# Evaluating Uncertainty-based Failure Detection for
Closed-Loop LLM Planners

Zhi Zheng∗,1,2, Qian Feng1,2, Hang Li1,2, Alois Knoll2, Jianxiang Feng∗,1,2 *:
Equal Contributions, {zhi.zheng<EMAIL_ADDRESS>Robot AG2
Department of Informatics, Technical University of Munich

###### Abstract

Recently, Large Language Models (LLMs) have witnessed remarkable performance
as zero-shot task planners for robotic manipulation tasks. However, the open-
loop nature of previous works makes LLM-based planning error-prone and
fragile. On the other hand, failure detection approaches for closed-loop
planning are often limited by task-specific heuristics or following an
unrealistic assumption that the prediction is trustworthy all the time. As a
general-purpose resoning machine, LLMs or Multimodal Large Language Models
(MLLMs) are promising for detecting failures. However, the appropriateness of
the aforementioned assumption diminishes due to the notorious hullucination
problem. In this work, we attempt to mitigate these issues by introducing a
framework for closed-loop LLM-based planning called KnowLoop, backed by an
uncertainty-based MLLMs failure detector, which is agnostic to any used MLLMs
or LLMs. Specifically, we evaluate three different ways for quantifying the
uncertainty of MLLMs, namely token probability, entropy, and self-explained
confidence as primary metrics based on three carefully designed representative
prompting strategies. With a self-collected dataset including various
manipulation tasks and an LLM-based robot system, our experiments demonstrate
that token probability and entropy are more reflective compared to self-
explained confidence. By setting an appropriate threshold to filter out
uncertain predictions and seek human help actively, the accuracy of failure
detection can be significantly enhanced. This improvement boosts the
effectiveness of closed-loop planning and the overall success rate of tasks.

##  I INTRODUCTION

Pre-trained Large language models (LLMs) have shown superior generalization
capability as zero-shot task planners in robot learning by transforming high-
level language instructions into low-level action plans [1, 2, 3, 4, 5].
However, these planners predominantly employ open-loop control, rigidly
following initial plans without incorporating environmental feedback.
Consequently, execution errors or plan deficiencies remain unaddressed,
potentially leading to task failure.

Recent works on facilitating closed-loop planning for LLM planners tend to be
less attentive toward the indispensable first step in closed-loop planning –
failure detection [6, 7, 8, 9]. They usually adopt either LLMs [6, 7, 8] or
Multimodal Large Language Models (MLLMs) [9] by assuming their predictions to
be trustworthy all the time.

Figure 1: A Sketch of our proposed closed-loop planning framework, KnowLoop
featured by an uncertainty-based failure detection module, allowing robots to
know when to close the loop properly during planning.

This simplified assumption might not hold due to the notorious hallucination
problem of LLMs/MLLMs [10], meaning the inconsistencies between the generated
contents and the factual truth or user input contexts. On the other hand, some
heuristic-based failure detection [11] for LLM planners is relatively accurate
but highly task-specific, restricting their general suitability to various
tasks and models. Uncertainty estimation [12, 13] has been considered a
promising technique to handle the hallucination issue for LLMs [10] and found
widespread adoption in robotics [14, 15, 16, 17]. More noteworthy, uncertainty
estimation for LLMs benefits from its characteristic of being model-agnostic,
hence facilitating wider suitability.

In light of this, we introduce KnowLoop (see Fig. 1), a framework backed up by
an uncertainty-based failure detection module, allowing robots to know when to
close the loop properly during planning. The proposed module is LLM/MLLM model
agnostic, meaning more general applicability when compared with the task-
specific approaches. Meanwhile, it is able to boost the performance of LLM-
based closed-loop planning by filtering out uncertain predictions from the
failure detector. When confronting predictions with low confidence, the
framework enables the robot to solicit human help actively in order to avoid
task failure during task execution. Specifically, we first estimate the
uncertainty for a MLLM, which we adopt as a failure detector given the task
description and an image depicting the current stage. Inspired by [18], we
evaluate three approaches for obtaining uncertainty estimates in MLLMs, namely
token probability, entropy, and self-explained confidence. In addition, to
comprehensively investigate the impact of different prompts to trigger the
MLLM-based failure detector, we carefully design three prompting strategies
spanning over two representative categories. The first category is direct
prompts, in which the question of success or failure appears, while the second
one is indirect prompts, where there is no this kind of question in the
context.

To study the effectiveness of the proposed idea, we build an LLM-based robot
system adapted from [4] and create a self-collected dataset for
success/failure analysis with five different manipulation tasks, including
both short-term and long-term complex actions, see Fig. 3. We verify the
benefits of exploiting uncertainty-based failure detection for closed-loop LLM
planning through empirical experiments. To assure comprehensiveness, the
experiments are conducted on both the dataset and real hardware with two
different MLLMs (LLaVA [19] and ChatGPT-4V). The contributions made in this
work are as follows:

  * •

We propose to exploit uncertainty estimation for more trustworthy and
effective failure detection based on LLMs/MLLMs.

  * •

We integrate the uncertainty-based MLLM failure detector into a zero-shot LLM-
based robot system to facilitate closed-loop planning.

  * •

With experiments on a self-collected dataset and real hardware, we show
promising results of leveraging uncertainty estimates in LLM/MLLM for closed-
loop planning.

##  II RELATED WORK

To lay the groundwork for a comprehensive understanding, we review the
literature in the following relevant areas:

####  II-1 Uncertainty Estimation in Robotics

Uncertainty estimation has a longstanding tradition in robotics, dating back
to the era of probabilistic probiotics [20] and robotic introspection [21].
With the advent of deep learning, uncertainty estimation is gaining more and
more attention in achieving trustworthy learning-enabled robotics due to the
black-box nature of neural networks [22, 12, 23]. The spectrum of applying
uncertainty estimation to empower robots spans from learning-based perception
[17, 16, 15] to control [24]. Moreover, the significance of uncertainty
estimation is increasingly prominent in the age of foundational models [14].
When it comes to LLMs for robotics, there is only a limited amount of work,
such as KnowNo [25], which proposes a method to estimate the uncertainty of an
LLM planer. It is done by generating multiple options as the next steps and
analyzing their corresponding token probabilities with conformal prediction.
However, they do not consider the failure of task execution itself.

####  II-2 Uncertainty Estimation in LLMs

LLMs have shown an inevitable tendency to create hallucinations, which refers
to the inconsistencies between the generated contents and the real-world truth
or user inputs [10]. The model’s uncertainty is demonstrated to be related to
the occurrence of LLM hallucinations [26]. Using uncertainty estimation to
predict LLM hallucination as a zero-resource setting prevents the need for
external knowledge resources. [27] attempt to use Prompt Engineering to have
the LLM output the confidence level of the answer simultaneously when speaking
it out, as well as obtaining the model’s confidence level based on the
Consistency base method. [18] propose an LLM self-evaluation method in the
form of multiple-choice questions or true/false statements to obtain a
quality-calibrated confidence score at the token level. [27] introduce a
confidence elicitation framework consisting of prompting, sampling, and
aggregation strategies. The framework is evaluated with confidence calibration
and failure prediction tasks. Their focuses are both on the general question-
answer settings instead of robotic applications.

####  II-3 Closed-loop planning with LLMs

LLMs can be harnessed to translate high-level, abstract task instructions into
actionable, step-by-step sequences for execution by agents. Recent studies
[11, 28, 29, 30, 31] have showcased the capabilities of LLMs for self-
reflection and correction in response to environmental feedback. Here they
often assume a ground truth environmental feedback. A prior work similar to
ours [8] utilizes multisensory data and hierarchical summary to detect,
reason, and correct failures in closed-loop robotic task planning in real-
world scenarios. They rely on dedicated hand-crafted external failure
detectors but we propose to estimate the uncertainty from a multimodal Large
Language Model (MLLM) for more general failure detection, providing a more
versatile and adaptable solution.

Figure 2: A flow diagram of the proposed framework for the task: opening the
drawer.

##  III The Proposed Closed-Loop Planner

In this chapter, we outline KnowLoop, the proposed closed-loop control
system’s framework. Our implementation of an interactive failure detector is
based on uncertainty derived from the MLLM. Specifically, we utilize open-
source MLLMs such as LLaVA [19] and ChatGPT-4V for failure detection within an
LLM planner based on ChatGPT-4. Note that the LLMs/MLLMs used in this work can
be merged into one in principle, which we leave for future work. The procedure
entails a decomposition of a language instruction into a series of sub-goals.
After executing each sub-goal, the failure detector is activated to collect
feedback from environment images and determine whether the sub-goal has been
achieved. In the following, we will articulate the major components of the
proposed uncertainty-based failure detector, including the prompting
strategies to activate the MLLM failure detector and approaches for
uncertainty quantification in LLMs. Finally, we integrate the uncertainty-
based failure detector into an LLM-based planning framework for closed-loop
planning.

###  III-A Uncertainty-based Failure Detection

To trigger an MLLM failure detector, a proper and informative prompt is
necessary. For this purpose, we specifically design two representative
categories of prompting strategies, namely a direct one and an indirect one.
Direct prompting strategy entails directly the question of success or failure,
while the indirect one does not encompace this kind of direct question of
failure or success in the context. Furthermore, to tackle the improper
assumption of the failure detector, we consequently estimate the uncertainty
of each response to these prompts. Based on the uncertainty estimates and a
predetermined threshold, the reliability of each response can be evaluated
quantitatively in a generic way. Responses with excessively high uncertainty
will be filtered out, and instead, human feedback will be requested to assess
the current state, as shown in Algorithm 1. More noteworthy, the proposed way
to detect failure cases is agnostic to LLM used in the planner. Therefore, it
enjoys the merit of general applicability when compared with other task-
specific or heuristic approaches.

####  III-A1 Prompting Strategy

The impact of utilizing diverse prompts for an LLM is substantial, as the
chosen prompt directly shapes the model’s output in manifold ways. To attain
improved detection outcomes and reasonable uncertainty, it is crucial to
meticulously select the prompts employed in the detection process. Toward this
objective, we have carefully designed three distinct prompt strategies,
articulated as follows.

Direct Prompts via Subgoal State Comparison (SSC). Given only background
information and environmental images, inferring the state description upon
completing an action is challenging for LLMs. Moreover, comparing this with
the environmental images to determine if the current state aligns with the
subgoal poses additional challenges. Consequently, we propose to succinctly
outline each subgoal or action, accompanied by an expected state description.
The MLLM will then compare this description against the current environmental
image to ascertain whether the present state matches the anticipated outcome.
This indicates the successful achievement of the subgoal.

Subgoal State Comparison (SSC) The robot arm is given a task: [task
instruction]. The robot arm just tried to execute [subtask at time t]. Q:
Based on the image, is the [expected state description] satisfied? A: Yes /
No.

Direct Prompts via Spatial Relationship Analysis (SRA). The ”Chain of Thought”
(CoT) [32] reasoning in LLMs epitomizes a sophisticated strategy for tackling
complex problem-solving tasks. This technique capitalizes on the model’s
capacity to generate intermediate steps or rationales leading to a conclusive
answer, thereby emulating a process akin to human reasoning. In the context of
robot manipulation tasks, the most profound transformations are observed in
the dynamics between the gripper and the spatial positioning of objects
following each action. Rather than directly evaluating the outcome as success
or failure based on state descriptions and images, the MLLM should incorporate
an intermediate cognitive process. This analyzes the current spatial positions
and conditions of objects within the environment, comparing these with the
anticipated states and subsequently adjudicating the action’s success or
failure.

Spatial Relationship Analysis (SRA) The robot arm is given a task: [task
instruction]. The robot arm just tried to execute [subtask at time t]. Q: To
tell whether [expected state description] is satisfied, first analyze the
spatial relationship between objects in the working space. A: [analysis] Q: Is
the [expected state description] satisfied? A: Yes / No.

Indirect Prompts via Next Action Prediction(NAC). To distinguish this strategy
from the preceding ones, success is no longer binary (a simple yes or no).
Instead, upon completing each action, we present all high-level plans and the
corresponding executable actions to the MLLM. Then, images are supplied to
illustrate the current state of the environment. Based on the current state,
the MLLM is asked to select the subsequent action from the presented options.
This approach reformulates the detection problem into a multiple-choice
scenario. Success is achieved when the MLLM’s selection aligns with the
subsequent step of the predefined plan. Conversely, the selection of any
alternative action is deemed an execution failure.

Next Action Prediction (NAC) The robot arm is given a task: [expected state
description]. The high-level plan for this task is [list of subtasks]. The
robot arm just tried to execute [subtask at time t]. Q: Based on the image,
which subtask should be the next step? A: B (One of the subtasks in the list).

####  III-A2 Uncertainty Quantification

The precision of the failure detector’s judgments is pivotal for the success
of the entire closed-loop control system. Should a true negative scenario
arise, it may cause the system to miss errors during execution, persisting
with the initial plan and leading to task failure. In contrast, a false
negative can interrupt the correct sequence of operations, leading to
erroneous states that hinder the task’s completion. Hence, the greater the
failure detector’s accuracy, the more effective the system. Nonetheless, even
the most advanced MLLMs, like ChatGPT-4V, fail to achieve flawless accuracy
and may produce hallucinations, generating responses arbitrarily. We aim to
quantify the uncertainty in the model’s responses to better gauge its
confidence in its answers. When the model is uncertain, expressing ”I don’t
know” is preferable to providing an arbitrary response.

Token probability. Prior work [25] suggests that, by formulating multiple-
choice questions and answering (MCQA) and having LLMs’ responses to them, it
is feasible to deduce the occurrence probability of each option token
$P(x_{i})$. This probability is interpreted as the likelihood of the
corresponding option being correct. In the context of failure detection, the
issue can be conceptualized as a binary question: whether the outcome of the
most recent action was successful or if the current state aligns with the
anticipated state. The response sought is either ’Yes’ or ’No’. In a similar
vein, deriving the token probability from the MLLM output can act as an
indicator of the probability of affirmative or negative responses.

Entropy. In information theory, entropy represents a measure of uncertainty or
disorder. For binary classification problems, the entropy of a model’s
predictions can be calculated using the formula:

$H(X)=-\sum_{i}P(x_{i})\log P(x_{i})$ (1)

where $P(x_{i})$ is the probability of class $i$ (in case of failure
detection, ”Yes” or ”No”). Within the output layer of the MLLM, each token is
assigned a score. When subjected to a softmax transformation, these scores
yield the probability associated with each token. The probability of a token
can thus be interpreted as the probability of a given classification. The
entropy derived from these token probabilities may reflect the model’s
uncertainty about its response to a certain extent. Higher entropy values
suggest greater model uncertainty. This relationship between entropy and model
uncertainty will be further explored and discussed in the subsequent sections.

Self-explained Confidence. In certain instances, LLMs can be induced to
articulate their confidence levels explicitly. For example, the model might be
prompted to deliver responses in the format: ”I am X% certain that the answer
is Y,” wherein X represents the model’s self-evaluated confidence level. This
approach is contingent upon meticulous prompt engineering. Given that it
hinges on the model’s inherent processes for comprehending and generating
confidence-related responses, it may not consistently produce precise
confidence estimations.

###  III-B Closed-loop Planning

In the final proposed closed-loop control framework, ChatGPT-4 is used as the
high-level task planner. The LLM planner understands language instructions and
decomposes them into multiple subtasks. GPT’s code generation function will be
called for each subtask, and combined with the robotic arm motion API, it will
generate corresponding Python code for each action. Subsequently, the robotic
arm will execute actions one by one according to the resultant task plan.
After each action execution, the detector will be called to assess the state
of the environment and detect the failure. If the assessment indicates
successful execution, the robotic arm will proceed with the following action
as planned.

Input: Prompts of the current subtask $i$: $l_{i}$, Image of the current state
of step $i$: $R_{i}$, a user-specified threshold: $\delta$.

Output: Whether the subtask was successfully excuted

Function  _failure_detect(_$l_{i}$ , $R_{i}$_)_:

uncertainty_est, response = MLLM($l_{i}$, $R_{i}$);

if  _uncertainty_est $<\delta$_ then

return _response_ ;

else

user_help $\leftarrow$ input("I am not sure! The current subtask is successful
or failed? ");

return _user_help_ ;

end if

Algorithm 1 Failure Detection.

If the assessment indicates failure, the robotic arm will stop the current
action and re-execute all actions as shown in Algorithm 2.

Input: A task planning prompt: $L$, maximum number of retrying: $k$.

LLM Code Generation based on $L$:

$subtask\\_list=[l_{1},l_{2},\ldots,l_{n}]$;

$retry=0$;

$index=0$;

while  _index $<$ len(Subtasks) and retry $<$ $k$_ do

$l_{i}=subtask\\_list[index]$;

$R_{i}=robot\\_execution(l_{i})$;

$prediction=failure\\_detect(l_{i},R_{i})$;

if  _prediction is success_ then

$index=index+1$;

else

$index=0$;

$retry=retry+1$;

end if

end while

Algorithm 2 Closed-loop Planning Framework.

##  IV Experiment

We first show our experimental setup including the self-collected dataset and
implementation details of the system and algorithm. Then we discuss the
experiment results. We first evaluate the performance of three different
uncertainty estimation methods along with different promoting strategies based
on a self-collected dataset.

Secondly, based on the results obtained in the uncertainty evaluation part, we
select the combination with the most encouraging performance for closed-loop
LLM planners on real hardware. To note that the test set includes slightly
different tasks111Task_1:pick up the sponge and place it on the notebook,
Task_2:pick up the smartphone and place it in the upper drawer, Task_3:open
the upper drawer but with similar items. The LLM initiates high-level planning
upon receiving language instructions, decomposing the tasks into several
subtasks. In scenarios devoid of external interference, the robotic arm’s
adherence to the high-level action plan results in a success rate ranging
between 70-80% for these tasks. To illustrate the significance of closed-loop
control more effectively, each experiment will introduce human
interference—such as displacing objects from their intended locations or
impeding operations. This interference leads to a success rate of 0% for these
tasks under open-loop control conditions. In this experiment, we also compare
ChatGPT-4V and LLaVA to show the general compatibility of our proposed
framework.

###  IV-A Dataset

Valid uncertainty implies that higher model uncertainty correlates with
decreased prediction accuracy. Conversely, as the model’s confidence
increases, its accuracy significantly improves. Additionally, the
applicability of this uncertainty in robotic manipulation tasks remains
uncertain. To investigate this more efficiently, we collected 142 samples
representing different execution states across five tasks222Task_1:pick up the
mouse and place it on the notebook, Task_2:pick up the sponge and place it in
the upper drawer, Task_3:pick up the smartphone and place it on the drawer,
Task_4:open the upper drawer, Task_5:close the upper drawer in Table I. The
primary failures encompass detection and operational errors, explicitly
excluding planning errors. The tasks primarily involve actions such as
grasping, placing, and pushing.

Figure 3: Example Images for different tasks in the self-collected dataset,
from left to right: picking up a mouse, placing a sponge on the notebook,
placing a smartphone in the drawer, opening the drawer, and closing the
drawer.

###  IV-B Evaluation Metrics

In this subsection, we explain the metrics used for evaluating different
variants in the experiment.

##### Uncertainty-Acuracy Curve and Area Under Curve (AUC) (Calibration-AUC)

The uncertainty-accuracy curve [18] depicts accuracy as a function of the
abstention uncertainty threshold. This threshold determines the point at which
samples are excluded from consideration if their uncertainty surpasses
$(1-threshold)$. Initially, no samples are omitted when the uncertainty
threshold is set to zero, allowing the accuracy measurement to reflect the
entire dataset’s performance. As the uncertainty threshold escalates, samples
exhibiting uncertainty above $(1-threshold)$ are selectively discarded.
Subsequently, the accuracy metric is recalculated to reflect the composition
of the residual dataset. Should uncertainty serve as a reliable indicator of
the model’s uncertainty, discarding samples marked by higher uncertainty
should theoretically enhance the quality of the retained samples.
Consequently, this selective exclusion is expected to improve the model’s
overall accuracy.

##### Selective Generation Curve and AUC (Selective-AUC)

The selective generation curve [18] depits accuracy as a function of the
abstention rate, denoted as $\alpha$. This approach involves ordering the
samples by their uncertainty and then abstaining from the top $\alpha\%$ of
samples based on their uncertainty values. No samples are abstained at
$\alpha=0\%$, meaning the curve initiates at the conventional accuracy metric.
Consequently, an increase in the curve is anticipated as $\alpha$ escalates.

##### Success Rate

For each real-world task, it is considered a success if the instruction
requirements are eventually met under our closed-loop control. If the task
remains incomplete after reaching the maximum number of attempts, it is deemed
a failure. The success rate is calculated by dividing the number of successful
experiments by the total number of experiments.

##### Detection Accuracy

Detection accuracy is the number of correct predictions divided by the total
number of detections made by the MLLM failure detector. Instances where humans
provide assistance are not included. This metric indicates the ability of MLLM
to make accurate predictions.

##### Human Involve Rate

The metric is calculated by dividing the number of times humans provided
assistance by the total number of detections performed during the experiment.
A higher human involvement rate indicates a larger effort to achieve a high
success rate, which further implies the limited performance of the MLLM
failure detector.

##### Generation Rate

This metric only applies to Self-explained Confidence, as not every attempt
results in a successful generation of confidence scores. It is calculated as
the number of successful confidence score generations divided by the total
number of tests.

###  IV-C System Setup

We use a Diana 7 robot arm and Robotiq 2f-140 gripper with a tabletop setup
(see Fig. 1). We mount one eye-to-hand Realsense D415 RGBD camera on the top
left from the top-down view. The camera returns the real-time RGB-D
observations at 30 Hz. Our LLM-based planner is adapted based on Voxposer [4].
We use Grounding-Dino [33] for open-set object detection, then feed it into
Segment Anything [34] to obtain a mask. The mask is used to crop the object
point cloud. We run our system on a single Nvidia RTX-4090 GPU.

TABLE I: Self-Collected Dataset Break Down | total | task_1 | task_2 | task_3 | task_4 | task_5
---|---|---|---|---|---|---
success | 72 | 12 | 18 | 12 | 18 | 12
failure | 70 | 14 | 19 | 10 | 14 | 13

###  IV-D Results

TABLE II: Results Comparison of three uncertainty estimation methods and three
prompting strategies on the self-collected dataset based on LLaVA. 1:
Calibration-AUC is not computable for self-explained confidence as the MLLM
might fail to provide a sensible response. Instead, we provide the generation
rate.

Token Proabbility Entropy Self-explained Confidence1 Prompting Strategies
Detection Accuracy Calibration- AUC Selective- AUC Detection Accuracy
Calibration- AUC Selective- AUC Generation Rate Detection Accuracy Selective-
AUC SSC 57.5% 0.58 0.65 57.5% 0.67 0.65 46.2% 57.1% 0.44 SRA 66.0% 0.64 0.68
66.0% 0.66 0.68 68.9% 59.0% 0.50 NAP 29.2% 0.58 0.49 29.2% 0.49 0.44 58.5%
3.2% 0.04

TABLE III: Results of uncertainty-based failure detection for closed-loop LLM
Planners on real hardware. Entropy is used as the uncertainty measure for both
ChatGPT-4V and LLaVA.

ChatGPT-4V LLaVA KnowLoop (LLaVA+Human Help) Tasks Success Rate Detection
Accuracy Success Rate Detection Accuracy Success Rate Detection Accuracy Human
Involve Rate Task_1 $10\%$ $38\%$ $30\%$ $57.6\%$ $70\%$ $78\%$ $28\%$ Task_2
$20\%$ $35.4\%$ $50\%$ $57.8\%$ $80\%$ $81.4\%$ $31.8\%$ Task_3 $40\%$
$34.1\%$ $10\%$ $25.8\%$ $70\%$ $76\%$ $21.8\%$

####  IV-D1 Uncertainty Estimation for MLLM Failure Detecor

We first evaluate the MLLM failure detector based on LLaVA, as we need token
probability for uncertainty calculation which is not supported by ChatGPT-4V
API at the moment. We document each sample’s prediction outcomes and token
probability for each prompting strategy. Upon quantification, the utility of
LLaVA’s entropy, as derived through the techniques mentioned above, in
differentiating between high-quality and low-quality predictions remains
ambiguous. Consequently, we employ the following criteria to assess the
correlation between the uncertainty derived from various prompting and
quantification methods and the accuracy of the predictions shown in Table II,
Fig. 4 and Fig. 5.

(a)

(b)

(c)

(d)

Figure 4: Uncertainty Accuracy Curve (left column) and Selective Generation
Curve (right column) for three prompting strategies based on entropy (top row)
and token probability (bottom row).

The increasing curves in Fig. 4 indicate that uncertainty quantification
through token probability and entropy yields promising performance. A distinct
trend emerges, showing increased accuracy as uncertainty diminishes. However,
as observed in Fig. 5, the correlation between self-reported uncertainty and
accuracy is weak, rendering the derived uncertainty measures impractical for
use.

In Table II, the AUC values for the strategies of SSC and SRA are relatively
similar and notably superior to that of the NAP strategy. This disparity is
primarily attributed to the lower accuracy of the NAP strategy in the absence
of filtering. The linear correlation between accuracy and entropy-based
uncertainty, particularly evident in the SSC prompt, is the most pronounced
among the measures examined.

For the promising performance of SSC with entropy, we adopt the curve produced
by the SSC prompting strategy with entropy for the real hardware experiment.

####  IV-D2 Uncertainty for Closed-Loop LLM Planners

In the real-world experiment, we chose ChatGPT-4V and LLaVA as baselines to
compare with the proposed framework KnowLoop. In both experiments with these
MLLMs, the detector fully trusts the MLLM’s prediction results without
considering uncertainty. For KnowLoop, we filter the failure predictions based
on uncertainty with a threshold of 0.6, above which the corresponding
prediction will not be trusted, and the model will actively seek human help
for failure detection. In Table III, as can be seen, when compared to the
baselines that completely trust the MLLM’s predictions, our framework can, to
some extent, improve the task success rate by around 50% compared to
ChatGPT-4v and by around 40% compared to LLaVA. The promising results
demonstrate that KnowLoop can enable the robot to actively ask for human help
when the prediction is uncertain.

ing

Figure 5: Selective Generation Curve for three prompting strategies based on
Self-explained Confidence.

##  V CONCLUSION

In this study, we have demonstrated that uncertainty quantified by token
probability, and entropy is able to reflect the predictive quality of the MLLM
LLaVA. We further integrate this into an MLLM failure detector within a zero-
shot LLM planner to facilitate closed-loop planning – a framework dubbed
KnowLoop. With this, lower-quality responses can be effectively identified
using a threshold. Our proposed framework can mitigate model illusions or
overconfidence, thereby enhancing model accuracy. Such improvements render it
suitable for the detection phase of closed-loop control systems. For future
work, it would be more resource-efficient to use the MLLM to substitute the
LLM in the task planner. Moreover, conformal prediction can be employed to
determine the threshold in the framework with a statistical guarantee [25]. In
addition, investigating how uncertainty estimation can be utilized for failure
reasoning and correction is an important next step in setting up a robust and
effective closed-loop system.

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# Basis of spinors expressed by differential forms and calculating its norm

Daisuke A. Takahashi1,2<EMAIL_ADDRESS><EMAIL_ADDRESS>of Physics1Department of Physics Chuo
University Chuo University 1-13-27 Kasuga 1-13-27 Kasuga Bunkyo-ku Bunkyo-ku
Tokyo 112-8551 Tokyo 112-8551 Japan
2Research and Education Center for Natural Sciences Japan
2Research and Education Center for Natural Sciences Keio University Keio
University Hiyoshi 4-1-1 Hiyoshi 4-1-1 Yokohama Yokohama Kanagawa 223-8521
Kanagawa 223-8521 Japan Japan

###### Abstract

The basis of spinors in three-dimensional Euclidean space is expressed by
differential forms. Its expression is found from the spectral decomposition of
the modified Hamiltonian describing Weyl semimetals where the wavenumber
parameters are replaced by the differential forms. The generalization of the
definitions of differential forms including not only fractional powers but
also any algebraic functions is justified by algebraic extension of the
symmetric Fock space isomorphic to multivariable polynomial ring. We
furthermore define the inner product for these fractional-order differential
forms written in two ways; the formal power series and the multiple integral.
While the norms of the powers $\mathrm{d}s^{2\nu}$ are reduced to the variant
of Dyson’s integral and have the value $2\nu+1$, possibly related to the
dimension of irreducible representation, the norm of two-component spinor is
given by the integral

$I=\frac{2}{\pi}\int_{x^{2}+y^{2}\geq 1\&0\leq x\leq 1\&0\leq y\leq
1}\frac{xy\mathrm{d}x\mathrm{d}y}{\sqrt{(1-x)(1-y)(x^{2}+y^{2}-1)}}=\frac{4\sqrt{2}}{\pi}\int_{0}^{1}\frac{(1+\\!\\!\sqrt{z})}{(1+z)^{3}}\big{[}2E(z)-(1-z)K(z)\big{]}\mathrm{d}z\simeq
1.774,$

where $K(z)$ and $E(z)$ are the complete elliptic integrals of the first and
second kind, respectively. The product can also be generalized to include one
parameter $p$ analogous to those in $L^{p}$ space, reducing to the original
one when $p=2$. The norm with $p=\infty$ is considered as an example and the
result for $\nu=-\frac{1}{2}$ is written by the Watson-Iwata integral. We also
discuss the ambiguity of the definition of the spinor under coordinate
rotation originating from Berry’s phase, and point out that if we
heuristically set $\mathrm{d}s=0$, the ambiguity disappears, though its
implication remains unclear.

In traditional textbooks of the theory of relativity, the vectors and tensors
are defined by their linear transformation rules under the coordinate
transformation, which, in today’s standard theory of manifold[1], are
understood as an expansion coefficient using the basis of the (co)tangent
space naturally associated with the choice of the coordinate. This knowledge
tells us a genuine mathematical identification of the vectors which we write
$\vec{e}_{x},\vec{e}_{y},\vec{e}_{z}$ in the undergraduate lecture of
electromagnetism and accept intuitively as an “arrow” without precise
definition; that is, their actual nature is the differential operators
$\partial_{x},\partial_{y},\partial_{z}$ for the tangent space and the
differential forms $\mathrm{d}x,\mathrm{d}y,\mathrm{d}z$ for the cotangent
space.
How about spinors? In fact, the representation theory of SO(3) found in the
textbooks of quantum mechanics also gives the “basisless” expression of
spinors, which is analogous to the above-mentioned traditional treatment of
vectors; if one gets the two-dimensional representation of rotation matrices,
then the spin-$\frac{1}{2}$ particle is defined by the transformation rule
under the multiplication of these $2\times 2$ representation matrices. We thus
reach the question that what is an actual entity of this geometrical object
possessing these two components as expansion coefficients. One may expect that
this question will be soon solved if we know how spinors have been formulated
in curved spacetime. However, the vielbein formalism[1] avoids this problem in
a tricky way, where one temporarily prepares a local orthonormal system by
diagonalizing the metric tensor point by point, and basisless spinors
formulated once in a flat space as they are are placed on it. This method
makes it possible to discuss physical phenomena concerning spinors in curved
space, but the problem of identification of spinor becomes obscured. One might
even think that this basisless formulation is the essential nature of spinor
and considering such question is meaningless. However, we in reality encounter
the spinor whose basis is explicitly written by the square root of the
differential form in conformal field theory[2], where, in the general
coordinate transformation $x=x(z)$, the fermion field transforms like
$\psi(z)=\left(\frac{\mathrm{d}x}{\mathrm{d}z}\right)^{1/2}\psi(x)$, which may
be formally written as
$\psi(z)(\mathrm{d}z)^{1/2}=\psi(x)(\mathrm{d}x)^{1/2}$, indicating that it is
a “rank-$\frac{1}{2}$ covariant tensor”. It simply suggests that the composite
of two spin-$\frac{1}{2}$ is a spin-1, i.e., a vector. In one dimension, the
dimensions of the (co)tangent space are also 1, so there remains no degree of
freedom in making the “square root of a basis”. However, if dimension gets
larger, what kind of combination of basis
$\mathrm{d}x^{1},\mathrm{d}x^{2},\dots,\mathrm{d}x^{n}$ is a suitable
definition becomes nontrivial. Here we provide a possible solution in three
dimension.

The starting observation is that the covariant vector
$A=\sum_{i=1}^{3}A_{i}\mathrm{d}x^{i}$ can be expressed as a trace of the
Pauli matrices:

$\displaystyle
A_{i}\mathrm{d}x^{i}=\tfrac{1}{2}\operatorname{Tr}\left[(A_{i}\sigma^{i})(\sigma_{j}\mathrm{d}x^{j})\right],$
(1)

where $\sigma_{i}=\sigma^{i}$ are Pauli matrices and henceforth summation
symbols for repeated indices are sometimes omitted. We also use the notation
$(\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3})=(\mathrm{d}x,\mathrm{d}y,\mathrm{d}z)$
depending on the situation. Let us now consider the column and row spinor
$\psi=\left(\begin{smallmatrix}\psi_{\uparrow}\\\
\psi_{\downarrow}\end{smallmatrix}\right)$ and
$\phi=\left(\begin{smallmatrix}\phi_{\uparrow}&\phi_{\downarrow}\end{smallmatrix}\right)$.
Their product $\psi\phi$ is a $2\times 2$ matrix possessing the irreducible
decomposition into spin-0 and spin-1, and the latter, following Eq. (1), is
given by

$\displaystyle\phi(\sigma_{j}\mathrm{d}x^{j})\psi=\operatorname{Tr}\left[\psi\phi(\sigma_{j}\mathrm{d}x^{j})\right]=(\phi_{\uparrow}\psi_{\downarrow}+\phi_{\downarrow}\psi_{\uparrow})\mathrm{d}x$
$\displaystyle\qquad\qquad\quad-\mathrm{i}(\phi_{\uparrow}\psi_{\downarrow}-\phi_{\downarrow}\psi_{\uparrow})\mathrm{d}y+(\phi_{\uparrow}\psi_{\uparrow}-\phi_{\downarrow}\psi_{\downarrow})\mathrm{d}z.$
(2)

Our present aim is to find the basis of spinor written by differential forms.
Looking at Eq. (2) carefully, we expect that a desired thing will be obtained
by “cutting” the form $(\sigma_{j}\mathrm{d}x^{j})$ into two parts. That is,
writing the spectral decomposition

$\displaystyle\sigma_{j}\mathrm{d}x^{j}=\mathrm{d}s\left(v_{0}v_{0}^{\dagger}-v_{1}v_{1}^{\dagger}\right),\quad\mathrm{d}s\coloneqq\\!\sqrt{\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}},$
(3)

where the decomposition is performed by regarding $\mathrm{d}x^{j}$’s as
ordinary real numbers (this will be justified later), then Eq. (2) is given by
$\phi(\sigma_{j}\mathrm{d}x^{j})\psi=\mathrm{d}s\left[\phi
v_{0}v_{0}^{\dagger}\psi-\phi v_{1}v_{1}^{\dagger}\psi\right]$, and the column
and row spinors with basis expressed by differential forms would be
constructed as

$\displaystyle\sqrt{\mathrm{d}s}\,v_{j}^{\dagger}\psi$
$\displaystyle=\psi_{\uparrow}\sqrt{\mathrm{d}s}\,v_{j,\uparrow}^{*}+\psi_{\downarrow}\sqrt{\mathrm{d}s}\,v_{j,\downarrow}^{*},$
(4) $\displaystyle\sqrt{\mathrm{d}s}\,\phi v_{j}$
$\displaystyle=\phi_{\uparrow}\sqrt{\mathrm{d}s}\,v_{j,\uparrow}+\phi_{\downarrow}\sqrt{\mathrm{d}s}\,v_{j,\downarrow},$
(5)

where we write $v_{j}=\left(\begin{smallmatrix}v_{j,\uparrow}\\\
v_{j,\downarrow}\end{smallmatrix}\right)$. Thus, the ket vectors
$\ket{\uparrow}$ and $\ket{\downarrow}$, which we learned in quantum
mechanics, would be identified as — or at least proportional to — the above
$\sqrt{\mathrm{d}s}\,v_{j,\uparrow}^{*}$ and
$\sqrt{\mathrm{d}s}\,v_{j,\downarrow}^{*}$.

Let us investigate the decomposition (3) in more detail. In fact, if the form
$\mathrm{d}x^{j}$ are replaced by an ordinary number $x^{j}$, it reduces to
the diagonalization of the Weyl Hamiltonian

$\displaystyle
H(\boldsymbol{x})=\sum_{i=1}^{3}x^{i}\sigma_{i}=\begin{pmatrix}x^{3}&x^{1}-\mathrm{i}x^{2}\\\
x^{1}+\mathrm{i}x^{2}&-x^{3}\end{pmatrix},$ (6)

which, if the parameter $\boldsymbol{x}=(x^{1},x^{2},x^{3})$ is interpreted as
a Bloch wavenumber $\boldsymbol{k}$, is equivalent to the model of Weyl
semimetals[3]. It is also a minimal model where the adiabatic change of the
system parameter yields a nonvanishing phase factor for quantum states, i.e.,
Berry’s phase [Ref. Nakahara, Chap. 10], thus exhibiting topological
phenomena.

In order to justify the diagonalization of the Weyl Hamiltonian where
differential forms are substituted $H(\mathrm{d}\boldsymbol{x})$ instead of
ordinary numbers, we must define algebraic operations for $\mathrm{d}x^{j}$’s
including divisions, fractional powers, etc. It is achieved by considering the
symmetric algebra, which is isomorphic to multivariable polynomial ring [Ref.
SatakeEng, Chap. V]. This algebra is also equivalent to the Fock space of
bosonic many-body systems. Then, the division is easily defined by that of
polynomial, and moreover, making an algebraic extension by adding zeros of
polynomials, we obtain the algebraic-function field. By this extension, we can
use the square root of differential forms unambiguously.
Next, we define the inner product for this space. For a while, we consider an
abstract $d$-dimensional vector space spanned by a basis $e_{1},\dots,e_{d}$.
We also write the basis of its dual space $f_{1},\dots,f_{d}$ satisfying
$\braket{e_{i},f_{j}}=\delta_{ij}$. Then, the inner product between rank-$r$
tensors is given by $\braket{e_{i_{1}}\otimes\dots\otimes
e_{i_{r}},f_{j_{1}}\otimes\dots\otimes
f_{j_{r}}}=\delta_{i_{1}j_{1}}\dots\delta_{i_{r}j_{r}}$. The correspondence
between polynomials and symmetric tensors is as follows[4]. Regarding
$e_{1},\dots,e_{d}$ as indeterminates, the monomial
$e_{1}^{n_{1}}e_{2}^{n_{2}}\dots e_{d}^{n_{d}},\
n_{1},n_{2},\dots,n_{d}\in\mathbb{Z}_{\geq 0}$, corresponds to
$e_{1}^{n_{1}}e_{2}^{n_{2}}\dots e_{d}^{n_{d}}=\mathcal{S}(e_{1}^{\otimes
n_{1}}\otimes e_{2}^{\otimes n_{2}}\otimes\dots\otimes e_{d}^{\otimes
n_{d}})$, where we write $e^{\otimes n}\coloneqq e\otimes\dots\otimes
e\,(\text{$n$ times})$ and $\mathcal{S}$ represents a projection operator to
the symmetric tensor space, satisfying $\mathcal{S}^{2}=\mathcal{S}$ and
acting as $\mathcal{S}(e_{i_{1}}\otimes\dots\otimes
e_{i_{r}})=\frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_{r}}e_{i_{\sigma(1)}}\otimes\dots\otimes
e_{i_{\sigma(r)}}$. The same correspondence also holds for
$f_{1}^{m_{1}}f_{2}^{m_{2}}\dots f_{d}^{m_{d}},\
m_{1},m_{2},\dots,m_{d}\in\mathbb{Z}_{\geq 0}$. The inner product of these two
is then calculated as

$\displaystyle\braket{e_{1}^{n_{1}}\dots e_{d}^{n_{d}},f_{1}^{m_{1}}\dots
f_{d}^{m_{d}}}=\delta_{n_{1}m_{1}}\dots\delta_{n_{d}m_{d}}\binom{\sum_{i=1}^{d}n_{i}}{n_{1},\dots,n_{d}}^{-1},$
(7)

where
$\binom{\sum_{i=1}^{d}n_{i}}{n_{1},\dots,n_{d}}=\frac{(\sum_{i=1}^{d}n_{i})!}{n_{1}!\dots
n_{d}!}$ is a multinomial coefficient.
Henceforth we only consider $d=3$ though keeping general $d$ is not hard. Let
$A(e_{1},e_{2},e_{3})$ and $B(f_{1},f_{2},f_{3})$ be $n$-th order homogeneous
functions, i.e., the functions s.t.
$e_{3}^{n}A(\frac{e_{1}}{e_{3}},\frac{e_{2}}{e_{3}},1)=A(e_{1},e_{2},e_{3})$
and
$f_{3}^{n}B(\frac{f_{1}}{f_{3}},\frac{f_{2}}{f_{3}},1)=B(f_{1},f_{2},f_{3})$.
First let us assume $n\in\mathbb{Z}_{\geq 0}$ and $A$ and $B$ are polynomial
functions. Then, writing $\xi_{i}=e_{i}/e_{3},\ \eta_{i}=f_{i}/f_{3},\ i=1,2$,
they have expansion $A=e_{3}^{n}\sum_{n_{1},n_{2}\in\mathbb{Z}_{\geq
0}}a_{n_{1},n_{2}}\xi_{1}^{n_{1}}\xi_{2}^{n_{2}}$ and
$B=f_{3}^{n}\sum_{m_{1},m_{2}in\mathbb{Z}_{\geq
0}}b_{m_{1},m_{2}}\eta_{1}^{m_{1}}\eta_{2}^{m_{2}}$ and the inner product is
given by

$\displaystyle\braket{A,B}=\sum_{n_{1},n_{2}\in\mathbb{Z}_{\geq
0}}a_{n_{1},n_{2}}b_{n_{1},n_{2}}\binom{n}{n_{1},n_{2},n-n_{1}-n_{2}}^{-1}.$
(8)

If $A$ and $B$ are polynomials the range of this summation is finite:
$\sum_{0\leq n_{1}+n_{2}\leq n}$. On the other hand, when
$n\not\in\mathbb{Z}_{\geq 0}$ and/or $A$ and $B$ include rational and/or
fractional-power functions, their Taylor series generally becomes infinite.
Even in this case, the inner product defined by Eq. (8) makes sense. However,
if we actually calculate this formal power series, the result often diverges.
So, we want another definition equivalent to this one but has a larger
convergence region. We can achieve it if we find a linear operator giving the
mapping

$\displaystyle
e_{1}^{n_{1}}e_{2}^{n_{2}}e_{3}^{n_{3}}f_{1}^{m_{1}}f_{2}^{m_{2}}f_{3}^{m_{3}}\to\delta_{n_{1}m_{1}}\delta_{n_{2}m_{2}}\delta_{n_{3}m_{3}}\binom{n}{n_{1},n_{2},n_{3}}^{-1}.$
(9)

It is indeed realized by setting
$e_{i}=\sqrt{\rho_{i}}\mathrm{e}^{\mathrm{i}u_{i}},\
f_{i}=\sqrt{\rho_{i}}\mathrm{e}^{-\mathrm{i}u_{i}}$ and applying the integral
operator
$\frac{1}{n!}\prod_{i=1}^{3}\int_{0}^{\infty}\mathrm{d}\rho_{i}\mathrm{e}^{-\rho_{i}}\int_{0}^{2\pi}\frac{\mathrm{d}u_{i}}{2\pi}$.
Thus, the integral expression of the inner product is given by

$\displaystyle\braket{A,B}=\frac{1}{n!}\prod_{i=1}^{3}\int_{0}^{\infty}\\!\\!\mathrm{d}\rho_{i}\mathrm{e}^{-\rho_{i}}\\!\int_{-\pi}^{\,\pi}\frac{\mathrm{d}u_{i}}{2\pi}$
$\displaystyle
A(\\!\\!\sqrt{\rho_{1}}\mathrm{e}^{\mathrm{i}u_{1}}\\!,\\!\\!\sqrt{\rho_{2}}\mathrm{e}^{\mathrm{i}u_{2}}\\!,\\!\\!\sqrt{\rho_{3}}\mathrm{e}^{\mathrm{i}u_{3}})B(\\!\\!\sqrt{\rho_{1}}\mathrm{e}^{-\mathrm{i}u_{1}}\\!,\\!\\!\sqrt{\rho_{2}}\mathrm{e}^{-\mathrm{i}u_{2}}\\!,\\!\\!\sqrt{\rho_{3}}\mathrm{e}^{-\mathrm{i}u_{3}}),\\!$
(10)

which often gives a finite value even when the series (8) diverges. Replacing
the quantum inner product by a classical multiple integral reminds us of path
integral. The two of the six-fold integral in Eq. (10) can be soon performed;
first, by shifting $u_{1}\to u_{1}+u_{3}$ and $u_{2}\to u_{2}+u_{3}$, the
integrand becomes $u_{3}$-independent, and next, introducing the polar
coordinate
$(\rho_{1},\rho_{2},\rho_{3})=\gamma^{2}(\hat{n}_{1}^{2},\hat{n}_{2}^{2},\hat{n}_{3}^{2})$
with
$(\hat{n}_{1},\hat{n}_{2},\hat{n}_{3})=(\sin\beta\cos\alpha,\sin\beta\sin\alpha,\cos\beta)$,
the integration w.r.t. $\gamma$ reduces to $\Gamma$ function. Therefore, the
integral which we must actually calculate is four-fold. We also note that if
the integrand in Eq. (10) only contains the even powers of
$\mathrm{e}^{\pm\mathrm{i}u_{i}}$’s, the replacement
$\mathrm{e}^{\pm\mathrm{i}u_{i}}\to\mathrm{e}^{\pm\mathrm{i}u_{i}/2}$ does not
change the result and simplifies the integrand. The examples considered below
all belong to this case and this replacement is made unless otherwise noted.
Note that this algebra can be equipped with another inner product. Let $p>0$
be a rational number. If the above-described algebraic extension starts from
the vector space spanned by $e_{1}^{p/2},\dots,e_{d}^{p/2}$ instead of
$e_{1},\dots,e_{d}$, the final algebraic field contains the same elements but
the product (7) is modified by

$\displaystyle\Braket{e_{1}^{n_{1}}\dots e_{d}^{n_{d}},f_{1}^{n_{1}}\dots
f_{d}^{n_{d}}}_{p}\coloneqq\binom{\frac{2}{p}\sum_{i=1}^{d}n_{i}}{\frac{2}{p}n_{1},\dots,\frac{2}{p}n_{d}}^{-1}.$
(11)

The original one (7) corresponds to $p=2$. The product itself can be defined
even for irrational $p$. In particular, the limit $p\to\infty$ is

$\displaystyle\Braket{e_{1}^{n_{1}}\dots e_{d}^{n_{d}},f_{1}^{n_{1}}\dots
f_{d}^{n_{d}}}_{\infty}=1,$ (12)

and the integral expression realizing this inner product is

$\displaystyle\\!\\!\\!\\!\braket{A,B}_{\infty}\\!=\\!\prod_{i=1}^{3}\\!\int_{-\pi}^{\,\pi}\\!\frac{\mathrm{d}u_{i}}{2\pi}\,A(\mathrm{e}^{\mathrm{i}u_{1}}\\!,\mathrm{e}^{\mathrm{i}u_{2}}\\!,\mathrm{e}^{\mathrm{i}u_{3}})B(\mathrm{e}^{-\mathrm{i}u_{1}}\\!,\mathrm{e}^{-\mathrm{i}u_{2}}\\!,\mathrm{e}^{-\mathrm{i}u_{3}}).\\!\\!$
(13)

The product $\braket{\cdot,\cdot}_{p}$ is invariant under the orthogonal
transformation $(e_{i}^{\prime})^{p/2}=R_{ij}e_{j}^{p/2},\
(f_{i}^{\prime})^{p/2}=R_{ij}f_{j}^{p/2}$ with $R$ an orthogonal matrix, which
is partially similar to those in $L^{p}$ space. Hence, the product with $p=2$
has prime importance in physical application.
Let us finish abstract algebra and go back to differential forms. Replacing
$e_{i}\to\mathrm{d}x^{i}$ and $f_{i}\to\frac{\partial}{\partial x^{i}}$, the
expressions shown above can be used. As a basic example, let us consider the
norm of
$\mathrm{d}s^{2\nu}=(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2})^{\nu}$.
The dual element is given by
$\nabla^{2\nu}=(\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2})^{\nu}$.
One can prove

$\displaystyle\braket{\mathrm{d}s^{2\nu},\nabla^{2\nu}}=2\nu+1,\quad(\nu>-\tfrac{3}{2}).$
(14)

The derivation of Eq. (14), expanding the integrand, reduces to the formula

${\displaystyle\prod_{i=1}^{3}\\!\int_{-\pi}^{\,\pi}\\!\frac{\mathrm{d}u_{i}}{2\pi}}\sin^{2l_{1}}\\!\frac{u_{2}-u_{3}}{2}\sin^{2l_{2}}\\!\frac{u_{3}-u_{1}}{2}\sin^{2l_{3}}\\!\frac{u_{1}-u_{2}}{2}\\!=\\!\frac{(l_{1}+l_{2}+l_{3})!(l_{1}-\frac{1}{2})!(l_{2}-\frac{1}{2})!(l_{3}-\frac{1}{2})!}{\pi^{3/2}(l_{1}+l_{2})!(l_{2}+l_{3})!(l_{3}+l_{1})!}\\!,$
(15)

which is a variant of Dyson’s integral[5] but eludes the proof by Good[6] (see
also Ref. AAR, §8.8); to calculate Eq. (15) we need the constant term of
$\prod_{j\neq l}(1-\frac{z_{j}}{z_{l}})^{a_{j,l}}$ with $a_{j,l}=a_{l,j}$ but
their theorem only treats the case $a_{j,l}=a_{j}$.
The $\infty$-norm $\braket{\mathrm{d}s^{2\nu},\nabla^{2\nu}}_{\infty}$ has
finite value when $\nu>-1$. In particular, the case $\nu=-\frac{1}{2}$ can be
explicitly evaluated as

$\displaystyle\braket{\tfrac{1}{\sqrt{\mathrm{d}s^{2}}},\tfrac{1}{\sqrt{\nabla^{2}}}}_{\infty}=\tfrac{2\sqrt{3}}{\pi^{2}}K(\sin^{2}\\!\tfrac{\pi}{12})^{2}=\tfrac{1}{6\pi^{2}}B(\tfrac{1}{2},\tfrac{1}{6})^{2}\simeq
0.896,$ (16)

where $K(z)$ is the complete elliptic integral of the first kind and $B(x,y)$
is the beta function. This integral reduces to those studied by Watson and
Iwata[8, 9, 10]. The power series given below suggests that all integrals with
other half-odd integers $\nu\geq-\frac{1}{2}$ reduce to elliptic integrals.
It is worth observing what happens if the formal power series (8) is used. We
consider general $p$-norm so we replace the multinomial coefficient by (11) or
(12). By binomial expansion
$\mathrm{d}s^{2\nu}=\mathrm{d}z^{2\nu}\sum_{l=0}^{\infty}\binom{\nu}{l}\sum_{r=0}^{l}\binom{l}{r}\xi_{1}^{2r}\xi_{2}^{2l-2r}$,
the power series of the norm is obtained but diverges. So, we temporarily
introduce an auxiliary variable $x^{l}$ in the summand, yielding

$\displaystyle\textstyle\braket{\mathrm{d}s^{2\nu},\nabla^{2\nu}}_{p}\mathop{\text{``$=$''}}\,{\displaystyle\sum_{l=0}^{\infty}\sum_{r=0}^{l}}\binom{\nu}{l}^{2}\binom{l}{r}^{2}\binom{\frac{4}{p}\nu}{\frac{4}{p}(\nu-l),\,\frac{4}{p}(l-r),\,\frac{4}{p}r}^{-1}x^{l},$
(17)

where $\mathop{\text{``$=$''}}$ means that the limit $x\to 1$ is taken at the
last stage. This summation can be carried out for $p=2$ and $\infty$:

$\displaystyle\braket{\mathrm{d}s^{2\nu},\nabla^{2\nu}}$
$\displaystyle\mathop{\text{``$=$''}}F^{2}_{1}\left(\begin{smallmatrix}1,-\nu\\\
\frac{1}{2}-\nu\end{smallmatrix};x\right),$ (18)
$\displaystyle\braket{\mathrm{d}s^{2\nu},\nabla^{2\nu}}_{\infty}$
$\displaystyle\mathop{\text{``$=$''}}F^{3}_{2}\left(\begin{smallmatrix}\frac{1}{2},-\nu,-\nu\\\
1,1\end{smallmatrix};4x\right),$ (19)

where $F^{p}_{q}\left(\begin{smallmatrix}a_{1},\dots,a_{p}\\\
b_{1},\dots,b_{q}\end{smallmatrix};z\right)\coloneqq\sum_{l=0}^{\infty}\frac{(a_{1})_{l}\dots(a_{p})_{l}}{(1)_{l}(b_{1})_{l}\dots(b_{q})_{l}}z^{l}$
is the generalized hypergeometric function. If $\nu\in\mathbb{Z}_{\geq 0}$,
both (18) and (19) return a correct answer since they reduce to a polynomial.
The expansion of Eq. (18),
$\frac{\sqrt{\pi}\,\Gamma(\frac{1}{2}-\nu)}{\Gamma(-\nu)}\frac{1}{\sqrt{1-x}}\\!+\\!(2\nu\\!+\\!1)\\!+\\!O(\\!\\!\sqrt{1-x})$,
reproduces Eq. (14) if the first divergent term is excluded. Equation (19)
substituted $x=1$ has a complex value, which is inconsistent with the integral
(13) whose result is real and positive, implying something wrong in analytic
continuation. More concrete information can be obtained for half-odd integer
$\nu$’s; using Clausen’s formula and the contiguous relations [Ref.
ErdelyiHTF1, Chap. IV], Eq. (19) reduces to the elliptic integral.
Furthermore, we can check that its real part coincides with the integral (13)
numerically. Indeed,
$\text{Eq.~{}(\ref{eq:dsfps2})}_{\nu=-1/2}=F^{3}_{2}\left(\begin{smallmatrix}\frac{1}{2},\frac{1}{2},\frac{1}{2}\\\
1,1\end{smallmatrix};4x\right)=\tfrac{4}{\pi^{2}}K\Big{(}\frac{1-\sqrt{1-4x}}{2}\Big{)}^{2}$,
with $x=1$ and Landen’s transformation, reduces to
$\frac{4}{\pi^{2}}K(\mathrm{e}^{-\mathrm{i}\pi/3})^{2}=\frac{4}{\pi^{2}}K(\sin^{2}\\!\frac{\pi}{12})\mathrm{e}^{-\mathrm{i}\pi/6},$
whose real part is (16).
Now we again consider writing the basis of spinors by differential forms.
Introducing the spherical-coordinate-like parametrization
$(\mathrm{d}x,\mathrm{d}y,\mathrm{d}z)=\\!\\!\sqrt{\mathrm{d}s^{2}}(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$,
which is now unambiguously defined by algebraic extension with
$\theta,\varphi$ being not ordinary numbers, the diagonalization of Eq. (6)
with differential forms substituted is performed in the same way as that of
ordinary numbers:

$\displaystyle H(\mathrm{d}\boldsymbol{x})$
$\displaystyle=\sqrt{\mathrm{d}s^{2}}V(\mathrm{d}\boldsymbol{x})\sigma_{3}V(\mathrm{d}\boldsymbol{x})^{\dagger},$
(20) $\displaystyle V(\mathrm{d}\boldsymbol{x})$
$\displaystyle\coloneqq\mathrm{e}^{\frac{-\mathrm{i}}{2}\varphi\sigma_{3}}\mathrm{e}^{\frac{-\mathrm{i}}{2}\theta\sigma_{2}}.$
(21)

Using the column and row spinors $\psi$ and $\phi$, we can make an
SU(2)-invariant $\phi H(\mathrm{d}\boldsymbol{x})\psi=\mathrm{d}s\phi
V(\mathrm{d}\boldsymbol{x})\sigma_{3}V(\mathrm{d}\boldsymbol{x})^{\dagger}\psi$.
From this expression, we introduce the covariant column spinor
$(\mathrm{d}s^{2})^{\frac{1}{4}}V(\mathrm{d}\boldsymbol{x})^{\dagger}\psi$ and
the covariant row spinor $(\mathrm{d}s^{2})^{\frac{1}{4}}\phi
V(\mathrm{d}\boldsymbol{x})$ as a spinor whose basis is written by
differential forms. We also simply call them the covariant ket and bra. If we
write $V=(v_{0},v_{1}),$ we easily find
$v_{j}(\theta\pm\pi)=\pm(-1)^{j}v_{1-j}(\theta)$. The above definition
includes an array of two possible bases $v_{0}$ or $v_{1}$; either of two can
be used, but we should carefully consider Berry’s phase discussed later. In
the spherical coordinate $(r,\theta,\varphi)$ of normal numbers, if we
formally allow negative $r$, the parametrization has the redundancy of
replacement $(r,\theta,\varphi)\to(-r,\theta+\pi,\varphi)$, reflecting the 1:2
correspondence of SO(3) and SU(2). By this replacement the exchange of the
eigenvalues and eigenvectors $r\leftrightarrow-r$ and $v_{0}\leftrightarrow
v_{1}$ occurs. Since we normally restrict the range of $\theta$ to $[0,\pi]$,
if $v_{0}$ is the “nearside” basis, $v_{1}$ is the “farside” one.
The contravariant ket and bra are also introduced in the same way:
$(\nabla^{2})^{1/4}V(\nabla)^{\dagger}\psi$ and $(\nabla^{2})^{1/4}\phi
V(\nabla)$. The norm density is then obtained by the product of contravariant
bra and covariant ket
$\psi^{\dagger}\Braket{(\nabla^{2})^{\frac{1}{4}}V(\nabla),V(\mathrm{d}\boldsymbol{x})^{\dagger}(\mathrm{d}s^{2})^{\frac{1}{4}}}\psi$.
If we write $\Braket{(\nabla^{2})^{\frac{1}{4}}V(\nabla),\
V(\mathrm{d}\boldsymbol{x})^{\dagger}(\mathrm{d}s^{2})^{\frac{1}{4}}}=I_{0}\boldsymbol{1}_{2}+\sum_{i=1}^{3}I_{i}\sigma_{i}$,
we can check that $I_{1},I_{2},I_{3}$ all vanish because the power series only
includes the oscillating terms. $\sqrt{\mathrm{d}s}\,v_{0}$ and
$\sqrt{\mathrm{d}s}\,v_{1}$ are orthogonal in this sense. The only non-
vanishing integral $I_{0}$ is

$\displaystyle
I_{0}=\\!\frac{1}{(\frac{1}{2})!}\prod_{j=1}^{3}\int_{0}^{\infty}\\!\mathrm{d}\rho_{i}\mathrm{e}^{-\rho_{i}}\\!\\!\\!\int_{-\pi}^{\pi}\\!\frac{\mathrm{d}u_{i}}{2\pi}\sqrt{N_{3}\left(\tfrac{1}{2}+\tfrac{\rho_{3}+N_{2}}{2N_{3}}\right)\left(\tfrac{1}{2}+\tfrac{\rho_{1}+\rho_{2}}{2N_{2}}\right)},$
(22)

where
$N_{3}=\left|\rho_{1}\mathrm{e}^{\mathrm{i}u_{1}}+\rho_{2}\mathrm{e}^{\mathrm{i}u_{2}}+\rho_{3}\mathrm{e}^{\mathrm{i}u_{3}}\right|,\
N_{2}=\left|\rho_{1}\mathrm{e}^{\mathrm{i}u_{1}}+\rho_{2}\mathrm{e}^{\mathrm{i}u_{2}}\right|$.
A little calculation reveals the following expressions:

$\displaystyle
I_{0}=\frac{2}{\pi}\int_{D}\frac{xy\mathrm{d}x\mathrm{d}y}{\sqrt{(1-x)(1-y)(x^{2}+y^{2}-1)}}$
(23)
$\displaystyle=\frac{4\sqrt{2}}{\pi}\int_{0}^{1}\frac{(1+\\!\\!\sqrt{z})}{(1+z)^{3}}\big{[}2E(z)-(1-z)K(z)\big{]}\mathrm{d}z\simeq
1.774,$ (24)

where $D$ is $x^{2}+y^{2}\geq 1\&0\leq x\leq 1\&0\leq y\leq 1$, and $K(z)$ and
$E(z)$ are the complete elliptic integral of the first and second kind. The
closed form of this integral is unknown.

Finally, we discuss the ambiguity originating from Berry’s phase under
coordinate rotation. This problem is more serious than the same problem for
the ordinary number (6); in the latter, we only observe the hysteresis as an
actual physical phenomenon under actual change of parameters, but in the
former, what we do is nothing but a change of frame, so the physical output
must be unchanged. Let $L_{1},L_{2},$ and $L_{3}$ be generators of rotation
with respect to $x,y,$ and $z$ axis in three dimension, e.g.,
$L_{3}=\sigma_{2}\oplus 0$, and write the element of SO(3) by the Euler angle
$R=\mathrm{e}^{-\mathrm{i}L_{3}\alpha}\mathrm{e}^{-\mathrm{i}L_{2}\beta}\mathrm{e}^{-\mathrm{i}L_{3}\gamma}.$
The corresponding element in SU(2) is
$U=\mathrm{e}^{-\mathrm{i}\frac{\sigma_{3}}{2}\alpha}\mathrm{e}^{-\mathrm{i}\frac{\sigma_{2}}{2}\beta}\mathrm{e}^{-\mathrm{i}\frac{\sigma_{3}}{2}\gamma}.$
By the coordinate rotation $\boldsymbol{x}^{\prime}=R\boldsymbol{x}$, the
corresponding differential form also changes as
$\mathrm{d}\boldsymbol{x}^{\prime}=R\mathrm{d}\boldsymbol{x}$. We can soon
check

$\displaystyle
H(R\mathrm{d}\boldsymbol{x})=UH(\mathrm{d}\boldsymbol{x})U^{\dagger}.$ (25)

On the other hand, if the diagonalization is performed from the beginning in
the $\boldsymbol{x}^{\prime}$-coordinate, we get

$\displaystyle
H(R\mathrm{d}\boldsymbol{x})=\sqrt{\mathrm{d}s^{2}}V(R\mathrm{d}\boldsymbol{x})\sigma_{3}V(R\mathrm{d}\boldsymbol{x})^{\dagger}.$
(26)

Since Eq. (25) with (20) must be equal to (26), and since the eigenvector with
nondegenerate eigenvalue is unique up to an overall factor, there exists a
diagonal matrix $D$ such that

$\displaystyle V(R\mathrm{d}\boldsymbol{x})D=UV(\mathrm{d}\boldsymbol{x}).$
(27)

Let $\theta^{\prime}$ and $\varphi^{\prime}$ be angle parameters for the
$\boldsymbol{x}^{\prime}$-coordinate. Then, if we write
$D=\mathrm{e}^{\mathrm{i}\xi\sigma_{3}},$ the expression for Berry’s phase
$\xi$ is found as

$\displaystyle\mathrm{e}^{2\mathrm{i}\xi}=\frac{\cos\theta\cos(\varphi+\gamma)\sin\beta+\cos\beta\sin\theta-\mathrm{i}\sin\beta\sin(\varphi+\gamma)}{\sin\theta^{\prime}}.$
(28)

Since
$\cos\theta^{\prime}=\cos\beta\cos\theta-\cos(\varphi+\gamma)\sin\beta\sin\theta$,
this is indeed a unit complex number ($|\mathrm{e}^{\mathrm{i}\xi}|=1$), but
not identically 1. However, if we rewrite this expression using
$\cos\theta=\frac{\mathrm{d}z}{\mathrm{d}s},\
\mathrm{e}^{\pm\mathrm{i}\varphi}=\frac{\mathrm{d}x\pm\mathrm{i}\mathrm{d}y}{\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}},$
and heuristically set $\mathrm{d}s\to 0$, we obtain

$\displaystyle\mathrm{e}^{2\mathrm{i}\xi}\\!=\\!\mbox{$\small\displaystyle\frac{\sin\beta(\frac{(\cos\gamma\mathrm{d}x-\sin\gamma\mathrm{d}y)\mathrm{d}z-\mathrm{i}\mathrm{d}y\mathrm{d}s}{\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}})+\cos\beta\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}}{\sqrt{\mathrm{d}s^{2}-(\cos\beta\mathrm{d}z-\cos\gamma\sin\beta\mathrm{d}x+\sin\gamma\sin\beta\mathrm{d}y)^{2}}}\overset{\mathrm{d}s\to
0}{\longrightarrow}1.$}$ (29)

Thus, if we would have some justification for this replacement, non-uniqueness
caused by Berry’s phase would be eliminated.
The above heuristic replacement also works for vectors obtained by the
composition of two spinors. The composition of ket and bra
$(\\!\\!\sqrt{\mathrm{d}s}v_{i}^{\dagger}\psi)(\\!\\!\sqrt{\mathrm{d}s}\phi
v_{j})$ with $i,j=0,1$ can be treated by matrix $V^{\dagger}\psi\phi
V\mathrm{d}s$ all at once; introducing the notation
$\psi\phi=\frac{1}{2}(n\boldsymbol{1}_{2}+\sum_{i=1}^{3}\Psi_{i}\sigma_{i})$,
we obtain $V^{\dagger}\psi\phi
V\mathrm{d}s=\tfrac{1}{2}n\left(\begin{smallmatrix}\mathrm{d}s&\\\
&\mathrm{d}s\end{smallmatrix}\right)+\tfrac{1}{2}\Psi_{3}\left(\begin{smallmatrix}\mathrm{d}z&-\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}\\\
-\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}&-\mathrm{d}z\end{smallmatrix}\right)+\tfrac{1}{2}\Psi_{1}\left(\begin{smallmatrix}\mathrm{d}x&\frac{\mathrm{d}x\mathrm{d}z+\mathrm{i}\mathrm{d}s\mathrm{d}y}{\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}}\\\
\frac{\mathrm{d}x\mathrm{d}z-\mathrm{i}\mathrm{d}s\mathrm{d}y}{\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}}&-\mathrm{d}x\end{smallmatrix}\right)+\tfrac{1}{2}\Psi_{2}\left(\begin{smallmatrix}\mathrm{d}y&\frac{\mathrm{d}y\mathrm{d}z-\mathrm{i}\mathrm{d}s\mathrm{d}x}{\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}}\\\
\frac{\mathrm{d}y\mathrm{d}z+\mathrm{i}\mathrm{d}s\mathrm{d}x}{\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}}&-\mathrm{d}y\end{smallmatrix}\right).$
While the diagonal components, which are the gauge-invariant combination
$v_{j}^{\dagger}$ and $v_{j}$ and Berry’s phase cancels out, successfully
yield the spin-0 and spin-1 components, the off-diagonal components, the
composition of $v_{j}^{\dagger}$ and $v_{1-j}$, depend on Berry’s phase and do
not provide an ordinary scalar and vector. However, if we set $\mathrm{d}s\to
0$, we get $V^{\dagger}\psi\phi
V\mathrm{d}s\to\frac{1}{2}\sum_{j=1}^{3}\Psi_{j}\mathrm{d}x^{j}\left(\begin{smallmatrix}1\\\
\mathrm{i}\end{smallmatrix}\right)\left(\begin{smallmatrix}1&\mathrm{i}\end{smallmatrix}\right)$,
thus all four components present vectors. The composition of two kets can also
be considered. Let $\psi,\chi$ be both column spinors. The composition of the
gauge-invariant combination $v_{0}^{\dagger}$ and $v_{1}^{\dagger}$
successfully generates the scalar and vector:
$(\\!\\!\sqrt{\mathrm{d}s}v_{0}^{\dagger}\psi)(\\!\\!\sqrt{\mathrm{d}s}v_{1}^{\dagger}\chi)=\tfrac{1}{\sqrt{2}}\Big{(}\Phi_{1,1}\tfrac{-(\mathrm{d}x+\mathrm{i}\,\mathrm{d}y)}{\sqrt{2}}+\Phi_{1,0}\mathrm{d}z+\Phi_{1,-1}\tfrac{\mathrm{d}x-\mathrm{i}\mathrm{d}y}{\sqrt{2}}+\Phi_{0,0}\mathrm{d}s\Big{)},$
where
$(\Phi_{1,1},\Phi_{1,0},\Phi_{1,-1},\Phi_{0,0})=(\psi_{\uparrow}\chi_{\uparrow},\frac{\psi_{\uparrow}\chi_{\downarrow}+\psi_{\downarrow}\chi_{\uparrow}}{\sqrt{2}},\psi_{\downarrow}\chi_{\downarrow},\frac{\psi_{\uparrow}\chi_{\downarrow}-\psi_{\downarrow}\chi_{\uparrow}}{\sqrt{2}})$
is the spin-triplet and singlet components. On the other hand, the composition
of the gauge-dependent pair, two $v_{j}^{\dagger}$’s, yields
$(\\!\\!\sqrt{\mathrm{d}s}v_{j}^{\dagger}\psi)(\\!\\!\sqrt{\mathrm{d}s}v_{j}^{\dagger}\chi)=\tfrac{1}{\sqrt{2}}\Big{[}\tfrac{(\mathrm{d}x+\mathrm{i}\mathrm{d}y)(\mathrm{d}s+(-1)^{j}\mathrm{d}z)}{\sqrt{2}\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}}\Phi_{1,1}+(-1)^{j}{\scriptstyle\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}}\Phi_{1,0}+\tfrac{(\mathrm{d}x-\mathrm{i}\mathrm{d}y)(\mathrm{d}s-(-1)^{j}\mathrm{d}z)}{\sqrt{2}\sqrt{\mathrm{d}s^{2}-\mathrm{d}z^{2}}}\Phi_{1,-1}\Big{]}$,
which does not reduce to the covariant vector, but if we set $\mathrm{d}s\to
0$, we obtain a vector
$\frac{\mathrm{i}(-1)^{j}}{\sqrt{2}}(\Phi_{1,1}\tfrac{-(\mathrm{d}x+\mathrm{i}\,\mathrm{d}y)}{\sqrt{2}}+\Phi_{1,0}\mathrm{d}z+\Phi_{1,-1}\tfrac{\mathrm{d}x-\mathrm{i}\mathrm{d}y}{\sqrt{2}})$.
A concluding remark is on the implication of the matrix
$\sigma_{j}\mathrm{d}x^{j}$ in Eq. (3). In physics, vectors and matrices
appear as an equivalent representation of bases of geometrical objects.
However, $\sigma_{j}\mathrm{d}x^{j}$ already includes the basis in its matrix
components. Therefore, these Pauli matrices are not an array of expansion
coefficients of physical quantities but a purely algebraic notion expressing
symmetry and structure of physical systems. In fact, in tensor networks and
matrix product states[12], matrices are already heavily used in this way,
where they consider matrices whose component itself is a ket vector; for
example, the ground state of the AKLT model is known to reduce the matrix
product state. Thus the spinor decomposition presented in this manuscript
might provide a novel dissection tool of quantum states in these fields.
The detailed calculations of the integrals will be submitted elsewhere.
Several closely related concepts would be found in other fields including,
e.g., spin geometry and twistor theory, and identifying these connections is
left for future work. The work was supported by JSPS Grant JP19H05821.

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