# Who Are All The Stochastic Parrots Imitating? They Should Tell Us!

Sagi Shaier∇ Lawrence E. Hunter† Katharina von der Wense∇♢♠
∇University of Colorado Boulder
†University of Colorado Denver
♢Johannes Gutenberg University Mainz
∇{sagi.shaier<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Both standalone language models (LMs) as well as LMs within downstream-task
systems have been shown to generate statements which are factually untrue.
This problem is especially severe for low-resource languages, where training
data is scarce and of worse quality than for high-resource languages. In this
opinion piece, we argue that LMs in their current state will never be fully
trustworthy in critical settings and suggest a possible novel strategy to
handle this issue: by building LMs such that can cite their sources – i.e.,
point a user to the parts of their training data that back up their outputs.
We first discuss which current NLP tasks would or would not benefit from such
models. We then highlight the expected benefits such models would bring, e.g.,
quick verifiability of statements. We end by outlining the individual tasks
that would need to be solved on the way to developing LMs with the ability to
cite. We hope to start a discussion about the field’s current approach to
building LMs, especially for low-resource languages, and the role of the
training data in explaining model generations.

$\spadesuit$$\spadesuit$footnotetext: Formerly: Katharina Kann

##  1 Introduction

Transformers Vaswani et al. (2017) and related models have been improving
rapidly, with applications in a surprisingly large number of domains, such as
natural language generation Zhang et al. (2019), machine translation Wang et
al. (2019), question answering Akermi et al. (2020), and code generation
Svyatkovskiy et al. (2020), based on the ability to generate sensible outputs
to prompts over a nearly limitless input domain.

Figure 1: An actual conversation with ChatGPT in Hebrew on the effects of not
drinking enough water. ChatGPT is unable to point the user to its sources and
instead falls back to a general answer (“I am ChatGPT, an OpenAI model based
on the GPT-3.5 deep learning model. I am powered by OpenAI’s learning set,
which has been raised with the help of machine learning techniques on Internet
culture, including websites, books, articles, quotes, and more”). We argue
that ChatGPT and similar models should be able to direct the user to the
sources of their information, which will have multiple benefits, such as quick
verifiability of model statements.

Despite impressive performance on a wide array of benchmark tasks, these
models are known to produce “AI-splaining,” confident sounding but incorrect
statements: “To the extent that a use case places importance on the truth of
the outputs provided, it is not a good fit for GPT-3” Dale (2021); see also
Church et al. (2022) and Marcus (2019).

This problem has proven to be especially true for models trained on low-
resource languages Guerreiro et al. (2023), where data may not only be scarce
Mager et al. (2018), but also not well curated with respect to correctness or
quality, in comparison to higher-resource languages Hedderich et al. (2021).
Furthermore, model hallucination in such settings can result in toxic patterns
that can be found in the training data Guerreiro et al. (2023).

In accordance with the large LMs and low-resource languages theme track, we
argue that while the performance and factuality of LMs has been improving,
both in high-resource and low-resource settings, in their existing state, LMs
will realistically never be fully trustworthy. Thus, in settings in which
factuality is required, such as medicine, they are dangerous and unemployable.
This is further noted in Menick et al. (2022), who state that users cannot
trust any claim a model makes without fact-checking.

Our proposal to address these concerns suggests both technical development and
a simple regulatory framework: as we often ask students, journalists and
scholars, we should ask our models to name their sources and provide evidence
for their assertions. Currently, even popular LMs often fail at this, as seen
on the ChatGPT OpenAI (2023) example in Figure 1. In the case of generative
models, either the model itself or a post-hoc procedure could – and, under
certain circumstances, _should be required to_ – be designed to produce
evidentiary justification for its output.

NLP tasks would benefit from such citation models, discuss the benefits they
would bring, and present a roadmap to develop such models. Our goal is to
motivate the field to start thinking about what is necessary to make current
models truly useful in all sorts of – potentially critical – scenarios.

##  2 Background

#### Factuality and the Lack Thereof

LMs store factual knowledge Dong et al. (2022); De Cao et al. (2021); Elazar
et al. (2021); Jiang et al. (2019) and previous work have shown that LMs can
act as knowledge bases Petroni et al. (2019); Sung et al. (2021). However,
there is no guarantee that the retrieved knowledge is indeed factual, and
unfortunately, often it is not. This can be seen in many areas, such as
question answering Xu et al. (2021), dialogue systems Dziri et al. (2021);
Shuster et al. (2021); Testoni and Bernardi (2021), image captioning Rohrbach
et al. (2018), text summarization Zhao et al. (2020b); Cao et al. (2022);
Maynez et al. (2020) and translation Raunak et al. (2021); Jeblick et al.
(2022). This is especially true in low-resource settings Guerreiro et al.
(2023). In order for LMs to be fully utilized as such knowledge bases and in
settings where factuality is crucial, the retrieved knowledge must first be
factual. But, without knowing the source of such the model’s knowledge,
verifying its factuality is a challenge.

#### Citation Generation

Although LMs, particularly those intended to produce scientific text, such as
Meta’s Galactica Taylor et al. (2022), already produce text that looks as if
it is a citation, frequently there is no document corresponding to the
apparent citation or the cited document does not support the statement
associated with it. Many existing approaches to citation recommendation offer
productive avenues to explore for factuality testing, post-hoc generation of
support, hybrid architectures, or creation of training data Ali et al. (2022);
Krasnova et al. (2023). There has also been work on citation generation, where
the task is either: 1) given two documents, generate an explanation for the
relation between them Luu et al. (2020), or 2) generate a citation for an
already existing text Gu and Hahnloser (2022); Xing et al. (2020); Wu et al.
(2021); Fetahu et al. (2016). This is different from our suggestion to
generate statements and citations simultaneously, and also not optimal: as LMs
are being trained on massive datasets, evaluating whether each statement came
from each of the potentially millions of article becomes impractical. Lastly,
many existing systems that can in fact provide citations are based on search
engines or retrieval models Menick et al. (2022); Glaese et al. (2022), see
also Perplexity AI\spadesuit\spadesuit$\spadesuit$https://www.perplexity.ai/,
YouChat\spadesuit\spadesuit$\spadesuit$https://you.com/, or the ALCE benchmark
Gao et al. (2023). This is problematic because 1) it is far more time
consuming than directly generating citations together with text; 2) access to
the information sources needs to be provided at all times; 3) in contrast to
our proposed approach, it does not increase model interpretability; and 4) for
low-resource languages the quantity and quality of the data is often limited,
and hence result in difficulties retrieving the relevant, factual source.

##  3 Citations and Their Pros and Cons

In this section, we will first discuss which NLP tasks – according to us –
require LMs with an ability to cite their sources. We will then discuss the
benefits and, subsequently, risks of such models.

###  3.1 Which Tasks Require Citations?

We propose to classify tasks via two questions: (1) Is the source of the
generated text obvious? (2) Is the generated text an objective truth or a
subjective statement? See Table 1 for examples.

If the answer to the first question is yes, no further citation is required.
This is the case, e.g., for machine translation Brants et al. (2007): the
content of the generated text comes from the input sentence. The same holds
true for summarization See et al. (2017) and paraphrase generation Zhou and
Bhat (2021). However, this is only partially the case for text simplification
Sheang and Saggion (2021): while most of the content comes from the original
text, simpler versions of text sometimes contain additional explanations,
which do require citations. In contrast, for many other tasks the input does
not act as the source for text generation – instead, the output comes from
information stored in the model parameters and, thus, originally from the
training data. An ideal system would be able to cite the part of its training
data responsible for any given output. This is the case for the popular NLP
tasks of closed-book free-text question answering Roberts et al. (2020),
dialogue generation Zhao et al. (2020a), or creative writing Xu et al. (2020).

For tasks for which the answer to Question 1 is no, we then turn to the second
aforementioned question and ask if the generated text without clear sources of
information in the input contains what should be objective truths. This is
typically true for closed-book free-text question answering, which, as a
consequence, according to our rules does require citations. However, this is
sometimes the case for other tasks too, such as the generation of additional
explanations during text simplification or image captioning. Similarly, for
dialogue generation, objective truths and subjective statements could be mixed
within the same conversation. As a result, some generated statements for those
tasks do require citations, while others are good without.

Task | Q1 | Q2 | Citation? |  Example
---|---|---|---|---
Creative writing | No | Sometimes | Sometimes |  Penguins are known for their ability to survive in harsh Antarctic conditions [CITATION], but few people know that they also possess the power of telekinesis which they use to build intricate nests out of ice blocks.
Dialogue generation | No | Sometimes | Sometimes |  Did you know that penguins can jump up to 6 feet out of water when leaping onto land or ice floes? [CITATION]. I think elephants can do the same.
Free-text QA | No | Yes | Yes |  The current president is not a penguin [CITATION].
Image captioning | No | Sometimes | Sometimes |  A group of penguins diving into the ocean to catch fresh fish for dinner, highlighting their impressive swimming abilities [CITATION], while one penguin emerges victorious with a giant fish twice its size.
Paraphrase generation | Yes | N/A | No |  Source text: Penguins are social animals who live in large colonies. Paraphrased sentence: Penguins thrive in community living
Summarization | Yes | N/A | No |  Source text: Emperor penguins are the largest species of penguin, standing up to 4 feet tall. They are skilled hunters, capable of catching fish and krill by diving hundreds of feet below the surface. Summary: Emperor penguins are notable for their size and hunting prowess, making them formidable predators in their environment.
Text simplification | Sometimes | Sometimes | Sometimes |  Source text: Penguins have evolved unique adaptations that allow them to survive in environments as harsh as Antarctica, such as their countershaded dark and white plumage, which camouflages them from predators above and below the ice. Simplified text: Penguins live in Antarctica, which is year-round one of the coldest places on Earth [CITATION], and they look different than other birds so they don’t get eaten.
Translation | Yes | N/A | No |  Source text: Penguins are cool. Translated text: Pinguine sind cool.
Table 1: An overview of natural language generation tasks together with our
opinion regarding if they require citations. Q1: Obvious source? Q2: Objective
truth?

###  3.2 Benefits of Citations

Citations allow us to verify the factuality of generated text easily. In
contrast, without knowing where the text came from we are often unable to
verify that it is correct. Moreover, knowing what portion of the text is
copied verbatim allows us to give credit to the author and prevent copyright
violations. Citations also increase the explainability of the answer and allow
users to learn more about interesting topics.

Additionally, recent work in prompt engineering have shown that models
providing justifications for their assertions (even when only partially
correct) can improve the correctness of the outputs Jung et al. (2022).
Trustworthiness judgments among people often include a social aspect, so by
doing a good job of identifying sources and influences has the potential to
increase both the trust in AI systems and their trustworthiness. For example,
human trustworthiness judgments about scientific claims are influenced by the
interests of the authors Gierth and Bromme (2020).

###  3.3 Risks of Citations

Unfortunately, citations also come with risks. Just by having a citation next
to a generated text, users are more likely to trust it Thornley et al. (2015).
However, it is likely that users will not examine each and every citation
manually to verify that the text is indeed factual, or that the source is
trustworthy Simkin and Roychowdhury (2002); Thornley et al. (2015). This will
be exacerbated by the fact that it is incredibly unlikely that any automated
system will ever produce $100\%$ correct citations at all times, and may
result in either users’ diminishing trust and usage of such systems or a
potential harm.

There is also the risk of decreased readability: backing up every statement
with many citations, as the text may appear in multiple places, will reduce
the readability of the text and may hinder users from reading or understanding
it. Lastly, privacy concerns also arise from the training process of LMs. For
example, state of the art LMs are often trained on a massive automatically
extracted text Radford et al. (2018). But, as manual examination of each text
is not feasible for its size, there is a possibility that it may contain
private user information, such as patient records. This may result in LMs cite
information that should stay private.

###  3.4 Citations vs. Explainability

The goal to understand why a model generates any given output is shared with
research on model explainability Danilevsky et al. (2020). However, in
contrast to the latter, we are not interested in the effect of certain input
on the output. In addition, we do not necessarily require that the model
describes its reasoning by providing citations – what we care about instead is
that the citations back up the model’s answer. This enables humans to verify
the output – even if the cited source should not actually in the technical
sense have been the reason for the model’s output.

##  4 Road Map

###  4.1 The Big Picture

#### Meta-information

Currently, the standard in the field is to train models on text, disjoint from
its origin. Even though some models are trained on data that contain text with
citations (e.g., Taylor et al. (2022)), the citations are only "attached" to
statements taken from other sources, while any other text, even taken from the
same article, does not have a citation attached to it. This results in LMs
that can only sometimes, on a limited text, produce citations. In order to
develop LMs that can cite their sources effectively, we need to give them the
metadata which contain citation information.

#### Retrieval

Say we trained a LM with the right data such that it has knowledge of which
statement came from which article. How would we extract text with citations?
One avenue for such knowledge extraction is to modify the pretraining, such
that citation information is being generated together with every piece of
generated text.

#### When To Cite?

The above strategy would result in LMs that would always produce a citation.
However, as mentioned in Section 3.1, not every task or statement requires a
citation. For tasks that do require citations, we can just let the model
always cite. For tasks that do not require citations, we can simply remove the
citations. For tasks in between, where citation is sometimes required, we
propose to utilize the existing subjectivity classification task Wiebe et al.
(1999).

###  4.2 Concrete Tasks to Master

Our goal is to lay out a roadmap for the community, which describes necessary
steps for the development of models that can cite their sources. This is not
trivial, as it requires improvements of models for existing tasks as well as
the development of systems for novel challenges.

#### Simultaneous Citation and Text Generation

As mentioned in Section 2, existing work mainly retrieve citations for already
generated text, which becomes intractable as models are trained on ever more
text and the number of possible source documents increases drastically. In
contrast, we propose STANCE: the task of Simultaneous Text ANd Citation
gEneration. As an additional challenge, future work should also focus on
MultiSTANCE: multihop citation generation, where the sources for a given text
are spread across multiple texts. As the number of citations can be
significant (though much smaller in the low-resource setting), we suggest to
use topic modeling, as a potential avenue to reduce such large search space.

#### Subjectivity Classification

As mentioned in Section 3.1, whether a task requires a citation partially
depends on if the text is objective or subjective. This is not a novel task as
the community has been working on subjectivity classification for quite some
time Wiebe et al. (1999); Wiebe and Riloff (2005). However, to the best of our
knowledge, models for this task have not been employed in the context of
citations.

#### Citation-Text Correctness

To ensure that the retrieval step (Section 4.1) is successful, we need to
identify whether the statement appears in the source. For that, two existing
tasks can be used: 1) identifying which part of the generated text refers to
the citation Wang et al. (2020). 2) Validate that the citation is appropriate
for the selected text span Karadzhov et al. (2017); Martín et al. (2021);
Honovich et al. (2022); Mihaylova et al. (2018); Lee et al. (2020). Using such
automatic methods instead of manually verifying citations will result in
faster model development.

#### Source Trustworthiness

We all know that Wikipedia is not a reliable source for citation. We propose
CUE (Citation qUality Evaluation), the task of evaluating the quality of the
source corresponding to a generated citation. We believe there are six main
sub-tasks for CUE, which consist of classifying 1) the time of publication, 2)
whether the source is credible, 3) how many times the source has been cited,
4) if the author is known, 5) if the source is unbiased, and 6) if the
statement and citation are still relevant. For example, answering that the
current US president is Barack Obama was _previously_ factual, and may still
show up in many source documents, but is not factual in 2023.

##  5 Conclusion

Language models (LMs) performance has been improving rapidly in a wide variety
of areas. However, there is the crucial issue of their generated text often
being nonfactual, especially for low-resource languages. We argue that, in
order for LMs to be fully trustworthy, they must cite their sources – i.e.,
point users to the parts of their training data that back up their outputs. In
this opinion piece, we discuss NLP tasks which would benefit from such
citation models, highlight the benefits and risks such models would bring, and
outline the individual tasks that would need to be solved on the way to
develop such LMs.

## Limitations

While developing the proposed language models that can cite their sources
increase their utility, there is a risk that people would trust them more
without actually verifying that the generated citations are actually correct.
Such increase in trust would be especially problematic in time-critical
scenarios where people cannot examine each citation manually. Ideally, our
proposal will result in an increase of data cleaning, such that each citation
is by default trustworthy. That being said, our approach does not solve
copyright issues.

## Ethics Statement

The main reason for this paper is to point out shortcomings of state-of-the-
art language models, which can have significant social, health-related, and
economic consequences. Future work should develop systems that can cite their
sources in order to facilitate a verification of the factuality of generated
statements.

## Acknowledgments

We would like to thank the reviewers for taking the time to provide feedback
on our work. Your insights have been valuable in helping us refine and enhance
our manuscript. The authors acknowledge financial support from NIH grants
OT2TR003422 and R01LM013400.

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# The Born Rule is a Feature of Superposition

David Ellerman
University of Ljubljana, Slovenia
<EMAIL_ADDRESS>
https://orcid.org/0000-0002-5718-618X

###### Abstract

Finite probability theory is enriched by introducing the mathematical notion
(no physics involved) of a superposition event $\Sigma S$–in addition to the
usual discrete event $S$ (subset of the outcome space
$U=\left(u_{1},...,u_{n}\right)$). Mathematically, the two types of events are
distinguished using $n\times n$ density matrices. The density matrix
$\rho\left(S\right)$ for a discrete event is diagonal and the density matrix
$\rho\left(\Sigma S\right)$ is obtained as an outer product
$\left|s\right\rangle\left\langle s\right|$ of a normalized vector
$\left|s\right\rangle\in\mathbb{R}^{n}$. Probabilities are defined using
density matrices as
$\Pr\left(T|\rho\right)=\operatorname*{tr}\left[P_{T}\rho\right]$ where
$T\subseteq U$ and $P_{T}$ is the diagonal projection matrix with diagonal
entries $\chi_{T}\left(u_{i}\right)$. Then for the singleton
$\left\\{u_{i}\right\\}\subseteq U$, the probability of the outcome $u_{i}$
conditioned by the superposition event $\Sigma S$ is
$\Pr\left(\left\\{u_{i}\right\\}|\Sigma S\right)=\left\langle
u_{i}|s\right\rangle^{2}$, the Born Rule. Thus the Born Rule arises naturally
from the mathematics of superposition when superposition events are added to
ordinary finite probability theory. No further explanation is required when
the mathematics uses $\mathbb{C}^{n}$ instead of $\mathbb{R}^{n}$ except that
the square $\left\langle u_{i}|s\right\rangle^{2}$ is the absolute square
$\left|\left\langle u_{i}|s\right\rangle\right|^{2}$.

Keywords: Born Rule, superposition, density matrices, finite probability
theory

##  0.1 Introduction

The purpose of this paper is to expand ordinary finite probability theory by
introducing superposition events in addition to the usual discrete events
(subsets of the outcome space) and then to show that the Born Rule naturally
arises mathematically with superposition events. A superposition event or
subset is a purely mathematical notion–although obviously inspired by the
notion of a superposition state in quantum mechanics (QM). As a mathematical
notion, it could have been (but was not) introduced centuries before QM. The
thesis is that the Born Rule is not a bug that needs to be “explained” or
“justified”; it is just a feature of the notion of a superposition event in
this expanded probability theory.

##  0.2 Superposition events

The outcome (or sample) space is a set $U=\left\\{u_{1},...,u_{n}\right\\}$
which has point probabilities $p=\left(p_{1},...,p_{n}\right)$. An event $S$
is a subset $S\subseteq U$ with the probability
$\Pr\left(S\right)=\sum_{u_{i}\in S}p_{i}$. For $T\subseteq U$, the
conditional probability of $T$ given $S$ is
$\Pr\left(T|S\right)=\frac{\Pr\left(T\cap S\right)}{\Pr\left(S\right)}$ so the
conditional probability of a singleton event $\left\\{u_{i}\right\\}$ is
$\Pr\left(\left\\{u_{i}\right\\}|S\right)=\frac{p_{i}}{\Pr\left(S\right)}$ if
$u_{i}\in S$, else $0$.

In an (ordinary) event $S$, the atomic outcomes or elements of $S$ are
considered as perfectly discrete and distinguished from each other; in each
run of the “experiment” or trial conditioned on $S$, one of the discrete
outcomes in $S$ is chosen. The intuitive idea of the corresponding
superposition state, denoted $\Sigma S$, is that the outcomes in the state are
not distinguished but are blobbed or cohered together as indefinite event. In
each run of the “experiment” or trial conditioned on $\Sigma S$, the
indefinite state is sharpened to a less indefinite state which maximally is
one of the outcomes in $S$ [4]. In the case of a singleton event
$S=\left\\{u_{i}\right\\}$, the ordinary event $S=\left\\{u_{i}\right\\}$ is
the same as the superposition event $\Sigma
S=\Sigma\left\\{u_{i}\right\\}=\left\\{u_{i}\right\\}=S$.

For a suggestive visual example, consider the outcome set $U$ as a pair of
isosceles triangles that are distinct by the labels on the equal sides and the
opposing angles.

Figure 1: Set of distinct isosceles triangles

The superposition event $\Sigma U$ is definite on the properties that are
common to the elements of $U$, i.e., the angle $a$ and the opposing side $A$,
but is indefinite where the two triangles are distinct, i.e., the two equal
sides and their opposing angles are not distinguished by labels..

Figure 2: The superposition event $\Sigma U$.

It might be noted that this notion of superposition and the notion of
abstraction are essentially flip-side viewpoints of the same idea of
extracting from a set an entity that is definite on the commonalities of the
elements of the set and indefinite on where the elements differ [3]. The two
flip-side viewpoints are like seeing a glass half-empty (superposition) or
seeing a glass half-full (abstraction).

What is a mathematical model that will distinguish between the ordinary event
$S$ and the superposition event $\Sigma S$? Using $n$-ary column vectors in
$\mathbb{R}^{n}$, the ordinary event $S$ could be represented by the column
vector, denoted $\left|S\right\rangle$, with the $i^{th}$ entry
$\chi_{S}\left(u_{i}\right)$, where
$\chi_{S}:U\rightarrow\left\\{0,1\right\\}$ is the characteristic function for
$S$, i.e., $\chi_{S}(u_{i})=1$ if $u_{i}\in S$, else $0$. But to represent the
superposition event $\Sigma S$ we need to add a dimension to use two-
dimensional $n\times n$ matrices to represent the blobbing together or
cohering of the elements of $S$ in the superposition even $\Sigma S$.

An incidence matrix for a binary relation $R\subseteq U\times U$ is the
$n\times n$ matrix $\operatorname*{In}\left(R\right)$ where
$\operatorname*{In}\left(R\right)_{jk}=1$ if $\left(u_{j},u_{k}\right)\in R$,
else $0$. The diagonal $\Delta S$ is the binary relation consisting of the
ordered pairs $\left\\{\left(u_{i},u_{i}\right):u_{i}\in S\right\\}$ and its
incidence matrix $\operatorname*{In}\left(\Delta S\right)$ is the diagonal
matrix with the diagonal elements $\chi_{S}\left(u_{i}\right)$. The
superposition state $\Sigma S$ could then be represented as
$\operatorname*{In}\left(S\times S\right)$, the incidence matrix of the binary
relation $S\times S\subseteq U\times U$, where the non-zero off-diagonal
elements represent the cohering or blobbing together of the corresponding
diagonal elements. That incidence matrix could be constructed as the outer
product
$\left|S\right\rangle\left(\left|S\right\rangle\right)^{t}=\left|S\right\rangle\left\langle
S\right|=\operatorname*{In}\left(S\times S\right)$ (where $\left\langle
S\right|=\left(\left|S\right\rangle\right)^{t}$ is the transpose).

If we divided $\operatorname*{In}\left(\Delta S\right)$ and
$\operatorname*{In}\left(S\times S\right)$ through by their trace (sum of
diagonal elements) $\left|S\right|$, then we obtain two density matrices
$\rho\left(S\right)=\frac{\operatorname*{In}\left(\Delta
S\right)}{\left|S\right|}$ and $\rho\left(\Sigma
S\right)=\frac{\operatorname*{In}\left(S\times
S\right)}{\left|S\right|}=\frac{1}{\sqrt{\left|S\right|}}\left|S\right\rangle\left\langle
S\right|\frac{1}{\sqrt{\left|S\right|}}$ over the reals $\mathbb{R}$. In
general, a density matrix $\rho$ over the reals $\mathbb{R}$ (or the complex
numbers $\mathbb{C}$) is a symmetric matrix $\rho=\rho^{t}$ (or conjugate
symmetric matrix $\rho=\left(\rho^{\ast}\right)^{t}$ in the case of
$\mathbb{C}$) with trace $\operatorname*{tr}\left[\rho\right]=1$ and all non-
negative eigenvalues. A density matrix $\rho$ is pure if $\rho^{2}=\rho$,
otherwise a mixture.

Consider the partition
$\pi=\left\\{B_{1},B_{2}\right\\}=\left\\{\left\\{\diamondsuit,\heartsuit\right\\},\left\\{\clubsuit,\spadesuit\right\\}\right\\}$
on the outcome set
$U=\left\\{\clubsuit,\diamondsuit,\heartsuit,\spadesuit\right\\}$ with
equiprobable outcomes like drawing cards from a randomized deck. For instance,
the superposition event associated with
$B_{1}=\left\\{\diamondsuit,\heartsuit\right\\}$, has a pure density matrix
since (rows and columns labelled in the order
$\left\\{\clubsuit,\diamondsuit,\heartsuit,\spadesuit\right\\}$):

$\rho\left(\Sigma
B_{1}\right)=\frac{1}{\sqrt{\left|B_{i}\right|}}\begin{bmatrix}0\\\ 1\\\ 1\\\
0\end{bmatrix}\begin{bmatrix}0&1&1&0\end{bmatrix}\frac{1}{\sqrt{\left|B_{1}\right|}}=\frac{1}{\left|B_{1}\right|}\begin{bmatrix}0&0&0&0\\\
0&1&1&0\\\ 0&1&1&0\\\ 0&0&0&0\end{bmatrix}=\begin{bmatrix}0&0&0&0\\\
0&\frac{1}{2}&\frac{1}{2}&0\\\ 0&\frac{1}{2}&\frac{1}{2}&0\\\
0&0&0&0\end{bmatrix}$

equals its square, but density matrix for the discrete set $B_{1}$:

$\rho\left(B_{1}\right)=\begin{bmatrix}0&0&0&0\\\ 0&\frac{1}{2}&0&0\\\
0&0&\frac{1}{2}&0\\\ 0&0&0&0\end{bmatrix}$

is a mixture since it does not equal its square.

Intuitively, the interpretation of the superposition event represented by
$\rho\left(\Sigma
B_{1}\right)=\rho\left(\Sigma\left\\{\diamondsuit,\heartsuit\right\\}\right)$
is that it is definite on the properties common to its elements, e.g., in this
case, being a red suite, but indefinite on where the elements differ. The
indefiniteness is indicated by the non-zero off-diagonal elements that
indicate that the diamond suite $\diamondsuit$ is blurred, cohered, or
superposed with the hearts suite $\heartsuit$ in the superposition state
$\Sigma\left\\{\diamondsuit,\heartsuit\right\\}$.

The next step is to bring in the point probabilities
$p=\left(p_{1},...,p_{n}\right)$ where those two real density matrices
$\rho\left(S\right)$ and $\rho\left(\Sigma S\right)$ defined so far correspond
to the special case of the equiprobable distribution on $S$ with $0$
probabilities outside of $S$.

##  0.3 Density matrices with general probability distributions

Let the outcome space $U=\left\\{u_{1},...,u_{n}\right\\}$ have the
probabilities $p=\left\\{p_{1},...,p_{n}\right\\}$. The conditional
probability distribution on $S\subseteq U$ is
$\Pr\left(\left\\{u_{i}\right\\}|S\right)=\frac{\Pr\left(\left\\{u_{i}\right\\}\cap
S\right)}{\Pr\left(S\right)}=\frac{p_{i}}{\Pr\left(S\right)}$ if $u_{i}\in S$,
else $0$. The density matrix $\rho\left(S\right)$ for the classical or
discrete event $S$ is then the diagonal matrix with the entries
$\Pr\left(\left\\{u_{i}\right\\}|S\right)\chi_{S}\left(u_{i}\right)=\frac{p_{i}}{\Pr\left(S\right)}\chi_{S}\left(u_{i}\right)$.

Given two column vectors
$\left|s\right\rangle=\left(s_{1},...,s_{n}\right)^{t}$ and
$\left|t\right\rangle=\left(t_{1},...,t_{n}\right)^{t}$ in $\mathbb{R}^{n}$,
their inner product is the sum of the products of the corresponding entries
and is denoted $\left\langle
t|s\right\rangle=\left(\left|t\right\rangle\right)^{t}\left|s\right\rangle=\sum_{i=1}^{n}t_{i}s_{i}$.
Their outer product is the $n\times n$ matrix denoted as
$\left|s\right\rangle\left\langle t\right|=$
$\left|s\right\rangle\left(\left|t\right\rangle\right)^{t}$. A vector
$\left|s\right\rangle$ is normalized if $\left\langle s|s\right\rangle=1$. Let
$\left|u_{i}\right\rangle$ be the $n$-ary column vector with the $i^{th}$
entry $1$ and otherwise $0$.

To construct the density matrix $\rho\left(\Sigma S\right)$ for the
superposition event $\Sigma S$, we may first construct the normalized column
vector $\left|s\right\rangle$ where the $i^{th}$ entry, denoted $\left\langle
u_{i}|s\right\rangle=\left(\left|u_{i}\right\rangle\right)^{t}\left|s\right\rangle$,
is the “amplitude”: $\sqrt{\frac{p_{i}}{\Pr\left(S\right)}}$ if $u_{i}\in S$,
else $0$. In the case of the equiprobable distribution
$\frac{1}{\left|S\right|}$ on $S$, then
$\left|s\right\rangle=\frac{1}{\sqrt{\left|S\right|}}\left|S\right\rangle$.
Then the density matrix for the superposition event $\Sigma S$ is the outer
product:

$\rho\left(\Sigma S\right)=\left|s\right\rangle\left\langle s\right|$.

The density matrices for the classical event $S$ and for the superposition
event $\Sigma S$ have quite different properties. The density matrix
$\rho\left(\Sigma S\right)$ is pure since $\rho^{2}=\rho$, while the density
matrix $\rho\left(S\right)$ is a mixture (except in the special case when
$S=\left\\{u_{i}\right\\}$ is a singleton so that
$\rho\left(\left\\{u_{i}\right\\}\right)=\rho\left(\Sigma\left\\{u_{i}\right\\}\right)$).
For $\rho\left(S\right)$, the eigenvalues are just the conditional
probabilities
$\Pr\left(\left\\{u_{i}\right\\}|S\right)=\frac{\Pr\left(\left\\{u_{i}\right\\}\cap
S\right)}{\Pr\left(S\right)}=\frac{p_{i}}{\Pr\left(S\right)}\chi_{S}\left(u_{i}\right)$.
But for the pure density matrix $\rho\left(\Sigma S\right)$, there is one
eigenvalue of $1$ with the rest of the eigenvalues being zeros (since the sum
of the eigenvalues is the trace). Given just $\rho\left(\Sigma S\right)$, the
vector $\left|s\right\rangle$ is obtained as the normalized eigenvector
associated with the eigenvalue of $1$ and $\rho\left(\Sigma
S\right)=\left|s\right\rangle\left\langle s\right|$.111This is by the spectral
decomposition of that density matrix as a Hermitian operator.

A partition $\pi=\left\\{B_{1},...,B_{m}\right\\}$ on $U$ is a set of non-
empty subsets, called blocks, $B_{j}\subseteq U$ that are disjoint and whose
union is $U$. Taking each block $B_{j}=S$, then there is the normalized column
vector $\left|b_{j}\right\rangle$ whose $i^{th}$ entry is
$\sqrt{\frac{p_{i}}{\Pr\left(B_{j}\right)}}\chi_{B_{j}}\left(u_{i}\right)$ and
the density matrix $\rho\left(\Sigma
B_{j}\right)=\left|b_{j}\right\rangle\left\langle b_{j}\right|$ for the
superposition subset $\Sigma B_{j}$. Then the density matrix
$\rho\left(\pi\right)$ for the partition $\pi$ is just the probability sum of
those pure density matrices for the superposition blocks:

$\rho\left(\pi\right)=\sum_{j=1}^{m}\Pr\left(B_{j}\right)\rho\left(\Sigma
B_{j}\right)$.

The eigenvalues for $\rho\left(\pi\right)$ are the $m$ probabilities
$\Pr\left(B_{j}\right)$ with the remaining $n-m$ values of $0$.

Given two partitions $\pi=\left\\{B_{1},...,B_{m}\right\\}$ and
$\sigma=\left\\{C_{1},...,C_{m^{\prime}}\right\\}$, the join is the partition
$\sigma\vee\pi$ whose blocks are the non-empty intersections $B_{j}\cap
C_{j^{\prime}}$ of the blocks of $\pi$ and $\sigma$. To construct the meet
$\sigma\wedge\pi$, form the undirected graph on $U$ where there is a link
between $u_{j}$ and $u_{k}$ if they are in the same block of $\pi$ or in the
same block of $\sigma$. Then the blocks of the meet are the connected
components of that graph. The partition $\pi$ refines the partition $\sigma$,
written $\sigma\precsim\pi$, if for each block $B_{j}\in\pi$, there is a block
$C_{j^{\prime}}\in\sigma$ such that $B_{j}\subseteq C_{j^{\prime}}$. Then the
partitions on $U$ form a lattice $\Pi\left(U\right)$ with the refinement
partial order. The maximal partition or top of the lattice is the discrete
partition $\mathbf{1}_{U}=\left\\{\left\\{u_{i}\right\\}\right\\}_{i=1}^{n}$
where all the blocks are singletons and the minimal partition or bottom is the
indiscrete partition $\mathbf{0}_{U}=\left\\{U\right\\}$ with only one block
$U$. Then the density matrices for these top and bottom partitions are just
the density matrices for the discrete set $U$ and the superposition set
$\Sigma U$:

$\rho\left(\mathbf{1}_{U}\right)=\rho\left(U\right)$ and
$\rho\left(\mathbf{0}_{U}\right)=\rho\left(\Sigma U\right)$.

##  0.4 Probabilities using density matrices

A (real-valued) random variable on the outcome space $U$ is a function
$f:U\rightarrow\mathbb{R}$ with distinct values of
$\left\\{\phi_{1},...,\phi_{m}\right\\}$. The inverse image of $f$ is a
partition $\pi=\left\\{B_{j}\right\\}_{j=1}^{m}$ where
$B_{j}=f^{-1}\left(\phi_{j}\right)$. In ordinary classical probability theory,
the conditional probability of getting the value $\phi_{j}$ given the event
$S$ in a trial is $\Pr\left(\phi_{j}|S\right)=\frac{\Pr\left(B_{j}\cap
S\right)}{\Pr\left(S\right)}$. But now we have two versions of $S$, the
discrete event and the superposition event. Since they have different density
matrices, we can take the given conditioning event as a density matrix $\rho$.
Let $P_{T}$ for $T\subseteq U$ be the diagonal projection matrix with the
diagonal entries $\left(P_{T}\right)_{ii}=\chi_{T}\left(u_{i}\right)$.
Projection matrices are idempotent, i.e., $P_{T}P_{T}=P_{T}$ and equal their
transpose $P_{T}=P_{T}^{t}$. The usual conditional probability of the
classical event $T$ given the classical event $S$ can be computed as:

$\Pr\left(T|S\right):=\frac{\Pr\left(S\cap
T\right)}{\Pr\left(S\right)}=\operatorname{tr}\left[P_{T}\rho\left(S\right)\right]$.

Starting with the conditioning event being the superposition event $\Sigma S$,
that probability is defined as:

$\Pr\left(\phi_{j}|\rho\left(\Sigma
S\right)\right):=\operatorname{tr}\left[P_{B_{j}}\rho\left(\Sigma
S\right)\right]=\frac{\Pr\left(B_{j}\cap
S\right)}{\Pr\left(S\right)}=\sum_{f\left(u_{i}\right)=\phi_{j}}\frac{\Pr\left(\left\\{u_{i}\right\\}\cap
S\right)}{\Pr\left(S\right)}$.

It is particularly notable that the probabilities for the values of a random
variable (or any given event $T$) are the same if the conditioning event is
the classical event $S$ represented by the mixed $\rho\left(S\right)$ or the
superposition event $\Sigma S$ represented by the pure $\rho\left(\Sigma
S\right)$:

$\Pr\left(\phi_{j}|\rho\left(\Sigma
S\right)\right)=\operatorname{tr}\left[P_{B_{j}}\rho\left(\Sigma
S\right)\right]=\frac{\Pr\left(B_{j}\cap
S\right)}{\Pr\left(S\right)}=\operatorname{tr}\left[P_{B_{j}}\rho\left(S\right)\right]=\Pr\left(\phi_{j}|\rho\left(S\right)\right)$.

But the interpretation is quite different. The classical trial starting with
the subset $S$ represented by $\rho\left(S\right)$ picks out the subset
$B_{j}\cap S$ with probability
$\Pr\left(\phi_{j}|S\right)=\operatorname{tr}\left[P_{B_{j}}\rho\left(S\right)\right]=\frac{\Pr\left(B_{j}\cap
S\right)}{\Pr\left(S\right)}$. However, the ‘measurement’ or trial conditioned
by the superposition event $\Sigma S$ represented by $\rho\left(\Sigma
S\right)$ ‘sharpens’ or projects that indefinite event to the more definite
superposition event $\Sigma\left(B_{j}\cap S\right)$ with probability
$\Pr\left(\phi_{j}|\rho\left(\Sigma
S\right)\right)=\operatorname{tr}\left[P_{B_{j}}\rho\left(\Sigma
S\right)\right]$.

In either case, the follow-up trial or ‘measurement’ returns the same value
$\phi_{j}$ with probability $1$, i.e., $\Pr\left(\phi_{j}|B_{j}\cap
S\right)=\operatorname{tr}\left[P_{B_{j}}\rho\left(B_{j}\cap
S\right)\right]=1=\operatorname{tr}\left[P_{B_{j}}\rho\left(\Sigma\left(B_{j}\cap
S\right)\right)\right]$. In the classical case, all the elements of a non-
empty $B_{j}\cap S$ have the value $\phi_{j}$ so $\phi_{j}$ occurs conditioned
on the classical event $B_{j}\cap S$ with probability $1$. In the
superposition case, the property of having the value $\phi_{j}$ is a
commonality, i.e., is definite, on the superposition event
$\Sigma\left(B_{j}\cap S\right)$ represented by
$\rho\left(\Sigma\left(B_{j}\cap S\right)\right)$, so no ‘sharpening’ occurs
and probability of a trial returning that definite value $\phi_{j}$ is $1$.

Let us illustrate this result with the case of flipping a fair coin. The
classical set of outcomes $U=\left\\{H,T\right\\}$ is represented by the
density matrix:

$\rho\left(U\right)=$ $\begin{bmatrix}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{bmatrix}$.

Figure 3: Classical event: trial picks out heads or tails

The superposition event $\Sigma U$, that blends or superposes heads and tails,
is represented by the density matrix:

$\rho\left(\Sigma U\right)=\begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}\end{bmatrix}$.

Figure 4: Superposition event: Trial sharpens to heads or tails.

The probability of getting heads in each case is:

$\Pr\left(H|\rho\left(U\right)\right)=\operatorname{tr}\left[P_{\left\\{H\right\\}}\rho\left(U\right)\right]=\operatorname{tr}\left[\begin{bmatrix}1&0\\\
0&0\end{bmatrix}\begin{bmatrix}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{bmatrix}\right]=\operatorname{tr}\begin{bmatrix}\frac{1}{2}&0\\\
0&0\end{bmatrix}=\frac{1}{2}$

$\Pr\left(H|\rho\left(\Sigma
U\right)\right)=\operatorname{tr}\left[P_{\left\\{H\right\\}}\rho\left(\Sigma
U\right)\right]=\operatorname{tr}\left[\begin{bmatrix}1&0\\\
0&0\end{bmatrix}\begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}\end{bmatrix}\right]=\operatorname{tr}\allowbreak\begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\
0&0\end{bmatrix}=\frac{1}{2}$

and similarly for tails. Thus the two conditioning events $U$ and $\Sigma U$
cannot be distinguished by performing an experiment or trial that
distinguishes heads and tails. This is a feature, not a bug, since the same
thing occurs in quantum mechanics. For instance, a spin measurement along,
say, the $z$-axis of an electron cannot distinguish between the superposition
state
$\frac{1}{\sqrt{2}}\left(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle\right)$
with a density matrix like $\rho\left(\Sigma U\right)$ and a statistical
mixture of half electrons with spin up and half with spin down with a density
matrix like $\rho\left(U\right)$ [1, p. 176].222In QM, they can only be
distinguished by measurement in a different observable basis.

##  0.5 Conclusion: The Born Rule

The Born Rule does not occur in ordinary classical probability theory because
that theory does not include superposition events. When superposition events
are introduced into the purely mathematical theory [2], then the Born Rule
naturally emerges as a feature of the mathematical treatment of superposition.

The pure density matrix $\rho\left(\Sigma S\right)$ can be constructed as the
outer product $\rho\left(\Sigma
S\right)=\left|s\right\rangle\left(\left|s\right\rangle\right)^{t}=\left|s\right\rangle\left\langle
s\right|$ where $\left|s\right\rangle$ is the $n$-ary column vector with the
$i^{th}$ entry as $\left\langle
u_{i}|s\right\rangle=\sqrt{\frac{p_{i}}{\Pr\left(S\right)}}\chi_{S}\left(u_{i}\right)=\sqrt{\frac{\Pr\left(\left\\{u_{i}\right\\}\cap
S\right)}{\Pr\left(S\right)}}$. Or starting with the pure density matrix
$\rho\left(\Sigma S\right)$, then $\left|s\right\rangle$ so that
$\rho\left(\Sigma S\right)=\left|s\right\rangle\left\langle s\right|$ is
obtained as the normalized eigenvector associated with the eigenvalue of $1$.

The probability of $u_{i}$ conditioned on the superposition event $\Sigma S$
is:

$\Pr\left(\left\\{u_{i}\right\\}|\rho\left(\Sigma
S\right)\right)=\operatorname{tr}\left[P_{\left\\{u_{i}\right\\}}\rho\left(\Sigma
S\right)\right]=\frac{\Pr\left(\left\\{u_{i}\right\\}\cap
S\right)}{\Pr\left(S\right)}$.

The point is that this same probability obtained conditioned by the two-
dimensional density matrix $\rho\left(\Sigma S\right)$ could also be obtained
from the one-dimensional vector $\left|s\right\rangle$ as:

$\left\langle
u_{i}|s\right\rangle^{2}=\frac{\Pr\left(\left\\{u_{i}\right\\}\cap
S\right)}{\Pr\left(S\right)}=\Pr\left(\left\\{u_{i}\right\\}|\rho\left(\Sigma
S\right)\right)=\operatorname{tr}\left[P_{\left\\{u_{i}\right\\}}\rho\left(\Sigma
S\right)\right]$.

The Born Rule

In the case of the random variable $f:U\rightarrow\mathbb{R}$,
$\Pr\left(\phi_{j}|\rho\left(\Sigma
S\right)\right)=\sum_{f\left(u_{i}\right)=\phi_{j}}\left\langle
u_{i}|s\right\rangle^{2}$.

The Born Rule does not occur in classical finite probability theory since the
events $S$ are all discrete sets that can be represented by $n$-ary column
vectors. The associated two-dimensional diagonal density matrix
$\rho\left(S\right)$ is not the outer product of a one-dimensional vector with
itself (except when S is a singleton). To accommodate the notion of a
superposition event $\Sigma S$, it is necessary to use two-dimensional density
matrices $\rho\left(\Sigma S\right)$ where the non-zero off-diagonal elements
indicate the blobbing or cohering together in superposition of the elements
associated with the corresponding diagonal entries. And mathematically those
density matrices $\rho\left(\Sigma S\right)$ can be constructed as the outer
product
$\left|s\right\rangle\left(\left|s\right\rangle\right)^{t}=\left|s\right\rangle\left\langle
s\right|$ of a one-dimensional vector $\left|s\right\rangle$ with itself. Then
the probability of the individual outcomes $u_{i}$ conditioned by the
superposition event $\Sigma S$ is given by the Born Rule:
$\Pr\left(\left\\{u_{i}\right\\}|\rho\left(\Sigma S\right)\right)=\left\langle
u_{i}|s\right\rangle^{2}$. Thus the Born Rule arises naturally out of the
mathematics of probability theory enriched by superposition events.333In the
vector spaces over the complex numbers $\mathbb{C}$ of quantum mechanics, the
square $\left\langle u_{i}|s\right\rangle^{2}$ is the absolute square
$\left|\left\langle u_{i}|s\right\rangle\right|^{2}$. It does not need any
more-exotic or physics-based explanation. No physics was used in the making of
this paper. The Born Rule is just a feature of the mathematics of
superposition.

## References

  * [1] Auletta, G., Fortunato, M., Parisi, G.: Quantum Mechanics. Cambridge University Press, Cambridge UK (2009)
  * [2] Ellerman, D.: Probability Theory with Superposition Events: A Classical Generalization in the Direction of Quantum Mechanics, https://arxiv.org/abs/2006.09918, (2020)
  * [3] Ellerman, D.: On Abstraction in Mathematics and Indefiniteness in Quantum Mechanics. Journal of Philosophical Logic. 50, 813–835 (2021). https://doi.org/10.1007/s10992-020-09586-1
  * [4] Ellerman, D.: Follow the Math!: The Mathematics of Quantum Mechanics as the Mathematics of Set Partitions Linearized to (Hilbert) Vector Spaces. Foundations of Physics. 52, (2022). https://doi.org/10.1007/s10701-022-00608-3


HU-EP-24/03

Conformal invariants of curves via those
for inscribed polygons with circular edges

Harald Dorn<EMAIL_ADDRESS>
Institut für Physik und IRIS Adlershof, Humboldt-Universität zu Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany

Abstract

The conformal nature of smooth curves in $\mathbb{R}^{3}$ is characterised by
conformal length, curvature and torsion. We present a derivation of these
conformal parameters via a limiting process using inscribed polygons with
circular edges . The procedure is based on elementary geometry in
$\mathbb{R}^{3}$ only and similar to the rectification of curves in the
metrical case. It seems to be not available in the literature so far.

##  1 Introduction

The conformal geometry of smooth curves in $\mathbb{R}^{3}$ and in higher
dimensions has been studied by mathematicians using various techniques, as
e.g. conformal derivatives, normal forms, group theoretical methods or the
kinematics of osculating spheres, see e.g. [1, 2, 3, 4, 5, 6, 7] and
references therein.
In analogy to the metrical invariants length $s$, curvature $\kappa$ and
torsion $\tau$ one finds for smooth 3-dimensional curves, see e.g. [3],
conformal length $\omega$,222The prime denotes derivative with respect to $s$.

$d\omega~{}=~{}\sqrt{\nu}ds,~{}~{}~{}~{}~{}~{}\nu~{}=~{}\sqrt{(\kappa^{\prime})^{2}+\kappa^{2}\tau^{2}}~{},$
(1)

conformal curvature $Q$

$Q~{}=~{}\frac{4(\nu^{\prime\prime}-\kappa^{2}\nu)\nu-5(\nu^{\prime})^{2}}{8\nu^{3}}~{},$
(2)

and conformal torsion $T$

$T~{}=~{}\frac{2(\kappa^{\prime})^{2}\tau+\kappa^{2}\tau^{3}+\kappa\kappa^{\prime}\tau^{\prime}-\kappa\kappa^{\prime\prime}\tau}{\nu^{\frac{5}{2}}}~{}.$
(3)

Our interest in this subject originates from studies of Wilson loops [8, 9,
10] related to polygons with circular edges. There the conformal length of the
edges is zero and both $Q$ and $T$ are ill-defined along the edges. As
conformal invariant parameters fixing these polygons, up to conformal
transformations, we identified the cross ratios formed out of the positions of
the corners, the angles between the edges (cusp angles) and angles between
circles fixed by three consecutive corners and adjacent edges (called torsion
angles). For these polygons conformal curvature and torsion is somehow located
at the corners in a distributional sense. So far we could not give this
comment a precise meaning.

But instead we were able to realise the opposite direction, i.e. to relate
$\omega,Q$ and $T$ for smooth curves to cusp and torsion angles of inscribed
polygons.333We did not find in the literature such a direct and vivid
derivation by means of elementary geometry in $\mathbb{R}^{3}$. Therefore, we
decided to write it down in this note. The procedure is analogously to the
derivation of metrical length, curvature and torsion via rectification by
ordinary polygons with straight edges. Such a rectification has also the
potential to be applied to more general irregular curves, for the metrical
case see e.g.[11]. Rectifications by ordinary polygons are not conformal
covariant, since in general straight lines are not mapped to straight lines.
But circles are mapped to circles, hence one can glue together pieces of
successive circles fixed by sets of three nearest neighbour points to a
conformal covariant approximation (rectification) of a given curve, see fig.1.

Figure 1: In blue is shown a certain smooth curve with five randomly chosen
points $x_{1},\dots,x_{5}$. Each set of three consecutive points
$(x_{j-1},x_{j},x_{j+1})$ fixes a circle $cc_{j}$. The arcs from $x_{j-1}$ to
$x_{j}$ are depicted in red, those from $x_{j}$ to $x_{j+1}$ in black. Both
the black and red polygon is a conformal covariant approximation
(rectification) of the respective piece of the blue smooth curve.

The paper is organised as follows. To emphasize the analogy to the metrical
case, section 2 gives a short recapitulation of curvature and torsion for
smooth curves via a limiting process with inscribed ordinary polygons. The
main result for the conformal invariants via inscribed polygons with circular
edges is derived in section 3. Finally in section 4 we add some by the way
obtained observations on cross ratios and circumcircles of triangles. An
appendix lists formulas from elementary geometry concerning circumcircles and
circumspheres.

##  2 A short recap of the metrical case

Let denote by $x(s)$ a smooth curve in $\mathbb{R}^{3}$ as a function of its
length parameter $s$ and

$x_{1}=x(s_{0}-\epsilon),~{}~{}~{}x_{2}=x(s_{0}),~{}~{}~{}x_{3}=x(s_{0}+\epsilon)$
(4)

three points on the curve, which at the same time are consecutive corners of
an inscribed polygon. Then the cosine of the angle $\bar{\alpha}$ between the
straight edges at $x_{2}$ is given by

$\mbox{cos}\,\bar{\alpha}~{}=~{}\frac{(x_{1}-x_{2})(x_{2}-x_{3})}{|x_{1}-x_{2}|\
|x_{2}-x_{3}|}~{}=~{}1-\frac{\kappa^{2}}{2}\
\epsilon^{2}+\big{(}\frac{1}{24}\kappa^{4}+\frac{1}{12}\kappa^{2}\tau^{2}-\frac{1}{12}\kappa\kappa^{\prime\prime}\big{)}\
\epsilon^{4}+{\cal O}(\epsilon^{6})~{}.$ (5)

In this expansion use has been made of the Frenet-Serret formula for the
derivatives of the unit vectors $t,n,b$ (note $t=x^{\prime}(s_{0})$) spanning
the orthogonal Frenet-Serret frame at $x(s_{0})$

$\displaystyle t^{\prime}$ $\displaystyle=$ $\displaystyle\kappa\,n~{},$
$\displaystyle n^{\prime}$ $\displaystyle=$
$\displaystyle-\kappa\,t+\tau\,b~{},$ $\displaystyle b^{\prime}$
$\displaystyle=$ $\displaystyle-\tau\,n~{}.$ (6)

In (5) the leading order $\epsilon^{2}$ reproduces the curvature, and the next
order $\epsilon^{4}$ is a certain function of curvature and torsion.

There is still another option to get the torsion, and this time directly from
the leading term of an expansion. Three consecutive corners of an inscribed
polygon define a plane. Let us then ask for the angle $\bar{\gamma}$ between
the normals between two such consecutive planes. To answer this, we look at
five consecutive points $x_{1},\dots,x_{5}$ on the curve with

$x_{1}=x(s_{0}-2\epsilon),~{}~{}~{}x_{2}=x(s_{0}-\epsilon),~{}~{}~{}x_{3}=x(s_{0}),~{}~{}~{}x_{4}=x(s_{0}+\epsilon),~{}~{}~{}x_{5}=x(s_{0}+2\epsilon)$
(7)

and find

$\mbox{cos}\,\bar{\gamma}~{}=~{}\frac{(x_{2}-x_{3})\times(x_{1}-x_{3})}{|x_{2}-x_{3}|\
|x_{1}-x_{3}|}\cdot\frac{(x_{5}-x_{3})\times(x_{4}-x_{3})}{|x_{5}-x_{3}|\
|x_{4}-x_{3}|}~{}=~{}1-2\tau^{2}\ \epsilon^{2}+{\cal O}(\epsilon^{4})~{}.$ (8)

##  3 The conformal case

To replace the approximation via ordinary polygons by an approximation which
is conformally covariant, we now describe in some more detail the use of
polygons with circular edges, sketched already in the introduction section. We
choose a set of points $\\{x_{j}\\}$ on the curve, ordered with respect to
increasing length parameter along the curve. Each triple
$(x_{j-1},x_{j},x_{j+1})$ fixes a circle $cc_{j}$. Let us call by $ca^{-}_{j}$
its arc from $x_{j-1}$ to $x_{j}$ and by $ca^{+}_{j}$ its arc from $x_{j}$ to
$x_{j+1}$. Then the sequence $(ca^{+}_{1},ca^{+}_{2},ca^{+}_{3},\dots)$
defines an inscribed polygon with circular edges. The same is true for the
analogous sequence using the arcs $ca^{-}_{j}$, compare fig.1.

For arbitrary choices of the ordered set of points $\\{x_{j}\\}$, under a
conformal transformation the inscribed circular polygon is mapped to the
circular polygon inscribed to the image of the original curve and fixed by the
image of the original points $\\{x_{j}\\}$. However, to make contact with the
conformal invariants for smooth curves (1)-(3), we should consider a situation
were consecutive points move closer and closer. Although metrical length is
not conformally invariant, let us start with a setup where consecutive point
on the curve have a uniform metrical distance $\epsilon$.

With the choice (7) and

$x(s_{0}+\epsilon)~{}=~{}x_{3}+y_{1}\,\epsilon+\frac{y_{2}}{2}\,\epsilon^{2}+\frac{y_{3}}{3!}\,\epsilon^{3}+\dots$
(9)

we want to calculate the conformally invariant angle between $ca^{+}_{3}$ and
$ca^{+}_{2}$ at $x_{3}$, i.e. the cusp angle of the black circular polygon at
the point $x_{3}$ in fig.1. Using the notation introduced in the appendix, it
is given by

$\mbox{cos}\,\alpha~{}=~{}t^{123}_{3}\cdot t^{234}_{3}~{}$ (10)

and via (36) by

$\mbox{cos}\,\alpha~{}=~{}\frac{x_{13}^{2}\,x_{24}^{2}+x_{34}^{2}\,x_{12}^{2}-x_{23}^{2}\,x_{14}^{2}}{2\,x_{12}\,x_{24}\,x_{13}\,x_{34}}~{}.$
(11)

Then with (7) we get, using the Frenet-Serret formula (6) to express the
coefficient vectors $y_{j}=\frac{d^{j}x(s)}{ds^{j}}|_{s=s_{0}}$ in terms of
the basis $(t,n,b)$

$\mbox{cos}\,\alpha~{}=~{}1-\frac{1}{8}\,(\kappa^{2}\tau^{2}+\kappa^{\prime
2})\,\epsilon^{4}~{}+~{}{\cal O}(\epsilon^{5})~{}.$ (12)

As stressed above, the metrical length is no conformal invariant. Therefore
the generic terms in $\epsilon$-expansions cannot be expected to be conformal
invariants. But there is for sure one exception: the leading nontrivial terms,
e.g. the ${\cal O}(\epsilon^{4})$ above. Hence

$(\kappa^{2}\tau^{2}+\kappa^{\prime 2})\,\epsilon^{4}$ (13)

is invariant, and we rediscover the differential of the conformal length (18).

To find more conformal invariants from higher orders, we now put the points
$x_{1}$ to $x_{5}$ uniformly equidistant with respect to the conformal length
(instead w.r.t. the metrical length). Then all orders in an expansion w.r.t.
this conformal distance will be conformal invariants.

From

$\omega(\epsilon)~{}=~{}\int_{s_{0}}^{s_{0}+\epsilon}f(s)\,ds,~{}~{}~{}~{}~{}f(s)=\sqrt{\nu(s)}=(\kappa^{2}\tau^{2}+\kappa^{\prime
2})^{\frac{1}{4}}$ (14)

and its related expansion

$\omega(\epsilon)~{}=~{}f(s_{0})\,\epsilon+\frac{1}{2}f^{\prime}\,\epsilon^{2}+\frac{1}{3!}f^{\prime\prime}\,\epsilon^{3}+\dots$
(15)

we get for the inverse function

$\epsilon(\omega)~{}=~{}g_{1}\,\omega+\frac{1}{2}\,g_{2}\,\omega^{2}+\frac{1}{3!}\,g_{3}\,\omega^{3}+\dots~{},$
(16)

with

$g_{1}=\frac{1}{f}~{},~{}~{}~{}~{}g_{j}=\frac{g^{\prime}_{j-1}}{f}~{},~{}~{}~{}~{}j\geq
2~{}.$ (17)

Now we repeat the calculation of the cusp angle $\alpha$ as an expansion in
$\omega$ for the setup

$\displaystyle x_{1}$ $\displaystyle=$ $\displaystyle
x(s_{0}+\epsilon(-2\omega)),~{}~{}~{}x_{2}~{}=~{}x(s_{0}+\epsilon(-\omega)),~{}~{}~{}x_{3}~{}=~{}x(s_{0}),$
$\displaystyle x_{4}$ $\displaystyle=$ $\displaystyle
x(s_{0}+\epsilon(\omega)),~{}~{}~{}x_{5}~{}=~{}x(s_{0}+\epsilon(2\omega))~{}.$
(18)

To get the result correctly up to order $\omega^{6}$ we used in the
corresponding Mathematica code the expansions (9) and (16) up to the seventh
order and obtained

$\mbox{cos}\,\alpha~{}=~{}1-\frac{1}{8}\
\omega^{4}~{}+~{}P\,\omega^{6}~{}+~{}{\cal O}(\omega^{7})~{},$ (19)

with

$\displaystyle P$ $\displaystyle=$
$\displaystyle\Big{(}2\kappa\tau\kappa^{\prime
2}(20\kappa^{\prime}\tau^{\prime}-19\tau\kappa^{\prime\prime})+2\kappa^{3}\tau^{3}(5\kappa^{\prime}\tau^{\prime}-4\tau\kappa^{\prime\prime})+\kappa^{\prime
2}(28\tau^{2}\kappa^{\prime 2}-5\kappa^{\prime\prime
2}+4\kappa^{\prime}\kappa^{\prime\prime\prime})$ (20)
$\displaystyle~{}+\kappa^{2}\big{(}19\tau^{4}\kappa^{\prime 2}-4\kappa^{\prime
4}+10\kappa^{\prime 2}\tau^{\prime
2}+2\tau\kappa^{\prime}(2\kappa^{\prime}\tau^{\prime\prime}-15\tau^{\prime}\kappa^{\prime\prime})+2\tau^{2}(5\kappa^{\prime\prime
2}+2\kappa^{\prime}\kappa^{\prime\prime\prime})\big{)}$
$\displaystyle~{}+\kappa^{4}\tau^{2}(6\tau^{4}-8\kappa^{\prime
2}-5\tau^{\prime
2}+4\tau\tau^{\prime\prime})-4\kappa^{6}\tau^{4}\Big{)}\,\Big{(}192\,(\kappa^{2}\tau^{2}+\kappa^{\prime
2})^{\frac{5}{2}}\Big{)}^{-1}~{}.$

At first glance this conformal invariant $P$ is not related in an obvious way
to conformal curvature $Q$ and torsion $T$. There is an analogous situation to
the metric case (5), where the second nontrivial order is also a certain
function of curvature and torsion.

For the moment, we leave this issue open and turn to the analogue of (8).
There we considered the angle between the normal vectors of two consecutive
planes, fixed in both cases by three consecutive points $x_{j}$. Now we look
at the angle between the normal vectors of two consecutive spheres, fixed both
by four consecutive points. Let us denote by $cs(x_{1},x_{2},x_{3},x_{4})$ the
sphere fixed by the points $x_{1},\dots x_{4}$. Then the angle between vectors
normal to $cs(x_{1},x_{2},x_{3},x_{4})$ and $cs(x_{2},x_{3},x_{4},x_{5})$ at
the point $x_{4}$ is given by

$\mbox{cos}\,\gamma~{}=~{}\frac{(A_{1}\,z_{1}+A_{2}\,z_{2}+A_{3}\,z_{3})\cdot(B_{2}\,z_{2}+B_{3}\,z_{3}+B_{5}\,z_{5})}{|A_{1}\,z_{1}+A_{2}\,z_{2}+A_{3}\,z_{3}|\,|B_{2}\,z_{2}+B_{3}\,z_{3}+B_{5}\,z_{5}|}~{}.$
(21)

We used the appendix (44) and

$B_{2}=A_{2}|_{x_{1}\rightarrow x_{5}}~{},~{}~{}B_{3}=A_{3}|_{x_{1}\rightarrow
x_{5}}~{},~{}~{}B_{5}=A_{1}|_{x_{1}\rightarrow x_{5}}~{}.$ (22)

Since our interest concerns the first nontrivial order only, the use of the
expansion for $\epsilon(\omega)$ as in (18) is not necessary. Then with the
setting

$x_{1}=x(s_{0}-3\epsilon),~{}~{}~{}x_{2}=x(s_{0}-2\epsilon),~{}~{}~{}x_{3}=x(s_{0}-\epsilon),~{}~{}~{}x_{4}=x(s_{0}),~{}~{}~{}x_{5}=x(s_{0}+\epsilon)~{}$
(23)

we obtain

$\mbox{cos}\,\gamma~{}=~{}1-\frac{1}{2}\,T^{2}\,(\kappa^{2}\tau^{2}+\kappa^{\prime
2})\,\epsilon^{2}~{}+~{}{\cal
O}(\epsilon^{3})~{}=~{}1-\frac{1}{2}\,T^{2}\,\omega^{2}~{}+~{}{\cal
O}(\omega^{3})~{}.$ (24)

After this straightforward recovering of the conformal torsion $T$, we come
back to the issue of relating the conformal invariant $P$ from (20) to $T$ and
$Q$. The denominator in (20) agrees with the square of that in (3), up to the
numerical factor $192$. In the nominator there is $6\kappa^{4}\tau^{6}$, as
first term in the third line, which shows up also in the square of the
numerator of $T$, again up to a numerical factor. Hence we try to relate
$P-\frac{T^{2}}{32}$ directly to $Q$ and find 444 A linear combination of $Q$
and $T^{2}$ also plays a role in some different approach [7].

$P~{}=~{}\frac{1}{24}\,\big{(}Q+\frac{3}{4}\,T^{2}\big{)}~{}.$ (25)

At the end of this section we still calculate a torsion angle $\beta$ as
defined in our papers [8, 9, 10], and now applied to the inscribed polygons
under discussion. Take e.g. for fig.1 the angle between the circumcircle
$cc_{4}$ and the circular edge $ca^{+}_{2}$ at the point $x_{3}$. In fig.1 it
is at $x_{3}$ the angle between the black arc $ca^{+}_{2}$ and the red arc
$ca^{-}_{4}$

$\mbox{cos}\,\beta~{}=~{}t^{123}_{3}\cdot t^{345}_{3}~{}.$ (26)

With (36) from the appendix this yields

$\mbox{cos}\,\beta~{}=~{}\frac{x_{13}^{2}\,x_{24}^{2}\,x_{35}^{2}+x_{15}^{2}\,x_{23}^{2}\,x_{34}^{2}-x_{13}^{2}\,x_{25}^{2}\,x_{34}^{2}-x_{14}^{2}\,x_{23}^{2}\,x_{35}^{2}}{2\,x_{12}\,x_{13}\,x_{23}\,x_{34}\,x_{35}\,x_{45}}~{}.$
(27)

For the setting (18) this implies

$\mbox{cos}\,\beta~{}=~{}1-\frac{1}{8}\,T^{2}\,\omega^{6}~{}+~{}{\cal
O}(\omega^{7})~{}.$ (28)

Obviously, we have two options to get $T^{2}$ from the leading nontrivial term
in an $\epsilon$-expansion. One using spheres for the angle $\gamma$ in (24)
and one using circles for the angle $\beta$ in (28).

But as a crucial message let us stress, that for extracting all three
conformal parameters $\omega,Q,T$, one can restrict to calculations of the
anglees $\alpha$ and $\beta$ with circles alone (19),(25),(28).

##  4 Cross ratios and circumcircles

As a byproduct of our manipulations with angles between crossing circles, we
add some comments on generic polygons 555The inscribed polygons used in the
previous section have edges, which are parts of circles fixed by three
consecutive corners. In the generic case the circular edges are not parts of
these circles. as considered in our Wilson loop papers [8, 9]. For the overall
conformal parameterisation of these polygons one needs also cross ratios of
the corner points, beyond the cusp and torsion angles. We use the elementary
geometry from the appendix to relate cross ratios to angles between certain
circumcircles.

We consider four generic points $x_{1},x_{2},x_{3},x_{4}\in\mathbb{R}^{3}$.
Their conformal properties are characterised by the cross ratios

$C(i,j,k,l)~{}=~{}\frac{x_{ik}}{x_{il}}~{}\frac{x_{jl}}{x_{jk}}~{}.$ (29)

Among them only two are independent. We follow the convention in [12] and
choose as independent parameters

$u~{}=~{}C(1,4,2,3)~{}=~{}\frac{x_{12}}{x_{13}}~{}\frac{x_{43}}{x_{42}}~{},~{}~{}~{}v~{}=~{}C(1,2,4,3)~{}=~{}\frac{x_{14}}{x_{13}}~{}\frac{x_{23}}{x_{24}}~{}.$
(30)

If we take out one of the four points, the remaining three determine a circle,
the circumcircle of the related triangle. In this manner we get four circles
$cc^{(i)}$, numerated just by the point omitted. By definition, each pair of
such circles has two points in common, always two points from the set
$(x_{1},x_{2},x_{3},x_{4})$.

Let us denote by $t^{(i)}_{j}$ the unit tangent vector to $cc^{(i]}$ at the
point $x_{j}$. Then the cosine of the crossing angle between e.g. $cc^{(1)}$
and $cc^{(2)}$is given by

$(cc^{(1)},cc^{(2)})~{}=~{}t^{(1)}_{3}t^{(2)}_{3}~{}=t^{(1)}_{4}t^{(2)}_{4}~{}.$
(31)

Our aim is now to express all the six $(cc^{(i)},cc^{(j)})$ in terms of the
two independent cross ratios $u$ and $v$. Then with (36) and suitable adaption
of indices we get

$\displaystyle p=\mbox{cos}\,\varphi$ $\displaystyle:=$
$\displaystyle(cc^{(1)},cc^{(2)})~{}=~{}(cc^{(3)},cc^{(4)})~{}=~{}\frac{1+v^{2}-u^{2}}{2\
v}~{},$ $\displaystyle q=\mbox{cos}\,\psi$ $\displaystyle:=$
$\displaystyle(cc^{(1)},cc^{(4)})~{}=~{}(cc^{(2)},cc^{(3)})~{}=~{}\frac{1+u^{2}-v^{2}}{2\
u}~{},$ $\displaystyle r=\mbox{cos}\,\chi$ $\displaystyle:=$
$\displaystyle(cc^{(1)},cc^{(3)})~{}=~{}(cc^{(2)},cc^{(4)})~{}=~{}\frac{1-u^{2}-v^{2}}{2\
u\ v}~{}.$ (32)

Using the above parametrisation of $p,q$ and $r$ by $u$ and $v$ we obtain

$r~{}=~{}p\,q-\sqrt{(1-p^{2})(1-q^{2})}~{},~{}\mbox{i.e.}~{}~{}~{}\mbox{cos}\,\chi=\mbox{cos}\,(\varphi+\psi)~{}.$
(33)

Figure 2: In yellow the boundary of the rounded tetrahedron (34) and in
yellow/blue that part, for which $(p,q,r)$ is allowed by (32) and (35).

A manifest symmetric form of the constraint666Interestingly, this cubic
constraint for three parameters appears also in the context of the conformal
geometry of triangles with circular edges [9] and the conformal properties of
null hexagons [13],[14]. between the three angles is

$p^{2}+q^{2}+r^{2}-2p\,q\,r~{}=~{}1~{}.$ (34)

However, one has to keep in mind, that not all points $(p,q,r)$ allowed by
(34) belong to a possible geometric setup for
$x_{1},x_{2},x_{3},x_{4}\in\mathbb{R}^{3}$. First of all, (34) includes also
points corresponding to the positive sign in front of the square root in (33).
Furthermore, since $p,q,r$ as cosines of angles have to have a modulus smaller
or equal to 1, the equations (32) imply for the by definition positive cross
ratios $u,v$

$\displaystyle u\geq 0~{},$ $\displaystyle v\geq 0~{},$ $\displaystyle u+v\geq
1~{},$ $\displaystyle|u-v|\leq 1~{}.$ (35)

Of course these constraints on the cross ratios are well-known. A
straightforward way to derive them without consideration of our circles is
given by conformal mapping e.g. $x_{4}$ to infinity. Then (35) is a
consequence of the triangle inequalities for the triangle with corners
$x_{1},x_{2},x_{3}$.

The surface in $(p,q,r)$-space, defined by (34) and $|p|,|q|,|r|\leq 1$ looks
like the boundary of a tetrahedron with rounded edges. Only one of the four
faces is allowed by (32) and (35). It is just the face opposite to the vertex
$(p,q,r)=(-1,-1,1)$, see fig.2.

The vertices $(1,1,1),~{}(1,-1,-1)$ and $(-1,1,-1)$ are via (32) the images of
the boundary of the allowed region (35) in the $(u,v)$-plane: $u+v=1,~{}u-v=1$
and $v-u=1$, respectively. They correspond to the limiting case, where all
four points are located on just one circle or one straight line. All crossing
angles are then equal to $0$ or $\pi$, depending on ordering issues generated
by the sign convention in (36).
Acknowledgement
I thank the Quantum Field and String Theory Group at Humboldt University for
kind hospitality.

## Appendix

Three points $x_{1},x_{2},x_{3}$ fix a circle uniquely. Its unit tangent
vectors at these points are given by

$\displaystyle t^{(123)}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{x_{23}}\big{(}\frac{x_{13}}{x_{12}}(x_{2}-x_{1})+\frac{x_{12}}{x_{13}}(x_{1}-x_{3})\big{)}~{},$
$\displaystyle t^{(123)}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{x_{13}}\big{(}\frac{x_{12}}{x_{23}}(x_{3}-x_{2})+\frac{x_{23}}{x_{12}}(x_{2}-x_{1})\big{)}~{},$
$\displaystyle t^{(123)}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{x_{12}}\big{(}\frac{x_{23}}{x_{13}}(x_{1}-x_{3})+\frac{x_{13}}{x_{23}}(x_{3}-x_{2})\big{)}~{},$
(36)

with $x_{ij}=|x_{i}-x_{j}|$.

Four points $x_{1},x_{2},x_{3},x_{4}$ fix a sphere uniquely. Its center
$x_{c}$ is given by

$x_{c}=x_{4}+\frac{1}{D}\,\big{(}A_{1}\,(x_{1}-x_{4})+A_{2}\,(x_{2}-x_{4})+A_{3}\,(x_{3}-x_{4})\big{)}~{},$
(37)

with $(z_{j}=x_{j}-x_{4})$

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{c}z_{1}z_{1},\,z_{1}z_{2},\,z_{1}z_{3}\\\
z_{2}z_{2},\,z_{2}z_{2},\,z_{2}z_{3}\\\
z_{3}z_{3},\,z_{3}z_{2},\,z_{3}z_{3}\end{array}\right|~{},~{}~{}~{}A_{2}~{}=~{}\left|\begin{array}[]{c}z_{1}z_{1},\,z_{1}z_{1},\,z_{1}z_{3}\\\
z_{2}z_{1},\,z_{2}z_{2},\,z_{2}z_{3}\\\
z_{3}z_{1},\,z_{3}z_{3},\,z_{3}z_{3}\end{array}\right|~{},$ (44)
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{c}z_{1}z_{1},\,z_{1}z_{2},\,z_{1}z_{1}\\\
z_{2}z_{1},\,z_{2}z_{2},\,z_{2}z_{2}\\\
z_{3}z_{1},\,z_{3}z_{2},\,z_{3}z_{3}\end{array}\right|~{},~{}~{}~{}~{}D~{}=~{}\left|\begin{array}[]{c}z_{1}z_{1},\,z_{1}z_{2},\,z_{1}z_{3}\\\
z_{2}z_{1},\,z_{2}z_{2},\,z_{2}z_{3}\\\
z_{3}z_{1},\,z_{3}z_{2},\,z_{3}z_{3}\end{array}\right|~{}.$ (51)

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