# Automatic and Human-AI Interactive Text Generation

Yao Dou1, Philippe Laban211footnotemark: 1, Claire Gardent3, Wei Xu1
1 Georgia Institute of Technology 2 Salesforce 3 CNRS/LORIA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Equal contribution.

### Note:

This tutorial proposal has been accepted and we will give the tutorial at ACL
2024.

##  1 Description

In this tutorial, we focus on text-to-text generation, a class of natural
language generation (NLG) tasks, that takes a piece of text as input and then
generates a revision that is improved according to some specific criteria
(e.g., readability or linguistic styles), while largely retaining the original
meaning and the length of the text. This includes many useful applications,
such as text simplification Xu et al. (2015a), paraphrase generation Narayan
et al. (2017); Iyyer et al. (2018); Goyal and Durrett (2020), style transfer
Xu et al. (2012); Rao and Tetreault (2018); Krishna et al. (2020),
neutralizing biased language Pryzant et al. (2020); Zhong et al. (2021),
decontextualization for question answering Choi et al. (2021a), factual
correction of human-written or machine-generated texts Gao et al. (2023), etc.

In contrast to summarization Liu and Lapata (2019) and open-ended text
completion (e.g., story) Fan et al. (2018) where the length or content between
input and output vary dramatically, the text-to-text generation tasks we
discuss in this tutorial are more constrained in terms of the semantic
consistency and targeted language styles. This level of control makes these
tasks ideal testbeds for studying the ability of models to generate text that
is both semantically adequate and stylistically appropriate. Moreover, these
tasks are interesting from a technical standpoint, as they require complex
combinations of lexical and syntactical transformations Heineman et al.
(2023a), stylistic control Kim et al. (2022), and adherence to factual
knowledge Devaraj et al. (2022) — all at once.

With a special focus on text simplification and revision, this tutorial aims
to provide an overview of the state-of-the-art natural language generation
research from four major aspects – Data, Models, Human-AI Collaboration, and
Evaluation – and to discuss and showcase a few significant and recent
advances: (1) the use of non-retrogressive approaches such as edit-based and
diffusion models; (2) the shift from fine-tuning to prompting with large
language models, which makes both human and automatic evaluation ever more
important; (3) the development of new learnable metric and fine-grained human
evaluation framework; (4) a growing body of studies and datasets on non-
English languages; (5) the rise of HCI+NLP+Accessibility interdisciplinary
research to create real-world writing assistant systems, bringing both
excitements and ethical implications.

We will start by introducing the preliminaries for text
simplification/revision, including the task formulation and high-quality
datasets. We will present several representative neural and language models
that excel in these tasks, and compare their strengths and weaknesses for
generating different types of text edits and errors. We will highlight the
newest developments in the Human-AI collaborative writing as well as in both
human and automatic evaluations. We will conclude with discussions on ethical
implications and future research directions.

Datasets For Text Revision Tasks. Text Simplification is one of the more
common text revision tasks and can serve as a good starting point for
attendees of the tutorial. More specifically, we will introduce the Newsela
corpus Xu et al. (2015a), a high-quality simplification resource in the news
domain, and resources based on Simple Wikipedia, including Turk Xu et al.
(2016), ASSET Alva-Manchego et al. (2020), CATS Štajner et al. (2018),
arXivEdits Jiang et al. (2022), and SWiPE Laban et al. (2023), gradually
building from sentence-level to document-level datasets. We will then survey
other text revision tasks, including tasks in the EditEval benchmark Dwivedi-
Yu et al. (2022), decontextualization Choi et al. (2021b), and iterative
codified editing Du et al. (2022b). Finally, we will showcase available
resources in different languages for text simplification, such as those in the
MultiSim benchmark Ryan et al. (2023), a growing collection of parallel
corpora in different languages constructed collectively by researchers all
over the world, which so far include: Arabic Khallaf and Sharoff (2022),
Basque Gonzalez-Dios et al. (2018), Brazilian Portuguese Aluísio and Gasperin
(2010), Danish Klerke and Søgaard (2012), English, French Gala et al. (2020);
Grabar and Cardon (2018); Cardon and Grabar (2020), German Mallinson et al.
(2020a); Säuberli et al. (2020); Aumiller and Gertz (2022); Trienes et al.
(2023); Naderi et al. (2019), Italian Miliani et al. (2022); Tonelli et al.
(2016); Brunato et al. (2016, 2015), Japanese Maruyama and Yamamoto (2018);
Katsuta and Yamamoto (2018); Goto et al. (2015), Russian Dmitrieva et al.
(2021); Dmitrieva and Tiedemann (2021); Sakhovskiy et al. (2021), Slovene
Gorenc and Robnik-Šikonja (2022), Spanish Orasan et al. (2013); Xu et al.
(2015b); Saggion et al. (2015), and Urdu Qasmi et al. (2020).

Neural and Language Models for Generating Semantically Adequate Texts. Various
modeling approaches have been used for text revision and simplification. We
begin with edit-based models, a family well-suited to text revision given the
substantial overlap in content between inputs and outputs. Mallinson et al.
(2020b) break down the task into sub-tasks of tagging tokens that are to kept
and inserting missing tokens. Mallinson et al. (2022) extend seq2seq models
for an end-to-end manner. Reid and Neubig (2022) propose multi-step editing
setup, while Du et al. (2022b) and Kim et al. (2022) incorporate edit
intention and paragraph-level edits into the training process. Agrawal and
Carpuat (2022) use a roll-in policy to generate realistic intermediate
sequences and apply curriculum training. Diffusion models Ho et al. (2020) is
another family in non-autoregressive generation. Adapted to text generation by
Li et al. (2022), these models are further tailored to conditional tasks such
as text simplification and paraphrase generation Gong et al. (2023); Yuan et
al. (2022). We then discuss the use of autoregressive models for text
revision. A simple and effective method is prepending of control tokens to the
input during fine-tuning. These tokens can be sentence-level attributes Martin
et al. (2020) or document-level edit plans Cripwell et al. (2023). Iterative
refinement with natural language instruction also gains attention, using
either fine-tuned language models Schick et al. (2022) or prompted LLMs111In
the InstructGPT paper Ouyang et al. (2022) which preceded ChatGPT, OpenAI
disclosed that “Rewrite” (text revision) made up 6.6% of use cases in their
API prompts, compared to 4.2% for “Summarization” and 8.4% for “Chat”. Madaan
et al. (2023). Moreover, Maddela et al. (2023) guide text simplification with
a metric using Minimum Bayes-Risk (MBR) decoding Kumar and Byrne (2004).

Human-AI Collaborative Writing. The recent progress in LLMs abilities has led
to many collaborative tools for writing, which involve realistic and diverse
text revision tasks. To provide a context on how human-AI collaboration are
evaluated, we will first introduce some HCI academic work on co-writing tools
prior to the existence of LLMs, such as Soylent Bernstein et al. (2010),
Wordcraft Coenen et al. (2021), and MIL Clark et al. (2018). Moving on to more
recent developments, several work use fine-tuned LLMs for collaborative
writing through iterative revision Du et al. (2022a) or interleaving planning
and generation Huot et al. (2023), while Lee et al. (2022) and Jakesch et al.
(2023) use GPT-3 Ouyang et al. (2022) with prompting. Finally, we will compare
and contrast new commercial LLM-based text revision tools (such as Grammarly
Go, QuillBot, ProWritingAid, etc.), and frame possible evaluation protocols
that could fairly compare such systems (and future academic systems) based on
the reviewed HCI work.

Automatic and Human Evaluation. Significant advancements have been made in
both human and automatic evaluation methods for text revision over the past
year. For automatic evaluation metrics, besides the commonly used SARI Xu et
al. (2016) and BERTScore Zhang et al. (2020), Maddela et al. (2023) introduce
the first learnable metric LENS that directly trained on human judgements. Liu
et al. (2023) use LLMs such as GPT-4 OpenAI (2023) for evaluation. Human
evaluation can be categorized into intrinsic or extrinsic methods. In terms of
intrinsic evaluation, most work prior to 2021 rely on 5-point Likert scale to
evaluate various aspects of text quality, while Alva-Manchego et al. (2021)
begin adopting direct assessment, rate from 0 to 100, for text simplification.
Maddela et al. (2023) incorporate ranking into direct assessment. Heineman et
al. (2023a) propose a fine-grained, edit-level framework, which is further
generalized into a platform for all types of fine-grained evaluations Heineman
et al. (2023b). On the extrinsic side, Laban et al. (2021) and others Angrosh
et al. (2014) evaluate text simplification through reading comprehension
questions.

##  2 Tutorial Outline

We will build in 10%-15% of time for Q&A in each part of the tutorial.

1\. Introduction (15 min.)

  * •

An overview of text generation research

  * •

An overview of the tutorial

  * •

Why text simplification and revision?

2\. Tasks and Datasets (30 min.)

  * •

Text Simplification

  * •

Other Text Revision Tasks

  * •

Multilingual Resources

3\. Neural and Language Models (50 min.)

  * •

Edit-based models

  * •

Diffusion models

  * •

Conditional language models

  * •

Large language Models with prompting

  * •

MBR decoding

4\. Automatic and Human Evaluation (25 min.)

  * •

Automatic Evaluation

  * •

Human Evaluation

5\. Human-AI Collaborative Writing (20 min.)

  * •

Co-Writing tools Prior LLMs

  * •

Co-Writing tools in the era of LLMs

  * •

Commercial tools

6\. Live Demos: Text Revision Models and Human-AI Collaboration (10 min.)

### 7\. Ethical Considerations (20 min.)

### 8\. Conclusions and Future Directions (10 min.)

##  3 Tutorial Information

Type of the Tutorial. Cutting Edge.

Length. This is a 3-hour tutorial.

Target Audience. The tutorial will be accessible to anyone who has a basic
knowledge of natural language processing. We think the topic will be of
interest to researchers, students, and practitioners in both academia and
industry. We will primarily target at the NLP community while bridging to a
broader audience from HCI, Linguistics, Accessibility, Education, and other
interdisciplinary fields.

Technical Equipment. Standard AC setup and access to the Internet to show
online demos.

Diversity Considerations. We see a significant rise in the number of
publications on the topic of Text Simplification in recent years, especially
by researchers who work on non-English languages in many different countries.
We will discuss resources and methods that support multilingual research,
different text domains (e.g., medical and legal texts), and special user
groups (e.g., deaf and hard-of-hearing users). The instructor team consists of
researchers at different career stages (50% female, 50% male; 25% Europe, 75%
U.S.): one senior researcher who is an ACL fellow, one professor, one Ph.D.
student, and one industrial researcher. We will reach out to academic
communities such as WiNLP,222https://www.winlp.org/ SIGEDU,333https://sig-
edu.org/ and Masakhane444https://www.masakhane.io/ to encourage participation
from more diverse audiences.

Estimated Size of Audience. Given the rising interests in text generation
(text simplification and revision in particular), we estimate the tutorial
will attract an audience of 100+, based on the recent success of Workshop on
Text Simplification, Accessibility, and Readability (TSAR) at EMNLP 2022 and
the birds-of-a-feather event (over 50 participants) on Text Simplification at
NAACL 2021.

Venues. We prefer ACL and NAACL due to travel constraints of some speakers.
EMNLP is our third option, and we currently do not consider EACL.

Breadth of Research Covered.  This tutorial is intended to cover recent
research advances from both academia and industry. We estimate that 20% of the
work covered in this tutorial will be from the presenters and the remaining
80% by others.

Open Access. We will build a website for the tutorial and share all teaching
material freely online. We also agree to allow the publication of slides and
video recordings by ACL.

##  4 Tutorial Instructors

Yao Dou (https://yao-dou.github.io/) is a Ph.D. student in the College of
Computing at the Georgia Institute of Technology, advised by Prof. Wei Xu. His
research interests lie in natural language processing and machine learning.
His recent work focuses on text simplification, text evaluation, and social
media privacy.

Philippe Laban (https://tingofurro.github.io/) is a Research Scientist at
Salesforce Research. His research is at the intersection of NLP and HCI,
focusing on several tasks within text generation, including text
simplification and summarization. He received his Ph.D. in Computer Science
from UC Berkeley in 2021. His thesis is titled “Unsupervised Text Generation
and its Application to News Interfaces”. His recent work has focused on
expanding the scope of text simplification to the paragraph and document-level
and evaluating text-editing interfaces. He publishes in both *ACL and HCI
conferences, including work on interactive user interface design for NLP
applications.

Claire Gardent (https://members.loria.fr/CGardent/) is a senior research
scientist at the French National Center for Scientific Research (CNRS), based
at the LORIA Computer Science research unit in Nancy, France. In 2022, she was
selected as an ACL Fellow and was awarded the CNRS Silver Medal. She works in
the field of NLP with a particular interest in Natural Language Generation. In
2017, she launched the WebNLG challenge, a shared task where the goal is to
generate text from Knowledge Base fragments. She has proposed neural models
for simplification and summarization; for the generation of long-form
documents such as multi-document summaries and Wikipedia articles; for
multilingual generation from Abstract Meaning Representations and for response
generation in dialog. She currently heads the AI XNLG Chair on multi-lingual,
multi-source NLG and the CNRS LIFT Research Network on Computational, Formal
and Field Linguistics.

Wei Xu (https://cocoxu.github.io/) is an Associate Professor in the College of
Computing at the Georgia Institute of Technology. Her recent research focuses
on text generation (including data construction, controllable model, human and
automatic evaluation), stylistics, analyzing and evaluating large language
models (including multilingual capability, cross-lingual transfer learning,
cultural bias, and cost efficiency). She is a recipient of the NSF CAREER
Award, CrowdFlower AI for Everyone Award, best paper award from COLING 2018,
and honorable mention from ACL 2023. She is an NAACL executive board member
and regularly serves as a (senior) area chair for *ACL conferences. She
frequently gives invited talks at universities and companies. She has given
tutorials on “NLP for Social Media and Text Analysis” and has organized
multiple workshops, including WNUT, GEM, and TSAR.

##  5 Ethical Statement

It is important to consider the ethical implications of text simplification
and revision research, especially in the realizations of them as assistive
technologies Gooding (2022). In the tutorial, we will cover many ethical
considerations of these technologies, including the distinctions and varied
needs of different targeted user groups (e.g., non-native speakers, children,
and people with dyslexia), best practices of crowdsourcing and user studies,
privacy preservation and safety concerns (e.g., when simplifying medical
texts), as well as relevant research findings in Alonzo et al. (2022); August
et al. (2023).

##  6 Reading list

  * •

SWiPE: A Dataset for Document-Level Simplification of Wikipedia Pages Laban et
al. (2023)

  * •

LENS: A Learnable Evaluation Metric for Text Simplification Maddela et al.
(2023)

  * •

RewriteLM: An Instruction-Tuned Large Language Model for Text Rewriting Shu et
al. (2023)

  * •

PEER: A Collaborative Language Model Schick et al. (2023)

  * •

DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models Gong et
al. (2023)

  * •

MUSS: Multilingual Unsupervised Sentence Simplification by Mining Paraphrases
Martin et al. (2022)

  * •

On the Ethical Considerations of Text Simplification Gooding (2022)

  * •

Making Science Simple: Corpora for the Lay Summarisation of Scientific
Literature Goldsack et al. (2022)

  * •

Findings of the TSAR-2022 Shared Task on Multilingual Lexical Simplification
Saggion et al. (2022)

  * •

Paragraph-level Simplification of Medical Texts Devaraj et al. (2021)

  * •

Neural CRF Model for Sentence Alignment in Text Simplification Jiang et al.
(2020)

  * •

CoAuthor: Designing a Human-AI Collaborative Writing Dataset for Exploring
Language Model Capabilities Lee et al. (2022)

  * •

Thresh: A Unified, Customizable and Deployable Platform for Fine-Grained Text
Evaluation Heineman et al. (2023b)

## Acknowledgements

The presenters’ research is supported in part by the NSF awards IIS-2144493
and IIS-2112633, ODNI and IARPA via the HIATUS program (contract
2022-22072200004). The views and conclusions contained herein are those of the
authors and should not be interpreted as necessarily representing the official
policies, either expressed or implied, of NSF, ODNI, IARPA, or the U.S.
Government. The U.S. Government is authorized to reproduce and distribute
reprints for governmental purposes notwithstanding any copyright annotation
therein.

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# Noether type formulation for space dependent polynomial symmetries

Rabin Banerjee
_S. N. Bose National Centre for Basic Sciences_ ,
_JD Block, Sector III, Salt Lake, Kolkata 700106, India_

###### Abstract

We develop a systematic algorithm, based on Noether’s theorem, for defining
the various currents in theories invariant under space dependent polynomial
symmetries. A master equation is given that yields all the conservation laws
corresponding to these currents. Explicit demonstration has been provided for
dipole and quadrupole conservation symmetries.

##  1 Introduction

For nearly a century, the search for novel quantum states of matter has been
an important and interesting scientific pursuit, both theoretically [1, 2, 3,
4] and experimentally [5, 6, 7]. Theoretical interests in novel classes of
quantum states of matter that do not fit well into any current framework have
recently grown [1, 2, 8, 9]. Among these are the fracton [8, 10] phases, and
they represent a new frontier of quantum physics. In contrast to other (quasi)
particles, quasi-excitation fractons are defined by their extremely low
mobility [8, 11]. As a result of their immobility, fractons are
extraordinarily slow to respond to an applied electric field. Fracton
composites, on the other hand, can move freely. On the theoretical side, the
fracton’s immobility in an isolated state and mobility in the presence of
additional fractons is a strange phenomenon [10, 11, 12]. Clearly, this
fracton phase of matter reveals an unknown (perhaps enlarged, beyond usual
space-time) symmetry which one can pursue to investigate using quantum field
theoretic methods [12, 13, 14, 15, 16, 17, 18, 19]. An essential tool is to
exploit, apart from charge conservation, higher order conservation laws
related to dipole, quadrupole and other n-pole symmetries [20]. These are
usually obtained by certain assumptions [14] or by inspection [12, 20].

In the present letter we give a systematic algorithm, based on the Noether
approach, to derive all the multiple currents and conservation laws related to
an n-pole symmetry. The parameter related to this symmetry is no longer
global. Rather, it is a polynomial of ($n-1$)th order in spatial coordinates
generating the various shift symmetries [21]. Thus we are able to generalise
the usual Noether approach, based on the first theorem, that is only valid for
a global symmetry leading to charge conservation.

We briefly introduce the Schroedinger theory, that has charge conservation, in
a way that is amenable to the generalisation that we are looking for. This is
followed by analysing, in successive sections, theories with linear, quadratic
and higher shift symmetries. Concrete realisations are provided.

##  2 Schroedinger Theory

The nonrelativistic Schroedinger theory is first discussed that illuminates
the various facets to be subsequently generalised. Consider an arbitrary
Lagrangian $\mathcal{L}(\Phi,\partial_{0}\Phi,\partial_{i}\Phi)$, where $\Phi$
is a complex scalar, that has a global $U(1)$ symmetry,

$\Phi\rightarrow e^{i\alpha}\Phi~{}~{},~{}~{}\Phi^{*}\rightarrow
e^{-i\alpha}\Phi^{*}$ (1)

Taking the infinitesimal version, $\delta\Phi=i\alpha\Phi$, the invariance of
the Lagrangian requires,

$\displaystyle\delta\mathcal{L}=i\alpha\frac{\partial\mathcal{L}}{\partial\Phi}\Phi+i\alpha\frac{\partial\mathcal{L}}{\partial(\partial_{0}\Phi)}\partial_{0}\Phi+i\alpha\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\partial_{i}\Phi+c.c.=0$
(2)

where $c.c.$ denotes complex conjugation.
The simplest (quadratic) Lagrangian satisfying (2) is just the Schroedinger
theory,

$\displaystyle\mathcal{L}=\frac{i}{2}\Phi^{*}\stackon[-4.2pt]{\partial}{\,\smash{\stackon[-1.95pt]{\mathchar
382\relax}{\rotatebox{180.0}{$\mathchar
382\relax$}}}}_{0}\Phi-\frac{1}{2m}\partial_{i}\Phi^{*}\partial_{i}\Phi$ (3)

with the global symmetry leading to the conservation law,

$\displaystyle\partial_{0}j_{0}-\partial_{i}j_{i}=0$ (4)

where the currents are defined by Noether’s first theorem,

$\displaystyle\alpha
j^{0}=\frac{\partial\mathcal{L}}{\partial(\partial_{0}\Phi)}\delta\Phi+c.c.~{}~{},~{}~{}\alpha
j^{i}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\delta\Phi+c.c.$
(5)

By using their explicit structures,

$\displaystyle
j^{0}=-j_{0}=-\Phi^{*}\Phi~{}~{};~{}~{}j^{i}=j_{i}=\frac{i}{2m}(\Phi^{*}\partial_{i}\Phi-\Phi\partial_{i}\Phi^{*})$
(6)

the conservation law (4) is easily verified, on shell. Conservation of charge
$Q=\int_{space}j_{0}$ also follows from (4).

The above discussion crucially hinges on the global nature of the parameter
$\alpha$. If $\alpha$ is taken to be local (i.e. spacetime dependent) then the
Lagrangian (3) is no longer invariant and the analysis breaks down. In the
next section we show how to modify the Noether procedure for a space dependent
parameter.

##  3 Theory with space-dependent linear shift symmetry

As a non-trivial example we try to pursue a similar analysis where the
Lagrangian is invariant under (1), but now with a space-dependent linear
shift,

$\alpha=\alpha_{0}+\alpha_{i}x_{i}$ (7)

with constant $\alpha_{0}$ and $\alpha_{i}$. Invariance under the above shift
symmetry for a Lagrangian that is a function of its fields and first
derivatives requires the following conditions,

$\displaystyle\frac{\partial\mathcal{L}}{\partial\Phi}(i\Phi)+\frac{\partial\mathcal{L}}{\partial(\partial_{0}\Phi)}(i\partial_{0}\Phi)+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}(i\partial_{i}\Phi)+c.c.=0$
(8a) $\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}(i\Phi)+c.c.=0$
(8b)

Both conditions cannot be satisfied simultaneously.111If we relax the
conditions modulo a total derivative so that the action is kept invariant,
then these can be satisfied. We discuss this later when considering explicit
realisations in section 3.1 Hence a more general Lagrangian has to be taken.
It turns out that it must have at least two space derivatives. Then,
invariance under (1) demands222The temporal contribution is not explicitly
written since the shift symmetry involves only spatial terms.,

$i(\alpha_{0}+\alpha_{k}x_{k})\bigg{[}\frac{\partial\mathcal{L}}{\partial\Phi}\Phi+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}(\partial_{i}\Phi)+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}(\partial_{i}\partial_{j}\Phi)\bigg{]}+c.c.=0$
(9a)
$\partial_{i}\alpha\bigg{[}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\Phi+2\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}(\partial_{j}\Phi)\bigg{]}+c.c.=0$
(9b)

While the first is associated with usual charge invariance, the second is new
and connected with dipole symmetry, which will be transparent as we go on.

Noether’s first theorem for a Lagrangian involving second derivatives leads to
a conserved current,

$\alpha
j_{i}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\delta\Phi+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}(\partial_{j}\delta\Phi)-\bigg{(}\partial_{j}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\bigg{)}\delta\Phi+c.c.$
(10)

where $\alpha$ is a constant. For the linearly shifted parameter $\alpha$ (7),
we posit the definition,

$\alpha
j_{i}-(\partial_{j}\alpha)j_{ji}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\delta\Phi+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}(\partial_{j}\delta\Phi)-\bigg{(}\partial_{j}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\bigg{)}\delta\Phi+c.c.$
(11)

This generalisation of the Noether current for a linear shift symmetry is
dictated by the fact that, for a constant (global) $\alpha$, it should reduce
to the usual definition; thus the correction must involve a linear derivative.
By appropriate contraction, a tensor current is introduced to achieve
consistency. We show it yields a viable definition. After an appropriate
reshuffling of derivatives, (11) is expressed as,

$\alpha
j_{i}-(\partial_{j}\alpha)j_{ji}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\delta\Phi+2\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}(\partial_{j}\delta\Phi)-\partial_{j}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\delta\Phi\bigg{)}+c.c.$
(12)

Using $\delta\Phi$ from (1), the condition (9b), and equating either side of
(12) yields,

$j_{i}=\partial_{j}j_{ji}$ (13)

where,

$j_{ji}=-i\bigg{[}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\Phi\bigg{]}$
(14)

The temporal part remains unaffected,

$\alpha
j_{0}=(\alpha_{0}+\alpha_{k}x_{k})j_{0}=\frac{\partial\mathcal{L}}{\partial(\partial_{0}\Phi)}\delta\Phi$
(15)

Relations (12) and (15) are used to deduce conservation laws analogous to (4),

$\partial_{0}(\alpha j_{0})-\partial_{i}[\alpha
j_{i}-(\partial_{j}\alpha)j_{ji}]=0$ (16)

which yields,

$\alpha[\partial_{0}j_{0}-\partial_{i}j_{i}]-\partial_{i}\alpha
j_{i}+\partial_{j}\alpha\partial_{i}j_{ji}=0$ (17)

Using (13) leads to the conservation law,

$\displaystyle\partial_{0}j_{0}-\partial_{i}j_{i}=0$ (18)

This derivation of the charge conservation hides the fact that there are two
conservation laws, related to charge and dipole symmetry associated with the
parameters $\alpha_{0}$ and $\alpha_{k}$ respectively. We thus equate
coefficients of $\alpha_{0}$ and $\alpha_{k}$ separately from (16). From
$\alpha_{0}$, the standard current conservation (18) is again reproduced.
Equating coefficients of $\alpha_{k}$ gives,

$\displaystyle\partial_{0}J_{0}^{k}-\partial_{i}J_{i}^{k}=0$ (19)

where the new currents are,

$\displaystyle J_{0}^{k}=x_{k}j_{0}$ (20a) $\displaystyle
J_{i}^{k}=x_{k}j_{i}-j_{ki}=x_{k}\partial_{j}j_{ji}-j_{ki}$ (20b)

All relations (13), (14), (18), (19) and (20) follow from the definition (11)
without any additional inputs. This shows the consistency of the formulation.

Relation (19) yields the familiar dipole conservation law,

$\displaystyle
Q^{k}=\int_{space}x^{k}j_{0}=\int_{space}J_{0}^{k}~{}~{}~{};~{}~{}\dot{Q}^{k}=0$
(21)

Since tensor sources $j_{ij}$ have appeared, it implies that these theories
admit symmetric tensor gauge fields. Their gauge transformation properties are
determined by using relation (13) to connect usual vector couplings with
tensor couplings,

$j_{i}A_{i}=\partial_{j}j_{ji}A_{i}=j_{ij}A_{ij}$ (22)

where a reshuffling of derivatives yields,

$A_{ij}=-\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})$ (23)

Usual gauge transformations for $A_{i}$,

$A_{i}\rightarrow A_{i}+\partial_{i}\beta$ (24)

gives the gauge transformation for $A_{ij}$,

$A_{ij}\rightarrow A_{ij}-\partial_{i}\partial_{j}\beta$ (25)

where $\beta$ is the gauge parameter. The matter sector can be coupled to
usual gauge fields with the standard Maxwell Lagrangian, or equivalently, to
symmetric tensor gauge fields with a nonstandard Lagrangian [13, 17].

###  3.1 Explicit realisations

Although an explicit realisation of a theory with charge and dipole
conservation has been given in [13, 18, 19] using fracton gauge principle or
an appropriate change of variables [14], it is nice to reproduce this result
in our approach and see its consequences.

It is possible to construct a viable Lagrangian, depending on two spatial
derivatives so that it satisfies the conditions (9). Especially (9b)
immediately suggests the following structure,

$\mathcal{L}_{t}=t|\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi|^{2}$
(26)

where $t$ is just a normalising factor. This Lagrangian trivially satisfies
(9a) also. By using (14) we obtain,

$\displaystyle
j_{ij}=-i\bigg{(}\Phi^{2}(\Phi^{*}\partial_{i}\partial_{j}\Phi^{*}-\partial_{i}\Phi^{*}\partial_{j}\Phi^{*})-{\Phi^{*}}^{2}(\Phi\partial_{i}\partial_{j}\Phi-\partial_{i}\Phi\partial_{j}\Phi)\bigg{)}$
(27)

The above current involves derivatives and transforms nontrivially under
rotations.

A rotationally invariant current, one that does not involve derivatives, can
also be found. In this case consider a Lagrangian that involves only single
derivatives,

$\mathcal{L}_{u}=u\,i\Big{(}{\Phi^{*}}^{2}\partial_{i}\Phi\partial_{i}\Phi-\Phi^{2}\partial_{i}\Phi^{*}\partial_{i}\Phi^{*}\Big{)}$
(28)

This Lagrangian is akin to conventional theories, not involving higher
derivatives. To see if it has both charge and dipole symmetries, we have to
now look for the conditions (8) since these are the relevant ones for this
case. We find that the first one, expectedly, is satisfied trivially, while
the second leads to,

$\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}(i\Phi)+c.c.=-\partial_{i}(|\Phi|^{4})$
(29)

Thus, if we allow for quasi invariance (the action being invariant), then this
is also an allowed form.

The Noether formulation has to account for this boundary term, apart from the
fact that only linear derivatives appear. In this case, therefore, the Noether
definition (11) is modified as,

$\alpha
j_{i}-(\partial_{j}\alpha)j_{ji}=\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\delta\Phi+c.c.+\alpha_{i}|\Phi|^{4}$
(30)

Comparing coefficients of $\alpha$ and $\alpha_{i}$, one is immediately led to
the results,

$j_{ij}=-\delta_{ij}|\Phi|^{4}\,\,;\,\,j_{i}=-\partial_{i}|\Phi|^{4}=\partial_{j}j_{ji}$
(31)

The tensor current is rotationally invariant and is connected to the vector
current in the same manner as discussed earlier for theories satisfying both
charge and dipole symmetries.

We next discuss the example of shift symmetry involving quadratic terms.

##  4 Theory with space-dependent quadratic shift symmetry

The analysis in the previous section is now extended to include quadratic
terms in the shift symmetry,

$\alpha=\alpha_{0}+\alpha_{i}x_{i}+\frac{1}{2}\alpha_{ij}x_{i}x_{j}$ (32)

Following previous arguments it is seen that the Lagrangian must contain terms
with triple derivatives. Then invariance of the Lagrangian
$\mathcal{L}(\Phi,\partial_{0}\Phi,\partial_{i}\Phi,\partial_{i}\partial_{j}\Phi,\partial_{i}\partial_{j}\partial_{k}\Phi)$
under the infinitesimal transformation $\delta\Phi=i\alpha\Phi$ requires,

$i\bigg{[}\frac{\partial\mathcal{L}}{\partial\Phi}\Phi+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\partial_{i}\Phi+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\partial_{i}\partial_{j}\Phi+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\partial_{i}\partial_{j}\partial_{k}\Phi\bigg{]}+c.c.=0$
(33a)
$i\bigg{[}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\Phi+2\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}(\partial_{j}\Phi)+3\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\partial_{j}\partial_{k}\Phi\bigg{]}+c.c.=0$
(33b)
$i\bigg{[}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\Phi+3\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\partial_{k}\Phi\bigg{]}+c.c.=0$
(33c)

These conditions are essential for deriving conservation laws related to
charge, dipole and quadrupole symmetries, respectively. The generalised
Noether current for this system with triple derivatives is given by following
the same logic used in writing (11),

$\displaystyle\alpha
j_{i}-(\partial_{j}\alpha)j_{ji}+(\partial_{j}\partial_{k}\alpha)j_{jki}$
$\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\delta\Phi-\partial_{j}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\bigg{)}\delta\Phi+\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\partial_{j}\delta\Phi+\partial_{j}\partial_{k}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\bigg{)}\delta\Phi$
(34) $\displaystyle+$
$\displaystyle\frac{{\partial\mathcal{L}}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\partial_{j}\partial_{k}\delta\Phi-\partial_{k}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\bigg{)}\partial_{j}\delta\Phi+c.c.$

where, expectedly, double derivatives appear while rank 2 and rank 3 tensor
currents are naturally introduced. After a rearrangement of the derivatives on
the right hand side, the above equation is put in the following form:

$\displaystyle\alpha
j_{i}-(\partial_{j}\alpha)j_{ji}+(\partial_{j}\partial_{k}\alpha)j_{kji}$
$\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial(\partial_{i}\Phi)}\delta\Phi+2\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\partial_{j}\delta\Phi+3\frac{{\partial\mathcal{L}}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\partial_{j}\partial_{k}\delta\Phi$
$\displaystyle-$
$\displaystyle\partial_{j}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\Phi)}\delta\Phi+2\frac{{\partial\mathcal{L}}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\partial_{k}\delta\Phi\bigg{)}+\partial_{j}\partial_{k}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\delta\Phi\bigg{)}+c.c.$

The advantage of this form is that it is readily generalisable to higher
orders, as will be discussed later. Further, the algebra simplifies on using
the conditions (33b,33c). Carrying out the complete computation yields,

$j_{i}=\partial_{j}\partial_{k}j_{jki}\,\,;\,\,j_{ji}=\partial_{k}j_{kji}$
(36)

where,

$j_{ijk}=i\frac{{\partial\mathcal{L}}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\Phi$
(37)

The tower of conservation laws may now be derived from (4), after including
the temporal contribution which remains unaffected,

$\displaystyle\partial_{0}(\alpha j_{0})-\partial_{i}\bigg{(}\alpha
j_{i}-(\partial_{j}\alpha)\partial_{k}j_{ijk}+(\partial_{j}\partial_{k}\alpha)j_{ijk}\bigg{)}=0$
(38)

Factoring out $\alpha$ yields,

$\displaystyle\alpha(\partial_{0}j_{0}-\partial_{i}j_{i})-\bigg{(}(\partial_{i}\alpha)j_{i}-(\partial_{i}\partial_{j}\alpha)\partial_{k}j_{ijk}-(\partial_{j}\alpha)\partial_{i}\partial_{k}j_{ijk}+(\partial_{j}\partial_{k}\alpha)\partial_{i}j_{ijk}+(\partial_{i}\partial_{j}\partial_{k}\alpha)j_{ijk}\bigg{)}=0$

Using (36) and the fact that $\alpha$ is a quadratic function in $x_{i}$ (32),
we conclude -

$\displaystyle\partial_{0}j_{0}-\partial_{i}j_{i}=0$ (40)

While this is an economical way of getting the basic (charge) conservation, it
conceals the effect of the other two, namely dipole and quadrupole symmetries.
These are obtained by independently considering the equalities related to
$\alpha$, opening it by using (32) and inserting in (38). Then $\alpha_{0}$
just leads to the charge conservation. Next comparing factors multiplying
$\alpha_{i}$ gives,

$\displaystyle\partial_{0}(\alpha_{j}x_{j}j_{0})-\partial_{i}(\alpha_{j}x_{j}j_{i}-\alpha_{j}\partial_{k}j_{ijk})=0$
(41)

Defining new currents,

$\displaystyle
J_{0j}=x_{j}j_{0}~{}~{},~{}~{}J_{ji}=x_{j}j_{i}-\partial_{k}j_{ijk}=x_{j}\partial_{l}\partial_{m}j_{lmi}-\partial_{k}j_{ijk}$
(42)

leads to the dipole conservation,

$\displaystyle\partial_{0}J_{0j}-\partial_{i}J_{ji}=0$ (43)

Finally, comparing factors multiplying $\alpha_{ij}$,

$\displaystyle\partial_{0}\bigg{(}\frac{1}{2}\alpha_{ij}x_{i}x_{j}j_{0}\bigg{)}-\partial_{i}\bigg{(}\frac{1}{2}\alpha_{jk}x_{j}x_{k}j_{i}-\frac{1}{2}\alpha_{jl}(x_{l}\partial_{k}j_{ijk}+x_{j}\partial_{k}j_{ilk})+\alpha_{jk}j_{jki}\bigg{)}=0$
(44)

where the second term in the second parenthesis has been suitably symmetrised.
Thus, the above equation provides the quadrupole conservation,

$\displaystyle\partial_{0}J_{ij0}-\partial_{k}J_{kji}=0$ (45)

where the new currents are defined by,

$\displaystyle J_{ij0}=\frac{1}{2}x_{i}x_{j}j_{0}$ (46)

and,

$\displaystyle
J_{kji}=\frac{1}{2}x_{i}x_{j}j_{k}-\frac{1}{2}(x_{i}\partial_{p}j_{kjp}+x_{j}\partial_{p}j_{kip})+j_{jik}=\frac{1}{2}\Big{(}x_{i}x_{j}\partial_{p}\partial_{q}j_{pqk}-x_{i}\partial_{p}j_{kjp}-x_{j}\partial_{p}j_{kip}\Big{)}+j_{jik}$
(47)

It is easy to verify, by substituting (46) and (47) in (45), that,

$\displaystyle
x_{i}x_{j}\bigg{(}\partial_{0}j_{0}-\partial_{k}j_{k}\bigg{)}=0$ (48)

on account of (40) and serves as a consistency check.

###  4.1 Explicit realisation

We now provide an explicit realisation of a theory having charge, dipole and
quadrupole conservations. A general term in the Lagrangian which will respect
the three conservation laws will involve three spatial derivatives. Hence it
can be written as,

$\displaystyle{{\cal
L}_{v}}=v|\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+b\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi+c\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)|^{2}$
(49)

where $b$ and $c$ are constant real numbers. Note that the coefficient of the
first term has been scaled to unity. These coefficients will be determined by
using the conditions (33). The first of these is trivially satisfied since it
conforms to the basic charge conservation. Nontrivial consequences follow from
the other two conditions. The last one, for instance, yields,

$\displaystyle
i(c+1)\bigg{(}|\Phi|^{4}\partial_{k}\Phi\partial_{ijk}\Phi^{*}+b\Phi^{2}\partial_{k}\Phi\partial_{i}\Phi^{*}\partial_{j}\Phi^{*}\partial_{k}\Phi^{*}$
$\displaystyle+|\Phi|^{2}\Phi\partial_{k}\Phi(\partial_{i}\Phi^{*}\partial_{j}\partial_{k}\Phi^{*}+\partial_{j}\Phi^{*}\partial_{k}\partial_{i}\Phi^{*}+\partial_{k}\Phi^{*}\partial_{i}\partial_{j}\Phi^{*})\bigg{)}+c.c.=0$
(50)

which gives $c=-1$. Taking the other condition now gives $b=2$. This
completely fixes the structure of the lagrangian 333After completing our paper
we found that a similar structure was given in [22], but without any
explanation regarding the method of its obtention.. The operator inside the
modulus in (49) transforms covariantly under the polynomial shift symmetry
[22]. Finally, including other possible terms, the Lagrangian looks like,

$\displaystyle\mathcal{L}=$ $\displaystyle
u|\Phi^{*3}\bigg{(}\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)\bigg{)}|^{2}$
$\displaystyle+v|\Phi^{2}\partial_{i}\partial_{j}\partial_{k}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{k}\Phi+\partial_{j}\Phi\partial_{k}\partial_{i}\Phi+\partial_{k}\Phi\partial_{i}\partial_{j}\Phi)|^{2}$
$\displaystyle+t|\Phi^{2}\partial_{i}\partial_{j}\partial_{j}\Phi+2\partial_{i}\Phi\partial_{j}\Phi\partial_{j}\Phi-\Phi(\partial_{i}\Phi\partial_{j}\partial_{j}\Phi+\partial_{j}\Phi\partial_{j}\partial_{i}\Phi+\partial_{j}\Phi\partial_{i}\partial_{j}\Phi)|^{2}-m^{2}\Phi^{\dagger}\Phi$

Now, the various currents corresponding to the $v$-th term in the Lagrangian
(4.1) are obtained from (36) where,

$\displaystyle j_{ijk}$ $\displaystyle=$ $\displaystyle
i\frac{\partial\mathcal{L}}{\partial(\partial_{i}\partial_{j}\partial_{k}\Phi)}\Phi+c.c.$
$\displaystyle=$ $\displaystyle
i\bigg{(}|\Phi|^{4}\Phi\partial_{i}\partial_{j}\partial_{k}\Phi^{*}+2\Phi^{3}\partial_{i}\Phi^{*}\partial_{j}\Phi^{*}\partial_{k}\Phi^{*}-|\Phi|^{2}\Phi^{2}(\partial_{i}\Phi^{*}\partial_{j}\partial_{k}\Phi^{*}+\partial_{j}\Phi^{*}\partial_{k}\partial_{i}\Phi^{*}+\partial_{k}\Phi^{*}\partial_{i}\partial_{j}\Phi^{*})\bigg{)}+c.c$

and likewise for other pieces in the Lagrangian.

##  5 Generalisation to n-pole symmetries

We now seek for a generalisation of our analysis which was carried out in the
preceding sections to include ($n-1$)th order spatial shift symmetries,

$\alpha=\alpha_{0}+\alpha_{i}x_{i}+\frac{1}{2}\alpha_{ij}x_{i}x_{j}+......+\frac{1}{n-1}\alpha_{i_{1}...i_{n-1}}x_{i_{1}}...x_{i_{n-1}}$
(53)

Then the general structure of Noether currents for a theory with
$\mathcal{L}(\Phi,\partial_{0}\Phi,\partial_{l_{1}}\Phi,\partial_{l_{1}}\partial_{l_{2}}\Phi,\partial_{l_{1}}\partial_{l_{2}}...\partial_{l_{n}}\Phi)$
is given in the following form,

$\displaystyle\alpha j_{l_{1}}$ $\displaystyle-$
$\displaystyle(\partial_{{l_{2}}}\alpha)j_{l_{1}l_{2}}+(\partial_{l_{2}}\partial_{l_{3}}\alpha)j_{l_{1}l_{2}l_{3}}+...+(-1)^{n-1}(\partial_{l_{2}}\partial_{l_{3}}..\partial_{l_{n}}\alpha)j_{{l_{1}}{l_{2}}..{l_{n}}}$
$\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial(\partial_{l_{1}}\Phi)}\delta\Phi+2\frac{\partial\mathcal{L}}{\partial(\partial_{l_{1}}\partial_{l_{2}}\Phi)}\partial_{l_{2}}\delta\Phi+..+n\frac{{\partial\mathcal{L}}}{\partial(\partial_{l_{1}}\partial_{l_{2}}..\partial_{l_{n}}\Phi)}\partial_{l_{2}}\partial_{l_{3}}..\partial_{l_{n}}\delta\Phi$
$\displaystyle-$
$\displaystyle\partial_{l_{2}}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{l_{1}}\partial_{l_{2}}\Phi)}\delta\Phi+.....+(n-1)\frac{{\partial\mathcal{L}}}{\partial(\partial_{l_{1}}\partial_{l_{2}}..\partial_{l_{n}}\Phi)}\partial_{l_{3}}..\partial_{l_{n}}\delta\Phi\bigg{)}$
$\displaystyle+$
$\displaystyle\partial_{l_{2}}\partial_{l_{3}}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{l_{1}}\partial_{l_{2}}\partial_{l_{3}}\Phi)}\delta\Phi+.....+(n-2)\frac{{\partial\mathcal{L}}}{\partial(\partial_{l_{1}}\partial_{l_{2}}..\partial_{l_{n}}\Phi)}\partial_{l_{4}}..\partial_{l_{n}}\delta\Phi\bigg{)}$
$\displaystyle+$
$\displaystyle.....+(-1)^{n-1}\partial_{l_{2}}\partial_{l_{3}}..\partial_{l_{n}}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{l_{1}}\partial_{l_{2}}..\partial_{l_{n}}\Phi)}\delta\Phi\bigg{)}+c.c.$

which is a natural extension of the $n=2$ (11) and $n=3$ (34) cases. All
tensor currents of rank less than $n$ are derivable from the highest rank $n$
tensor current,

$j_{l_{1}}=\partial_{l_{2}}\partial_{l_{3}}..\partial_{l_{n}}j_{l_{1}l_{2}...l_{n}}\,\,;\,\,j_{l_{1}l_{2}}=\partial_{l_{3}}..\partial_{l_{n}}j_{{l_{1}}{l_{2}}..{l_{n}}}\,\,;\,\,j_{l_{1}l_{2}l_{3}}=\partial_{l_{4}}..\partial_{l_{n}}j_{{l_{1}}{l_{2}}..{l_{n}}}\,\,;\,\,.......$
(55)

where the $n$-th rank fully symmetric tensor current is given by,

$j_{l_{1}l_{2}...l_{n}}=(-1)^{n-1}i\frac{{\partial\mathcal{L}}}{\partial(\partial_{l_{1}}\partial_{l_{2}}...\partial_{l_{n}}\Phi)}\Phi$
(56)

The multiple conservation laws now follow from the following “master
equation”,

$\displaystyle\partial_{0}(\alpha j_{0})-\partial_{i}\bigg{(}\alpha
j_{i}-(\partial_{{l_{2}}}\alpha)j_{il_{2}}+(\partial_{l_{2}}\partial_{l_{3}}\alpha)j_{il_{2}l_{3}}+...+(-1)^{n-1}(\partial_{l_{2}}\partial_{l_{3}}..\partial_{l_{n}}\alpha)j_{{i}{l_{2}}..{l_{n}}}\bigg{)}=0$

If we factor out $\alpha$ then there there is a pairwise cancellation of the
various terms, effected by exploiting (55) and we are left with the charge
conservation law,

$\displaystyle\alpha(\partial_{0}j_{0}-\partial_{i}j_{i})=0$ (58)

Now, note that this procedure hides the additional conservation laws which can
be uncovered, as before, by opening up $\alpha$ using (53) and inserting in
the “master equation” (5) to yield (58) together with the following tower of
($n-1$) additional conservation laws,

$\displaystyle\partial_{0}J_{0}^{k}-\partial_{i}J_{i}^{k}=0$ (59)
$\displaystyle\partial_{0}J_{0}^{jk}-\partial_{i}J_{i}^{jk}=0$ (60)
$\displaystyle.....................=0$ (61)
$\displaystyle\partial_{0}J_{0}^{l_{1}l_{2}..l_{n-1}}-\partial_{i}J_{i}^{l_{1}l_{2}..l_{n-1}}=0$
(62)

obtained by separately equating the coefficients of $\alpha_{0}$,
$\alpha_{k}$, $\alpha_{jk}$ till $\alpha_{l_{1}l_{2}..l_{n-1}}$ respectively.
As an illustration the dipole conservation law for a system exhibiting the
n-pole symmetry in the present formulation is given by (59) where the currents
are defined in (20). The rank 2 current $j_{ik}$ appearing there is obtained
from (55).

##  6 Conclusions

Theories with polynomial space dependent shift symmetries, originally
introduced in [21], have recently been studied extensively due to their wide
applications in various branches of physics, especially in unravelling and
understanding features brought about by the imposition of new conservation
laws that go beyond usual charge conservation [17, 18, 19, 20, 22]. These
conservation laws were usually obtained by certain assumptions or by
inspection.

Polynomial space dependent shift symmetries involve, apart from standard
invariance under global (constant) transformations yielding charge
conservation, higher $n$-pole conservation laws. A natural question is whether
Noether’s first theorem, that is strictly valid for global symmetries, can be
extended to include such space dependent shift symmetries.

In this paper we have provided an appropriate generalisation of Noether’s
first theorem. A definition has been given that yields all the $n$-pole
currents. Also, a master equation has been provided from which various
conservation laws associated with these currents are derived. The case of
quasi-invariance where the Lagrangian changes by a boundary term under the
transformation has also been accommodated.

A consequence of these shifted symmetries is the occurrence of higher
derivative theories. This has raised issues in the literature [20] regarding
their incorporation within the conventional fold of field theories. The
successful generalisation of Noether’s theorem presented in this paper would
be a welcome step in this direction.

As a future possibility we hope to extend the present formulation to Noether’s
second theorem by carrying out a suitable gauging of the theories given here.

##  7 Acknowledgements

This work has been supported by a (DAE) Raja Ramanna Fellowship.

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