# A Layered Architecture Enabling Metaverse Applications in Smart
Manufacturing Environments

Armir Bujari, Alessandro Calvio, Andrea Garbugli, Paolo Bellavista  University
of Bologna
Department of Computer Science and Engineering
Bologna, Italy
<EMAIL_ADDRESS>
###### Abstract

The steady rollout of Industrial IoT (IIoT) technology in the manufacturing
domain embodies the potential to implement smarter and more resilient
production processes. To this end, it is expected that there will be a strong
reliance of manufacturing processes on cloud/edge services so as to act
intelligently and flexibly. While automation is necessary to handle the
environment’s complexity, human-in-the-loop design approaches are paramount.
In this context, Digital Twins play a crucial role by allowing human operators
to inspect and monitor the environment to ensure stability and reliability.
Integrating the IIoT with the Metaverse enhances the system’s capabilities
even further, offering new opportunities for efficiency and collaboration
while enabling integrated management of assets and processes. This article
presents a layered conceptual architecture as an enabler for smart
manufacturing metaverse environments, targeting real-time data collection and
representations from shopfloor assets and processes. At the bottom layer, our
proposal relies on middleware technology, serving differentiated Quality of
Service (QoS) needs of the Operation Technology (OT) monitoring processes. The
latter contributes to feeding a virtual layer where data processes reside,
creating representations of the monitored phenomena at different timescales.
Metaverse applications can consume data by tapping into the metaverse engine,
a microservice-oriented and accelerated Platform as a Service (PaaS) layer
tasked with bringing data to life. Without loss of generality, we profile
different facets of our proposal by relying on two different proof-of-concept
inspection applications aimed at real-time monitoring of the network fabric
activity and a visual asset monitoring one.

###### Index Terms:

Smart Manufacturing, Digital Twin, Metaverse, Industrial IoT, Middleware

##  I Introduction

We are in the middle of an industrial revolution, i.e., of Industry 4.0
(I4.0)-based intelligent and cooperative Cyber-Physical Systems (CPSs),
underpinned by a digital transformation that will affect all industries. At
the core of this revolution is the convergence of flexible and multi-faceted
communication technologies, novel computing techniques such as cloud/edge
computing, and the application of the IoT vision to industrial manufacturing
systems [1]. As a result, industrial devices and machines will rely on
heterogeneous wireless/wired technologies to communicate with applications
running on global cloud/local edge platforms, thus enabling new efficient
solutions, e.g., for predictive maintenance and process optimization.

The application of big data and AI has been successfully adopted to streamline
and optimize non-manufacturing processes and is now being expanded into the
industrial sector. Today’s manufacturing environment involves machines that
can communicate independently and generate data at a rapid pace. This
information can be utilized proactively to enhance control, and business
processes in manufacturing, engineering, supply chain, and product life cycle
management [2]. A significant challenge in realizing this goal is the outdated
and inflexible division between technology departments involved in product
manufacturing and those focused on management tasks. Indeed, industrial
automation has taken a traditional approach, opting for a strict separation
between Operation and Information Technology domains (OT & IT) [3].

Until now, IT technologies such as cloud/edge computing, Service-Oriented
Architectures (SOA), and virtualization have been used in the industrial
domain in limited ways, only where stringent requirements were not necessary
[4]. However, it is becoming clear that smart manufacturing environments will
have a significant impact only with a complete convergence of OT/IT, allowing
for full utilization of data intelligence and recent computing and
communication technologies.

While recognizing the OT/IT synergy as a major milestone in tackling the
complexity of this evolving environment, human-in-the-loop design approaches
are paramount. Despite the many benefits that data intelligence can provide to
society and the economy, the technology behind it can also have negative or
unpredictable impacts, bringing about new risks to individuals. A synergetic
development of both AI-driven and human-centric approaches is key to driving a
safe, resource-efficient, and sustainable manufacturing domain [5].

To this end, a central component toward this objective is the concept of the
digital twin; a virtual replica of a physical asset that can be monitored and
optimized in real-time. Data streams often connect this digital replica and
the physical counterpart, feeding and continuously updating the digital model,
used as a descriptive or predictive tool for planning and operational purposes
[6]. Integrating the shopfloor with the Metaverse enhances the system’s
capabilities even further, offering new opportunities for efficiency and
collaboration while enabling integrated management of assets and processes
[7]. In a Metaverse, digital replicas of real-life objects and processes can
be used for various purposes such as testing, monitoring, and collaboration
through AR/VR technology.

In this work, we present a layered conceptual architecture and accompanying
software prototype conceived to enable a human-in-the-loop manufacturing
environment enriched thanks to real-time data sourced from shopfloor
processes. Data are stored at different layers of the architecture, having
different scopes, and are brought to life via virtual representations of
assets, depending on the layer and the application used to access it. In
synthesis, the architecture consists of multiple layers, including a bottom
layer that utilizes middleware technology to serve the different Quality of
Service (QoS) requirements of the OT monitoring processes, implementing a
beyond state-of-the-art OT/IT convergence mechanism.

Moving up is the virtual layer tasked with the creation of multi-faceted
representations of the monitored phenomena at different timescales. Herein are
logically collocated the repositories containing the virtual representations
of assets at different levels of granularity, e.g., individual machines or
production cells within the shopfloor, etc. Finally, at the application layer
reside the immersive and collaborative applications. The applications leverage
the metaverse engine, a microservice-oriented and accelerated Platform as a
Service (PaaS) layer that fuses and renders synchronized information sources
from the virtual layer. Without loss of generality, we profile different
facets of our proposal by relying on two different proof-of-concept inspection
applications aimed at real-time monitoring of the network fabric activity and
a physical asset monitoring one.

##  II Background

In this section, we provide a concise survey on the paradigm shift advocated
by Industry 4.0, motivating the need for a human-centric approach, as recently
advocated by many nation-state initiatives on their next industrial vision.
Next, we provide a concise survey on the technology behind some of the adopted
design choices.

###  II-A Towards a Human-in-the-loop Manufacturing Environment

The automation pyramid serves as a reference model for dealing with the
challenges posed by complex and heterogeneous manufacturing systems and
consists of different levels, each representing a specific stage of automation
[8]. A simplified specification consists of four different hierarchy levels
(see left part of Fig. 1).

The sensing and actuation units are located on Level 1. They commonly operate
within milliseconds or seconds to control the physical manufacturing processes
to reach the required quality demands. Monitoring, supervision, and control is
the task of Level 2 systems. Production environments mainly use programmable
logic controllers (PLCs). Depending on the individual production system, PLCs
operate within hours down to less than periods of seconds. As Level 2 directly
controls the actuators, PLCs embedded logic may implement short-term adaptions
of the process(es).

Level 3 of the automation pyramid encompasses activities related to the
management and control of manufacturing operations. This includes dynamic
scheduling of jobs, optimization of production processes, and data aggregation
and distribution, all encompassed in a Manufacturing Execution System (MES),
which operates within one day or shift and must be capable of reacting to
unexpected events, e.g., machine breakdowns. The level also includes SCADA
systems that collect real-time data, data-driven analysis, and decision
support systems to optimize processes. This level bridges lower levels of
control and monitoring and higher levels of strategic decision-making and
planning.

Figure 1: Transition to Industry 4.0.

The top Level 4 consists of an Enterprise Resource Planning (ERP) tool, which
executes the required tasks and manages inventory and resources. As the ERP
defines the long-term production utilization, the long-term operations
(optimizations) are also determined here.

Cyber-physical systems, which merge the physical and virtual world through
embedded hardware and software systems, present a new, non-hierarchical
approach to production (see right part of Fig. 1). This change is due to the
need to better highlight and distinguish the networks that connect the layers
and to reflect that some technologies may no longer reside in the facility but
in local/remote edge-cloud computing environments. This has been made possible
thanks to the convergence of the so-called disaggregation trend in the IT
industry and the market penetration of deterministic and high-throughput
communication technologies such as Time Sensitive Networking (TSN) and 5G [9,
10]. In this _automation pillar_ , processes that do not require hard, real-
time control can now be run in the cloud with virtual PLCs, as shown in the
graphic. The graphic also depicts how, in the IIoT, data can now be shared
more easily through all levels rather than sequentially from one layer to the
next.

The hierarchical architecture of the automation pyramid is still present and
very common in production systems. This is also indicated by the adoption of
the automation pyramid within the more recent RAMI 4.0 reference model for
Industry 4.0 [11]. However, the paradigm shift has already begun by
recognizing the OT/IT convergence problem, paving the way for the use of
descriptive and predictive Digital Twin technology in manufacturing systems.
The availability of data and recent advances in AI technology could pave the
way for smarter manufacturing environments, but this evolution should go in
pace with a human-centric design approach. In this context, the notion of the
Metaverse could enhance an operator’s capabilities even further, offering new
opportunities for efficient collaboration while enabling integrated management
of assets and processes in the manufacturing domain. Our proposal goes in this
direction, providing the basic building blocks for enabling an immersive,
human-in-the-loop manufacturing environment.

###  II-B Data Homogenization and Transport from the Shopfloor

The OPC Unified Architecture (OPC UA) is an industrial automation standard
that aims to solve the issue of incompatibility among different Ethernet
technologies (such as PROFINET, EtherCAT, and Modbus-TCP) by providing a
unified and consolidated view of assets and processes [12, 13]. This platform-
independent standard can facilitate interoperability among vendors.

The initial OPC UA standard employed a Client/Server paradigm: an OPC UA
server offers access to data and functions structured in an object-oriented
information model, where clients interacted with the information model through
standardized services. Based on the request-response method, this
communication model does not satisfy the needs of some industrial (control)
applications as it creates a strong coupling between different system
components and does not meet the performance demands of (hard) real-time
systems.

To overcome these limitations, Part 14 of the OPC UA specification introduces
an extension based on the Publish/Subscribe (Pub/Sub) communication paradigm
[14]. In this model, an application can act as either a publisher or a
subscriber (or both), where the publisher sources the data, and the subscriber
consumes it. Communication between publishers and subscribers is message-
based, with publishers sending messages to a message-oriented middleware
without prior knowledge of the subscribers. The latter express interest in
certain types of data without having specific knowledge of the publishers.
This message-based communication is ideal for applications that value location
independence and scalability.

OPC UA is a significant IIoT protocol for addressing communication needs at
the OT layer, but it does not fully meet the requirements of the IT layer.
Here, there is a need for solutions and frameworks that can handle high-volume
data transfers in a secure and reliable manner, which are not priorities of
the OPC UA standard. To meet these requirements, we rely on Apache Kafka, an
open-source Message Oriented Middleware that uses a Pub/Sub communication
model enabling a many-to-many communication pattern [15]. Kafka is well-suited
for scenarios where scalable and loosely-coupled systems must work together.
In this setting, producers publish messages or batches of messages on a
channel called topic, while consumers can read the messages by subscribing to
a specific topic. Topics in Kafka make use of partitions, which can be thought
of as an infinite log file that is not immediately flushed to disk, leading to
highly efficient I/O messaging and having the capability to force strong
ordering constraints on message delivery.

In the following, we present a detailed overview of the proposed layered
architecture, discussing some design choices and technological building
blocks.

##  III A Layered Architectural Approach

The conceptual architecture, depicted in Fig. 2, takes a structured approach
to integrate the virtual and the physical realms of next-generation
manufacturing systems. In the following, we provide an in-depth description of
the various layers, discussing their functional components and some concrete
technological choices.

###  III-A Operational Technology Layer

The _Physical layer_ encompasses Level 1 of the automation pyramid, including
controllers, machinery, sensors, and processes that make up the industrial
shop floor. In addition to these physical components, the physical layer
includes users and human operators. Moreover, the layer includes various
interfaces, such as haptic technology, VR/AR headsets, and touchscreens,
enabling interaction between the physical and the virtual world.

One of the key aspects of our proposal is the implementation of a scalable and
reliable solution addressing OT/IT convergence. This convergence is made
possible by relying on an OT middleware, which serves as a bridge to gather
and elaborate data before they are sent to the virtual layer. One of the main
problems when dealing with the convergence of OT and IT is that OT systems
often employ a variety of communication protocols, as previously discussed.
Therefore, the role of the OT middleware is to standardize the data collection
process, abstracting from the details of the specific protocol.

Another common challenge is data format heterogeneity and resource
configuration modeling. We address these challenges by using a standard and
common information model (object model) available in the OPC UA dictionary. We
anticipate our testbed uses a deterministic network fabric (TSN), and
communication on the shopfloor relies on the OPC UA PubSub profile.

As part of the OT middleware, one or more Gateway components are responsible
for listening to OPC UA PubSub endpoints and managing data flows between the
physical and virtual layers. The Gateway uses different topics and partition
levels to prioritize specific traffic flows, such as monitoring data or
controlling data traffic in the network. To increase the reliability of the
data collection process, the Gateway implements a replication mechanism. For
example, monitoring and control topics are assigned a single partition with a
high degree of replication. In contrast, data topics from raw sensor elements
are assigned multiple partitions with a lower degree of replication.

The OT middleware is also responsible for forwarding the data to the IT level,
which logically spans across the upper layers of the architecture. To achieve
this, our solution relies on Apache Kafka, a message-oriented middleware,
well-suited for scenarios where scalable and loosely coupled systems must
interoperate.

Figure 2: Proposed conceptual architecture.

###  III-B Virtual Layer

The virtual layer is conceptually located above the physical layer and serves
as the metaverse engine’s primary data access layer and support component. It
integrates essential functions utilized by the upper layers in the
construction of virtual worlds. Going into more detail, the virtual layer is
composed of three main sub-blocks: _digital twin_ , _digital assets_ , and
_metadata repository_.

####  III-B1 Digital Twin

The digital twin subcomponent is a crucial part of the virtual layer in the
factory environment. It creates and maintains digital twins of all the
physical machines, products, and facility assets. This allows for a
comprehensive representation of the factory’s operations.

In addition to creating digital twins of individual assets, this subcomponent
maintains a global view of the facility, including both the plant and the
network. This allows for a holistic understanding of the entire factory’s
operations and enables the identification of interdependencies and bottlenecks
in the production process.

Here, the processing module collects data from the lower layers and transforms
it with varying spatial and temporal resolution levels, e.g., allowing for a
more detailed and accurate representation of the physical assets and
processes. For example, the processing module can collect and aggregate data
at a high pace, providing real-time monitoring of machine performance. On the
other hand, it can be dynamically reconfigured to adjust the scope of the data
forwarding processes, feeding data from single production units to an entire
production cell, and providing a detailed representation of current processes
in the factory.

Combining the various data perspectives generated by the processing layer to
create a unified and multi-faceted view of the operations within the factory
can represent a challenge. To overcome this, the data fusion layer utilizes
various techniques, such as statistical, rule-based, and ontology-aided
methods, to synthesize the data into a single, consistent representation of a
particular phenomenon. For example, data from different parts of a machine or
the machine and its products can be combined to identify the root cause of
process issues or inefficiencies. Furthermore, simulation engines are used in
this layer to perform several kinds of analysis to gain deeper insights into
the potential future evolution of the (sub)systems.

####  III-B2 Digital Asset

This part consists of an indexed repository, which includes digital
representations of physical assets. This data encompasses a wide range of
information, including geometric structures, physical attributes, and
technical specifications, and may consist of both structured and unstructured
information. The current prototype employs various storage technologies to
enable data injection into the data lake to promote efficient data management.

For instance, we rely on a NoSQL database technology such as MongoDB to store
device configurations or other static information. In contrast, data sourced
from the shopfloor are stored in time-series databases such as Prometheus.
Additionally, object storage solutions such as MinIO or Ceph are deployed for
storing digital resources such as CAD/CAE and 3D models and multimedia data
(i.e., audio/video files).

####  III-B3 Metadata Repository

An important data repository that allows the association of different sources
of information, at different levels of spatial and temporal granularity,
between objects. Specifically, the _Static Binding_ stores the binding between
the data sources (Kafka topics) and the actual assets generating them. To
uniquely identify assets and their local in the RAMI 4.0 hierarchy model, we
adopt IEC 62264 identifiers. This allows us to keep track of assets and their
deployment context, e.g., the production cell where they are located. Another
essential piece of information in this hierarchical binding is the Kafka
topic, a human-readable string associated with the asset identifier.

Last is the _App Bindings_ module, which stores information about the
resources that an application can access, thus allowing the retrieval of data
streams related to a particular resource and its model from the _Digital
Asset_ repository and to overlay this information on the application itself.
This wiring presents a flexible structure, allowing Metaverse applications to
interact with objects in space, performing zoom-in and out operations while
retaining context and overlaying real-time data of interest.

###  III-C The Metaverse Engine

This component acts as a bridge between the virtual world and the application
layer. It comprises a series of subsystems typically found in engines for
augmented or virtual reality experiences and interactive applications. These
subsystems, depicted in Fig. 3 include physics simulators, rendering engines,
streaming services, and other similar components deployed as a series of
microservices that can be combined to form more complex services or entire
applications. To ensure that each service can meet its QoS/E requirements, the
engine can be replicated and distributed across multiple infrastructure nodes.

The second part of the Metaverse Engine is a resource-aware orchestrator that
enables the deployment of the microservices and infrastructure components
based on the requirements of individual microservices [16]. The orchestrator
leverages a deployment strategy that maps service requirements to the
underlying resources and guarantees end-to-end QoS/E specifications of various
applications. For instance, in the case of shared applications, where multiple
users can interact with the virtual world simultaneously, the services can be
associated with groups of players to ensure a better overall experience.

The engine has synchronization and state management mechanisms to support
various deployment options. The synchronization mechanism manages interactions
between the metaverse engine services and the virtual world services, as well
as between platform users and these services. The state manager ensures that
the consistency of the state is maintained, both for individual applications
and for the various microservices, especially in the case of their replication
across multiple infrastructure nodes.

Figure 3: An example of a service chain that utilizes the metaverse engine,
where the rendering service acquires data inputs in the form of a time-series
data which are superimposed on a video stream (Flow #1), spatial contextual
information (Flow #2), and user inputs. The service either forwards the raw
frame directly to the client so as to visualize the scene through an AR
headset or employs a supplementary service for encoding and transmitting the
frame using widely adopted streaming protocols (e.g., RTP, WebRTC).

The engine is underpinned by a communication bus that enables the exchange of
data between the engine’s microservices and the applications running within
the platform through an event or message system. For instance, a rendering
service may be notified of modifications to a scene following a simulation
executed by a physics service. The communication bus can utilize different
communication and acceleration or memory/storage access technologies, based on
node availability and the requirements of individual services, in a manner
that is transparent to them. On the network communication side, the bus can
utilize technologies such as DPDK, and RDMA, rely on the classic TCP/IP
network stack, or IPC mechanisms such as shared memory [17]. Concerning
memory/storage access, the engine can employ acceleration technologies such as
SPDK or GPUDirect to enhance access and transfer of large data and models.
These elements and mechanisms, such as orchestration, synchronization, and the
communication bus, are crucial to support the application hub model (Sec.
III-D), where different metaverse applications can coexist and be utilized
seamlessly by users and operators of the smart factory.

###  III-D Application Layer and Use-cases

The central component of the application layer is the application hub, a
logically centralized repository of applications that can exploit the
metaverse engine. The hub can be accessed by industrial plant operators,
offering a user-friendly interface that facilitates interaction with the
metaverse engine and the ability to switch between applications seamlessly. A
notable feature of the hub and its applications is the capability to display
information selectively in the user interface, such as runtime data streams
and network performance. The application context is established at application
design time, and the state is stored and retrieved from the _App Bindings_
available at the Virtual Layer.

Furthermore, the hub grants access to individual machines and controllers,
enabling real-time monitoring of ongoing production processes. In essence, the
hub and its applications serve as an _enriched_ access point to the virtual
layer and its subcomponents outlined in Sec. III-B.

Figure 4: The image represents a smart factory where data sourced from various
machines and processes is collected to feed the Virtual Plane. The digital
twins are local instances deployed at edge/cloud nodes and can be transferred
across them on demand. The upper layer shows a representation of two virtual
applications: (i) a network (observability) and (ii) a mixed reality app. used
to inspect potential problems on a shopfloor machine.

Fig. 4 shows a simplified instance of the proposed architecture for two use
cases to understand better how applications interact with the metaverse
engine. The first proof-of-concept application is an AR solution for
visualizing the real-time network fabric servicing control and best-effort
flows on the shopfloor network. The app. provides the means to monitor the
behavior of the network and/or appliances, overlaying and updating the
rendered image via real-time data from the control plane. The pictorial
representation showcases a graph-like structure, displaying the different data
flows, such as control and sensor data. Visualizing the network in an AR
environment allows operators and maintenance personnel to quickly identify and
understand the relationships between various devices and data flows. This can
lead to improved decision-making and increased operational efficiency,
reducing downtime and improving the overall performance of the critical
industrial network. In addition to visualizing the network fabric and real
activities therein, the application provides the means for operators to manage
the network in real-time.

The second proof-of-concept application aims to enhance the efficiency and
accuracy of maintenance tasks in industrial settings through video streaming
rendered through an augmented reality (AR) appliance. The solution involves
streaming high-resolution video data from industrial cameras to an edge node
for further processing. The processed video is then transmitted to an AR
headset worn by an operator performing maintenance on industrial shopfloor
appliances. AR technology enhances the operator’s situational awareness by
providing real-time, hands-free access to machine information and data.
Operators can visualize machine components and systems in real-time, reducing
the risk of human error. As an example, in Fig. 4 is shown the visual
inspection application which is deployed at the edge, receiving and processing
the video frames acquired by the industrial camera(s) deployed on the shop
floor. The metaverse engine and optimized middleware support ensure that the
video is of sufficient quality for display via the AR headset. In this
scenario, both video and metric data sourced by the digital twin of the asset
are combined so to show a more representative view of the phenomenon.

##  IV Preliminary Evaluation

This section presents a preliminary evaluation of our proposal in a teal
testbed. We first discuss the experimental settings and successively present
the obtained results.

###  IV-A Settings

The deployment environment consists of five nodes, each serving different
functionalities within the OT and IT layers. The IT layer includes the virtual
and application layers. The OT layer comprises three interconnected nodes,
linked by a mesh topology using 1Gbit links. Each OT node is equipped with an
Ubuntu 22.04.1 LTS operating system, an Intel Core i5-2400 CPU @ 3.10GHz
processor, and 8GB of RAM. Nodes 1 and 2 are dedicated to traffic simulation,
utilizing software packages to emulate realistic industrial machine traffic,
as described in a previous work [18]. On the other hand, Node 3 hosts a
Gateway entity, acting as a bridge between the OT and IT layers. The edge
nodes, Node 4 and Node 5 share the same operating system and have an 18-core
Intel i9-10980XE CPU @ 3.00GHz, along with 64GB of memory. They are directly
connected via two 100Gbps Mellanox DX-6 NICs, ensuring minimal network
overhead. Additionally, Nodes 4 and 5 serve as the deployment location for the
IT subsystems.

The first scenario involves transmitting operational data from a simulated
industrial asset at Node 1 via the OPC UA PubSub protocol to the OPC UA
Subscriber at Node 2. This scenario simulates a typical sensor-to-controller
scenario in an industrial environment, where the internal operational status
of the industrial asset is extracted and transmitted using the OPC UA
protocol. The Gateway component at Node 3 subscribes to these messages (OPC UA
PubSub), then transmits them to the Kafka deployment at Node 4, which the
custom Kafka consumer will eventually receive at Node 5.

The second scenario focuses on simulating the streaming of a dedicated video
surveillance system designed to monitor a workshop. Within this scenario, the
rendering and streaming services of the Metaverse Engine are deployed on Node
4, while a client application for receiving the video stream is located on
Node 5. The objective of this scenario is to evaluate the engine’s ability to
provide immersive user experiences. Notably, the direct 100Gbps link
connecting the two nodes allows for an assessment of the performance metrics
influenced by the engine’s acceleration technology.

The outcome of these two scenarios will provide crucial insights into the
performance and efficiency of the proposed architecture in managing
operational data and its suitability for delivering immersive experiences. In
both scenarios, to accurately measure the time taken for the transmission of
the messages, the nodes are synchronized by using the Precision Time Protocol
(PTP). This allows us to extract fine-grained metrics, and ensure that the
results are accurate and reliable.

###  IV-B Results

The proposed system’s efficacy is assessed by measuring the message latency
between the OT and the application layer, under varying traffic loads. The
results, corresponding to the network observability application, are presented
in Fig. 5 and Fig. 6.

Fig. 5 illustrates the latency between two simulated machines, Node 1 and Node
2, in the OT layer, as the number of messages/second (real-time diagnostic
information) is increased from $400$ to $1200$. The latency is calculated as
the time difference between receiving and sending messages at the application
level. Our results show that the latency between the two machines remains
stable and in the sub-millisecond range, which is the required latency for
communication between machines or PLCs in the OT layer.

Figure 5: Machine-to-consumer communication latency under varying message load
of the IT layer. Figure 6: Machine-to-machine communication latency under
varying message load of the OT layer.

Fig. 6 displays the end-to-end latency between the OT layer and the Kafka
consumer at the IT layer (Node 5) for the same message rate. The results show
a latency that is an order of magnitude higher than the latency in the OT
layer. This increase in latency is expected due to the number of software
components the message must traverse, especially the latency introduced by the
Kafka MOM topics, which have been configured to ensure a reliable and ordered
delivery of messages conveyed from the OT. This configuration is particularly
important for safety-critical data. The effects of this feature are more
pronounced when increasing the number of messages/second, leading to an ever-
increasing number of queued messages. To address this rate imbalance, the OT
layer can be equipped with selective pre-processing capabilities, such as
filtering and aggregation, reducing the burden at the OT/IT interface [19].

It is worth noting that although the latency at the IT layer is higher than in
the OT layer, it is still less than $10\text{\,}\mathrm{ms}$, an acceptable
threshold for many interactive applications, and sufficient to meet most use
case requirements. Nevertheless, the fact that the latency grows significantly
when the message rate increases highlights the need for further optimization
and tuning, particularly in critical applications where low latency and data
integrity are essential.

In the second experiment, we evaluate the performance of the streaming
component (Metaverse engine) of the testbed by comparing two communication
mechanisms for transmitting images of varying resolutions (from HD to 8K). One
mechanism utilizes DPDK acceleration technology, while the other utilizes UDP
traffic to simulate RTP-based communication [17]. We then analyze two
performance metrics: (i) the number of frames per second (FPS) and (ii) the
average end-to-end latency per frame transmission. The obtained results
indicated an excellent performance of the streaming service in terms of both
latency (Fig. 7) and supported FPS (Fig. 8), particularly in the case when
relying on DPDK. The system can support frame rates above 100 FPS for images
up to 4K resolution and exceed 1000 FPS for lower-quality images. Moreover,
the observed end-to-end latency never exceeds $10\text{\,}\mathrm{ms}$ for
images up to 4K resolution. These findings demonstrate that the system
satisfies the requirements of next-generation interactive applications.

Figure 7: End-to-end latency for the streaming service as the resolution of
the transmitted image increases (from HD to 8K). Figure 8: FPS for the
streaming service as the resolution of the transmitted image increases (from
HD to 8K).

##  V Related Work

To the best of our knowledge, the design and implementation of a comprehensive
solution targeting Metaverse applications in the smart manufacturing domain
are entirely novel in the existing literature. This section summarizes the
prior research that influenced the authors’ proposal, inspiring some
architectural and technological choices.

The European Reference Architectural Model Industry 4.0 (RAMI 4.0) [11] is a
widely recognized standard that emphasizes the need for a close alignment
between IT and OT. RAMI 4.0 provides a high-level reference architecture
encompassing various Industry 4.0 scenarios. The reference communication layer
of RAMI 4.0 leverages the OPC UA standard as the sole solution to ensure
interoperability in the realm of OT [13].

In [20], the authors conducted a systematic literature review to assess the
impact and effectiveness of using Augmented Reality (AR) in real industrial
processes, investigating the applicability and usefulness of AR technology in
the industry. The study found that AR is a growing trend and is being employed
so to improve process flexibility, monitoring, and inspection capabilities,
streamlining operations.

In [18], the authors propose a multi-layer architecture for monitoring
industrial equipment in customer plants. The architecture uses two Apache
Kafka installations, one in OT and one in IT, to gather near-real-time data.
Dedicated software components called _hmi-forwarders_ interface with Modbus-
TCP machinery, exporting data to the OT Kafka instance, which then forwards it
to the appropriate Kafka topic in the IT layer.

The CODEG framework for cloud-oriented gaming, proposed by DeGiovanni et al.
[21], addresses the limitations of conventional monolithic game engines,
advocating for a distributed framework that can exploit the full potential of
the heterogeneous resources in the cloud continuum. The implementation of the
CODEG framework involves the integration of multiple game engine modules into
the available network infrastructure, ranging from core to edge resources.

##  VI Conclusion and Future Work

In this article, we presented a conceptual architecture and discussed the
accompanying software prototype and functional components. The proposal aims
at enabling an immersive smart manufacturing environment, embodying a human-
in-the-loop approach. A preliminary evaluation was presented, validating the
framework components as a whole by relying on two proof-of-concept immersive
application scenarios. Our ongoing efforts are directed toward expanding the
Metaverse engine and the applications described in this study to enable the
creation of shared virtual environments where multiple users can engage in
synchronous sessions.

## References

  * [1] zVEI - German Electrical and Electronic Manufacturers, “5G for Connected Industries and Automation,” White Paper, 2019, [Online] Accessed Dec. 2022.
  * [2] World Economic Forum, “An introduction to the Digital Transformation of Industries initiative,” http://reports.weforum.org/digital-transformation/an-introduction-to-the-digital-transformation-initiative/, 2022, [Online] Accessed Dec. 2022.
  * [3] Gartner, “The 2020 Top Strategic Technology Trends for Manufacturing Operations,” https://www.gartner.com/en/documents/3991206/the-2020-top-strategic-technology-trends-for-manufacturi, 2020, [Online] Accessed Dec. 2022.
  * [4] P. Pop, M. L. Raagaard, M. Gutierrez, and W. Steiner, “Enabling Fog Computing for Industrial Automation Through Time-Sensitive Networking (TSN),” _IEEE Communications Standards Magazine_ , vol. 2, no. 2, pp. 55–61, 2018\.
  * [5] EU Directorate-General for Research and Innovation, “Industry 5.0 - Towards a sustainable, human-centric and resilient European industry,” Report, January 2021, [Online] Accessed Dec. 2022.
  * [6] M. Grieves and J. Vickers, _Digital Twin: Mitigating Unpredictable, Undesirable Emergent Behavior in Complex Systems_. Cham: Springer International Publishing, 2017, pp. 85–113.
  * [7] L.-H. Lee, P. Zhou, T. Braud, and P. Hui, “What is the Metaverse? An Immersive Cyberspace and Open Challenges,” 2022.
  * [8] A. Willner and V. Gowtham, “Toward a Reference Architecture Model for Industrial Edge Computing,” _IEEE Communications Standards Magazine_ , vol. 4, no. 4, pp. 42–48, 2020.
  * [9] M. Gutiérrez, A. Ademaj, W. Steiner, R. Dobrin, and S. Punnekkat, “Self-configuration of IEEE 802.1 TSN Networks,” in _Proc. of IEEE ETFA_ , 2017.
  * [10] 5G ACIA, “Integration of 5G with Time-Sensitive Networking for Industrial Communications,” https://www.5g-acia.org/publications/integration-of-5g-with-time-sensitive-networking-for-industrial-communications/, 2020, [Online] Accessed Dec. 2022.
  * [11] RAMI4.0, “Reference Architecture Model Industrie 4.0,” https://www.beuth.de/en/technical-rule/din-spec-91345/250940128, January 2021, [Online] Accessed Dec. 2022.
  * [12] M. Wollschlaeger, T. Sauter, and J. Jasperneite, “The Future of Industrial Communication: Automation Networks in the Era of the Internet of Things and Industry 4.0,” _IEEE Industrial Electronics Magazine_ , vol. 11, no. 1, pp. 17–27, 2017.
  * [13] J. Pfrommer, A. Ebner, S. Ravikumar, and B. Karunakaran, “Open Source OPC UA PubSub Over TSN for Realtime Industrial Communication,” in _Proc. of IEEE ETFA_ , vol. 1, 2018, pp. 1087–1090.
  * [14] OPC Foundation, “OPC UA Specification Part 14–PubSub 1.04,” https://reference.opcfoundation.org/v104/Core/docs/Part14/, February 2018, [Online] Accessed Jan. 2023.
  * [15] Apache Kafka, “Apache Kafka Middleware,” https://github.com/apache/kafka, February 2018, [Online] Accessed Nov. 2022\.
  * [16] L. Mazzuca, A. Garbugli, A. Sabbioni, A. Bujari, and A. Corradi, “Towards a Resource-Aware Middleware Support for Distributed Game Engine Design,” in _Proc. of ACM GoodIT_ , New York, NY, USA, 2022, p. 409–413.
  * [17] L. Rosa and A. Garbugli, “Poster: INSANE – A Uniform Middleware API for Differentiated Quality using Heterogeneous Acceleration Techniques at the Network Edge,” in _Proc. of IEEE ICDCS_ , 2022, pp. 1282–1283.
  * [18] F. Bosi _et al_., “Cloud-enabled Smart Data Collection in Shop Floor Environments for Industry 4.0,” in _Proc. of IEEE WFCS_ , 2019, pp. 1–8.
  * [19] A. Bujari, A. Corradi, L. Foschini, L. Patera, and A. Sabbioni, “Enhancing the Performance of Industry 4.0 Scenarios via Serverless Processing at the Edge,” in _Proc. of IEEE ICC_ , 2021, pp. 1–6.
  * [20] L. F. de Souza Cardoso, F. C. M. Q. Mariano, and E. R. Zorzal, “A Survey of Industrial Augmented Reality,” _Computers & Industrial Engineering_, vol. 139, 2020.
  * [21] L. De Giovanni _et al_., “Revamping Cloud Gaming with Distributed Engines,” _IEEE Internet Computing_ , vol. 26, no. 6, pp. 88–95, 2022.


∎

11institutetext: Kostas Glampedakis
Departamento de Física, Universidad de Murcia, Murcia, E-30100, Spain
11email<EMAIL_ADDRESS>
# A Machian reformulation of Quantum Mechanics

Kostas Glampedakis

(Received: date / Accepted: date)

###### Abstract

The widely known but also somewhat esoteric Mach principle envisages a fully
relational formulation of physical theories without any reference to a concept
of ‘absolute space’. When applied to classical mechanics, under the guise of
an extended symmetry group, this procedure is known to lead to an equation of
motion with inertial-like forces that are sourced by the mass distribution of
the system itself. In this paper we follow a similar procedure and reformulate
the Schrödinger equation of non-relativistic quantum mechanics in a fully
Machian way. Just like its classical counterpart, the resulting quantum theory
is fully relational in the positions and momenta of the bodies comprising a
given physical system, leaving no room for the notion of absolute space.

###### Keywords:

Mach’s principle Quantum mechanics Relational mechanics

††journal: Foundations of Physics

##  1 Introduction

The late 19th century principle formulated by Ernst Mach MachOriginal  was
born out of the criticism to Newton’s famous buck experiment and the
associated concept of absolute space in classical mechanics (for modern
reviews of Mach’s principle see Refs. MachBook ; review ). The same principle
is known to have played an important role in the early-stage development of
Einstein’s General Relativity (GR) although, as it turned out, the final
theory failed to be fully Machian.

Among the subsequent discussions of the principle and of the closely related
concept of the ‘origin of inertia’ stand out the phenomenological model of
Sciama Sciama53 , Lynden-Bell’s attempt to build a Machian extension of GR
LyndenB67  and Barbour & Bertotti’s first implementation of Mach’s principle
as a dynamical theory BB82 . In parallel, a significant body of work has
studied the possibility of a Machian origin of the relativistic dragging of
inertial frames effect; early calculations based on a rotating shell model
appear to lend support to this connection Brill66  but later work has claimed
frame dragging to be anti-Machian Rindler94  (a modern review on this subject
can be found in Ref. Pfister07 ).

Although never truly achieving front-stage actor status, Mach’s principle and
the concepts associated with it are still subjects of investigation in
cosmology and gravitational physics (for a very recent example see astro21 )
and the question to what extent our Universe is ‘Machian’ is still an open
one.

Mach’s principle is far easier conceptualised and implemented within non-
relativistic physics. The reformulation of classical mechanics without an
absolute space reference frame by Lynden-Bell & Katz LyndenB95  provides a
prime example of this assertion. This theory (which is equivalent to that of
Ref. BB82 ) is designed to be invariant under a larger group of symmetries
than Newtonian mechanics and as a consequence it is fully _relational_ , that
is, its Lagrangian depends only on the relative positions and velocities of
the bodies that make up the ‘universe’ in this model. More recent work on the
same model Ferraro16  has established that the familiar inertial forces of
Newtonian mechanics are determined by the global properties of the system
itself rather than the motion with respect to an inertial frame of ‘fixed
stars’; this is the very essence of Mach’s principle.

Although Mach’s principle has traditionally been considered in relation with
inertial forces, relativistic frame dragging and cosmology, there is no deep
reason for it to be confined within the realm of classical physics. After all,
standard non-relativistic quantum mechanics (QM) is formulated in the same
inertial frame ‘absolute space’ as Newtonian mechanics and shares the same
group of space-time symmetries. To the best of our knowledge no such Machian
formulation of QM exists in the literature and it is the purpose of this paper
to develop one. Our approach is based on the extension of the classical model
of Lynden-Bell, Katz & Ferraro LyndenB95 ; Ferraro16  and the final product is
the derivation of a fully relational Schrödinger equation. This is the only –
but crucial – Machian modification of standard QM.

The rest of the paper is organised as follows. Section 2 sets the stage by
summarising the relational-Machian model of classical mechanics developed in
Refs. LyndenB95 ; Ferraro16 . Section 2.1 provides the link between the
extended symmetry group of the new framework and its Machian character. In
Section 2.2 the relevant Lagrangian is constructed and a new ‘Mach-Newton’
equation of motion is derived. Some implications of this equation are
discussed in Section 2.3. The QM portion of the paper can be found in Section
3.1 (construction of the classical relational Hamiltonian), Section 3.2
(definition of quantum operators) and Section 3.3 (derivation of a ‘Mach-
Schrödinger’ equation). Our concluding remarks can be found in Section 4.
Throughout the paper the indices $\\{i,j\\}$ (with or without primes) are
exclusively used as ‘body labels’. All other latin indices represent
vector/tensor components. Calligraphic symbols refer to parameters in the
system’s center of mass frame. An overdot (hat) denotes a time derivative
(quantum operator).

##  2 Setting the stage: classical mechanics without absolute space

###  2.1 From absolute space to relational space

This section summarises previous work by Lynden-Bell & Katz LyndenB95  (as
well as its later extension by Ferraro Ferraro16 ) who were among the first to
develop a rigorous and fully Machian reformulation of classical Newtonian
mechanics without any tethers to the concept of absolute space. As pointed out
in the introduction, this model serves as the necessary stepping stone towards
a Machian treatment of QM and for that reason we believe it should be
discussed here in some detail.

The ‘universe’ in this model comprises a collection of $N$ point-like bodies
of mass $m_{i}$ and position $\mathbf{x}_{i}(t)$ (where $i=1,2,...N$). Each
pair $ij$ of bodies is assumed to interact via a potential,

$V_{ij}=V_{ij}(\mathbf{x}_{ij}),\qquad$ (1)

which is only a function of the pair’s _relative_ position
$\mathbf{x}_{ij}\equiv\mathbf{x}_{i}-\mathbf{x}_{j}$. For the specific example
of gravitationally interacting bodies we would have

$V=\sum_{i<j}\sum_{j=1}^{N}V_{ij}\equiv-\sum_{i<j}V_{ij},$ (2)

with

$V_{ij}(\mathbf{x}_{ij})=-\frac{Gm_{i}m_{j}}{x_{ij}}.$ (3)

The system’s Lagrangian is defined in the usual way,

$L=T-V=\frac{1}{2}\sum_{i}m_{i}\mathbf{v}_{i}\cdot\mathbf{v}_{i}-V(\mathbf{x}_{ij}).$
(4)

where $\mathbf{v}_{i}=\dot{\mathbf{x}}_{i}$. This functional encapsulates the
familiar symmetries attached to Newtonian mechanics: (i) invariance with
respect to time-independent rigid spatial translations and rotations,

$\mathbf{x}_{i}\to\mathbf{x}_{i}+\bm{\xi},\qquad\mathbf{x}_{i}\to\mathbf{x}_{i}+(\mathbf{A}\times\mathbf{x}_{i}),$
(5)

where $\mathbf{A}$ is a constant vector along the axis of rotation and (ii)
invariance with respect to the Galilean transformation (GT),

$t\to t+\epsilon,\qquad\mathbf{x}_{i}\to\mathbf{x}_{i}+\mathbf{V}t,$ (6)

where $\epsilon$ and $\mathbf{V}$ are constants. Indeed, the privileged family
of inertial frames defined by the GT represents the modern incarnation of
Newton’s ‘absolute space’ where the theory is supposed to be valid.

Newtonian mechanics can be converted to a fully relational theory (i.e. free
of the concept of absolute space) provided it is defined via a Lagrangian that
is invariant with respect to a wider set of gauge symmetries, namely,
infinitesimal time-dependent rigid translations (also known as ‘extended GT’
in the literature Green01 ) and rigid rotations,

$\displaystyle\mathbf{x}_{i}\to\mathbf{x}_{i}+\bm{\xi}(t)~{}\Rightarrow~{}\mathbf{v}_{i}\to\mathbf{v}_{i}+\dot{\bm{\xi}},$
(7)
$\displaystyle\mathbf{x}_{i}\to\mathbf{x}_{i}+\left(\mathbf{a}(t)\times\mathbf{x}_{i}\right)~{}\Rightarrow~{}\mathbf{v}_{i}\to\mathbf{v}_{i}+(\bm{\omega}\times\mathbf{x}_{i})+(\mathbf{a}\times\mathbf{v}_{i}),$
(8)

where $\bm{\omega}(t)\equiv\dot{\mathbf{a}}$. These transformations only
affect the kinetic energy $T$ in (4); the transformation (7) produces a
variation

$\delta T=\dot{\bm{\xi}}\cdot\mathbf{P},$ (9)

where $\mathbf{P}$ is the system’s total momentum. This variation can be
eliminated if we instead work with the kinetic energy in the center of mass
(CM) frame,

${\cal T}\equiv T-\frac{1}{2}Mu_{c}^{2},$ (10)

where $M=\sum_{i}m_{i}$ is the total mass and

$\mathbf{u}_{\rm
c}=\frac{1}{M}\sum_{i}m_{i}\mathbf{v}_{i}=\frac{1}{M}\mathbf{P},$ (11)

is the CM velocity. The new kinetic energy is invariant, $\delta{\cal T}=0$,
with respect to (7) and can be rewritten in the equivalent forms LyndenB95 ;
Ferraro16

${\cal T}=\frac{1}{2}\sum_{i}m_{i}|\mathbf{v}_{i}-\mathbf{u}_{\rm
c}|^{2}=\frac{1}{2}\sum_{i<j}m_{ij}v_{ij}^{2}.$ (12)

Notice that ${\cal T}$ resembles the potential energy term, being a double sum
and depending on the relative velocities
$\mathbf{v}_{ij}\equiv\mathbf{v}_{i}-\mathbf{v}_{j}$ instead of the original
absolute velocities. At the same time the single-body masses have been
replaced by the pair masses $m_{ij}\equiv m_{i}m_{j}/M$ which are a
generalisation of the familiar two-body system’s reduced mass.

The kinetic energy ${\cal T}$ is still not invariant with respect to the
second transformation (8); the associated variation is found to be LyndenB95 ;
Ferraro16 ,

$\delta{\cal T}=\bm{\omega}\cdot\bm{{\cal J}},$ (13)

where $\bm{{\cal J}}$ is the system’s total angular momentum with respect to
the CM frame. This can be written in the three equivalent forms,

$\bm{{\cal
J}}\equiv\sum_{i}m_{i}(\mathbf{x}_{i}\times\mathbf{v}_{i})-(\mathbf{x}_{\rm
c}\times\mathbf{P})=\sum_{i}m_{i}(\mathbf{x}_{ic}\times\mathbf{v}_{ic})=\sum_{i<j}m_{ij}(\mathbf{x}_{ij}\times\mathbf{v}_{ij}),$
(14)

where we have defined the relative to the CM position and velocity,

$\mathbf{x}_{ic}\equiv\mathbf{x}_{i}-\mathbf{x}_{\rm
c},\qquad\mathbf{v}_{ic}\equiv\mathbf{v}_{i}-\mathbf{u}_{\rm c}.$ (15)

In particular, according to the last equation in (14) we have $\bm{{\cal
J}}=\bm{{\cal J}}(\dot{\mathbf{x}}_{ij},\mathbf{x}_{ij})$.

As in the previous case with the modification (10), $\delta{\cal T}$ can be
eliminated after removing a CM-related rotational term from the kinetic
energy. The new, fully gauge-invariant, kinetic energy is found to be

${\cal T}^{*}={\cal T}-\frac{1}{2}\bm{{\cal
J}}\cdot\bm{\Omega}=\frac{1}{2}\sum_{i<j}m_{ij}v_{ij}^{2}-\frac{1}{2}{\cal
I}_{nk}^{-1}{\cal J}_{n}{\cal J}_{k}.$ (16)

The symmetric tensor ${\cal I}_{kn}$ is the system’s intrinsic moment of
inertia with respect to the CM. This is,

${\cal
I}_{nk}=\sum_{i}m_{i}\left(x_{ic}^{2}\delta_{nk}-x^{n}_{ic}x^{k}_{ic}\right)=\sum_{i<j}m_{ij}\left(\,x_{ij}^{2}\delta_{kn}-x_{ij}^{k}x_{ij}^{n}\,\right).$
(17)

As evident from the last equation this moment of inertia is a fully relational
quantity, ${\cal I}_{nk}={\cal I}_{nk}(\mathbf{x}_{ij})$. As a result of its
definition ${\cal I}_{nk}$ is related in the usual way to the absolute space
moment of inertia $I_{nk}$,

$I_{nk}=\sum_{i}m_{i}\left(x_{i}^{2}\delta_{nk}-x^{n}_{i}x^{k}_{i}\right)={\cal
I}_{nk}+M\left(x_{\rm c}^{2}\delta_{nk}-x_{\rm c}^{n}x_{\rm c}^{k}\right).$
(18)

The vector $\bm{\Omega}=\bm{\Omega}(\mathbf{x}_{i},\mathbf{v}_{i})$ in (16) is
given by,

${\cal J}_{n}={\cal I}_{nk}\Omega_{k}~{}\Rightarrow~{}\Omega_{k}={\cal
I}_{nk}^{-1}{\cal J}_{n},$ (19)

from which it follows that $\bm{\Omega}$ is the system’s angular velocity with
respect to the CM.

###  2.2 The new relational Lagrangian & the ‘Mach-Newton’ equation of motion

We can now assemble the previous results and define a fully relational
Lagrangian that describes the present Machian reformulation of Newtonian
mechanics. This Lagrangian, first derived in Ref. LyndenB95 , is:

${\cal L}(\mathbf{x}_{ij},\dot{\mathbf{x}}_{ij})\equiv{\cal
T}^{*}-V=\frac{1}{2}\sum_{i<j}m_{ij}\dot{\mathbf{x}}_{ij}^{2}-\frac{1}{2}{\cal
I}_{nk}^{-1}{\cal J}_{n}{\cal J}_{k}-V(\mathbf{x}_{ij}).$ (20)

The same Lagrangian can be written in several equivalent ways. For example,
two forms that show explicitly the CM’s central role in this Machian framework
are,

${\cal L}=L-\left(\frac{1}{2}Mu_{\rm c}^{2}+\frac{1}{2}\bm{{\cal
J}}\cdot\bm{\Omega}\right)=\frac{1}{2}\sum_{i}m_{i}\left|\mathbf{v}_{ic}-\bm{\Omega}\times\mathbf{x}_{ic}\right|^{2}-V(\mathbf{x}_{ij}).$
(21)

The $i$-body’s equation of motion can be derived from the above Lagrangian in
the standard way; working with the conjugate parameters
$\\{\mathbf{x}_{ic},\dot{\mathbf{x}}_{ic}\\}$ we first find the canonical
momentum

$\bm{{\cal P}}_{i}=\frac{\partial{\cal
L}}{\partial\dot{\mathbf{x}}_{ic}}=m_{i}\left(\mathbf{v}_{ic}-\bm{\Omega}\times\mathbf{x}_{ic}\right).$
(22)

The corresponding Euler-Lagrange equation is,

$\frac{d\bm{{\cal P}}_{i}}{dt}=\frac{\partial{\cal
L}}{\partial\mathbf{x}_{ic}},$ (23)

and leads to the equation of motion

$m_{i}\ddot{\mathbf{x}}_{ic}=-\bm{\nabla}_{ic}V+m_{i}\left(\dot{\bm{\Omega}}\times\mathbf{x}_{ic}+\bm{\Omega}\times\dot{\mathbf{x}}_{ic}\right)-\frac{1}{2}\bm{\nabla}_{ic}\left(\bm{{\cal
J}}\cdot\bm{\Omega}\right),$ (24)

where the gradient is taken with respect to $\mathbf{x}_{ic}$. After
manipulating the last term this becomes,

$m_{i}\ddot{\mathbf{x}}_{ic}=-\bm{\nabla}_{ic}V+m_{i}\left[\dot{\bm{\Omega}}\times\mathbf{x}_{ic}+2\bm{\Omega}\times\dot{\mathbf{x}}_{ic}-\bm{\Omega}\times\left(\bm{\Omega}\times\mathbf{x}_{ic}\right)\,\right].$
(25)

We would have obtained an identical result had we used the variables
$\\{\mathbf{x}_{i},\dot{\mathbf{x}}_{i}\\}$ (e.g. note that
$\bm{\nabla}_{ic}V(\mathbf{x}_{ij})=\bm{\nabla}_{ic}V(\mathbf{x}_{ic}-\mathbf{x}_{jc})$
is equal to
$\bm{\nabla}_{i}V(\mathbf{x}_{ij})=\bm{\nabla}_{i}V(\mathbf{x}_{i}-\mathbf{x}_{j})$).

The ‘Mach-Newton’ equation (25) is the central result of this section and was
previously obtained in Ferraro16 . It is structurally identical to the
textbook Newton’s law in a non-inertial frame attached to the system’s CM and
rotating with angular frequency $-\bm{\Omega}$ with the usual inertial forces
present. However, appearances are deceiving because these forces are, in fact,
generated by the spatial distribution and motion of the rest of the ‘universe’
(through the function $\bm{\Omega}$). This property of the new theory is
highly non-trivial because these Machian terms are present even when there is
_no_ direct interaction, $V=0$, between the bodies; one could claim that these
terms represent the ‘backreaction’ of space to the presence of matter.

All distances and velocities in Eq. (25) are measured relative to the CM. The
motion of the CM itself is left unconstrained since the new theory does not
supply an equation of motion for it. This is in contrast to standard Newtonian
mechanics where the CM’s motion is prescribed relative to ‘absolute space’.
The true Newtonian limit of (25) can be easily identified from (21) and
corresponds to $\dot{\mathbf{u}}_{\rm c}=\bm{\Omega}=0$, that is, uniformly
moving CM and vanishing total angular momentum in the CM frame. The privileged
frames where these conditions are met can be rightly called ‘Newtonian frames’
and their definition implies that these frames are related to each other
through time-independent rigid translations and rotations (plus the usual GT).

###  2.3 A gravitational bucket experiment

No discussion of a Machian theory would be complete without a remark on the
famous rotating bucket experiment. In Ref. Ferraro16  the ‘bucket’ is modelled
as a binary system of two point masses $m$ moving in a circular orbit of
radius $R$ about their CM. As a proxy for the ‘fixed stars’ this model assumes
a distant spherical shell centered at the binary’s CM.

Following Ref. Ferraro16 , we work in the binary’s corotating frame and find
that Eq. (25) reduces to the following balance between the gravitational and
‘centrifugal’ forces (the net gravitational force exerted by the shell on the
binary is of course equal to zero),

$\frac{Gm^{2}}{4R^{2}}=m\Omega^{2}R.$ (26)

The shell’s angular momentum is $\bm{{\cal J}}=I_{0}\bm{\Omega}_{\rm b}$ where
$I_{0}$ is the shell’s isotropic moment of inertia and $\bm{\Omega}_{\rm
b}=\Omega_{b}\mathbf{\hat{z}}$ is the binary’s orbital frequency. At the same
time ${\cal J}_{n}={\cal I}_{nk}\Omega_{k}$ and we find,

$\bm{\Omega}=\frac{I_{0}}{{\cal I}_{zz}}\bm{\Omega}_{\rm b}.$ (27)

The principal axis component ${\cal I}_{zz}$ can be obtained from (18),
leading to ${\cal I}_{zz}=I_{0}+2mR^{2}$. Inserting this in (26)

$\frac{Gm}{4R^{2}}=\Omega_{\rm
b}^{2}R\left(1+\frac{2mR^{2}}{I_{0}}\right)^{-2}$ (28)

This key expression illustrates how the standard centrifugal acceleration is
related to the large-scale distribution of matter. Once the shell is removed,
$I_{0}\to 0$, the inertial force vanishes altogether in agreement with Mach’s
principle. The more realistic situation is the limit $2mR^{2}/I_{0}\ll 1$
which leads to the familiar centrifugal force of Newtonian mechanics.

An alternative interpretation of (28) presents itself if we write it as the
standard Kepler’s third law with an effective non-local gravitational constant

$G_{\rm eff}\equiv G\left(1+\frac{2mR^{2}}{I_{0}}\right)^{2}.$ (29)

As any practitioner of GR would instantly spot, this relation signals a
violation of the weak equivalence principle. The loss of the universality of
free fall in a gravitational field is a completely general property of the new
Machian theory and a clear demonstration of its non-GR character. This can be
easily seen by writing down the equation of motion (25) for the case of
gravitationally interacting bodies,

$\ddot{\mathbf{x}}_{ic}=-\sum_{j\neq
i}\frac{Gm_{j}}{x_{ij}^{3}}\mathbf{x}_{ij}+\dot{\bm{\Omega}}\times\mathbf{x}_{ic}+2\bm{\Omega}\times\dot{\mathbf{x}}_{ic}-\bm{\Omega}\times\left(\bm{\Omega}\times\mathbf{x}_{ic}\right).$
(30)

Unlike its Newtonian counterpart, this equation does depend on the body’s mass
$m_{i}$ as it enters the global variable $\bm{\Omega}$.

##  3 Quantum mechanics ‘without absolute space’

###  3.1 The classical Hamiltonian

The next order of business following the successful Machian reconstruction of
classical mechanics is the formulation of a non-relativistic QM theory without
the concept of absolute space.

The first step towards that goal is the construction of an appropriate
classical Hamiltonian. Starting from the definition,

${\cal H}\equiv\sum_{i}\bm{{\cal P}}_{i}\cdot\mathbf{v}_{ic}-{\cal
L}(\mathbf{v}_{ic},\mathbf{x}_{ic}),$ (31)

we can subsequently manipulate the first term,

$\sum_{i}\bm{{\cal P}}_{i}\cdot\mathbf{v}_{i}=\sum_{i}\frac{{\cal
P}_{i}^{2}}{m_{i}}+\mathbf{u}_{\rm c}\cdot\sum_{i}\bm{{\cal
P}}_{i}+\bm{\Omega}\cdot\sum_{i}(\mathbf{x}_{ic}\times\bm{{\cal P}}_{i}).$
(32)

It is straightforward to verify that the canonical momenta obey the
conditions,

$\sum_{i}\bm{{\cal
P}}_{i}=0,\qquad\sum_{i}\left(\mathbf{x}_{ic}\times\bm{{\cal
P}}_{i}\right)=0.$ (33)

These carry a clear physical meaning: the system’s total canonical momentum
and angular momentum (with respect to the CM) are equal to zero. As a result
of these conditions the Hamiltonian takes the simple ‘single particle’ form

${\cal H}=\sum_{i}\frac{{\cal
P}^{2}_{i}}{2m_{i}}+V(\mathbf{x}_{ij})=\sum_{i}\frac{|\mathbf{p}_{ic}-m_{i}\bm{\Omega}\times\mathbf{x}_{ic}|^{2}}{2m_{i}}+V(\mathbf{x}_{ij}),$
(34)

where $\mathbf{p}_{ic}=m_{i}\mathbf{v}_{ic}$ is the $i$-body’s kinematical
momentum relative to the CM. As expected, ${\cal E}={\cal H}$ represents the
system’s conserved energy:

$\displaystyle\frac{d{\cal H}}{dt}$
$\displaystyle=\sum_{i}\left(\dot{\bm{{\cal
P}}}_{i}\cdot\mathbf{v}_{ic}+\bm{{\cal
P}}_{i}\cdot\dot{\mathbf{v}}_{ic}-\frac{\partial{\cal
L}}{\partial\mathbf{x}_{ic}}\cdot\mathbf{v}_{ic}-\frac{\partial{\cal
L}}{\partial\mathbf{v}_{ic}}\cdot\dot{\mathbf{v}}_{ic}\right)$
$\displaystyle=\sum_{i}\left(\dot{\bm{{\cal P}}}_{i}-\frac{\partial{\cal
L}}{\partial\mathbf{x}_{ic}}\right)\cdot\mathbf{v}_{ic}=0.$ (35)

It is worth pointing out that, modulo the previous subtraction of the
rotational term $\bm{{\cal J}}\cdot\bm{\Omega}/2$ in (16), the Hamiltonian
(34) looks structurally identical to the classical mechanics Hamiltonian in a
rotating frame (e.g. see Ref. Gulshani78 ). In reality the two cases are
physically distinct due to the presence of the Machian function
$\bm{\Omega}(\mathbf{x}_{ij},\mathbf{v}_{ij})$ in (34).

###  3.2 The position and momentum operators

Having at our disposal the classical relational Hamiltonian (34) we are in a
position to build a relational theory of QM. For simplicity, we will work in
the position representation; following the established textbook approach
Messiah  we first introduce the usual operators for the fundamental triad of
position, energy and (kinematical) momentum:

$\hat{\mathbf{x}}_{ic}=\mathbf{x}_{ic},\qquad\hat{{\cal
E}}=i\hbar\frac{d}{dt},\qquad\mathbf{\hat{p}}_{ic}=-i\hbar\bm{\nabla}_{ic}.$
(36)

These operators obey the fundamental commutation relation,

$\left[x_{ic}^{k},\hat{p}_{jc}^{n}\right]=i\hbar\,\delta_{nk}\delta_{ij}.$
(37)

A new operator that needs to be defined is that of the canonical momentum (22)
which is the Hamiltonian’s main variable. For convenience we rewrite it in
index notation,

${\cal P}_{i}^{k}=p_{ic}^{k}+m_{i}\epsilon_{k\ell m}x_{ic}^{\ell}{\cal
I}_{mq}^{-1}\epsilon_{qsn}\sum_{j}x_{jc}^{s}p_{jc}^{n}.$ (38)

The direct substitution recipe $p_{ic}^{k}\to\hat{p}_{ic}^{k}$ produces a non-
hermitian operator111For the conjugate operator we find $(\hat{{\cal
P}}_{i}^{k})^{\dagger}=\hat{p}_{ic}^{k}+m_{i}\epsilon_{k\ell
m}\epsilon_{qsn}\sum_{j}\hat{p}_{jc}^{n}\left[x_{ic}^{\ell}{\cal
I}_{mq}^{-1}x_{jc}^{s}\right].$ The last term in this expression contains
products that do not commute, e.g. $\hat{p}_{ic}^{n}x_{ic}^{n}$ and
$\hat{p}_{ic}^{n}{\cal I}_{mq}^{-1}$. Therefore $(\hat{{\cal
P}}_{i}^{k})^{\dagger}\neq\hat{{\cal P}}_{i}^{k}$.. The proper method Messiah
for defining the hermitian operator of a variable with a non-trivial
dependence on $\mathbf{p}$ is to write the classical expression in a
$\mathbf{p}$-symmetric form. In the case at hand this is,

$\bm{{\cal
P}}_{i}=\mathbf{p}_{ic}+\frac{1}{2}m_{i}\left(\bm{\Omega}\times\mathbf{x}_{ic}-\mathbf{x}_{ic}\times\bm{\Omega}\right),$
(39)

and we define the corresponding operator as

$\hat{{\cal P}}_{i}^{k}=\hat{p}_{ic}^{k}+\frac{1}{2}m_{i}\epsilon_{k\ell
m}\epsilon_{qsn}\sum_{j}x_{jc}^{s}\left(\,x_{ic}^{\ell}{\cal
I}_{mq}^{-1}\hat{p}_{jc}^{n}+\hat{p}_{jc}^{n}[x_{ic}^{\ell}{\cal
I}_{mq}^{-1}]\,\right),$ (40)

where in the last term $x_{jc}^{s}$ has been factored out as it always
commutes with $\hat{p}_{jc}^{n}$ (unless $s=n$ in which case the term is
zero). This operator is indeed hermitian:

$(\hat{{\cal
P}}_{i}^{k})^{\dagger}=\hat{p}_{ic}^{k}+\frac{1}{2}m_{i}\epsilon_{k\ell
m}\epsilon_{qsn}\sum_{j}x_{jc}^{s}\left(\,\hat{p}_{jc}^{n}[x_{ic}^{\ell}{\cal
I}_{mq}^{-1}]+x_{ic}^{\ell}{\cal I}_{mq}^{-1}\hat{p}_{jc}^{n}\,\right)={\cal
P}^{k}_{i}.$ (41)

###  3.3 The ‘Mach-Schrödinger’ equation

The action of the canonical momentum operator on an arbitrary wavefunction
$\Psi$ results in,

$\hat{{\cal
P}}_{i}^{k}\Psi=-i\hbar\left[\nabla_{ic}^{k}\Psi+\frac{1}{2}m_{i}\epsilon_{k\ell
m}\epsilon_{qsn}\sum_{j}x_{jc}^{s}\left(\,x_{ic}^{\ell}{\cal
I}_{mq}^{-1}\nabla_{jc}^{n}\Psi+\nabla_{jc}^{n}[x_{ic}^{\ell}{\cal
I}_{mq}^{-1}\Psi]\,\right)\right].$ (42)

The calculation of $\hat{{\cal P}}_{i}^{2}\Psi$ involves several lines of
algebra and the final result is rather unwieldy. However, it can be simplified
if we notice that

$\nabla^{n}_{ic}({\cal I}^{-1}_{mq})\sim\frac{m_{i}x_{ic}}{I^{2}}\ll 1,$ (43)

where $I$ is the typical size of the system’s total moment of inertia. Working
to first order in the small parameter $I^{-1}$ we find the much simpler
expression,

$\displaystyle\hat{{\cal P}}_{i}^{2}\Psi$
$\displaystyle=-\hbar^{2}\left[\nabla^{2}_{ic}+2m_{i}\epsilon_{klm}x_{ic}^{l}{\cal
I}_{mq}^{-1}\Big{(}\epsilon_{qkn}\nabla^{n}_{ic}+\epsilon_{qsn}\sum_{j}x_{jc}^{s}\nabla_{ic}^{k}\nabla_{jc}^{n}\Big{)}\right]\Psi+{\cal
O}\left(\frac{m_{i}^{2}}{I^{2}}\right).$ (44)

The theory’s ‘Mach-Schrödinger’ equation is

$\hat{{\cal H}}\Psi=\left[\sum_{i}\frac{\hat{{\cal
P}}_{i}^{2}}{2m_{i}}+V(\mathbf{x}_{ij})\right]\Psi=i\hbar\frac{d\Psi}{dt},$
(45)

where $\Psi=\Psi(\mathbf{x}_{ij},t)=\Psi(\mathbf{x}_{ic}-\mathbf{x}_{jc},t)$
is the system’s _global_ wavefunction. After inserting (44) this becomes,

$i\hbar\frac{d\Psi}{dt}=-\hbar^{2}\sum_{i}\left[\frac{\nabla^{2}_{ic}}{2m_{i}}+\epsilon_{klm}x_{ic}^{l}{\cal
I}_{mq}^{-1}\Big{(}\epsilon_{qkn}\nabla^{n}_{ic}+\epsilon_{qsn}\sum_{j}x_{jc}^{s}\nabla_{ic}^{k}\nabla_{jc}^{n}\Big{)}\right]\Psi+V\Psi.$
(46)

This fully relational equation, which describes a system of interacting
quantum particles in their CM frame, is the main new result of this paper. It
is markedly more complicated than the standard multi-particle Schrödinger
equation in inertial space Messiah  and, unlike that equation, it may not be
separable. Our equation bears some resemblance to the non-inertial frame
Schrödinger equation Gulshani78  but fails to be equivalent to it due to the
non-uniform function $\bm{\Omega}(\mathbf{x}_{jc},\hat{\mathbf{p}}_{jc})$. As
it was the case with the classical equation of motion, the new Machian terms
are present even in the absence of an interaction potential, simply existing
as a result of the presence of the bodies themselves. These terms however,
considering that $I\gg m_{i}x_{ic}^{2}$ , are typically much smaller than the
kinetic energy term.

The rest of the theory can be axiomatised as in standard QM. For instance, the
expectation value $\langle\hat{Q}\rangle$ of any observable $Q$ is

$\langle\hat{Q}\rangle=\int dV\Psi^{*}\hat{Q}\Psi,\qquad
dV=\prod_{i=1}^{N}d^{3}\mathbf{x}_{ic}.$ (47)

where the integral is to be evaluated in the full $N$-dimensional space of the
system.

The hermitianity of $\hat{{\cal H}}$ together with $i\hbar d\Psi/dt=\hat{{\cal
H}}\Psi$ leads to the textbook time evolution law

$i\hbar\frac{d\langle\hat{Q}\rangle}{dt}=\langle[\hat{Q},\hat{{\cal
H}}]\rangle.$ (48)

With the help of this formula and the commutation relations (accurate to
${\cal O}\left(I^{-1}\right)$),

$\displaystyle[\,x^{k}_{jc},\hat{{\cal P}}^{n}_{i}\,]$
$\displaystyle=i\hbar\left[\delta_{kn}\delta_{ij}+m_{i}\epsilon_{nlm}\epsilon_{qsk}{\cal
I}^{-1}_{mq}x_{jc}^{s}x_{ic}^{l}\right],$ (49)
$\displaystyle[\,\hat{p}^{k}_{jc},\hat{{\cal P}}^{n}_{i}\,]$
$\displaystyle=-\frac{i\hbar}{2}m_{i}{\cal
I}_{mq}^{-1}\Big{[}\epsilon_{nlm}\epsilon_{qkb}\left(x^{l}_{ic}\hat{p}^{b}_{jc}+\hat{p}^{b}_{jc}x_{ic}^{l}\right)+2\delta_{ij}\epsilon_{nkm}\epsilon_{qsb}\sum_{j^{\prime}}x_{j^{\prime}c}\hat{p}^{b}_{j^{\prime}c}\Big{]}$
$\displaystyle=-\hbar^{2}m_{i}{\cal
I}_{mq}^{-1}\Big{[}\epsilon_{nlm}\epsilon_{qkb}x_{ic}^{l}\nabla_{jc}^{b}+\delta_{ij}\epsilon_{nkm}\epsilon_{qsb}\sum_{j^{\prime}}x_{j^{\prime}c}^{s}\nabla_{j^{\prime}c}^{b}$
$\displaystyle\quad+\frac{1}{2}\delta_{ij}\left(\delta_{mq}\delta_{nk}-\delta_{mk}\delta_{nq}\right)\Big{]},$
(50)

we can establish the validity of the Ehrenfest theorem in the new formulation.
For example, setting $\hat{Q}=x_{j}^{k}$

$i\hbar\frac{d\langle x_{jc}^{k}\rangle}{dt}=\langle[x_{jc}^{k},\hat{{\cal
H}}]\rangle=\sum_{i}\frac{1}{2m_{i}}\langle[\,\hat{x}_{jc}^{k},\hat{{\cal
P}}^{2}_{ic}\,]\rangle+\langle[\,\hat{x}_{jc}^{k},V\,]\rangle.$ (51)

For the commutators we have,

$[\,\hat{x}_{jc}^{k},V(x_{ij})\,]=0,\qquad[\,\hat{x}_{jc}^{k},\hat{{\cal
P}}^{2}_{i}\,]=[\,\hat{x}_{jc}^{k},\hat{{\cal P}}^{n}_{i}\,]\hat{{\cal
P}}^{n}_{i}+\hat{{\cal P}}^{n}_{i}[\,\hat{x}_{jc}^{k},\hat{{\cal
P}}^{n}_{i}\,],$ (52)

and we find

$\displaystyle\frac{d\langle x_{jc}^{k}\rangle}{dt}$
$\displaystyle=\frac{\langle\hat{{\cal
P}}^{k}_{jc}\rangle}{m_{j}}+\frac{1}{2}\Big{\langle}\epsilon_{ksq}x_{jc}^{s}{\cal
I}_{mq}^{-1}\epsilon_{mln}\sum_{i}x_{ic}^{l}\hat{p}_{ic}^{n}-\epsilon_{kqs}x_{jc}^{s}\epsilon_{mln}{\cal
I}_{mq}^{-1}\sum_{i}x_{ic}^{l}\hat{p}_{ic}^{n}\Big{\rangle}.$ (53)

This result indeed coincides with the averaged value of the classical
expression (39). A similar but significantly longer calculation establishes
the same result for the classical equation of motion.

##  4 Concluding remarks

The new physics of the relational Schrödinger equation (46) lies in its non-
local character. This is a striking property not present in the non-inertial
frame formulation of the standard Schrödinger equation Gulshani78 . Even in
the case of a vanishing interaction potential, each body is simultaneously
affected by the rest of the system. Although the non-local terms are expected
to be tiny compared to the usual kinetic terms they are, nevertheless, always
present, thus forbidding (at least in a generic sense) the separability of the
multi-body equation into decoupled single-body equations.

It should be noted that the relational QM formulated in this paper is distinct
from the more well known ‘relational QM’ of Ref. Rovelli96 . In this latter
theory standard QM is reinvented as a relational framework with respect to the
observer-observed system pair and the concept of an observer’s reality. In
contrast, our framework is more ‘classical’, replacing the standard absolute
space Schrödinger equation with a fully Machian equation. As pointed out
earlier, our QM model is relational with respect to the bodies’ positions and
momenta.

Several different directions present themselves for future work. The non-
locality of the model proposed in this paper may have interesting
ramifications for the non-local physics of entanglement and the EPR argument.
Following the example of the classical mechanics model, the physics behind the
Mach-Schrödinger equation could be unveiled with the help of a suitable
‘quantum bucket experiment’ which should be designed to facilitate an exact
solution of the aforementioned equation. We hope to be able to study some of
these problems in the near future.

###### Acknowledgements.

The author is grateful to his transatlantic colleagues Daniel Kennefick, Aaron
Johnson and Daniel Oliver at the Univ. of Arkansas for discussions that
rekindled his interest in Machian physics and eventually led to this work. He
also acknowledges support from a Fundación Seneca (Region de Murcia) grant No.
20949/PI/18.

## Conflict of interest

The author declares that he has no conflict of interest.

## References

  * (1) Mach, E.: The Science of Mechanics. Open Court, La Salle (1960)
  * (2) Barbour, J., Pfister, H., (editors). Mach’s Principle: From Newton’s Bucket to Quantum Gravity. Birkhäuser, Boston (1995)
  * (3) Lichtenegger, H., Mashhoon, B. Mach’s Principle, Chapter 2 in The Measurement of Gravitomagnetism: A Challenging Enterprise (edited by L. Iorio). Nova Science, New York, (2007). [preprint arXiv: 0407078]
  * (4) Sciama, D. W.: On the origin of inertia. Mon. Not. R. Astron. Soc. 113, 34-42 (1953)
  * (5) Lynden-Bell, D.: On the origins of space-time and inertia. Mon. Not. R. Astron. Soc. 135, 413-428 (1967)
  * (6) Barbour, J. B., Bertotti, B.: Mach’s Principle and the Structure of Dynamical Theories. Proc. R. Soc. Lond. A 382, 295-306 (1982)
  * (7) Brill, D. R., Cohen, J. M.: Rotating Masses and their Effect on Inertial Frames. Phys. Rev. 143, 1011-1015 (1966)
  * (8) Rindler, W.: The Lense-Thirring effect exposed as anti-Machian. Phys. Lett. A 187, 236-238 (1994)
  * (9) Pfister, H.: On the history of the so-called Lense-Thirring effect. Gen. Relativy Gravit. 39, 1735-1748 (2007)
  * (10) Szapudi, I.: Constraining Mach’s principle with high precision astrometry. Preprint arXiv: 2105.05337
  * (11) Lynden-Bell, D., Katz, J.: Classical mechanics without absolute space. Phys. Rev. D 52, 7322-7324 (1995)
  * (12) Ferraro, R.: Relational mechanics as a gauge theory. Gen. Relativy Gravit. 48, 23 (2016)
  * (13) Greenberger, D. M.: Inadequacy of the Usual Galilean Transformation in Quantum Mechanics. Phys. Rev. Lett. 87, 100405 (2001)
  * (14) Gulshani, P., Rowe, D. J.: Quantum mechanics in rotating frames. I. The impossibility of rigid flow. Can. J. Phys. 56, 468-479 (1978)
  * (15) Messiah, A.: Quantum Mechanics (2nd edition). Dover Publications, New York (2014)
  * (16) Rovelli, C.: Relational Quantum Mechanics. Int. J. Theoret. Phys. 35, 1637-1678 (1996)