condition for $\phi_{t}$ to be a group of isometries is
$\mathfrak{L}_{\xi}g_{ab}=0$. Using the expression above for the action of a
Lie derivative on a tensor field,

$\displaystyle\mathfrak{L}_{\xi}g_{ab}=\xi^{c}\nabla_{c}g_{ab}+g_{cb}\nabla_{a}\xi^{c}+g_{ac}\nabla_{b}\xi^{c}$
$\displaystyle=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}$

Thus, we come upon Killing’s equation:

$\nabla_{a}\xi_{b}+\nabla_{a}\xi_{b}=0$

For any particular Killing vector field $\xi^{a}$, along a geodesic $\gamma$
with tangent vector $U^{a}$ one finds

$\displaystyle
U^{b}\nabla_{b}(\xi_{a}U^{a})=U^{b}U^{a}\nabla_{b}\xi_{a}+\xi^{a}U^{b}\nabla_{b}U^{a}$
$\displaystyle=\frac{1}{2}U^{a}U^{b}[\nabla_{b}\xi_{a}+\nabla_{a}\xi_{b}]+\xi^{a}U^{b}\nabla_{b}U^{a}=0$

where the first term vanishes by Killing’s equation and the second by the
geodesic equation (72). Meaning, along $\gamma$, $\xi^{a}U_{a}$ is constant
(Noether’s theorem).

Using our affine connection component terms, Killing’s equation takes the form

$\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}=2\sum_{\rho}\Gamma^{\rho}_{\mu\nu}\xi_{\rho}$

which gives for each pair ($\mu,\nu$),

$\displaystyle(t,t):\partial_{t}\xi_{t}=\Lambda
r\xi_{r}\indent(t,r):\partial_{t}\xi_{r}+\partial_{r}\xi_{t}=\frac{2}{r}\xi_{t}\indent(t,\theta):\partial_{t}\xi_{\theta}+\partial_{\theta}\xi_{t}=2\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\xi_{\theta}$
$\displaystyle(t,\phi):\partial_{t}\xi_{\phi}+\partial_{\phi}\xi_{t}=2\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\xi_{\phi}\indent(r,r):\partial_{r}\xi_{r}=0\indent(r,\theta):\partial_{\theta}\xi_{r}+\partial_{r}\xi_{\theta}=\frac{2}{r}\xi_{\theta}$
$\displaystyle(r,\phi):\partial_{\phi}\xi_{r}+\partial_{r}\xi_{\phi}=\frac{2}{r}\xi_{\phi}\indent(\theta,\phi):\partial_{\theta}\xi_{\phi}+\partial_{\phi}\xi_{\theta}=2\cot{\theta}\xi_{\phi}$
$\displaystyle(\theta,\theta):\partial_{\theta}\xi_{\theta}=\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\xi_{t}-r\cosh^{2}(\sqrt{\Lambda}t)\xi_{r}$
$\displaystyle(\phi,\phi):\partial_{\phi}\xi_{\phi}=\sin^{2}{\theta}\bigg{[}\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\xi_{t}-r\cosh^{2}(\sqrt{\Lambda}t)\xi_{r}\bigg{]}-\sin{\theta}\cos{\theta}\xi_{\theta}$

Immediately we notice from the $(t,t)$ equation that
$\xi_{t}=0\Longrightarrow\xi_{r}=0$ and from the $(\theta,\theta)$ equation
that $\xi_{r}=\xi_{\theta}=0\Longrightarrow\xi_{t}=0$. For
$\xi_{t}=\xi_{r}=0$, we find the three rotational Killing vector fields:

$\displaystyle\Omega^{\mu}_{1}\rightarrow\langle
0,0,\cos{\phi},-\cot{\theta}\sin{\phi}\rangle$
$\displaystyle\Omega^{\mu}_{2}\rightarrow\langle
0,0,\sin{\phi},\cot{\theta}\cos{\phi}\rangle$
$\displaystyle\psi^{\mu}\rightarrow\langle 0,0,0,1\rangle$

For $\xi_{\theta}=\xi_{\phi}=0$, we have a time and radial Killing vector
field:

$\rho^{\mu}\rightarrow\langle\frac{1}{\sqrt{\Lambda}r}\cosh(\sqrt{\Lambda}t),-\sinh(\sqrt{\Lambda}t),0,0\rangle$

For $\xi_{r}=\xi_{\phi}=0$, a time and $\theta$ Killing vector field:

$\Theta^{\mu}\rightarrow\langle\frac{1}{\sqrt{\Lambda}}\cos{\theta},0,-\sin{\theta}\tanh(\sqrt{\Lambda}t),0\rangle$

For $\xi_{\phi}=0$, a time, radial, and $\theta$ Killing vector field:

$\xi^{\mu}_{(t,r,\theta)}=\langle-\frac{\sinh(\sqrt{\Lambda}t)\cos{\theta}}{\sqrt{\Lambda}r},\cosh(\sqrt{\Lambda}t)\cos{\theta},-\frac{\sin{\theta}}{r\cosh(\sqrt{\Lambda}t)},0\rangle$

In addition, for only $\xi_{r}=0$, we find two Killing vector fields:

$\displaystyle\xi^{\mu}_{(t,\theta,\phi),1}\rightarrow\langle\frac{1}{\Lambda}\sin{\theta}\sin{\phi},0,\frac{1}{\sqrt{\Lambda}}\tanh(\sqrt{\Lambda}t)\cos{\theta}\sin{\phi},\frac{1}{\sqrt{\Lambda}\sin{\theta}}\tanh(\sqrt{\Lambda}t)\cos{\phi}\rangle$
$\displaystyle\xi^{\mu}_{(t,\theta,\phi),2}\rightarrow\langle-\frac{1}{\Lambda}\sin{\theta}\cos{\phi},0,-\frac{1}{\sqrt{\Lambda}}\tanh(\sqrt{\Lambda}t)\cos{\theta}\cos{\phi},\frac{1}{\sqrt{\Lambda}\sin{\theta}}\tanh(\sqrt{\Lambda}t)\sin{\phi}\rangle$

Finally, taking all components to be non-zero, we have the last two Killing
vector fields:

$\displaystyle\xi^{\mu}_{(t,r,\theta,\phi),1}\rightarrow\langle-\frac{1}{\sqrt{\Lambda}r}\sinh(\sqrt{\Lambda}t)\sin{\theta}\sin{\phi},\cosh(\sqrt{\Lambda}t)\sin{\theta}\sin{\phi},\frac{1}{r\cosh(\sqrt{\Lambda}t)}\cos{\theta}\sin{\phi},$
$\displaystyle\frac{1}{r\cosh(\sqrt{\Lambda}t)\sin{\theta}}\cos{\phi}\rangle$
$\displaystyle\xi^{\mu}_{(t,r,\theta,\phi),2}\rightarrow\langle-\frac{1}{\sqrt{\Lambda}r}\sinh(\sqrt{\Lambda}t)\sin{\theta}\cos{\phi},\cosh(\sqrt{\Lambda}t)\sin{\theta}\cos{\phi},\frac{1}{r\cosh(\sqrt{\Lambda}t)}\cos{\theta}\cos{\phi},$
$\displaystyle-\frac{1}{r\cosh(\sqrt{\Lambda}t)\sin{\theta}}\sin{\phi}\rangle$

Summarizing, we have ten linearly independent Killing vector fields for this
metric.

##  Appendix D Symplectic structure

Within this appendix, we’ll briefly address and apply the concepts presented
in [43] for the formulation of a quantum field theory of a real scalar field
with a general background metric, where we shall not concern ourselves with
the interaction between matter and space-time at the quantum level and instead
treat the metric as non-dynamic (hence the term “background”). However, we
strongly encourage the reader to review [43] in order to fully understand all
the material presented below.

The information associated with the dynamical evolution of a physical system
can be conveyed within the symplectic structure $\Omega$ for a particular
action $S$ given by

$\Omega([\phi_{1},\pi_{1}],[\phi_{2},\pi_{2}])=\int_{\Sigma_{0}}d^{3}x\bigg{(}\pi_{1}\phi_{2}-\pi_{2}\phi_{1}\bigg{)}$

where $\Omega$ is a non-degenerate antisymmetric bilinear map from the
solutions of the equation of motion associated with our action to the real
numbers. In addition, a point in phase-space (Hamiltonian formalism)
corresponds to the specification of our field solution $\phi$ and its
conjugate momentum $\pi=\partial\mathcal{L}/\partial\dot{\phi}$ on a space-
like hypersurface $\Sigma_{0}$ associated with our “initial value”
configuration ($\mathcal{L}$ is the Lagrangian density associated with the
action $S$ which we’ll give below). The fundamental Poisson brackets in
classical theory can then be expressed as

$\\{\Omega([\phi_{1},\pi_{1}],\cdot),\Omega([\phi_{2},\pi_{2}],\cdot)\\}=-\Omega([\phi_{1},\pi_{1}],[\phi_{2},\pi_{2}])$

where $\Omega(y,\cdot)$ is a linear function assuming our choice of $y$ does
not vary (our input argument is only ‘$\cdot$’). If we arbitrarily choose
$[\phi_{1},\pi_{1}]=[0,f_{1}]$ and $[\phi_{2},\pi_{2}]=[f_{2},0]$, our
classical Poisson brackets reduce to

$\bigg{\\{}\int d^{3}xf_{1}(x)\phi(x),\int d^{3}yf_{2}(y)\pi(y)\bigg{\\}}=\int
d^{3}xf_{1}(x)f_{2}(x)$

which we can think of as the more familiar canonical relations

$\\{\phi(x),\pi(y)\\}=\delta(x-y)$

Then to construct our quantum theory of a scalar field, we extend the
functions $\Omega([\phi,\pi],\cdot)$ to operators
$\hat{\Omega}([\phi,\pi],\cdot)$ satisfying the commutation relations

$[\hat{\Omega}([\phi_{1},\pi_{1}],\cdot),\hat{\Omega}([\phi_{2},\pi_{2}],\cdot)]=-i\hbar\Omega([\phi_{1},\pi_{1}],[\phi_{2},\pi_{2}])\hat{\mathrm{I}}$

($\hat{\mathrm{I}}$ denotes the identity operator) and introduce the inner
product associated with this system

$(\psi^{+},\chi^{+})=-i\Omega(\bar{\psi}^{+},\chi^{+})$

where $\bar{\psi}^{+}$ represents the complex conjugate of $\psi^{+}$ and we
have decomposed our full solutions $\psi,\chi\in\mathcal{S}$ of the equation
of motion for our action into

$\psi=\psi^{+}+\psi^{-}$

such that our inner product with respect to these “positive frequency”
solutions $\psi^{+},\chi^{+}$ is positive-definite. We denote the solution
space spanned by these “positive frequency” parts as
$\mathcal{S}^{\mathbb{C}+}$. In addition, we have expressed our inner product
only in terms of solutions to our equation of motion $\psi\in\mathcal{S}$ as
for each solution there corresponds a point in phase space
$[\psi,\pi_{\psi}]$. One proceeds to “Cauchy-complete” in the norm defined by
this inner product to obtain our complex Hilbert space $\mathcal{H}$ (see
Chapter 3.2 and Appendix A.1 of [43]). Thus, we represent our classical
observables $\Omega(\psi,\cdot)$ for each solution $\psi$ by the operator

$\hat{\Omega}(\psi,\cdot)=i\hat{a}(\bar{\mathrm{K}\psi})-i\hat{a}^{\dagger}(\mathrm{K}\psi)$

where $\mathrm{K}:\mathcal{S}\rightarrow\mathcal{H}$ is a map from the full
solutions to our complex Hilbert space. As well, $\hat{a}(\cdot)$ and
$\hat{a}^{\dagger}(\cdot)$ denote the annihilation and creation operators,
respectively, which act on a general state $\Psi$ in the symmetric Fock space
$\mathcal{F}_{s}(\mathcal{H})$ in the following manner. For a general state
$\Psi=\langle\psi,\psi^{a_{1}},\psi^{a_{1}a_{2}},\ldots,\psi^{a_{1}\ldots
a_{n}},\dots\rangle$ representing our “n-particle amplitudes” where for scalar
theory $\psi^{a_{1}\ldots a_{n}}=\psi^{(a_{1}\ldots a_{n})}$ $\forall n$
(round parantheses denote symmetrization when dealing with abstract indices
here and below) and $\xi^{a}\in\mathcal{H}$,
$\bar{\xi}_{a}\in\bar{\mathcal{H}}$, we have

$\displaystyle\hat{a}(\bar{\xi})\Psi=\langle\bar{\xi}_{a}\psi^{a},\sqrt{2}\bar{\xi}_{a}\psi^{aa_{1}},\sqrt{3}\bar{\xi}_{a}\psi^{aa_{1},a_{2}},\ldots\rangle$
$\displaystyle\hat{a}^{\dagger}(\xi)\Psi=\langle
0,\psi\xi^{a_{1}},\sqrt{2}\xi^{(a_{1}}\psi^{a_{2})},\sqrt{3}\xi^{(a_{1}}\psi^{a_{2}a_{3})},\ldots\rangle$

Indices on $\bar{\xi}_{a}$ and $\xi^{a}$ are dropped in our expressions on the
left-hand side of these equations for notational convenience. In this paper,
we operate under the assumption that the norms of these two expressions are
finite. In addition, the inner product of two vectors $\xi,\eta\in\mathcal{H}$
is denoted by

$(\xi,\eta)=\bar{\xi}_{a}\eta^{a}$

In this notation, $\psi\in\otimes^{n}\mathcal{H}$ is denoted
$\psi^{a_{1}\ldots a_{n}}$ and $\bar{\psi}\in\otimes^{n}\bar{\mathcal{H}}$ as
$\bar{\psi}_{a_{1}\ldots a_{n}}$ where
$\otimes^{n}\mathcal{H}=\mathcal{H}_{1}\otimes\ldots\otimes\mathcal{H}_{n}$
for $\mathcal{H}_{1}=\ldots=\mathcal{H}_{n}=\mathcal{H}$ (n-fold tensor
product space). As well,
$\mathcal{F}_{s}(\mathcal{H})=\oplus_{n=0}^{\infty}(\otimes^{n}_{s}\mathcal{H})$
where $\otimes^{n}_{s}\mathcal{H}$ is the symmetric n-fold tensor product
space and $\otimes^{0}\mathcal{H}$ is defined to be the complex numbers
$\mathbb{C}$ (see Appendix A of [43]).

To clarify further with regard to Fock space notation, we relate back to Dirac
“bra-ket” notation:

$|0\rangle_{\hat{a}}\equiv\langle 1,0,0,\ldots\rangle$

where

$\hat{a}(\bar{\xi})|0\rangle_{\hat{a}}=0$

for some general $\xi^{a}$ given $|0\rangle_{\hat{a}}$ denotes the vacuum
state associated with the creation and annihilation operators on our Fock
space (i.e. the $\hat{a}$’s). Given our general state $\Psi$, the probability
of finding only a single ‘$\hat{a}$ particle’ in state $\beta\in\mathcal{H}$
is taken to be $|\bar{\beta}_{a}\psi^{a}|^{2}$ (i.e. $\psi^{a}$ is the “one-
particle amplitude”, $\psi^{a_{1}a_{2}}$ is the “two-particle amplitude”,
etc.). Here, the term ‘particle’ really refers to an excitation of the
particular field associated with our $\hat{a},\hat{a}^{\dagger}$ operators
(quanta). Therefore, one can think of $\hat{a}(\bar{\xi})$ as an operator
annihilating a quantum of state $\xi^{a}$ from each of the “n-particle” states
in the general state $\Psi$, and analogously $\hat{a}^{\dagger}(\xi)$ as
creating a quantum in each. Our annihilation and creation operators also
satisfy the commutation relation:

$[\hat{a}(\bar{\xi}),\hat{a}^{\dagger}(\eta))]=\bar{\xi}_{a}\eta^{a}\hat{\mathrm{I}}$

Note that our use of abstract index notation in this paragraph does not refer
to the metric. In other words, when working with Hilbert space vectors, we
always assume contraction occurs over the inner product of the respective
Hilbert space as defined earlier in this appendix and not with regard to the
metric.

Then for our theory of inertial centers, the action associated with the
equation of motion for our Klein-Gordon extension in a particular inertial
system (60) takes the form

$S=-\frac{1}{2}\int
d^{4}x\sqrt{|g|}\bigg{(}\nabla^{a}\phi\nabla_{a}\phi+{\tilde{\mu}}^{2}r^{2}\phi^{2}\bigg{)}$
(73)

where one can verify this by extremizing the action ($\delta S=0$) to obtain
our equation of motion. We proceed with our formulation by “slicing” our
manifold $M$ into space-like hypersurfaces each indexed by a time parameter
$t$ ($\Sigma_{t}$). Then, we introduce a vector field on $M$ associated with
our time evolution and defined by $t^{a}\nabla_{a}t=1$, which we can decompose
in the following manner:

$t^{a}=Nn^{a}+N^{a}$

(in contrast with our previous paragraph, abstract index notation here employs
the metric). $n^{a}$ is the future-directed unit normal vector field to our
space-like hypersurfaces $\Sigma_{t}$ (future-directed in the sense that
$n^{a}$ lies in the same direction as $t^{a}$), and $N^{a}$ represents the
remaining tangential portion of $t^{a}$ to $\Sigma_{t}$. In addition, we
introduce coordinates $t,x^{1},x^{2},x^{3}$ such that $t^{a}\nabla_{a}x^{i}=0$
for $i=1,2,3$ which allows $t^{a}=(\partial/\partial t)^{a}$. Our action in
(73) can then be rewritten in terms of the integral of a Lagrangian density
$\mathcal{L}$ over our time parameter $t$ and our space-like hypersurface
$\Sigma_{t}$:

$S=\int dt\int_{\Sigma_{t}}d^{3}x\mathcal{L}$

with

$\mathcal{L}=\frac{1}{2}N\sqrt{|h|}\bigg{(}(n^{a}\nabla_{a}\phi)^{2}-h^{ab}\nabla_{a}\phi\nabla_{b}\phi-{\tilde{\mu}}^{2}r^{2}\phi^{2}\bigg{)}$

where $h_{ab}$ is the induced Riemannian metric on $\Sigma_{t}$ and
$h=\mathrm{det}(h_{\beta\nu})$. Yet, since

$n^{a}\nabla_{a}\phi=\frac{1}{N}(t^{a}-N^{a})\nabla_{a}\phi=\frac{1}{N}\dot{\phi}-\frac{1}{N}N^{a}\nabla_{a}\phi$

where $\dot{\phi}=t^{a}\nabla_{a}\phi$, we find that our conjugate momentum
density on $\Sigma_{t}$ takes the form

$\pi=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=(n^{a}\nabla_{a}\phi)\sqrt{|h|}$

as it does with our original Klein-Gordon action. Consequently, our symplectic
structure for a free scalar field in the theory of inertial centers is given
by

$\displaystyle\Omega([\phi_{1},\pi_{1}],[\phi_{2},\pi_{2}])=\int_{\Sigma_{0}}d^{3}x(\pi_{1}\phi_{2}-\pi_{2}\phi_{1})$
$\displaystyle=\int_{\Sigma_{0}}d^{3}x\sqrt{|h|}[\phi_{2}n^{a}\nabla_{a}\phi_{1}-\phi_{1}n^{a}\nabla_{a}\phi_{2}]$

where $\Sigma_{0}$ is the space-like hypersurface associated with our “initial
value” configuration at $t=0$.

##  Appendix E Divergence of a vector field

Following Chapter 3.4 of [24], the divergence of a vector field $v^{a}$ is
given by

$\nabla_{a}v^{a}=\partial_{a}v^{a}+\Gamma^{a}_{ab}v^{b}$

where we have used our knowledge from Appendix B to expand this expression.
However, in component form

$\Gamma^{a}_{a\nu}=\sum_{\mu}\Gamma^{\mu}_{\mu\nu}=\frac{1}{2}\sum_{\mu,\rho}g^{\mu\rho}[\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}]$

Yet the first and last of these terms cancel as we are summing over both $\mu$
and $\rho$ and $g^{ab}=g^{ba}$. This leaves us with

$\Gamma^{a}_{a\nu}=\frac{1}{2}\sum_{\mu,\rho}g^{\mu\rho}\partial_{\nu}g_{\mu\rho}$

but if we think in terms of the matrix form of our components, $(g_{\mu\nu})$,
we have

$\sum_{\mu,\rho}g^{\mu\rho}\partial_{\nu}g_{\mu\rho}=\frac{\partial_{\nu}g}{g}$

where $g=\mathrm{det}(g_{\mu\nu})$. Therefore,

$\Gamma^{a}_{a\nu}=\frac{1}{2}\frac{\partial_{\nu}g}{g}=\partial_{\nu}\ln{\sqrt{|g|}}$

Plugging in above for our divergence term,

$\nabla_{a}v^{a}=\sum_{\mu}\bigg{[}\partial_{\mu}v^{\mu}+v^{\mu}\partial_{\mu}\ln{\sqrt{|g|}}\bigg{]}=\sum_{\mu}\frac{1}{\sqrt{|g|}}\partial_{\mu}(\sqrt{|g|}v^{\mu})$

Then for a scalar field $f$,

$\nabla_{a}\nabla^{a}f=\sum_{\mu}\frac{1}{\sqrt{|g|}}\partial_{\mu}(\sqrt{|g|}g^{\mu\nu}\partial_{\nu}f)$

as $\nabla_{a}f=\partial_{a}f$.

##  Appendix F Scalar field solutions

We look for solutions ($\phi_{i}$) to our extension of the Klein-Gordon
equation:

$(\nabla_{a}\nabla^{a}-\tilde{\mu}^{2}r^{2})\phi_{i}=0$

As shown in Appendix E, for a real scalar field

$\nabla_{a}\nabla^{a}\phi_{i}=\sum_{\nu,\beta}\frac{1}{\sqrt{|g|}}\partial_{\nu}(\sqrt{|g|}g^{\nu\beta}\partial_{\beta}\phi_{i})$

where for our purposes $g_{\nu\beta}$ refers to the radial Rindler metric
components and
$\sqrt{|g|}=\sqrt{\Lambda}r^{3}\cosh^{2}(\sqrt{\Lambda}t)\sin{\theta}$.
Expanding (60),

$\displaystyle 0=-\tilde{\mu}^{2}r^{2}\phi_{i}-\frac{1}{\Lambda
r^{2}\cosh^{2}(\sqrt{\Lambda}t)}\partial_{t}(\cosh^{2}(\sqrt{\Lambda}t)\partial_{t}\phi_{i})+\frac{1}{r^{3}}\partial_{r}(r^{3}\partial_{r}\phi_{i})$
$\displaystyle+\frac{1}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\sin{\theta}}\partial_{\theta}(\sin{\theta}\partial_{\theta}\phi_{i})+\frac{1}{\sin^{2}{\theta}}\partial^{2}_{\phi}\phi_{i}\bigg{]}$
(74)

We look for separable solutions of the form, $\phi_{i}=Z_{i}\cdot g(t)\cdot
h(r)\cdot Y_{l}^{m}(\theta,\phi)$, where $Z_{i}$ is a normalization constant
and the $Y_{l}^{m}$ are spherical harmonics satisfying

$\frac{1}{\sin{\theta}}\partial_{\theta}(\sin{\theta}\partial_{\theta}Y_{l}^{m})+\frac{1}{\sin^{2}{\theta}}\partial^{2}_{\phi}Y_{l}^{m}=-l(l+1)Y_{l}^{m}$
(75)

where

$Y_{l}^{m}(\theta,\phi)=\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos{\theta})\cdot
e^{im\phi}$ (76)

with $l$ as a non-negative integer, $|m|\leq l$, and $m$ also as an integer
(see Chapter 3.6 of [56] or Chapter 15.5 of [45]). $P_{l}^{m}$ is an
associated Legendre function which satisfies the differential equation

$\bigg{[}(1-x^{2})\partial^{2}_{x}-2x\partial_{x}+\bigg{(}l[l+1]-\frac{m^{2}}{1-x^{2}}\bigg{)}\bigg{]}P^{m}_{l}(x)=0$

and can be expressed in terms of Rodrigues’ formula [46]:

$P_{l}^{m}(x)=\frac{(-1)^{m}}{2^{l}l!}(1-x^{2})^{m/2}\frac{d^{l+m}}{dx^{l+m}}(x^{2}-1)^{l}$

with

$P_{l}^{-m}=(-1)^{m}\frac{(l-m)!}{(l+m)!}P_{l}^{m}$

The spherical harmonics $Y_{l}^{m}$ as expressed above obey the orthogonality
relation:

$\int_{0}^{\pi}\sin{\theta}d\theta\int_{0}^{2\pi}d\phi
Y_{l}^{m}\bar{Y}_{l^{\prime}}^{m^{\prime}}=\delta_{ll^{\prime}}\delta_{mm^{\prime}}$

Plugging in and dividing by $\phi_{i}$,

$0=-\tilde{\mu}^{2}r^{2}-\frac{1}{g\Lambda
r^{2}\cosh^{2}(\sqrt{\Lambda}t)}\partial_{t}(\cosh^{2}(\sqrt{\Lambda}t)\partial_{t}g)+\frac{1}{hr^{3}}\partial_{r}(r^{3}\partial_{r}h)-\frac{l(l+1)}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)}$

Multiplying through by $r^{2}$ and grouping functions of $t$ and $r$:

$\frac{1}{hr}\partial_{r}(r^{3}\partial_{r}h)-\tilde{\mu}^{2}r^{4}=\frac{1}{g\Lambda\cosh^{2}(\sqrt{\Lambda}t)}\partial_{t}(\cosh^{2}(\sqrt{\Lambda}t)\partial_{t}g)+\frac{l(l+1)}{\cosh^{2}(\sqrt{\Lambda}t)}=-(4\alpha^{2}+1)$

where $\alpha$ is a constant. Thus, we have two differential equations:

$\displaystyle
0=r^{2}\partial^{2}_{r}h+3r\partial_{r}h+\bigg{[}(4\alpha^{2}+1)-\tilde{\mu}^{2}r^{4}\bigg{]}h$
(77) $\displaystyle
0=\frac{1}{\Lambda\cosh^{2}(\sqrt{\Lambda}t)}\partial_{t}(\cosh^{2}(\sqrt{\Lambda}t)\partial_{t}g)+\bigg{[}\frac{l(l+1)}{\cosh^{2}(\sqrt{\Lambda}t)}+(4\alpha^{2}+1)\bigg{]}g$
(78)

Focusing on our radial equation first, we set $\rho=\sqrt{\tilde{\mu}}r$:

$0=\rho^{2}\partial^{2}_{\rho}h+3\rho\partial_{\rho}h+\bigg{[}(4\alpha^{2}+1)-\rho^{4}\bigg{]}h$
(79)

Letting $h(\rho)=z(\rho)/\rho$

$0=\rho^{2}\partial^{2}_{\rho}z+\rho\partial_{\rho}z+\bigg{[}4\alpha^{2}-\rho^{4}\bigg{]}z$

and setting $y=\rho^{2}/2$, we find

$0=y^{2}\partial^{2}_{y}z+y\partial_{y}z+[\alpha^{2}-y^{2}]z$

But this is just the modified Bessel equation of pure imaginary order (see
Chapter 3 of [47]). Choosing the physically realistic solution (we expect $h$
to decay for large $r$ since in our classical analysis massive objects are
“confined” to motion about their inertial centers), our full expression takes
the form

$h_{\alpha}(\rho)=\frac{K_{i\alpha}(\frac{\rho^{2}}{2})}{\rho}$ (80)

where $K_{i\alpha}$ is the Macdonald function (modified Bessel function) of
imaginary order $\alpha$ given in integral form (for $y>0$):

$K_{i\alpha}(y)=\int_{0}^{\infty}d\eta\cos(\alpha\eta)e^{-y\cosh{\eta}}$

and $\alpha$ is restricted to the range: $0\leq\alpha<\infty$ (see Chapter
4.15 of [48] and [42] for its application to quantization in the classic
Rindler case with the Klein-Gordon equation).

The Macdonald function of imaginary order obeys an orthogonality relation
which will be useful to us for determining part of our normalization constant.
From [49],

$\int_{0}^{\infty}dy\frac{K_{i\alpha}(y)K_{i\alpha^{\prime}}(y)}{y}=\frac{\pi^{2}}{2\alpha\sinh(\pi\alpha)}\delta(\alpha-\alpha^{\prime})$

Later we’ll need:

$\displaystyle\int_{0}^{\infty}d\rho\frac{K_{i\alpha}(\frac{\rho^{2}}{2})K_{i\alpha^{\prime}}(\frac{\rho^{2}}{2})}{\rho}=\int_{0}^{\infty}d\eta\int_{0}^{\infty}d\eta^{\prime}\cos(\alpha\eta)\cos(\alpha^{\prime}\eta^{\prime})\int_{0}^{\infty}\frac{d\rho}{\rho}e^{-(\rho^{2}/2)(\cosh{\eta}+\cosh{\eta^{\prime}})}$
$\displaystyle=\int_{0}^{\infty}d\eta\int_{0}^{\infty}d\eta^{\prime}\cos(\alpha\eta)\cos(\alpha^{\prime}\eta^{\prime})\int_{0}^{\infty}\frac{dy}{2y}e^{-y(\cosh{\eta}+\cosh{\eta^{\prime}})}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}dy\frac{K_{i\alpha}(y)K_{i\alpha^{\prime}}(y)}{y}=\frac{\pi^{2}}{4\alpha\sinh(\pi\alpha)}\delta(\alpha-\alpha^{\prime})$

Examining our second differential equation, we let
$\eta=\tanh(\sqrt{\Lambda}t)$:

$0=(1-\eta^{2})^{2}\partial^{2}_{\eta}g+[l(l+1)(1-\eta^{2})+(4\alpha^{2}+1)]g$
(81)

where $\eta^{2}<1$. Before proceeding any further, we must make one remark
that will be crucial for evaluating our inner product. Taking the complex
conjugate of (81)

$0=(1-\eta^{2})^{2}\partial^{2}_{\eta}\bar{g}+[l(l+1)(1-\eta^{2})+(4\alpha^{2}+1)]\bar{g}$

Multiplying (81) by $\bar{g}$ and subtracting by $g$ times the complex
conjugate of (81), we find

$\bar{g}\partial^{2}_{\eta}g-g\partial^{2}_{\eta}\bar{g}=0$

or

$\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}={\rm constant}$ (82)

Returning to (81), we divide through by $1-\eta^{2}$ and make the substitution
$g(\eta)=\sqrt{1-\eta^{2}}\cdot p(\eta)$:

$0=(1-\eta^{2})\sqrt{1-\eta^{2}}\bigg{[}\partial^{2}_{\eta}p-\frac{2\eta}{1-\eta^{2}}\partial_{\eta}p-\frac{p}{(1-\eta^{2})^{2}}\bigg{]}+\bigg{[}l(l+1)+\frac{4\alpha^{2}}{1-\eta^{2}}+\frac{1}{1-\eta^{2}}\bigg{]}\sqrt{1-\eta^{2}}\cdot
p$

which reduces to

$0=(1-\eta^{2})\partial^{2}_{\eta}p-2\eta\partial_{\eta}p+\bigg{[}l(l+1)+\frac{4\alpha^{2}}{1-\eta^{2}}\bigg{]}p$

But the solution to this differential equation is the Legendre function of the
first kind [46] (since our domain is restricted to $\eta^{2}<1$) which can be
expressed in the following manner:

$P^{\pm 2i\alpha}_{l}(\eta)=\frac{1}{\Gamma(1\mp
2i\alpha)}\bigg{[}\frac{1+\eta}{1-\eta}\bigg{]}^{\pm
i\alpha}\,_{2}F_{1}(-l,l+1;1\mp 2i\alpha,\frac{1-\eta}{2})$ (83)

where $\,{}_{2}F_{1}$ is the hypergeometric function which for our parameters
can take the form

$\,{}_{2}F_{1}(-l,l+1;1\mp 2i\alpha,\frac{1-\eta}{2})=\frac{\Gamma(1\mp
2i\alpha)}{\Gamma(-l)\Gamma(l+1)}\sum_{k=0}^{\infty}\frac{\Gamma(k-l)\Gamma(k+l+1)}{k!\cdot\Gamma(k+1\mp
2i\alpha)}\bigg{(}\frac{1-\eta}{2}\bigg{)}^{k}$ (84)

and $\Gamma(z)$ is the gamma function which can be written in integral form
for $\Re(z)>0$ where $z$ is a complex variable as

$\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt$

Note that this Legendre function, $P_{\nu}^{\mu}$, is in fact a generalization
of the Legendre function used earlier for our angular dependence where the
parameters $\mu$, $\nu$ here are allowed to be complex numbers instead of
solely real integers. Our full expression for $g$ is then

$g(\eta)=\sqrt{1-\eta^{2}}\cdot P_{l}^{-2i\alpha}(\eta)$ (85)

where as we’ll see below, our choice of $-2i\alpha$ is necessary in order to
ensure that our inner product is positive-definite with respect to our
solutions for $\phi_{i}$ so that we may properly construct our field operator
(i.e. we take the “positive frequency” solutions; see Chapter 3.2 of [43]).

To find our normalization constant, we’ll need to evaluate (82). For the rest
of our analysis in this appendix, we use Chapter 8 of [46] as a reference for
our general expressions. Beginning with $\partial_{\eta}g$:

$\partial_{\eta}g=-\frac{\eta}{\sqrt{1-\eta^{2}}}P_{l}^{-2i\alpha}+\sqrt{1-\eta^{2}}\partial_{\eta}P_{l}^{-2i\alpha}$

But

$\partial_{\eta}P_{\nu}^{\mu}=-\frac{\nu\eta}{1-\eta^{2}}P_{\nu}^{\mu}+\frac{\mu+\nu}{1-\eta^{2}}P_{\nu-1}^{\mu}$

Plugging in above

$\displaystyle\partial_{\eta}g=-\frac{\eta}{\sqrt{1-\eta^{2}}}P_{l}^{-2i\alpha}+\sqrt{1-\eta^{2}}\bigg{(}-\frac{l\eta}{1-\eta^{2}}P_{l}^{-2i\alpha}+\frac{l-2i\alpha}{1-\eta^{2}}P_{l-1}^{-2i\alpha}\bigg{)}$
$\displaystyle=-\frac{\eta(1+l)}{\sqrt{1-\eta^{2}}}P_{l}^{-2i\alpha}+\frac{l-2i\alpha}{\sqrt{1-\eta^{2}}}P_{l-1}^{-2i\alpha}$

Then

$\displaystyle\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}=\sqrt{1-\eta^{2}}\bigg{[}\bar{P}_{l}^{-2i\alpha}\bigg{(}-\frac{\eta(1+l)}{\sqrt{1-\eta^{2}}}P_{l}^{-2i\alpha}+\frac{l-2i\alpha}{\sqrt{1-\eta^{2}}}P_{l-1}^{-2i\alpha}\bigg{)}$
$\displaystyle-
P_{l}^{-2i\alpha}\bigg{(}-\frac{\eta(1+l)}{\sqrt{1-\eta^{2}}}\bar{P}_{l}^{-2i\alpha}+\frac{l+2i\alpha}{\sqrt{1-\eta^{2}}}\bar{P}_{l-1}^{-2i\alpha}\bigg{)}\bigg{]}$
$\displaystyle=(l-2i\alpha)\bar{P}_{l}^{-2i\alpha}P_{l-1}^{-2i\alpha}-(l+2i\alpha)P_{l}^{-2i\alpha}\bar{P}_{l-1}^{-2i\alpha}$

However, as one can tell from (83),
$\bar{P}_{l}^{-2i\alpha}=P_{l}^{2i\alpha}$. Therefore,

$\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}=(l-2i\alpha)P_{l}^{2i\alpha}P_{l-1}^{-2i\alpha}-(l+2i\alpha)P_{l}^{-2i\alpha}P_{l-1}^{2i\alpha}={\rm
constant}$

Yet since this expression must be constant, we can evaluate for any particular
value of $\eta$. Because we have expressions for the Legendre functions at
$\eta=0$, we’ll make this convenient choice where

$P_{\nu}^{\mu}(0)=2^{\mu}\pi^{-1/2}\cos\bigg{[}\frac{\pi}{2}(\nu+\mu)\bigg{]}\frac{\Gamma(\frac{1}{2}+\frac{1}{2}\nu+\frac{1}{2}\mu)}{\Gamma(1+\frac{1}{2}\nu-\frac{1}{2}\mu)}$

For $\eta=0$:

$\displaystyle P_{l-1}^{-2i\alpha}(0)\cdot
P_{l}^{2i\alpha}(0)=2^{-2i\alpha}\pi^{-1/2}\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}\frac{\Gamma(\frac{1}{2}+\frac{l-1}{2}-i\alpha)}{\Gamma(1+\frac{l-1}{2}+i\alpha)}$
$\displaystyle\cdot
2^{2i\alpha}\pi^{-1/2}\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}\frac{\Gamma(\frac{1}{2}+\frac{l}{2}+i\alpha)}{\Gamma(1+\frac{l}{2}-i\alpha)}$
$\displaystyle=\frac{1}{\pi}\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}\frac{\Gamma(\frac{l}{2}-i\alpha)}{\Gamma(1+\frac{l}{2}-i\alpha)}$

But from properties of the gamma function (see Chapter 6 of [46]),

$\Gamma(1+z)=z\Gamma(z)$

Using this property above,

$\displaystyle P_{l-1}^{-2i\alpha}(0)\cdot
P_{l}^{2i\alpha}(0)=\frac{1}{\pi(\frac{l}{2}-i\alpha)}\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}$

And with the expressions

$\displaystyle\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}=\frac{1}{2}\bigg{[}e^{i(\pi/2)(l-1-2i\alpha)}+e^{-i(\pi/2)(l-1-2i\alpha)}\bigg{]}$
$\displaystyle\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}=\frac{1}{2}\bigg{[}e^{i(\pi/2)(l+2i\alpha)}+e^{-i(\pi/2)(l+2i\alpha)}\bigg{]}$

we have

$\cos\bigg{[}\frac{\pi}{2}(l-1-2i\alpha)\bigg{]}\cos\bigg{[}\frac{\pi}{2}(l+2i\alpha)\bigg{]}=\frac{1}{4}\bigg{[}e^{i\pi(l-1/2)}+e^{-i\pi(l-1/2)}+e^{i\pi(2i\alpha+1/2)}+e^{-i\pi(2i\alpha+1/2)}\bigg{]}$

Thus,

$(l-2i\alpha)P_{l-1}^{-2i\alpha}P_{l}^{2i\alpha}|_{\eta=0}=\frac{1}{2\pi}\bigg{[}e^{i\pi(l-1/2)}+e^{-i\pi(l-1/2)}+e^{i\pi(2i\alpha+1/2)}+e^{-i\pi(2i\alpha+1/2)}\bigg{]}$

Applying similar logic to the second term in our expression above, we find

$(l+2i\alpha)P_{l}^{-2i\alpha}P_{l-1}^{2i\alpha}|_{\eta=0}=\frac{1}{2\pi}\bigg{[}e^{i\pi(l-1/2)}+e^{-i\pi(l-1/2)}+e^{i\pi(2i\alpha-1/2)}+e^{-i\pi(2i\alpha-1/2)}\bigg{]}$

Then

$\displaystyle\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}=\frac{1}{2\pi}\bigg{[}e^{i\pi/2-2\pi\alpha}+e^{-i\pi/2+2\pi\alpha}-e^{i\pi/2+2\pi\alpha}-e^{-i\pi/2-2\pi\alpha}\bigg{]}$
$\displaystyle=\frac{1}{2\pi}\bigg{[}e^{i\pi/2}\bigg{(}e^{-2\pi\alpha}-e^{2\pi\alpha}\bigg{)}+e^{-i\pi/2}\bigg{(}e^{2\pi\alpha}-e^{-2\pi\alpha}\bigg{)}\bigg{]}$
$\displaystyle=-\frac{(e^{2\pi\alpha}-e^{-2\pi\alpha})(e^{i\pi/2}-e^{-i\pi/2})}{2\pi}$

But

$e^{i\pi/2}-e^{-i\pi/2}=2i\sin(\pi/2)=2i$

and

$e^{2\pi\alpha}-e^{-2\pi\alpha}=2\sinh(2\pi\alpha)=4\sinh(\pi\alpha)\cosh(\pi\alpha)$

Therefore, plugging in above

$\bar{g}\partial_{\eta}g-g\partial_{\eta}\bar{g}=-\frac{4i}{\pi}\sinh(\pi\alpha)\cosh(\pi\alpha)$

Addressing our inner product where our solutions are of the form
$\phi_{i}=Z_{i}\cdot g\cdot h\cdot Y_{l}^{m}$

$\displaystyle(\phi_{1},\phi_{2})=-\frac{i}{\sqrt{\Lambda}}Z_{1}Z_{2}\cosh^{2}(\sqrt{\Lambda}t)[g_{2}\partial_{t}\bar{g}_{1}-\bar{g}_{1}\partial_{t}g_{2}]|_{t=0}\int_{0}^{\pi}\sin{\theta}d\theta\int_{0}^{2\pi}d\phi\bar{Y}_{l_{1}}^{m_{1}}Y_{l_{2}}^{m_{2}}$
$\displaystyle\cdot\bigg{[}\int_{0}^{\infty}r\bar{h}_{\alpha_{1}}h_{\alpha_{2}}dr-\int_{-\infty}^{0}r\bar{h}_{\alpha_{1}}h_{\alpha_{2}}dr\bigg{]}$
$\displaystyle=iZ_{1}Z_{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}[\bar{g}_{1}\partial_{\eta}g_{2}-g_{2}\partial_{\eta}\bar{g}_{1}]|_{\eta=0}\cdot\bigg{[}\int_{0}^{\infty}dr\frac{K_{i\alpha_{1}}(\frac{\tilde{\mu}r^{2}}{2})K_{i\alpha_{2}}(\frac{\tilde{\mu}r^{2}}{2})}{\tilde{\mu}r}$
$\displaystyle-\int_{-\infty}^{0}dr\frac{K_{i\alpha_{1}}(\frac{\tilde{\mu}r^{2}}{2})K_{i\alpha_{2}}(\frac{\tilde{\mu}r^{2}}{2})}{\tilde{\mu}r}\bigg{]}$
$\displaystyle=iZ_{1}Z_{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}[\bar{g}_{1}\partial_{\eta}g_{2}-g_{2}\partial_{\eta}\bar{g}_{1}]|_{\eta=0}\cdot\frac{2}{\tilde{\mu}}\int_{0}^{\infty}d\rho\frac{K_{i\alpha_{1}}(\frac{\rho^{2}}{2})K_{i\alpha_{2}}(\frac{\rho^{2}}{2})}{\rho}$
$\displaystyle=iZ_{1}Z_{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta(\alpha_{1}-\alpha_{2})\cdot\frac{\pi^{2}}{2\tilde{\mu}\alpha_{1}\sinh(\pi\alpha_{1})}\cdot[\bar{g}_{1}\partial_{\eta}g_{2}-g_{2}\partial_{\eta}\bar{g}_{1}]|_{\eta=0}$
$\displaystyle=iZ_{1}^{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta(\alpha_{1}-\alpha_{2})\cdot\frac{\pi^{2}}{2\tilde{\mu}\alpha_{1}\sinh(\pi\alpha_{1})}\cdot[\bar{g}_{1}\partial_{\eta}g_{1}-g_{1}\partial_{\eta}\bar{g}_{1}]|_{\eta=0}$
$\displaystyle=iZ_{1}^{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta(\alpha_{1}-\alpha_{2})\cdot\frac{\pi^{2}}{2\tilde{\mu}\alpha_{1}\sinh(\pi\alpha_{1})}\bigg{[}-\frac{4i\sinh(\pi\alpha_{1})\cosh(\pi\alpha_{1})}{\pi}\bigg{]}$
$\displaystyle=Z_{1}^{2}\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta(\alpha_{1}-\alpha_{2})\cdot\frac{2\pi\cosh(\pi\alpha_{1})}{\tilde{\mu}\alpha_{1}}$

where we keep in mind the fact that

$\frac{\partial}{\partial t}=\frac{\partial\eta}{\partial
t}\frac{\partial}{\partial\eta}=\frac{\sqrt{\Lambda}}{\cosh^{2}(\sqrt{\Lambda}t)}\cdot\frac{\partial}{\partial\eta}$

Thus, our normalization constant is found to be

$Z_{\alpha}=\sqrt{\frac{\tilde{\mu}\alpha}{2\pi\cosh(\pi\alpha)}}$ (86)

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## Figure Legends

Figure 1: Geodesic paths in Minkowski coordinates Figure 2: Local
approximation of the inertial frame of reference Figure 3: Plots of
$h_{\alpha}(\rho)$ for $\alpha=0,1,5,$ and $20$

## Tables

Table 1:  Redshift from objects within the Local Group

Object | RA (J2000.0) | Dec (J2000.0) | Redshift | Distance Mod (mag)
---|---|---|---|---
Andromeda V | 01h10m17.10s | +47d37m41.0s | -0.001344 | 24.52
Andromeda I | 00h45m39.80s | +38d02m28.0s | -0.001228 | 24.46
Andromeda VI | 23h51m46.30s | +24d34m57.0s | -0.001181 | 24.58
Andromeda III | 00h35m33.78s | +36d29m51.9s | -0.001171 | 24.38
IC 0010 | 00h20m17.34s | +59d18m13.6s | -0.001161 | 24.57
Andromeda VII | 23h26m31.74s | +50d40m32.6s | -0.001024 | 24.7
MESSIER 031 | 00h42m44.35s | +41d16m08.6s | -0.001001 | 24.46
Draco Dwarf | 17h20m12.39s | +57d54m55.3s | -0.000974 | 19.61
Pisces I | 01h03m55.00s | +21d53m06.0s | -0.000956 | 24.5
UMi Dwarf | 15h09m08.49s | +67d13m21.4s | -0.000824 | 19.3
MESSIER 110 | 00h40m22.08s | +41d41m07.1s | -0.000804 | 24.5
IC 1613 | 01h04m47.79s | +02d07m04.0s | -0.000781 | 24.33
NGC 0185 | 00h38m57.97s | +48d20m14.6s | -0.000674 | 24.13
MESSIER 032 | 00h42m41.83s | +40d51m55.0s | -0.000667 | 24.42
NGC 0147 | 00h33m12.12s | +48d30m31.5s | -0.000644 | 24.3
Andromeda II | 01h16m29.78s | +33d25m08.8s | -0.000627 | 24.03
Pegasus Dwarf | 23h28m36.25s | +14d44m34.5s | -0.000612 | 26.34
MESSIER 033 | 01h33m50.89s | +30d39m36.8s | -0.000597 | 24.69
Aquarius dIrr | 20h46m51.81s | -12d50m52.5s | -0.00047 | 26
WLM | 00h01m58.16s | -15d27m39.3s | -0.000407 | 25.09
Cetus Dwarf Spheroidal | 00h26m11.03s | -11d02m39.6s | -0.00029 | 24.51
SagDIG | 19h29m59.58s | -17d40m51.3s | -0.000264 | 25.03
NGC 6822 | 19h44m57.74s | -14d48m12.4s | -0.00019 | 23.41
Leo A | 09h59m26.46s | +30d44m47.0s | 0.000067 |
Fornax Dwarf Spheroidal | 02h39m59.33s | -34d26m57.1s | 0.000178 | 20.7
Phoenix Dwarf | 01h51m06.34s | -44d26m40.9s | 0.000187 | 23.08
Leo B | 11h13m28.80s | +22d09m06.0s | 0.000264 | 21.67
Sculptor Dwarf Elliptical | 01h00m09.36s | -33d42m32.5s | 0.000367 | 19.67
Sagittarius Dwarf Spheroidal | 18h55m19.50s | -30d32m43.0s | 0.000467 | 17.17
Small Magellanic Cloud | 00h52m44.78s | -72d49m43.0s | 0.000527 | 18.95
Tucana Dwarf | 22h41m49.60s | -64d25m10.0s | 0.000647 | 24.74
Sextans Dwarf Spheroidal | 10h13m02.96s | -01d36m52.6s | 0.000747 | 19.73
Carina Dwarf | 06h41m36.69s | -50d57m58.3s | 0.000764 | 20.02
Large Magellanic Cloud | 05h23m34.53s | -69d45m22.1s | 0.000927 | 18.46
Leo I | 10h08m28.10s | +12d18m23.0s | 0.000951 | 21.91

Data retrieved from NASA/IPAC Extragalactic Database (NED):
http://ned.ipac.caltech.edu

Table 2:  Redshift from galaxies within $\approx\pm 2$ h of 0 h in RA within
Local Group

Object | RA (J2000.0) | Dec (J2000.0) | Redshift | Distance Mod (mag)
---|---|---|---|---
Andromeda V | 01h10m17.10s | +47d37m41.0s | -0.001344 | 24.52
Andromeda I | 00h45m39.80s | +38d02m28.0s | -0.001228 | 24.46
Andromeda VI | 23h51m46.30s | +24d34m57.0s | -0.001181 | 24.58
Andromeda III | 00h35m33.78s | +36d29m51.9s | -0.001171 | 24.38
IC 0010 | 00h20m17.34s | +59d18m13.6s | -0.001161 | 24.57
Andromeda VII | 23h26m31.74s | +50d40m32.6s | -0.001024 | 24.7
MESSIER 031 | 00h42m44.35s | +41d16m08.6s | -0.001001 | 24.46
Pisces I | 01h03m55.00s | +21d53m06.0s | -0.000956 | 24.5
MESSIER 110 | 00h40m22.08s | +41d41m07.1s | -0.000804 | 24.5
IC 1613 | 01h04m47.79s | +02d07m04.0s | -0.000781 | 24.33
NGC 0185 | 00h38m57.97s | +48d20m14.6s | -0.000674 | 24.13
MESSIER 032 | 00h42m41.83s | +40d51m55.0s | -0.000667 | 24.42
NGC 0147 | 00h33m12.12s | +48d30m31.5s | -0.000644 | 24.3
Andromeda II | 01h16m29.78s | +33d25m08.8s | -0.000627 | 24.03
Pegasus Dwarf | 23h28m36.25s | +14d44m34.5s | -0.000612 | 26.34
MESSIER 033 | 01h33m50.89s | +30d39m36.8s | -0.000597 | 24.69
WLM | 00h01m58.16s | -15d27m39.3s | -0.000407 | 25.09
Cetus Dwarf Spheroidal | 00h26m11.03s | -11d02m39.6s | -0.00029 | 24.51
Fornax Dwarf Spheroidal | 02h39m59.33s | -34d26m57.1s | 0.000178 | 20.7
Phoenix Dwarf | 01h51m06.34s | -44d26m40.9s | 0.000187 | 23.08
Sculptor Dwarf Elliptical | 01h00m09.36s | -33d42m32.5s | 0.000367 | 19.67
Small Magellanic Cloud | 00h52m44.78s | -72d49m43.0s | 0.000527 | 18.95

Data retrieved from NASA/IPAC Extragalactic Database (NED):
http://ned.ipac.caltech.edu