[style=chinese]

1]organization=Dept. of Computer Engineering, ZHCET, AMU, city=Aligarh,
postcode=202002, state=UP, country=India

[type=editor, auid=000,bioid=1]

[1]

2]organization=Marine Engineering Technology Department in a joint affiliate
appointment with Electrical & Computer Engineering and Computer Science &
Engineering Department, Texas A&M University, city=Galveston, postcode=TX
77573, country=USA

[cor1]Corresponding author

# Cyber Attacks on Maritime Assets and their Impacts on Health and Safety
Aboard: A Holistic View

Mohammad Ammar<EMAIL_ADDRESS>[  Irfan Ahmad Khan<EMAIL_ADDRESS>[

###### Abstract

There has been an unprecedented digitization drive in the industrial sector,
especially in the maritime industry. The profusion of intelligent electronic
devices and IOT-enabled cyber-physical systems (CPS) has helped in the
efficient use of resources and increased convenience. CPS has enabled real-
time remote command and control of industrial assets. Unlike the relatively
isolated legacy systems, the intertwined nature of Information Technology(IT)
and Operations Technology(OT) brought by Industry 4.0 has increased the
complexity of the systems, thereby increasing the attack surface. This work
explores the possible consequences of these attacks from a more holistic view,
focusing on high-risk assets such as offshore oil rigs, offshore wind farms,
and autonomous vessels. The attacks have become more aggressive with the
proliferation of such technologies, disrupting the physical process, causing
fire and explosion hazards, and endangering human life and environmental
health. The possible attack scenarios, the attack vectors, and their physical
consequences have been discussed from the perspective of personnel safety and
health, along with known security breaches of such nature. To the best of the
authors’ knowledge, seldom has any work been done that accentuates the
possible human and environmental impacts of such attacks.

###### keywords:

Cyber-Attacks Cyber-Physical Systems Industry 4.0 IoT Maritime Assets
Personnel Safety

The Cyber-attacks on maritime infrastructures are growing rapidly, targeting
production, transportation and distribution networks.

The profusion of OT and IoT based systems paired with legacy equipment
exacerbate the physical effects of cyber-attacks, opening a new era of cyber-
terrorism.

This study explores the impacts of cyber-attacks in terms of physical damage
to the assets, risks to human life and environmental health.

##  1 Introduction

In recent years, the digitization drive has been unprecedented, replacing
legacy systems with smart devices, interconnecting almost everything with the
internet, from massive data centers to personal wearables. Energy
infrastructure is not alien to these developments; it too, has gotten
interconnected. The use of smart devices such as Intelligent Electronic
Devices (IEDs), IOT-enabled and Cyber-Physical systems, and the adoption of
Industry 4.0, in general, has enabled efficient use of resources, less human
intervention and increased convenience[1][2]. Maritime assets, such as ships,
containers, port infrastructure, offshore wind farms, and oil rigs, can be
equipped with sensors and connected to the Internet, enabling real-time data
collection and monitoring. IoT enables remote tracking of assets, predictive
maintenance, and improved operational efficiency. CPS enables remote real-time
command and control of on-site systems and equipments. Ships integrated with
IoT devices can be interconnected with each other and the internet, sending
and receiving information like location, system status, weather, and ocean
currrents[3]. Autonomous vessels and smart port utilities like automated quay
cranes, automated guided vehicles (AGVs) for container movement, and automated
gate systems for efficient cargo clearance use technologies like machine
learning, computer vision, and advanced navigation systems employing less
human intervention and increased cost-effectiveness[4].
Similarly, Offshore Wind Farms employ IoT-enabled sensors to monitor and track
real-time data across different parameters. Real-time data from these sensors
are transmitted to monitoring systems, allowing intelligent systems or manual
operators to continuously monitor the performance of wind turbines, discern
faults and anomalies, and take appropriate actions accordingly[5] thereby
minimizing on-site inspection. The offshore Oil and Gas Industry is embracing
the Industry 4.0 paradigm across its multi-faceted assets, which include
offshore production facilities, drill ships, storage and offloading units, and
complex control and monitoring systems. The integration of Industry 4.0
technologies in these assets brings numerous benefits, including increased
efficiency, improved safety, and enhanced decision-making[6].
Of all the greatness which embodies in embracing these smart solutions, there
is this huge vulnerability to attacks and disruption by adversaries. As the
system becomes more digitized and interconnected, the complexity of the system
increases, and so does the threat from cyber-attacks and consequently the
potential human and financial risks. Oil and Gas Industries are fairly
acquainted with cyber attacks for the past 30 years. There has been a 400%
increase in cyber attacks in the Oil and Gas industry since February 2020[7].
The February 2020 timeline coincides with the covid 19 lockdown and
restrictions as it increased the use of remotely operated and monitoring
technologies. Some of the attacks include disruption of offshore operational
technology (OT) platforms resulting in the failure of operational capability
and safety measures. False data injection in the critical monitoring and
controlling facilities, severing of vital communication systems, and
sabotaging critical safety infrastructure like gas detectors, flame detectors,
and safety triggering mechanisms can entail catastrophic human and financial
costs.
Like any other interconnected and digitally-driven infrastructure, offshore
wind farms face cybersecurity risks that can have significant consequences.
Attackers can exploit the vulnerabilities in the supervisory control and data
acquisition (SCADA) system and can trip multiple wind turbines, thus
undermining the integrity of the whole system[8]. Offshore wind farms can be
susceptible to malware and ransomware attacks, where malicious software is
introduced into the system, often through phishing emails or compromised
software updates. Once infected, the malware can disrupt operations, encrypt
data, or demand ransom payments for the restoration of systems. These attacks
can cause operational downtime, financial losses, and reputation damage.
Similarly, autonomous shipping vessels pose a serious security threat, this
potential threat is exacerbated when combined with other maritime
infrastructures and shipping vessels. Unauthorized access to the vessel’s
control system can have devastating consequences, as the intruder can steer
the vessel off-course, which can hit other stationary and moving maritime
assets, using it as a battering ram. This will not only cause financial costs
but can entail serious human loss.
There are very few cases of these attacks being publicized. Most of it, if not
all, stays confidential as it may undermine the reputation at both the
business and state levels[9][10]. Further, most studies in the literature
about cyberattacks on marine infrastructure consider its societal, financial,
and supply chain loss impacts, among others. To the best of the authors’
knowledge, only a few studies have been performed on the impacts of
cyberattacks on the health and safety of the personnel working on marine
infrastructure. Therefore, the purpose of this study is to identify and
explore the less studied possible human risks associated with the automation
and digitization of these maritime assets. Also, it will dig into the impacts
and consequences of cyberattacks on the health and safety, human life, and
environmental health of personnel and marine infrastructure.

Figure 1: Division of Maritime Assets

The rest of the paper is structured as follows: In the following section, we
shall get acquainted with the maritime infrastructure covering Offshore Wind
Farms, Upstream Oil and Gas Infrastructure, Smart ports, and Autonomous
Shipping Vessels. In section III, we shall briefly explain possible types of
cyber attacks on maritime infrastructure along with some previous known
attacks on CPS. In section IV, we will hypothesize some possible attack
scenarios and evaluate their possible human and health costs and finally,
section V will conclude the paper.

##  2 Maritime Energy Infrastructure

Maritime energy infrastructure constitutes multifaceted assets that can be
broadly classified into three categories 1). Production 2). Transportation and
3). Distribution. Although Ports and Autonomous Shipping vessels are not
directly considered under the energy infrastructure, they are often used for
Transportation and Distribution.

###  2.1 Production

####  2.1.1 Oil Rigs and Offshore Platforms

These structures are located in offshore waters used to explore and produce
oil and gas resources. Offshore platforms can be fixed structures (such as
platforms built on seabed or elevated platforms) or floating structures (such
as floating production storage and offloading vessels or floating platforms).
These platforms are equipped with drilling equipment, including a drilling
rig, drill pipe, blowout preventers, and well control systems.

The drilling process involves drilling a wellbore into the seabed to access
oil or gas reservoirs[11]. Software systems are used to monitor and control
drilling parameters, such as drilling speed, pressure, and mud circulation.

Figure 2: Types of fixed offshore platform (Yew 2014)

Once a well is drilled, offshore platforms facilitate the extraction and
processing of hydrocarbons. Oil and gas production systems, such as wellheads,
production trees, and risers, are used to control the flow of fluids from the
wells to the platform. Production processing facilities, including separators,
pumps, compressors, and treatment systems, are employed to separate oil, gas,
and water and prepare them for transportation. Offshore platforms incorporate
sophisticated Industrial Cyber-Physical Systems (ICPS) which allows for real-
time remote monitoring and control[12][13]. Supervisory Control and Data
Acquisition (SCADA) systems and Distributed Control Systems (DCS) are used to
monitor and control various processes, equipment, and safety systems on the
platform. These systems collect data from sensors and instruments distributed
throughout the platform and provide operators with real-time information on
operational parameters[14]. Previously, these Industrial Control Systems were
proprietary and were used on local networks[15] fairly isolated from the
outside world[16]. However, with recent advancements in Operational
Technologies(OT) and the need to make better and faster decisions, these
systems are now being integrated with various IT based industrial
technologies[17]. This has led to a greater integration of these ICS with the
outside network. This integration has opened a larger potential attack
frontier for the adversaries. Any attack targeting the plant can have serious
consequences. Failure of the Blowout Prevention system can result in a sudden
and powerful release of hydrocarbons. Blowouts can result in explosions,
fires, and the release of toxic substances into the environment, endangering
both human workers and marine life. Cramped living quarters can hamper the
evacuation procedures, again jeopardizing the rescue operation.

####  2.1.2 Off-shore Wind Farms

Off-shore wind farms consist of multiple wind turbines installed in strategic
locations. The turbines are anchored to the seabed using foundation
structures, such as monopiles, jacket foundations, or floating platforms. Each
wind turbine comprises several components, including the rotor, nacelle,
tower, and foundation.

Figure 3: Off-shore Wind farm

The wind turbines convert the kinetic energy of the wind into mechanical
energy through the rotation of their rotor blades.

Figure 4: Wind Turbine

The rotor blades spin a generator within the nacelle, which produces
electricity. Power electronics and transformers within the nacelle convert the
generated electricity to a suitable voltage for transmission. Offshore wind
farms utilize sophisticated communication and control systems to monitor and
manage the turbines, substations, and overall farm operations. The task of
gathering grid data—measured electrical variables at the wind turbines—is
carried out by the wind farm SCADA systems. This data is transmitted to a
control center for monitoring and control purposes[8]. The control system can
alter various turbine parameters like pitch and yaw of the turbine blades, it
can also set the maximum rotor speed and can apply emergency breaks in case of
fast winds[18]. In the case of floating wind turbines, the ballast tanks can
also be controlled. Vulnerabilities in the Supervisory Control System can
result in unauthorized access resulting in the errant behaviour of turbines,
which can have potential human and financial risks.

###  2.2 Transportation

####  2.2.1 Autonomous Vessels

Autonomous vessels, also known as unmanned or autonomous ships, are maritime
vessels that operate without an onboard crew or with minimal human
intervention. These vessels leverage advanced technologies, software systems,
and hardware components to navigate, operate, and communicate. Autonomous
vessels rely on a combination of technologies for navigation, situational
awareness, and decision-making. They are equipped with various sensors, such
as radar, lidar, sonar, GPS, and cameras, to perceive their surroundings and
collect real-time data about the environment. Control systems, including
autopilot and dynamic positioning systems, enable autonomous vessels to
control their propulsion, steering, and maneuvering. These systems use data
from sensors, navigation algorithms, and feedback mechanisms to maintain
course, avoid collisions, and perform docking operations. The relation of
autonomous vessels to maritime energy transportation lies in their potential
to improve efficiency, safety, and sustainability in the movement of energy
resources. Autonomous vessels can be used for transporting various forms of
energy, such as oil, gas, LNG, or renewable energy components, across maritime
routes[19]. By leveraging autonomous technology, these vessels can optimize
navigation, reduce human errors, enhance fuel efficiency, and enable
continuous operations.

####  2.2.2 Underwater Gas Pipelines

Underwater gas pipelines are critical infrastructure that enables natural gas
transportation across bodies of water, such as oceans, seas, and lakes. These
pipelines play a significant role in the maritime energy transportation
sector. They are typically made of steel or high-density polyethylene (HDPE)
pipes, designed to withstand the harsh marine environment. The pipelines
consist of a continuous length of interconnected pipe segments that form a
route from the gas source (e.g., offshore gas field) to the receiving terminal
onshore or to other interconnecting pipelines. Various software and hardware
systems are used to monitor and maintain the integrity of underwater gas
pipelines. These systems include:

  * •

Pipeline Inspection Gauges (PIGs)

  * •

Subsea Sensors

  * •

Remote Operated Vehicles (ROVs)

  * •

Fiber Optic Systems

  * •

Emergency Shutdown Systems (ESD)

###  2.3 Distribution

####  2.3.1 Offshore Substations

The foundation of offshore wind farms is offshore substations, commonly called
offshore electrical platforms. They act as centers for gathering, converting,
and distributing the electricity produced by wind turbines. Large steel
constructions are generally used for offshore substations since they are made
to endure the severe sea environment. They are made up of numerous levels or
decks that can hold different systems and equipment. The design considers
stability, buoyancy, and resistance to wave, wind, and ice loads. As with any
other substation, the offshore substation uses SCADA systems for monitoring
and maintenance. Various relays, circuit breakers, and other safety devices
are used to protect the station from electrical faults.

####  2.3.2 Floating Storage and Regasification Units (FSRUs)

Liquefied natural gas (LNG) is imported, stored, and regasified using Floating
Storage and Regasification Units (FSRUs), which are specialized ships. They
offer a versatile and affordable way to receive LNG from LNG carriers, store
it onboard, and then transform it back into gas for distribution. FSRUs are
anchored near the shore and receive LNGs from the carrier ships in liquid form
through cryogenic arms and flexible hoses. The LNG is stored at 109K - 110K.
Before distribution, the LNG is converted into gas by heating it using
seawater and then pushed into pressurized pipelines. Due to the inherently
hazardous nature of the LNG, FSRUs incorporate advanced safety systems to
ensure the secure handling of LNG and to prevent accidents. These systems
include fire and gas detection systems, emergency shutdown systems, and
containment measures to prevent leakage or spills.

##  3 A Background on Cyber Security

Cybersecurity is the practice of preventing unauthorized access, damage, and
theft to computer systems, networks, and data. It involves a variety of
safeguards, technologies, and best practices meant to stop cyberattacks and
protect information.

###  3.1 Types of Cyber-Attacks

There are many types of cyber attacks, but in the scope of our study, we are
most interested in cyber attacks on marine energy systems and their impacts on
health and safety. Although cyber-attacks cause great monetary loss, a cyber
attack on a marine energy system entails a loss of human life and other health
hazards.

Figure 5: General overview of cyber-physical systems and the attack vectors.

####  3.1.1 Zero-day Attacks

A zero-day attack on marine energy systems refers to exploiting previously
unknown vulnerabilities or weaknesses in the software or hardware that
controls physical devices, infrastructure, or industrial systems. These
attacks can have severe consequences as they target the integration between
digital and physical components, affecting critical infrastructure and
industrial operations. Some of the famous Zero-day attacks are Stuxnet Worm
(2010); Stuxnet was a sophisticated computer worm that exploited zero-day
vulnerabilities in Microsoft Windows and Siemens SCADA (Supervisory Control
and Data Acquisition) systems, which were used to control and monitor
industrial processes in the nuclear enrichment facility. Stuxnet manipulated
the centrifuges’ speed control system, causing physical damage to the uranium
enrichment process and sabotaging the facility’s operations. Another recent
one is the Triton/Trisis Malware (2017). It targeted a petrochemical plant in
the Middle East, sabotaging the safety instrumented system (SIS). This system
was responsible for shutting down the processes in case of an emergency. The
malware allowed the attackers to reprogram the system to cause catastrophic
physical consequences. Similarly, Maroochy Water Services Attack (2000), while
technically it was not considered a Zero-day attack, it serves as an example
of how cyber-physical systems can be targeted. Australian hacker named Vitek
Boden gained unauthorized access to a sewage treatment plant’s control system.
He used radio transmissions to manipulate pump controls, causing millions of
liters of raw sewage to spill into waterways and parks[20]. This attack can
serve as an analogue scenario to the cataclysm that can be caused on the
maritime energy infrastructure by these types of attacks.

####  3.1.2  Eavesdropping Attacks

While eavesdropping attacks do not have imminent physical consequences, it
does threaten the information integrity of the system. Eavesdropping attacks
are typically passive. The attacker does not actively modify or disrupt the
communication but listens to the exchanged data. While sounding harmless, this
sensitive data can actually be used to stage a large-scale attack on the
system.

####  3.1.3 Denial of Service Attacks (DoS)

DoS attacks aim to disrupt the availability of computer systems, networks, or
websites by overwhelming them with a flood of requests or traffic[21]. This
prevents legitimate users from accessing the targeted resource. In maritime
energy systems, DoS can have severe consequences, disrupting the integration
between digital and physical components. There are many different types of DoS
attacks, such as Ping Flood Attack, SYN Flood Attack, HTTP/HTTPS Flood Attack,
NTP Amplification Attack, DNS Amplification Attack, and Slowloris Attack. In
the Industrial Control System (ICS) domain, these attacks can prohibit the
legitimate user from enforcing countermeasures to protect the machinery from
erratic behavior. In case of an unforgiving marine environment, these failures
can result in potential harm and life-threatening consequences for the on-site
crew.

####  3.1.4 Data Injection Attack

Data Injection Attack is also known as Data Spoofing Attack or Falsification
Attack. In these types of attacks, the communication channel is compromised,
whereby the attacker can manipulate or insert false data into a system or
database. This is done by exploiting the vulnerabilities in the communication
protocol, software, and device itself. The attacker can go through the
physical layer using frequencies to manipulate the communication or through
the cyber-layer, assuming he has access to the credentials[22]. In both cases,
altering the signals sent to the actuators and disrupting the nominal state of
the plant. FDI can also target the sensor reading of the plant, thereby
deceiving the system or the remote operator. Although there are measures to
counter these attacks, sophisticated and highly coordinated attacks can still
bluff the security measures. Mo et al.[23] provides optimal conditions under
which the attacker can bypass the failure detection systems, Zhang el al.[24]
proposed a stealthy FDI attack which is robust to residual-based detectors.

###  3.2 Some Notable Cyber- Attacks on CPS

There are few attacks that are in the public domain, and most, if not all, are
confidential. One of the most notable attacks, as mentioned earlier, was the
Maroochy Water Service attack. A disgruntled employee manipulated the control
pumps and was able to release large amounts of sewage, flooding the parks and
waterways[20][25]. The Stuxnet attack of 2010 targeted Iran’s uranium
enrichment facility. The malware exploited four zero-day vulnerabilities to
sabotage the centrifuges’ nominal operations, thereby halting the enrichment
process[26][27]. The Stuxnet worm infected the facility computers through a
USB stick and was able to infect the windows operating systems and siemens
step7 software used to program industrial control system. The worm caused the
centrifuge to spin above its nominal speed and injected false data into the
supervisory systems causing the operators to be oblivious to the attack. In
2014, the German Steel Mill’s network was compromised, and the attackers were
able to disrupt the shutting down process of the blast furnace, causing
widespread destruction to the facility[28]. A similar incident occurred in the
U.K. where two people lost their lives, and 13 were injured. However, this was
caused by a human error instead of a cyber attack at the German Plant. These
two events serve as a testament to the dangers posed by compromised Cyber-
Physical systems[29]. The Ukrainian power grid attack of 2015 caused a
blackout across Ukraine, affecting around 200K civilians. The attack targeted
the SCADA systems, which controlled the distribution substations[30]. One of
the most recent attacks was on Oldsmar’s water treatment facility. The
attacker gained access to one of the consoles of the facility using a
compromised credential of a publicly available remote access application
namely TeamViewer. The attacker logged on to the console and, using the Human
Machine Interface(HMI) increased the concentration of Sodium Hydroxide, also
known as Lye, from 100ppm to 11100ppm, a hundred times increase in the
concentration[31]. Lye in these amounts can cause severe burns, permanent
damage to the organs, and even death[32]. This attack coincided with the Super
Bowl event in the area. Thankfully, the attack was foiled. If it had been
successful, thousands of people would have been affected. This attack serves
as a wake-up call for the government as well as the industry at large.

Table 1: Some of the Cyber-Attacks from 2000 Onwards Year |  Country |  Details
---|---|---
2000 |  Australia |  Causing sewage to spill in waterways
2010 |  Iran |  Stuxnet attack destroying core controllers of the industries
2014 |  Germany |  Attacks led to the disruption of the shutting down process of blast furnace
2015 |  Ukraine |  BlackEnergy attack on power grid leading to massive power outage
2017 |  Russia, Ukraine India, China |  WannaCry attack aiming to encrypt data and demand ransom payments
2020 |  Brno University Hospital Czech Republic |  A cyber attack that shut down IT network of a Czech hospital
2020 |  US Dept. Health Human Services |  Unspecified attack on servers
2021 |  Colonial Pipeline, US |  A ransomware attack on a US fuel pipeline, leading to shutdown of a critical fuel network
2021 |  United States |  Attacker gains access to the water treatment facility’s systems and increased the concentration of NaOH in the supply water

##  4 Potential Hazard to Humans and Environment in Maritime Infrastructure

The proliferation of cyber physical systems and IoT based control and
monitoring technology, in the maritime industry has made it highly susceptible
to cyber physical attacks. The complexity and hazardous operational nature
coupled with the high asset value of the infrastructure all together attract
perpetrators to target these systems. Due to the socio-technical nature of the
system, any attack targeted at the physical process can have serious
consequences, Coupled with the remote and unforgiving environment of the sea,
these attacks can have life threating outcomes. In the upcoming sections, we
shall discern the potential effects on human life of these attacks in each of
the maritime infrastructures.

###  4.1 Offshore Oil & Gas

The Offshore Oil Facility is composed of a drilling rig, transportation
system, processing plant, accommodation and utilities for staff, and the
structure itself. The process plant is where the production of oil and gas
takes place before transporting it to the onshore facilities. The processing
plant has several levels of processing units which include separation,
compression, and dehydration as shown in fig 6. There are quite a few types of
risks associated with the Oil and Gas facilities but we shall only focus on
the risks associated with the process itself. The process facility at the oil
rig is a small dense chamber with many machines and equipment. These areas are
overcrowded with compressors, separators, and high-pressure pipelines carrying
highly flammable materials. Any mishap can have catastrophic consequences and
can also lead to a domino effect. To illustrate the hazard of cyber attacks on
these facilities a simple example is adapted from Khakzad et al.[33]. A
propane injecting control system feeds a scrubber with propane from a gas
pipeline. To maintain the required amount of pressure in the scrubbing chamber
the pipeline is equipped with an actuator operated automatic valve. In the
Industry 4.0 era it is plausible to assume it as a cyber-physical system
connected over the internet. The control system has an actuator, a pressure
transmitter, and a pressure controller. A MitM attack if successfully executed
can stealthily alter the values of the pressure transmitter. The insertion of
malicious readings will cause the pressure to increase in the scrubbing
chamber. The altered readings displayed on the monitoring station will cause
the entire system to be oblivious to the attack, hence forestalling any
remedial measures.

Taking forward the analogy a similar incident is comprehensible at the
offshore O&G platforms. As mentioned earlier, the processing facility houses
high-pressure units and flammable materials. The fluid from the well is passed
through the first separator which separates crude oil, water, and gas followed
by a second separator which separates gas condensate and water. This is
followed by compressor units, a flash drum, and a drier. The pressure in the
separator is controlled by the flow valve. A PLC command injection attack
targeting the flow valve controller can cause the pressure to increase in the
separator. This increase in pressure if not mitigated can cause the Separator
to fail as Boiling Liquid Expanding Vapour Explosion (BLEVE). These explosions
produce a fireball, whose emissive power typically ranges from 180kW/$m^{2}$
[34]to 350kW/$m^{2}$[35]. The shock wave velocity is typically 753m/s with
sub-sonic fragment velocity. The fragments resulting from the explosion are
hard to simulate, but many researchers have used deterministic models and
stochastic simulations to determine its

Figure 6: Schematic Representation of process facility and possible attack
vectors.

characteristics[36][37][38]. The direct exposure to the fireball can cause
severe burns, while the shock wave and peak overpressure can cause barotrauma,
resulting in tympanic membrane rupture and middle ear damage[39], injury to
the lung parenchyma, injury to brain parenchyma and injury causing abdominal
hemorrhage and perforation. These are categorized as the primary injuries
caused by the explosion[40].

The secondary injuries include those which are caused by the fragments and
projectiles like of metals and concrete. The tertiary category accounts for
injuries caused by human transposition and impacts, like hitting a wall or
furniture by being thrown away. Injuries can include fractures, amputation,
and multiple types of soft tissue injuries. Quaternary injuries result from
the toxin released by the explosion and other effects which are not covered by
the above three categories. The effect of BLEVE will not stop here as it may
cause a domino effect, critically damaging other infrastructure and causing
fires[41]. Other critical process plants can also be affected in a similar
way, either by injecting erroneous values in PLC controllers, by spoofing
sensor readings or sabotaging the automated safety mechanism by changing the
safety values in the PLC controllers or the manual intervention by denial of
service attacks (DoS). Table 2 summarises some of these attacks and their
impact. All in all, it is of utmost importance to identify these
vulnerabilities, quantify the threat probabilities, and to mitigate the risks.

Table 2: Some of the Cyber-Attacks in O&G process facilities and its impacts Process |  Attack |  Target Component |  Consequence |  Environment & Human Impact
---|---|---|---|---
High Pressure Separator |  FDI, Malicious Command Injection |  Flow Valve PLC controller, Pressure and Temp Sensor |  Over-Pressure, BLEVE |  Burns, Damage to Air-filled organs, penetration by projectile, pool of fire
Emergency Shutdown and Disaster Mitigation Systems |  DoS attack |  PLC controller, actuators, Safety Instrumented System |  Operation Under Undesirable Conditions |  Damage to Equipement, Possibility of Spillage, Potential Loss of Marine or Human Life
Low Pressure Separator |  FDI, Malicious Command Injection |  PLC controller, Pressure and Temp Sensor |  Over-Pressure, BLEVE |  Burns due to fireball explosion, Release of Toxin, pool of fire
Compressor Unit |  FDI, Malicious Command Injection |  PLC controller, Pressure and Temp Sensor |  Over-Pressure, Leakage |  Jet Stream of fire causing burns and potential loss of life
Storage Unit |  Spoofing Attack |  Level Sensor |  Spillage or Tank Rupture |  Burns due to fire, Oil Spill affecting marine life.

###  4.2 Offshore Wind Farm

The offshore wind farms(OWF) are a cluster of wind turbines located off the
shore, these structures harness energy from the wind that blows over the sea.
The collected energy is transmitted over to the substation either offshore or
onshore which is then further distributed. Unlike the oil rig structures which
has a permanent human presence, these structures are unattended. Most of the
systems are automated and the monitoring is done through software like SCADA.
However maintenance is done physically, this may involve climbing the nacelle
or grappling down the turbine itself. The sheer scale of the wind farm and the
remoteness of it makes any rescue attempt futile. The OWF employs a flat
logical control structure with multiple turbines connected in a ring topology.
An array of sensors are used to gather data on the status of the turbines,
these information is relayed to the control station which uses a combination
of IT and OT to control various parameters to optimize the energy output. 5
shows the structure of wind turbines, the nacelle holds the major components
of the turbine including the generator and gear assembly. Most of the turbine
systems are autonomous– adjusting the blade angles and nacelle direction in
order to optimize the energy output. One major component of the system is the
braking mechanism, this helps to ensure that the turbine operates under the
required RPM. An attack on the PLC controllers can compromise the control
system of the turbine. This can give the attacker the privilege to change the
direction of the nacelle, the orientation of the blades and the brakes itself.
These parameters are vital for the optimization of energy output. However
these three control surfaces can be used to push the wind turbine beyond safe
operational RPMs. The wind turbines are designed to cut off in high gusty
winds or when the RPM reaches beyond a certain threshold. These systems ensure
that the turbine experiences mechanical and aerodynamic stresses in acceptable
ranges.

An attack on these surfaces can compromise these safety systems. A turbine
operating at extreme overload can experience multiple failures. Various
publications have studied the failure rate of these systems, mainly focusing
on the technical components. From the perspective of personnel safety, blade
flew off, partial blade failure are of the utmost interest. The blade flew
off, or the partial blade flew off, is the structural failure of the blade,
causing the entire blade to fly off or sometimes in pieces as projectiles.
[42] provides an in-depth analysis of the throwback region. These high-energy
projectiles can cause severe damage to the human body. These can include cuts,
lesions, amputations, and even death. Operation of the wind turbines beyond
safe limits can also lead to complete structure failure. In the case of
offshore turbines a service vessel is also present. A complete structure
collapse can cause the nacelle to fall. The weight of offshore wind turbine
nacelle typically ranges from 100 to 600 Ton. An impact of this energy on the
vessel deck can be devastating.

###  4.3 Underwater Oil Pipeline

The sub-sea crude oil pipeline is an important part of oil transportation
network along with oil transporter and ships. These pipelines are continuously
monitored by sensors and computers to maintain the flow rate and pressure of
the oil. Some of these sensors are used to detect oil leakages. A sensor
spoofing attack can render the sensor useless, hence keeping the whole system
oblivious to the leakage. This can cause large amount of crude oil to leak
before detection, causing mayhem in ocean habitats. Attacks can also effect
the PLC based PID controllers which are used to control flow rate through
SCADA software. These changes in operational parameters can cause erratic
pressure and flows in the pipeline causing it to leak or explode. In 2009 a
disgruntled employee was found guilty of disabling the computer systems that
were designated to detect oil leaks in pipeline off the coast of Southern
California[43]. In another event in 2008 a section of the Baku-Tbilisi-Ceyhan
(BTC) pipeline exploded, an investigation into the explosion later by the U.S.
shed light on a cyber-attack. The hacker got access to the pipeline monitoring
system and shut down the alarm warning system and was able to super pressurise
the pipeline causing it to explode[44].

###  4.4 Autonomous Ships and Unmanned Vessels

Autonomous Ships and Vessels possess unmatched potential to cause destruction
in the maritime infrastructure. Currently, their use is restricted to small
and local vessels. However, the trend shows that they will be used in deep-sea
voyages and general shipping by 2030[45]. These vessels can either be
completely autonomous or commanded by a shore control center. Relying on
multiple sensors and communication protocols for navigation and actuators for
control, these systems are vulnerable to attacks as much as any other cyber-
industrial system. However, unlike static assets like wind turbines and oil
rigs, these assets pose a wider threat to the maritime industry. The attacks
on the shipping industry are not new and have been happening for quite a long
time. One attack targeted the Electronic Chart Display and Information System
(ECDIS), resulting in the loss of critical information[46]. Another attack
compromised the GPS system, indicating the ship to be 32 km inland[47]. With
the advent of autonomous ships, new attack surfaces are to be recognized. A
compromised container can be used as a battering ram by the perpetrators–
opening a new kind of terrorism. One scenario could be to use the hijacked
vessel to ram it against other stationery assets like wind farms, oil rigs,
and other regasification units. Another scenario could be to use these ships
to ram other vessels and ports. A vessel colliding with an oil rig can have
severe consequences. However, the extent of damage depends on various factors
such as vessel size, speed, vessel design, type of oil rig and other factors.
[48] has done a comprehensive analysis of these collisions. Considering the
worst-case scenario, a small vessel of 5000 ton at a modest speed of 12 knots
possess around 200 MJ of kinetic energy, and most of the offshore structures
have kinetic resistance of less than 200 MJ. An impact with such energy can
cause the oil platform structure to fail. The most imminent damage will be to
the oil riser which can snap under such impact load. This can cause huge
amounts of crude oil to be released into the ocean. Just like the Deep Water
Horizon, these blowouts can cause the destruction of marine life both on the
surface and in the deep waters. Oil spills can affect microbial life such as
phytoplanktons as well as fisheries[49], the impact on the ecosystem is highly
complex to determine and is out of the scope of this study. Apart from the
environmental impact, there is also a high fatality risk associated with
topside events such as fires and explosions. Secondary fatalities and injuries
are from structural impacts such as debris falling and other flying shrapnel.

##  5 Conclusion

This work attempts to accentuate the possible consequences of cyber attacks in
the maritime industry on the health and safety of personnel working abroad. In
this paper, effort is made to shed light on those consequences, which are
often eclipsed by the monetary and economic factors – namely the human and
environmental impacts. We started with a brief description of the trend in the
maritime industry, the proliferation of digital technologies, sensors, cyber-
physical systems, and the adoption of Industry 4.0 in general. The maritime
sector is evolving rapidly, with new technologies being introduced regularly.
The industry is ever more susceptible to cyber-attacks. Attacks can now impact
the physical systems, causing financial losses and endangering human lives and
the environment, causing great destruction. The attack surface will continue
to grow as the technology advances, becoming more intertwined and ever more
complex, engendering new risks and consequences. At the time of publication,
to the best of our knowledge, there is seldom any work that draws attention to
the possible human and environmental factors of such attacks. There is an
eminent need for the qualitative and quantitative assessment of risks
associated with these attacks from a much more holistic viewpoint.

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fig/P Mohammad Ammar is currently a final year undergraduate at the Dept. of
Computer Engineering, Zakir Husain College of Engineering and Technology
(ZHCET), Aligarh Muslim University, Aligarh. He is also the Chairperson for
the Marine Technology Society Autonomous Underwater Vehicle Club, ZHCET, AMU.
His interests lie in Artificial intelligence, Deep-learning, and Underwater
Marine Robotics.

fig/IrfanSir Dr. Irfan Khan is an Assistant Professor at the Department of
Marine Engineering Technology with a courtesy joint appointment with the
Electrical and Computer Engineering at Texas A&M College Station. Before
joining TAMU in 2018, he received a Ph.D. in Electrical and Computer
Engineering from Carnegie Mellon University USA. He is also the director of
the Clean And Resilient Energy Systems (CARES) Lab, which focuses on the cyber
security, reliability, and sustainability of cyber-physical systems, including
electric energy systems, drones, marine vessels, and biomedical systems.
Recently, he has been presented with several prestigious awards and honors,
such as the 2021 Jim Leonard Outstanding Member Award from IEEE Region 5, the
Gulf Research Program’s Early-Career Research Fellowship: Offshore Energy
Safety (Track 3) from the National Academies, and the 2021 IEEE Region 5
Director’s Award Technical Conference Co-Chair.


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# Note on an extended chiral bosons system contextualized in a modified gauge-
unfixing formalism

Gabriella V. Ambrósio<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Juiz de Fora, Juiz de Fora – 36036-330, MG, Brazil
Cleber N. Costa<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Juiz de Fora, Juiz de Fora – 36036-330, MG, Brazil  Paulo R. F.
Alves<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de
Juiz de Fora, Juiz de Fora – 36036-330, MG, Brazil  Jorge Ananias Neto
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Juiz de
Fora, Juiz de Fora – 36036-330, MG, Brazil  Ronaldo Thibes<EMAIL_ADDRESS>Universidade Estadual do Sudoeste da Bahia, DCEN, Rodovia BR 415, Km 03,
Itapetinga – 45700-000, BA, Brazil

###### Abstract

We analyze the Hamiltonian structure of an extended chiral bosons theory in
which the self-dual constraint is introduced via a control $\alpha$-parameter.
The system has two second-class constraints in the non-critical regime and an
additional one in the critical regime. We use a modified gauge unfixing
formalism to derive a first-class system, disclosing hidden symmetries. To
this end, we choose one of the second-class constraints to build a
corresponding gauge symmetry generator. The worked out procedure converts
second-class variables into first-class ones allowing the lifting of gauge
symmetry. Any function of these GU variables will also be invariant. We obtain
the GU Hamiltonian and Lagrangian densities in a generalized context
containing the Srivastava and Floreanini-Jackiw models as particular cases.
Additionally, we observe that the resulting GU Lagrangian presents
similarities to the Siegel invariant Lagrangian which is known to be suitable
for describing chiral bosons theory with classical gauge invariance, however
broken at quantum level. The final results signal a possible equivalence
between our invariant Lagrangian obtained from the modified GU formalism and
the Siegel invariant Lagrangian, with a distinct gauge symmetry.

Chiral bosons (CB) are basic ingredients in the description of two-dimensional
scalar field models Sevrin:2013nca ; Mezincescu:2022hnb ; epl3 ;
Arvanitakis:2022bnr  with applications to modified gravity theories
Giaccari:2008zx ; Sen:2015nph ; Merbis:2023uax , noncommutativity epl3 ;
Miao:2003ab ; Das:2004vc  and string/brane theories Tseytlin:1990va ; bar ;
Hull:2004in ; Phonchantuek:2023iao , besides providing building blocks for
fundamental concepts in modern theoretical physics Abreu:2000sn ;
Upadhyay:2011ph ; Shukla:2013baa . Additionally, CB theories also present rich
mathematical canonical structures per si, among which we underline the typical
chiral constraint algebra that appears in the Floreanini-Jackiw (FJ) theory
fjprl ; bw ; ea . Interestingly enough, the original FJ chiral constraint
manifests itself in a single second-class form, with a non-trivial commutator
along the real line. In higher dimensions, the chirality condition generalizes
to chiral $p$-form theories with self-dual field strength, from which we
mention the interesting Past-Sorokin-Tonin (PST) approach Pasti:1995ii ;
Pasti:1995tn ; Pasti:1996vs , recently refined by Mkrtchyan Mkrtchyan:2019opf
, where gauge-invariant self-dual models containing infinitely many auxiliary
fields are related to equivalent ones with fewer finite auxiliary fields in
non-polynomial form. Neither infinite set of fields nor non-polynomiality
being welcome properties, they seemed at first necessary for a consistent
first-class theory. Building on established string theory compactification
ideas, over the past years, Ashoke Sen has shown that for specific space-time
dimensions it is possible to avoid both situations obtaining a finite Lorentz
covariant action containing an extra set of free unphysical fields Sen:2015nph
; Sen:2019qit . Correspondingly, a partition function for Sen’s action has
been recently evaluated in Andriolo:2021gen ; Lambert:2023qgs  providing
further novel developments on the duality relations. Following a different
path, in the present note, we focus on the peculiarities of the two-
dimensional case from the gauge-unfixing formalism viewpoint searching for
hidden symmetries and clues to the quantization process. A pioneering proposal
for including the chirality condition in a two-dimensional scalar theory using
an auxiliary Lagrange multiplier field was developed by Srivastava in sriva .
Unfortunately, Srivastava’s model does not seem to be equivalent to the FJ one
and does not even appear to be unitary since its reduced Hamiltonian cannot be
positive definite harada ; ggr . One way of circumventing this difficulty is
by introducing an additional open free parameter $\alpha$ along with the
Lagrange multiplier field, leading to the Kim, Kim and Park (KKP) model kkp
which embodies the chiral constraint linearly into the Lagrangian. On that
case, depending on the value of the parameter $\alpha$, one can reproduce the
FJ or the Srivastava models. That extended model amounts to a second-class
Dirac-Bergmann system with two second-class constraints. On the other hand,
the Siegel version for the CB sig  contains the chirality condition
quadratically and enjoys gauge invariance, which is known to be related to
first-class systems. Inspired by those facts, in this current note, we propose
to convert the second-class KKP model into first-class in order to analyze the
extended CB system in the context of gauge theories. For that purpose, we
employ the modified gauge-unfixing (GU) formalism epl3 ; proto ; epl4  to
construct a broader gauge theory encompassing the extended CB as a particular
gauge-fixing realization. In addition to playing a key role in the description
of all fundamental interactions, gauge theories are also technically
significant as the quantization of second-class systems is substantially more
complex than first-class ones mes . Furthermore, by exposing new symmetries,
the conversion to first class often sheds light to open problems from
different perspectives as it allows the application of a powerful machinery of
functional quantization methods Mandal:2022xuw . Along the last decades, two
main second-to-first-class system conversion lines have evolved in the
literature, leading to different formalisms. The first one comes from
pioneering works by Wess, Zumino, Faddeev and Shatashvili Wess:1971yu ; fasha
in which extra variables are introduced in phase space in order to convert
second-class constraints into first-class and has led to the systematic
Batalin-Fradkin-Tyutin (BFT) approach bfto1 ; bfto2  with corresponding
subsequent modifications and improvements Banerjee:1994pp ; Kim:2005mf ;
Pandey:2021myh . Since then, there have been countless applications of the BFT
formalism. Closer to our present topic, we can particularly mention the work
of Amorim and Barcelos-Neto in which the BFT formalism is directly applied to
CB models Amorim:1994np ; ab . The second line, which we shall be following
here, originated from works by Mitra, Rajamaran and Vytheeswaran mr ; vyt1 ;
vyt2 ; vyt3  and is based on more clean economical grounds as it does not rely
on the introduction of new variables but rather views the second-class system
as a gauge-fixed version of a more general first-class theory. Bringing more
understanding to the intricacies of singular Dirac-Bergmann systems, this
second approach has grown in depth along the literature finding many
applications and has come to be known as the GU formalism. As far as we know,
apart from the recent work on noncommutativity epl3 , the GU formalism has
never been applied to CB theories before. Concerning the existing first and
second class CB descriptions along with their corresponding mentioned
concurring issues, the current GU approach to the quantization of an extended
CB model encompassing the previous situations as particular limiting cases
certainly brings new relevant features for CB theories.

In the original Vytheeswaran works vyt1 ; vyt2 ; vyt3 , only already existing
phase space variables are used to convert second-class systems into first-
class ones. No Wess-Zumino terms are introduced and a Lie projection operator
is directly applied to the second-class Hamiltonian to reveal hidden
symmetries. The modified GU formalism epl3 ; proto ; epl4  consists in
redefining the second-class phase space variables as first-class ones. It
turns out that the construction of gauge-invariant variables from start leads
to considerable simplifications when compared with the usual GU formalism.
Given a second-class system, we split the constraints into two families, the
first of which shall provide the gauge symmetry generators while the second is
actually discarded in a broader gauge-invariant theory context in which they
play a role of possible gauge choices among others. Consider, for example, a
second-class phase-space function $T(A_{\mu},\pi_{\mu})$, with the index $\mu$
running through all phase space variables. The strategy is to write a
corresponding first-class version $\tilde{T}({A}_{\mu},{\pi}_{\mu})$ obtained
from the second-class one $T$ with the same functional form as

$\tilde{T}({A}_{\mu},{\pi}_{\mu})\equiv{T}(\tilde{A}_{\mu},\tilde{\pi}_{\mu})\,,$
(1)

by redefining the original phase space variables

$A_{\mu}\longrightarrow\tilde{A}_{\mu}({A}_{\mu},{\pi}_{\mu})\,,$ (2)
$\pi_{\mu}\longrightarrow\tilde{\pi}_{\mu}({A}_{\mu},{\pi}_{\mu})\,,$ (3)

such that

$\delta\tilde{A}_{\mu}=\alpha\left\\{\tilde{A}_{\mu},\psi\right\\}=0\,,$ (4)
$\delta\tilde{\pi}_{\mu}=\alpha\left\\{\tilde{\pi}_{\mu},\psi\right\\}=0\,,$
(5)

where $\alpha$ are the infinitesimal gauge parameters and $\psi$ the gauge
symmetry generators built from the second-class constraints. The new deformed
variables $\tilde{A}_{\mu}$, $\tilde{\pi}_{\mu}$ are known as GU variables. It
is clear that all functions of the GU variables, in particular $\tilde{T}$,
will be gauge invariant since

$\displaystyle\left\\{\tilde{T},\psi\right\\}=\left\\{\tilde{A},\psi\right\\}\frac{\partial{T}}{\partial\tilde{A}}+\frac{\partial{T}}{\partial\tilde{\pi}}\left\\{\tilde{\pi},\psi\right\\}=0\,.$
(6)

Consequently, we can obtain a gauge invariant function from the replacement

$\displaystyle T(A_{\mu},\pi_{\mu})\rightarrow
T(\tilde{A_{\mu}},\tilde{\pi_{\mu}})=\tilde{T}({A_{\mu}},{\pi_{\mu}})\,.$ (7)

For the sake of argument clearness, assume for the moment that the system has
only two second-class constraints $Q_{1}$ and $Q_{2}$. The phase space gauge
invariant GU variables, collectively denoted by
$\tilde{\Lambda}\equiv(\tilde{A}_{\mu},\tilde{\pi}_{\mu})$, can be constructed
as a power series in the discarded second-class constraint $Q_{2}$ as

$\displaystyle\tilde{\Lambda}(x)=\Lambda(x)+\int dyb_{1}(x,y)Q_{2}(y)+\iint
dydzb_{2}(x,y,z)Q_{2}(y)Q_{2}(z)+...\,.$ (8)

This assures that we recover the original second-class system back when
$Q_{2}=0$ as the GU variables satisfy the boundary condition

$\displaystyle{\tilde{\Lambda}}_{\big{|}Q_{2}=0}=\Lambda$ (9)

on the constraint surface $Q_{2}=0$. The unknown coefficient functions $b_{n}$
present in Eq. (8) are then determined by the GU gauge invariant condition

$\delta\tilde{\Lambda}=\alpha\left\\{\tilde{\Lambda},Q_{1}\right\\}=0\,.$ (10)

The general equation for $b_{n}$ obtained from (10) reads

$\displaystyle\delta\tilde{\Lambda}(x)=\delta\Lambda(x)+\delta\int
dyb_{1}(x,y)Q_{2}(y)+\delta\iint dydzb_{2}(x,y,z)Q_{2}(y)Q_{2}(z)+...=0\,,$
(11)

with the infinitesimal transformations being given by

$\displaystyle\delta\Lambda(x)=\int
dy\,\alpha(y)\left\\{\Lambda(x),\psi(y)\right\\},$ (12) $\displaystyle\delta
b_{1}(x)=\int dy\,\alpha(y)\left\\{b_{1}(x),\psi(y)\right\\},$ (13)
$\displaystyle\delta Q_{2}(x)=\int
dy\,\alpha(y)\left\\{Q_{2}(x),\psi(y)\right\\}\,.$ (14)

So, for the first order correction term $(n=1)$, we have from Eq. (11)

$\displaystyle\delta\Lambda(x)+\int dyb_{1}(x,y)\delta Q_{2}(y)=0\,.$ (15)

From Eq. (15) we can determine $b_{1}$. For the second order correction term
$(n=2)$, we have

$\displaystyle\int dy\delta b_{1}(x,y)Q_{2}(y)+2\iint dydzb_{2}(x,y,z)\delta
Q_{2}(y)Q_{2}(z)=0\,.$ (16)

Then, from Eq. (16) we can determine $b_{2}$ and the same procedure described
so far is used to determine all the remaining $b_{n}$ coefficient functions.
Therefore, from the GU variables power series defined in Eq. (8), we can
derive a corresponding gauge invariant theory.

We proceed now to our main course in which we consider an extended CB model
described by a Lagrangian density obtained from the sum of a free scalar field
theory with the modified linear self-dual constraint kkp

$\mathcal{L}_{\alpha}=\frac{1}{2}(\dot{\phi}^{2}-{\phi}^{\prime
2})+\lambda(\dot{\phi}-{\phi}^{\prime})+\frac{1}{2}\alpha\lambda^{2}\,.$ (17)

The overdot and prime mean differentiation with respect to time $t$ and space
$x$, respectively, $\lambda$ stands for an auxiliary field working as a
Lagrange multiplier and $\alpha$ is a convenient regularizing parameter which
may, as we shall soon see, succeed in solving the difficulties present in
Srivastava’s model sriva . The point is that (17) smoothly interpolates among
some interesting relevant values of $\alpha$. For $\alpha\rightarrow 0$, it
describes Srivastava’s chiral boson with the chirality condition arising
naturally as the equation of motion for $\lambda$ while, as we shall see
throughout the article, for $\alpha\rightarrow 1$ we have a FJ chiral boson
whilst in the improper limit $\alpha\rightarrow\infty$ the chirality condition
is suppressed leading to a usual free scalar field.

From the Lagrangian density (17), we get the momenta field variables

$\displaystyle\pi_{\lambda}=\frac{\partial\mathcal{L}_{\alpha}}{\partial\dot{\lambda}}$
$\displaystyle=0\,,$ (18)
$\displaystyle\pi_{\phi}=\frac{\partial\mathcal{L}_{\alpha}}{\partial\dot{\phi}}$
$\displaystyle=\dot{\phi}+\lambda\,,$ (19)

which lead to the primary constraint

$\displaystyle\Omega_{1}\equiv\pi_{\lambda}\approx 0\,.$ (20)

Performing a Legendre transformation

$\displaystyle\mathcal{H}_{\alpha}=\pi_{\phi}\dot{\phi}-\mathcal{L}_{\alpha}\,,$
(21)

we obtain the canonical Hamiltonian

$\displaystyle\mathcal{H}_{\alpha}=\frac{1}{2}\pi^{2}_{\phi}+\frac{1}{2}{\phi}^{\prime
2}-\lambda(\pi_{\phi}-{\phi}^{\prime})-\frac{1}{2}(\alpha-1)\lambda^{2}\,.$
(22)

Following the usual additional steps of the Dirac-Bergmann algorithm
Dirac:1950pj ; Anderson:1951ta , the dynamical stability of $\Omega_{1}$
generates a further secondary constraint given by

$\displaystyle\Omega_{2}\equiv\pi_{\phi}-{\phi}^{\prime}+(\alpha-1)\lambda\approx
0\,.$ (23)

Since $\lambda$ has a non-null Poisson bracket with $\pi_{\lambda}$, the
Dirac-Bergmann iterative process comes to an end with $\Omega_{1}$ and
$\Omega_{2}$ representing the complete set of constraints for the model. The
nature of the system is revealed from the corresponding Poisson bracket
relations among the constraint functions

$\displaystyle\left\\{\Omega_{2}(x),\Omega_{2}(y)\right\\}$
$\displaystyle=-2\delta^{\prime}(x-y)\,,$ (24)
$\displaystyle\left\\{\Omega_{1}(x),\Omega_{2}(y)\right\\}$
$\displaystyle=-(\alpha-1)\delta(x-y)\,.$ (25)

For $\alpha\neq 1$, the constraints $\Omega_{1}$ and $\Omega_{2}$ satisfy a
second-class algebra. From the previous brackets, defining
$C_{ij}(x,y)\equiv\left\\{\Omega_{i}(x),\Omega_{j}(y)\right\\}$, with
$i,j=1,2$, and $C^{-1}_{ij}(x,y)$ by

$\int
dz\,\left\\{\,\Omega_{i}(x)\,,\,\Omega_{k}(z)\,\right\\}C_{kj}^{-1}(z,y)=\int
dz\,C_{ik}^{-1}(x,z)\left\\{\,\Omega_{k}(z)\,,\,\Omega_{j}(y)\,\right\\}=\delta_{ij}\delta(x-y)\,,$
(26)

we obtain the Dirac matrix $C_{ij}(x,y)$ and its corresponding inverse
$C^{-1}_{ij}(x,y)$ as

$\displaystyle C_{ij}(x,y)=\begin{bmatrix}0&1-\alpha\\\
\alpha-1&-2\partial_{x}\end{bmatrix}\delta(x-y)$ (27)

and

$\displaystyle
C^{-1}_{ij}(x,y)=\frac{1}{\alpha-1}\begin{bmatrix}-2\partial_{x}&1\\\
-1&0\end{bmatrix}\delta(x-y)\,,$ (28)

which are needed to compute de Dirac brackets (DB) algebra. In fact, by
inserting the inverse Dirac matrix (28) into the general definition

$\displaystyle\\{F(x),G(y)\\}_{D}=\\{F(x),G(y)\\}-\int
dzd\bar{z}\,\\{F(x),\Omega_{i}(z)\\}C^{-1}_{ij}(z,\bar{z})\\{\Omega_{j}(\bar{z}),G(y)\\}\,,$
(29)

the fundamental DB’s among the phase space variables can be computed as

$\displaystyle\left\\{\phi(x),\pi_{\phi}(y)\right\\}_{D}=\delta(x-y)\,,$ (30)
$\displaystyle\left\\{\phi(x),\phi(y)\right\\}_{D}=\left\\{\pi_{\phi}(x),\pi_{\phi}(y)\right\\}_{D}=0\,,$
(31)
$\displaystyle\left\\{\phi(x),\lambda(y)\right\\}_{D}=-\frac{1}{\alpha-1}\delta(x-y)\,.$
(32)

At this moment, one can proceed with the canonical quantization program by
ensuring that the corresponding operators adhere to commutation relations as
established by the DB algebra obtained above. In addition, other functional
quantization framework lines can be used. A key point in the GU formalism
amounts to reproducing the precise above DB structure by means of ordinary PBs
among the GU variables. We shall come to that fine result in a moment.

In order to apply the GU technique to the current model, it is convenient to
rescale the primary constraint $\Omega_{1}=\pi_{\lambda}$ to construct a
proper gauge symmetry generator

$\displaystyle\psi\equiv-\frac{\Omega_{1}}{(\alpha-1)}=-\frac{\pi_{\lambda}}{(\alpha-1)}\,,$
(33)

so that $\left\\{\psi(x),\Omega_{2}(y)\right\\}=\delta(x-y)$. The
infinitesimal gauge transformations generated by $\psi$ are

$\displaystyle\delta\phi(x)$ $\displaystyle=\int
dy\varepsilon(y)\left\\{\phi(x),\psi(y)\right\\}=0\,,$ (34)
$\displaystyle\delta\pi_{\phi}(x)$ $\displaystyle=\int
dy\varepsilon(y)\left\\{\pi_{\phi}(x),\psi(y)\right\\}=0\,,$ (35)
$\displaystyle\delta\lambda(x)$ $\displaystyle=\int
dy\varepsilon(y)\left\\{\lambda(x),\psi(y)\right\\}=-\frac{1}{(\alpha-1)}\varepsilon(x)\,,$
(36) $\displaystyle\delta\Omega_{2}(x)$ $\displaystyle=\int
dy\varepsilon(y)\left\\{\Omega_{2}(x),\psi(y)\right\\}=-\varepsilon(x)\,.$
(37)

Thus, we can see that precisely $\lambda$ and $\Omega_{2}$ are not invariant.
The corresponding GU gauge invariant variable $\tilde{\lambda}$ can be
constructed in terms of a power series in $\Omega_{2}$

$\displaystyle\tilde{\lambda}(x)=\lambda(x)+\int
dyb_{1}(x,y)\Omega_{2}(y)+\iint
dydzb_{2}(x,y,z)\Omega_{2}(y)\Omega_{2}(z)+...\,.$ (38)

Applying the GU invariant condition $\delta\tilde{\lambda}=0$, we can compute
all the coefficient functions $b_{n}$ corresponding to the correction terms in
the series (38). For the $\Omega_{2}$ linear order correction, we obtain

$\displaystyle\delta\lambda(x)+\int dyb_{1}(x,y)\delta\Omega_{2}(y)=0\,,$ (39)

from which, by using Eqs. (36) and (37), follows

$\displaystyle b_{1}(x,y)=-\frac{1}{(\alpha-1)}\delta(x-y)\,.$ (40)

For the quadratic term we have $b_{2}=0$ and it is clear that all the
remaining correction coefficient functions $b_{n}$ are null for $n\geq 2$.
Consequently, considering this fact and plugging Eq. (40) into (38), we can
derive the GU variable $\tilde{\lambda}$ as

$\displaystyle\tilde{\lambda}(x)=\lambda(x)-\frac{1}{(\alpha-1)}\Omega_{2}(x)$
$\displaystyle=-\frac{1}{\alpha-1}(\pi_{\phi}-\phi^{\prime})\,.$ (41)

Therefore, we have achieved our first task obtaining the deformed GU variables

$\displaystyle\tilde{\phi}=\phi\,,$ (42)
$\displaystyle\tilde{\pi}_{\phi}=\pi_{\phi}\,,$ (43)
$\displaystyle\tilde{\lambda}=-\frac{1}{\alpha-1}(\pi_{\phi}-\phi^{\prime})\,.$
(44)

The canonical Poisson bracket relations among the GU variables are given by

$\displaystyle\left\\{\tilde{\phi}(x),\tilde{\pi}_{\phi}(y)\right\\}=\delta(x-y)=\left\\{\phi(x),\pi_{\phi}(y)\right\\}_{D}\,,$
(45)
$\displaystyle\left\\{\tilde{\phi}(x),\tilde{\phi}(y)\right\\}=\left\\{\tilde{\pi}_{\phi}(x),\tilde{\pi}_{\phi}(y)\right\\}=0=\left\\{\phi(x),\phi(y)\right\\}_{D}=\left\\{\pi_{\phi}(x),\pi_{\phi}(y)\right\\}_{D}\,,$
(46)
$\displaystyle\left\\{\tilde{\phi}(x),\tilde{\lambda}(y)\right\\}=-\frac{1}{\alpha-1}\delta(x-y)=\left\\{\phi(x),\lambda(y)\right\\}_{D}\,.$
(47)

As we can see, the above Poisson brackets structure satisfied by the GU
variables completely agree with the previous results obtained in terms of DB’s
calculated from the definition (29) using the original phase space variables.
This fine result, as anticipated in db , provides an important calculational
shortcut and confirms the consistency of the improved GU formalism.

Replacing the second-class variables $\phi,\pi_{\phi}$ and $\lambda$ by the GU
variables $\tilde{\phi},\tilde{\pi}_{\phi}$ and $\tilde{\lambda}$, given by
Eqs. (42), (43) and (44) respectively, in the canonical Hamiltonian, Eq. (22),
we can derive the gauge invariant Hamiltonian density

$\displaystyle\tilde{\cal H}_{\alpha}$
$\displaystyle=\frac{1}{2}\pi_{\phi}^{2}+\frac{1}{2}\phi^{\prime}\,{}^{2}+\frac{1}{2}(\alpha-1)^{-1}(\pi_{\phi}-\phi^{\prime})^{2}$
$\displaystyle=\frac{1}{2}\frac{\alpha}{(\alpha-1)}\left(\pi_{\phi}-\phi^{\prime}\right)^{2}+\pi_{\phi}\phi^{\prime}.$
(48)

The GU Hamiltonian obtained above, Eq. (48), coincides with the second-class
reduced Hamiltonian obtained in kkp  and, as we can observe, it is not
possible to guarantee that $\tilde{\cal H}_{\alpha}$ will be positive for
$\alpha<1$. This result is in line with those previously discussed in harada ;
ggr ; kkp , regarding the fact that it is not possible to secure a positive
definite Hamiltonian for Srivastava’s model ($\alpha=0$).

In order to obtain a well-behaving limit $\alpha\rightarrow 1$ for the gauge
symmetry generator, we have to consider the vanishing of the numerator,
$\pi_{\lambda}$, in the corresponding term in Eq. (33), which is precisely the
primary constraint $\Omega_{1}\equiv\pi_{\lambda}\approx 0$. For the GU
Hamiltonian, we have to consider the vanishing of the numerator,
$(\pi_{\phi}-\phi^{\prime})^{2}$, in the corresponding term in Eq. (48), to
obtain a well-behaving limit $\alpha\rightarrow 1$, which results the
secondary constraint, Eq. (23),

$\displaystyle\Omega_{2(\alpha\rightarrow
1)}\equiv\pi_{\phi}-\phi^{\prime}\approx 0\,.$ (49)

Actually, condition (49) above is already necessary for a consistent behavior
of the GU variable $\tilde{\lambda}$ in that critical limit, c.f. Eq. (44).
The time evolution of (49) produces one additional constraint

$\displaystyle\Omega_{3}\equiv\lambda\approx 0\,.$ (50)

Further time evolution of $\Omega_{3}$ results in no additional constraints.
In our GU model, we have the fields $\phi,\pi_{\phi},\lambda,\pi_{\lambda}$,
three second-class constraints, which are Eqs. (20), (49) and (50), resulting
in the number of degrees of freedom (DOF) being equal to 1 which confirms
that, for $\alpha=1$, our theory is equivalent to the FJ model. This same
result was already obtained by KKP kkp  in a second-class extended CB system.

If we consider $\alpha=0$ in Eq. (48), we obtain the particular Hamiltonian

$\displaystyle\tilde{\mathcal{H}}_{0}=\pi_{\phi}\phi^{\prime}\,.$ (51)

In this case, we have the same four fields, namely
$\phi,\pi_{\phi},\lambda,\pi_{\lambda}$, and one first-class constraint,
$\pi_{\lambda}$, resulting in DOF=2 and confirming that, for $\alpha=0$, our
model is equivalent to Srivastava’s. Still in the $\alpha=0$ case, by using
the Hamilton equations of motion and performing an inverse Legendre
transformation
$\tilde{\mathcal{L}}_{0}=\pi_{\phi}\dot{\phi}-\tilde{\mathcal{H}}_{0}\,$ in
Eq. (51), we obtain the null result $\tilde{\mathcal{L}}_{0}=0$, plainly
agreeing with reference ggr .

Back to the general case, calculating the Hamilton equation of motion for an
arbitrary value of the parameter $\alpha$

$\displaystyle\frac{\delta\tilde{\mathcal{H}}_{\alpha}}{\delta\pi_{\phi}}=\frac{\alpha}{(\alpha-1)}\left(\pi_{\phi}-\phi^{\prime}\right)+\phi^{\prime}=\dot{\phi}\,,$
(52)

and using an inverse Legendre transformation, we can derive the GU Lagrangian

$\displaystyle\tilde{\mathcal{L}}_{\alpha}$
$\displaystyle=\frac{1}{2\alpha}\left[(\alpha-1)\dot{\phi}\dot{\phi}+2\dot{\phi}\phi^{\prime}-(1+\alpha)\phi^{\prime}\phi^{\prime}\right]$
$\displaystyle=\frac{1}{2}\left[(\dot{\phi}^{2}-{\phi^{\prime}}^{2})-\frac{1}{\alpha}(\dot{\phi}-\phi^{\prime})^{2}\right]\,.$
(53)

Here it can be mentioned that, although the initial Lagrangian, Eq. (17),
depends on the auxiliary field $\lambda$, the invariant Lagrangian, Eq. (Note
on an extended chiral bosons system contextualized in a modified gauge-
unfixing formalism), does not. Moreover, the GU invariant Lagrangian obtained
above presents a quadratic term on the chirality condition and, consequently,
the GU invariant Lagrangian has the same functional form as the invariant
Lagrangian proposed by Siegel to describe chiral bosons sig ; is . However,
the gauge transformations for the GU variables, c.f. Eqs. (34), (35) and (36),
are clearly different from Siegel’s, which is known to be anomalous due to its
gauge invariance breaking at quantum level. With the choice of $\Omega_{1}$
for constructing the gauge-symmetry generator in (33), we see that while
$\phi$ and $\pi_{\phi}$ do not transform, we have freedom to neatly gauge away
the Lagrange multiplier from the theory. The quest of whether such process can
be done at quantum level in a non-anomalous way is currently under
investigation.

Another way to understand the GU Lagrangian follows from the procedure where
the original second-class variables in the initial Lagrangian, Eq. (17), which
are $\phi$ and $\lambda$, are replaced directly with the GU variables which
are $\tilde{\phi}$ and $\tilde{\lambda}$, given by Eqs. (42) and (44)
respectively. Then, using the Hamiltonian equation of motion, Eq. (52), with
the aim of eliminating the momentum variable $\pi_{\phi}$, in which the GU
variable $\tilde{\lambda}$, as we can see, is written in terms of it, we again
obtain the GU Lagrangian, Eq. (Note on an extended chiral bosons system
contextualized in a modified gauge-unfixing formalism). Thus, we have
established a remarkable condition on the invariant Lagrangian within the
extended chiral boson model and modified GU formalism. Specifically, when
employing the transformation-invariant variable $\phi$ and the transformation-
exhibiting variable $\lambda$, the sole Lagrangian that preserves invariance
seems to be uniquely determined by Eq. (Note on an extended chiral bosons
system contextualized in a modified gauge-unfixing formalism).

For the limit $\alpha\rightarrow 0$, if we consider the general GU Lagrangian
(Note on an extended chiral bosons system contextualized in a modified gauge-
unfixing formalism), we have

$\displaystyle\tilde{\mathcal{L}}_{\alpha\rightarrow
0}=\lim_{\alpha\rightarrow
0}\;\frac{1}{2}\left[(\dot{\phi}-\phi^{\prime})(\dot{\phi}+\phi^{\prime})-\frac{1}{\alpha}(\dot{\phi}-\phi^{\prime})^{2}\right]\,.$
(54)

From Eq. (54), we can see that we have to impose the chirality condition
$\dot{\phi}=\phi^{\prime}$ in order to have the second term in the GU
Lagrangian well-behaved. This chirality condition just corresponds to
Srivastava’s model sriva . In this way, Eq. (54) reduces to

$\displaystyle\tilde{\mathcal{L}}_{\alpha\rightarrow
0}=-\lim_{\alpha\rightarrow
0}\;\frac{1}{2\alpha}(\dot{\phi}-\phi^{\prime})^{2}\,.$ (55)

Applying L’Hôpital’s rule to calculate the limit in (55), we have
$\tilde{\mathcal{L}}_{\alpha\rightarrow 0}=0$. This result agrees with the one
previously obtained performing a direct Legendre transformation to the
$\alpha=0$ invariant Hamiltonian, Eq. (51).

If we set $\alpha=1$, then the general GU Lagrangian, Eq. (Note on an extended
chiral bosons system contextualized in a modified gauge-unfixing formalism),
results in

$\tilde{\mathcal{L}}_{1}=\dot{\phi}\phi^{\prime}-\phi^{\prime 2}\,,$ (56)

which is precisely the Lagrangian density for the Floreanini-Jackiw model
fjprl . For the case $\alpha=\pm\infty$ in (Note on an extended chiral bosons
system contextualized in a modified gauge-unfixing formalism), we retrieve a
free scalar theory or the left-right chiral boson theory. It is worth
mentioning here that the secondary constraint, Eq. (23), due to the
noncommutative property, Eq. (24), is a single second-class constraint.
Therefore, it is not possible to use it to build another gauge symmetry
generator in the GU formalism, as it was done for the primary constraint
$\Omega_{1}$, Eq. (20).

In this note, we have converted an extended CB theory with a modified linear
constraint into a first-class system. We have used a modified GU formalism
where the phase space variables are redefined as first-class variables without
introducing any Wess-Zumino terms. Consequently, any function of these first-
class variables is gauge invariant. In general, the procedure of obtaining GU
variables simplifies the derivation of invariant quantities when compared with
the usual GU formalism. The general GU invariant Hamiltonian and Lagrangian
densities, respectively Eqs. (48) and (Note on an extended chiral bosons
system contextualized in a modified gauge-unfixing formalism), were obtained
in terms of the parameter $\alpha$. For $\alpha$ equal to zero, we have the
Srivastava model. For $\alpha$ equal to one, we have the Floreanini-Jackiw
model. Finally we can mention that once the GU variables are defined then they
will be the ones that will reveal the properties of chiral bosons theory in a
gauge invariant form. As a natural sequel of the present work, we are
currently investigating whether the GU procedure applied to the extended
chiral boson system leads to a theory that can be free or not of quantum
anomalies.

We would like to thank the anonymous Referee for valuable suggestions. We
would also like to acknowledge CAPES (Coordenação de Aperfeiçoamento de Peso
de Nível Superior) and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de
Minas Gerais) for financial support. Jorge Ananias Neto thanks CNPq (Conselho
Nacional de Desenvolvimento Científico e Tecnológico), Brazilian scientific
support federal agency, for partial financial support, CNPq-PQ, Grant number
307153/2020-7.

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