# COMPRER: A Multimodal Multi-Objective Pretraining Framework for Enhanced
Medical Image Representation

Guy Lutsker1
<EMAIL_ADDRESS>
&Hagai Rossman2
<EMAIL_ADDRESS>
&Nastya Godiva1
<EMAIL_ADDRESS>
&Eran Segal1,3
<EMAIL_ADDRESS>

###### Abstract

Substantial advances in multi-modal Artificial Intelligence (AI) facilitate
the combination of diverse medical modalities to achieve holistic health
assessments. We present COMPRER , a novel multi-modal, multi-objective
pretraining framework which enhances medical-image representation, diagnostic
inferences, and prognosis of diseases. COMPRER employs a multi-objective
training framework, where each objective introduces distinct knowledge to the
model. This includes a multimodal loss that consolidates information across
different imaging modalities; A temporal loss that imparts the ability to
discern patterns over time; Medical-measure prediction adds appropriate
medical insights; Lastly, reconstruction loss ensures the integrity of image
structure within the latent space. Despite the concern that multiple
objectives could weaken task performance, our findings show that this
combination actually boosts outcomes on certain tasks. Here, we apply this
framework to both fundus images and carotid ultrasound, and validate our
downstream tasks capabilities by predicting both current and future
cardiovascular conditions. COMPRER achieved higher Area Under the Curve (AUC)
scores in evaluating medical conditions compared to existing models on held-
out data. On the Out-of-distribution (OOD) UK-Biobank dataset COMPRER
maintains favorable performance over well-established models with more
parameters, even though these models were trained on $75\times$ more data than
COMPRER. In addition, to better assess our model’s performance in contrastive
learning, we introduce a novel evaluation metric, providing deeper
understanding of the effectiveness of the latent space pairing.

11footnotetext: Department of Math and Computer Science, Weizmann Institute of
Science, Rehovot, Israel22footnotetext: Pheno.AI, Tel Aviv,
Israel33footnotetext: Department of Machine Learning, Mohamed Bin Zayed
University of Artificial Intelligence, Abu Dhabi, UAE

##  1 Background

Figure 1: Schematic representation of COMPRER, our Contrastive Multi-objective
Pretraining approach for Multi-modal Representation. We utilize ViT-Base
encoders equipped with DINOV2 pre-trained weights for processing each imaging
modality, accompanied by a linear projection head. Our method is defined by a
combination of multiple loss objectives: (1) an ID-centric multi-modal
contrastive loss that bridges features between fundus images and carotid
ultrasound images; (2) a patient visit-based contrastive loss that discerns
temporal discrepancies across repeat visits for each patient; (3) a
contrastive scheme for the bilateral fundus images to ensure coupling of right
and left eye data per patient; (4) a decoding objective to restore original
images from condensed latent representations; and (5) a predictive mechanism
to estimate general medical measures directly from modality-specific
embeddings.

The evolution of AI within healthcare is promoting an era of precision
medicine, marked by enhanced diagnostic accuracy, improved prognostic
evaluations, and personalized treatment strategies (Bajwa et al., 2021; Esteva
et al., 2019), where deep learning is increasingly central to medical imaging
analysis (Huang et al., 2023). Technologies such as fundus imaging and carotid
ultrasound are pivotal in cardiovascular health assessments, granting insights
into micro and macrovascular structures and pathologies (Poplin et al., 2018;
Spence, 2006). Fundus imaging, a non-invasive procedure, reveals the retinal
microvasculature and is used to detect early manifestations of diseases like
diabetes and hypertension (Dai et al., 2021; Yan et al., 2019). Such
microvascular changes are significant indicators of systemic conditions,
enabling broader health monitoring. Carotid ultrasound complements fundus
imaging by providing a structural assessment of the carotid arteries, crucial
for identifying risks of stroke and atherosclerosis through blood flow
dynamics and plaque visualization (Yu et al., 2021; Siontis et al., 2021). The
integration of both the fundus imaging, and carotid ultrasound modalities,
offers a comprehensive representation of cardiovascular health, capitalizing
on their individual strengths to assess conditions at various scales of the
vascular system. However, the full potential of AI in medical imaging is
challenged by the limited availability of large, annotated datasets necessary
for traditional supervised learning (Zhou et al., 2023). This dataset scarcity
is addressed by initiatives such as the Human Phenotype Project (HPP), which
embraces a multi-modal deep-phenotyping approach, capturing a vast range of
data modalities, from high-resolution images to comprehensive clinico-
pathological records (Shilo et al., 2021). Such datasets are ideal for
investigating and improving AI models that surpass existing boundaries in
medical diagnostic capabilities. (Moor et al., 2023). To overcome these
limitations self-supervised learning (SSL) has become a key tool in this
field. (Huang et al., 2023). SSL circumvents the need for extensive labeled
datasets by utilizing the data itself to derive informative features through
the resolution of proxy tasks. It enables the extraction of significant
patterns intrinsic to the data, fostering a model’s ability to generalize
robustly to unseen data (Grill et al., 2020; Chen et al., 2020). SSL with
multi-modal data are particularly potent, with each modality enhancing the
model’s capabilities and utility across various health signals (Radhakrishnan
et al., 2023).

##  2 Introduction

In this paper, we present COMPRER (COntrastive Multi-objective PREtraining for
multi-modal Representation), a multi-modal, multi-objective pretraining
framework that can be used in numerous downstream tasks and applications
through the analysis of fundus imaging and carotid ultrasound. These include:
diagnosing current patient diseases, predicting clinically significant medical
features, and prognosing the probability of developing a medical condition in
the future. As seen in Figure 1, COMPRER using a multi-modal approach, where
it integrates distinct but complementary data sources— fundus imaging and
carotid ultrasound imaging. Each modality offers a unique glimpse into the
cardiovascular health of patients, capturing a diverse array of medical
measures that, when combined, provide a comprehensive assessment framework.
Our framework leverages ViTs (Dosovitskiy et al., 2020), specifically the
DINOV2-Base pre-trained model (Oquab et al., 2023), as our architectural
backbone, augmented by a multi-objective learning strategy that incorporates
reconstruction via an image decoder, predictive heads, and contrastive
learning losses.

###  2.1 COMPRER Training Objectives

In this section, we describe the training objectives utilized for our model.
Our approach employs paired batches of fundus images and carotid ultrasound
images to learn a joint embedding space, inspired by the CLIP training
paradigm (Radford et al., 2021). This multimodal training maximizes the
similarity of embeddings from matching image pairs and minimizes it for non-
matching pairs within a batch.

The contrastive loss function specific to a set of two embeddings types $u$
and $v$ is defined as:

$\mathcal{L}_{\text{contr}}(u,v)=-\frac{1}{N}\sum_{i=1}^{N}\log\left(\frac{\exp\left(\frac{\text{sim}(u_{i},v_{i})}{\tau}\right)}{\sum_{j=1}^{N}\exp\left(\frac{\text{sim}(u_{i},v_{j})}{\tau}\right)}\right)$

where $\text{sim}(u,v)\equiv\frac{u^{\top}v}{\|u\|\|v\|}$ and $N$ is the batch
size. The contrastive loss is as in CLIP:
$\mathcal{L}_{\text{contr\\_CLIP}}(u,v)=\frac{1}{2}\left(\mathcal{L}_{\text{contr}}(u,v)+\mathcal{L}_{\text{contr}}(v,u)\right)$
Several contrastive losses based on the relationship between the embeddings
compared: For fundus ($f$) and carotid ($c$) image embeddings:
$\mathcal{L}_{\text{contr\\_fc}}=\mathcal{L}_{\text{contr\\_CLIP}}(f,c)$.

For fundus images across different visits $t$ and $t^{\prime}$:
$\mathcal{L}_{\text{contr\\_fv}}=\mathcal{L}_{\text{contr\\_CLIP}}(f^{t},f^{t^{\prime}})$
For carotid images over time $t$ and $t^{\prime}$:
$\mathcal{L}_{\text{contr\\_cv}}=\mathcal{L}_{\text{contr\\_CLIP}}(c^{t},c^{t^{\prime}})$
For right ($R$) and left ($L$) eye fundus images:
$\mathcal{L}_{\text{contr\\_eye}}=\mathcal{L}_{\text{contr\\_CLIP}}(f^{R},f^{L})$

Predictive accuracy for fundus and carotid embeddings is assessed with Mean
Squared Error (MSE) losses:

$\mathcal{L}_{\text{pred}}(m,\hat{m})=\frac{1}{N}\sum_{i=1}^{N}(m_{i}-\hat{m}_{i})^{2}$

Yielding $\mathcal{L}_{\text{pred\\_r}}$ for fundus with $N=N_{r}$, and
$\mathcal{L}_{\text{pred\\_c}}$ for carotid with $N=N_{c}$, with their
respective medical measurements predictions. Finally, the total loss
$\mathcal{L}$ combines individual contrastive and predictive components:

$\mathcal{L}=\mathcal{L}_{\text{contr\\_fc}}+\mathcal{L}_{\text{contr\\_fv}}+\mathcal{L}_{\text{contr\\_cv}}+\mathcal{L}_{\text{contr\\_eye}}+\mathcal{L}_{\text{pred\\_r}}+\mathcal{L}_{\text{pred\\_c}},$

###  2.2 Model Summary and Key Contributions

We evaluate COMPRER through validation across individual learning objectives
and demonstrate its capabilities not only in extracting meaningful
representations but also in projecting these representations into actionable
clinical insights. Notably, we show that our multi-objective framework results
in enhanced diagnostic and prognostic accuracy. Moreover, the model’s ability
to outperform not only a baseline pretrained DINOV2 but also dedicated models
with substantial advantages in terms of parameters and data scale demonstrates
the efficacy of our approach. In conclusion, our contributions are:

  1. 1.

We introduce COMPRER, a novel deep learning framework that leverages multi-
modal, multi-objective pretraining to forecast and predict the development of
future diseases from medical imaging data.

  2. 2.

We provide evidence for the efficacy of our modeling approach through an
internal validation scheme, showing that our embeddings are capable of
predicting medical measures with high R2 scores.

  3. 3.

We introduce a novel, understandable metric for assessing the performance of
contrastive learning, offering an approach to measure the quality of
embeddings in identifying correct image pairs across different modalities.

  4. 4.

We substantiate the translational value of our model through its application
in predicting cardiovascular health conditions, in both our cohort as well as
in external cohorts.

##  3 Related Work

The combination of AI with healthcare represents a significant shift toward
redefining clinical methodology and patient care. At the heart of this
transformation, deep learning, particularly through convolutional neural
networks (CNNs), has played a pivotal role in enhancing medical diagnostics.
CNNs have demonstrated their efficacy in detection and classification tasks
across a range of medical imaging modalities, including dermatology
(Kwasigroch et al., 2020), radiology (Tiu et al., 2022; Jaiswal et al., 2019),
and neuroradiology (Pereira et al., 2016). The profound pattern recognition
capabilities of CNNs have thus become instrumental in medical image
interpretation. Recently, the success of SSL methods, such as SimCLR (Chen et
al., 2020) and BYOL (Grill et al., 2020), has redirected the focus from
supervised learning reliant on extensive labeled datasets to SSL in extracting
features from unlabeled data (Huang et al., 2023). SSL’s resilience to dataset
imbalances (Liu et al., 2021) particularly proves its adaptability in medical
contexts, making it a cornerstone for foundation models, designed for broad
application across multiple tasks (Bommasani et al., 2021; Moor et al., 2023).
Zhou et al. (Zhou et al., 2023) exemplified this adaptability in a self-
supervised masking strategy applied to a vast array of unlabeled fundus
images, yielding a foundation model with an ability to perform disease
detection across multiple scenarios. The breakthrough with OpenAI’s
Contrastive Language-Image Pretraining (CLIP) system has provided a novel
perspective on utilizing versatile architectures, specifically transformers
(Vaswani et al., 2017), for a wide range of modalities (Radford et al., 2021;
Ramesh et al., 2022, 2021; Brown et al., 2020). CLIP’s revolutionary approach
to interpreting images through a natural language lens has revealed the
potential of transformer architectures to tokenize and process multimodal data
efficiently. Our COMPRER framework adopts a similar stance, leveraging the
transformer architecture for both image encoders in a contrastive mechanism to
align information across medical imaging types. The methodology rooted in
CLIP’s cross-modal learning inspired both the multi-visit and multi-modal
contrastive losses of COMPRER, allowing it to not only decode spatial
characteristics but also trace temporal patterns indicative of disease
progression. In the era of multimodal data, significant strides have been made
in cross-modal representations (Radhakrishnan et al., 2023) and feature
extraction (Holmberg et al., 2020), revealing the intersecting pathways of SSL
and multimodal methodologies. Complementing this trend are ViTs, which have
remodeled the AI landscape with their extraordinary image processing
capabilities (Dosovitskiy et al., 2020) and interpretability (Chefer et al.,
2020). The advent of multi-task learning frameworks further amplifies these
models’ ability to assimilate diverse data and objectives, showcasing their
robustness across a spectrum of tasks relevant to cardiovascular health
analytics (Ruder, 2017; Crawshaw, 2020).

##  4 Methodology

In our study, we present a multimodal, multi-objective deep learning
architecture designed to create a versatile pretrained model suitable for a
wide range of health-related tasks. This architecture can generate adaptable
embeddings for predicting a multitude of medical features or be fine-tuned for
diverse medical applications. The model achieves this through the analysis of
both fundus images and carotid ultrasounds. This part delves into the
intricacies of our approach, which capitalizes on the robust capabilities of
ViTs. Initially, we assembled a dataset encompassing approximately 11.5K
participants’ fundus and carotid ultrasound images, of which 1.5K have
returned for a follow-up visit after two years. The dataset is divided into
training (80% of data), validation (validation is 20% of the training set),
and test sets (20% of the data), with the latter consisting solely of new
participants arriving after the start of this research to ensure the integrity
of our evaluation. Our preprocessing protocols ensure high-quality, artifact-
free images using AutoMorph (Zhou et al., 2022) for fundus images and custom
preprocessing to isolate relevant regions in carotid ultrasounds. Both image
types are standardized to a resolution of 280x280 pixels, facilitating uniform
processing where even grayscale ultrasound images are converted to three-
channel format to align with the fundus images. The structural backbone of our
model derives from the pretrained DINOV2-Base ViT, a vision transformer by
Meta that has shown great performance in image representation tasks, as well
as multiple vision downstream tasks (Caron et al., 2021; Oquab et al., 2023).
As DINOV2 was trained on millions of images, we can capitalize on its
extensive pretrained ability to represent image datasets and we can fine-tune
it to our unique medical imaging context for enhanced efficiency. To
complement the ViT, a linear projection head condenses high-dimensional
embeddings to a more manageable state, serving a dual purpose: reducing
computational demands and assisting with stability and shown to be essential
by Balestriero et al. (2023). A transposed convolution neural network,
comprising of transposed convolutional layers with gaussian error linear unit
(GELU) activations, reconstructs the original images from latent embeddings,
introducing a regularization effect that underpins the self-supervised
learning within our framework. Lastly, a small 2-layer Multi-Layer Perceptron
(MLP) is added to predict medical measures from the latent space embeddings.
To train the model we use multiple different optimization objectives
simultaneously. These include the multimodal and multi-visit contrastive
losses used as shown in CLIP by OpenAI (Radford et al., 2021), which fortify
the model’s time-awareness and cross-modality inference capabilities. A
classic mean squared error (MSE) decoder loss ensures fidelity in image
reconstruction, and another MSE loss is applied when predicting medical
measures from the embeddings. We trained the model over four days, distributed
across 8 NVIDIA A40 GPUs. We trained with an AdamW optimizer, with learning of
$3\times 10^{-4}$ and weight decay of $0.5$ with a StepLR scheduler, and a
batch size of 9 per GPU. We found that the model does not overfit the train
set in this time period, but due to limited resources we chose to run for only
4 days (even though the model might not have saturated the training set during
this period). Addressing missing data, we introduce four parallel data loaders
that guarantee optimal usage of the available data by including every sample
where possible in the training, even when dealing with missing modalities or
visit data. During training, we employ the validation set to discern the
model’s evolving accuracy. To evaluate performance on the validation set, we
used different metrics for the different losses we employed. For the medical
measures task, we relied on the R2 to gauge the medical measure predictions
from latent embeddings. For the decoding task, we relied on the decreasing MSE
loss, as well as human evaluation of the resulting reconstructions. To
evaluate the contrastive learning performance, we have crafted a novel metric,
assessing the proximity of paired image embeddings and optionally adjusting
for random chance, thus providing an intuitive measure of the model’s
learning. Essentially, by viewing the contrastive task as a classification
task, we can view this metric as top-K accuracy.

Algorithm 1 Top-K Metrics for Contrastive Learning

procedure TopK($sim\\_mat$, $k$)

$correct\leftarrow 0$

for $i=0$ to $|sim\\_mat|$ :

if index $i$ in top-$k$ similar items then

$correct\leftarrow correct+1$

return $correct\text{ }/\text{ }|sim\\_mat|$

$kVals\leftarrow[5,25,100,\ldots]$

$emb_{i},emb_{j}\leftarrow\text{embeddings}$

$norm_{i}\leftarrow\text{l2\\_normalize}(emb_{i})$

$norm_{j}\leftarrow\text{l2\\_normalize}(emb_{j})$

$cosSimMat\leftarrow norm_{i}\cdot norm_{j}^{\top}$

for $k\in kVals$ :

$rand\\_base\leftarrow k\text{ }/\text{ }|cosSimMat|$

$metricScore\leftarrow\textsc{TopK}(cosSimMat,k)$

$angleTopK\leftarrow metricScore$

$multAngleK\leftarrow metricScore/randBase$

In contrastive learning, the central goal is to learn representations such
that similar or "paired" samples are brought closer together in the embedding
space, while dissimilar samples are pushed apart. As we deal with batches of N
samples, we inherently face an N-way classification problem during training.
Achieving perfect performance is often challenging, and a binary assessment of
model proficiency via top-1 prediction accuracy may not sufficiently capture
the nuances in the embeddings the model has learned. In practice, it may
appear that the model is underperforming when, in fact, it has developed a
representation where correct matches are amongst the nearest neighbors, not
necessarily the immediate first. By introducing a Top-K metric (Algorithm 1)
specifically tailored for contrastive learning, we extend the single-label
evaluation to a multi-neighbor perspective, which is analogous to considering
a set of K nearest neighbors in the embedding space. Selecting different
values of K enables us to explore the depth of the model’s understanding of
data relationships. Lower values of K can indicate fine-grained discriminatory
power, while larger values suggest a broader comprehension of sample
similarity. Furthermore, by adjusting for random chance in our metric — by
dividing the raw Top-K score by the expected score under random matching — we
gain insight into how much more effectively our model is at reconciling these
pairs compared to a trivial random embedding model. Notably, this also allows
us to use our hardware efficiently, as to avoid evaluating our model directly
using the downstream tasks, we can evaluate the contrastive task performance
at pretraining time.

Figure 2: Interval Validation, 2.A (left panel). Prediction accuracy of
various medical measures over iterative steps. This figure illustrates the
test R2 values achieved during the training of predictive models for four
different medical measures: fractal dimension, age, vessel density, and artery
average width. The x-axis is the training steps iterations (in tens of
thousands), while the y-axis indicates the R2 value observed on held-out test
data. 2.B (right panel). Evolution of Top-K Test Accuracy in Contrastive
Learning for Multimodal Image Matching. This figure depicts the test accuracy
of a contrastive learning model for two different values of k: 25 and 100, as
labeled Top-25 (orange line) and Top-100 (blue line), respectively. The x-axis
is, again, training steps iterations, and the y-axis denotes the Top-K Test
Accuracy. The Top-100 accuracy increases as training goes on, reaching top
performance of 0.65. Similarly, the Top-25 accuracy ends around 0.35.

##  5 Results

###  5.1 Internal Validation

Our exploration of COMPRER’s results began with internal validation metrics.
Interval validation metrics are composed of metrics that validate the
pretraining phase of our model with multiple objectives. For each one of these
objectives, we report their score on a held-out test set.

####  5.1.1 Medical Measures Prediction

To evaluate the generalization capabilities of the medical measures prediction
head of COMPRER, we engaged in predicting measures from the test set that have
direct clinical applicability. Among the predicted medical measurements were
age, fundus image fractal dimension, vessel density, and artery average width.
Predicting age from fundus images is particularly intriguing, as it suggests a
correlation between ocular characteristics and biological aging, which can
have various medical implications (Ahadi et al., 2023). The Fundus Image
Fractal Dimension is a measure of the complexity and branching patterns of the
retinal vasculature, indicative of overall vascular health(Dinesen et al.,
2021; Macgillivray et al., 2007) . Vessel Density refers to the proportion of
the retina occupied by blood vessels, a crucial factor in assessing retinal
and systemic circulatory health. Artery Average Width provides insights into
vascular caliber, important for understanding cardiovascular risks. The
ability to predict these measures from fundus images is noteworthy, indicating
that our model retains spatial understanding of the images despite the
multiple objectives enforced on this image representation. As detailed in
Figure 2, the generalization performance, quantified by R2 score — showed
differing degrees of success across the medical measures. An R2 score of
approximately 0.6 was observed for most medical measures, indicating a
meaningful predictive relationship between the learned representations and the
clinical measures. However, the prediction for Artery Average Width presented
a lower R2 score of around 0.3, signifying a less robust prediction
capability, or a harder prediction task.

####  5.1.2 Evaluation of Contrastive Learning

A critical component of the COMPRER framework is the multimodal contrastive
loss, which plays a pivotal role in aligning features across distinct imaging
modalities—namely, fundus and carotid ultrasound images. In fact, in the whole
multiple objective framework, this loss is the only connection between the two
distinct image encoders. To measure the effectiveness of contrastive learning,
we devised a scaleless, interpretable metric that provides a concrete
understanding of model performance. While the loss term itself provides an
indication of model learning during training, it lacks direct translatability
to practical outcomes. In Figure 3.a, we show the Top-K test accuracy of
$k\in[5,25,100]$, and in Figure 3.b we show the same plot for the
multiplicative metric, which highlights the model getting better than random
performance. The plots show the metric as calculated from algorithm 1. The
model exhibited non-trivial, non-random results in the multimodal matching
task, which is notable considering the inherent challenge of this problem -
one that even skilled clinicians do not typically address. As part of our
experimental setup, we developed an ablation model, the Multi-Modal
Contrastive Learning (MMCL) model, which was trained on the same data as
COMPRER. Unlike COMPRER, MMCL was trained using a single objective with
pretraining - focusing exclusively on multi-modal contrastive learning. This
approach allowed us to evaluate the impact of the other objectives on our main
model’s performance. Interestingly, the COMPRER model outperforms the Multi-
Modal Contrastive Learning (MMCL) model, which has trained on the same data,
with only the multi-modal contrastive loss, in multimodal matching accuracy,
emphasizing the advantage of a multiple objective training strategy.

Figure 3: Comparative Analysis of Top-K Contrastive Metric on COMPRER and
MMCL. We see in green, purple, and red the Top-100, Top-25, Top-5 contrastive
metric respectively. We see in a dotted line the MMCL model, which trained
solely on the multi-modal contrastive loss (trained using only
$\mathcal{L}_{\text{contr\\_rc}}$ ), and in the straight line COMPRER which
trained on multiple objectives, including the multi-modal contrastive loss. In
3.A We see the multiplicative tok-K metric, and in 3.B we see the top-K
metric. We see that COMPRER consistently outperforms MMCL.

In figure 3.b we see that all start off with random performance (mult top-K
score of 1 for all K) , and rise as the optimization starts. In COMPRER, the
observed Top-100 accuracy reached 0.65, which is higher than the baseline set
by MMCL of 0.56. For the more stringent Top-25 accuracy, COMPRER obtained an
accuracy of 0.35, while the MMCL achieved a lower score of 0.2. This
difference not only underpins our model’s superior matching capability but
also implies the potential benefits of multi-objective training in augmenting
the feature space for more nuanced discriminatory powers.

####  5.1.3 Image Reconstruction Capability

Both fundus image and carotid image reconstructions, while losing some fine
details, remain structurally similar to the original general structure, an
encouraging sign for the model’s comprehension of microvascular features.
While select losses in high-frequency details were observed—likely
attributable to the inherent information compression within the network—the
structural integrity was maintained.

Figure 4: Comparative Analysis of Model Performance in Predicting
Cardiovascular Conditions from Fundus Images. 4.A illustrates the AUC results
for various models fully finetuned to predicting current cardiovascular
condition based on fundus images taken at the participants’ first visit. The
models compared include, our proposed COMPRER, MMCL, DINOV2-Base, and a Retina
Foundation Model. The boxplots represent the distribution of AUC scores
achieved on the test set after selection of top 5 hyperpameters for each model
based on validation performance. 4.B depicts the AUC results for the same set
of models, but focuses on predicting the future onset of cardiovascular
conditions. This analysis only includes participants who were healthy at
baseline, with the goal of determining if the models can predict the
development of cardiovascular conditions at a follow-up visit. While the AUC
scores on their own are not high, COMPRER consistently outperforms the
competition. It’s also worth observing that MMCL has also some gains with
respect to DINOV2 and the retina foundation model, which goes to show that
even just applying multi-modal contrastive learning on its own can increase
downstream performance.

###  5.2 Predictive Performance on Cardiovascular Conditions in the HPP Cohort

While internal validation schemes during the pretraining phase provide
essential insights into the immediate learning dynamics of our COMPRER model,
the true test of its effectiveness lies in its clinical application. Thus, our
goal is to demonstrate that our model pretraining not only captures intricate
data patterns but also translates into significant improvements in real-world
clinical diagnostics. To this end, we focused our attention on fine-tuning
COMPRER to predict cardiovascular health conditions. Figure 4 shows the
model’s capacities in both a diagnostic and prognostic context, providing
valuable insights into cardiovascular health. The performance metrics
presented in Figure 4 were constructed using the models, which was fitted with
a 1-layer MLP regression head. To find appropriate hyperparameters, we
employed a systematic hyperparameter search on the validation set, from which
the top 5 models of each type were identified. These leading models were then
assayed on an independent test set, yielding a distribution of results, which
denotes the robustness and consistency of performance across model instances.
The MMCL model is an ablation model, representing a version of our
architecture and data trained with only multimodal contrastive loss, to
provided a baseline to quantify the value added by the multi-objective
learning. The Retina Foundation Model embodies a high-parameter (300M
parameters, which is $3.5\times$ larger than all other competitor models)
alternative, leveraging a considerably larger latent space (1024, which
$\approx 1.3$ times larger than all other competitors) and trained on an
extensive dataset of 1.6M fundus images (which is $75\times$ larger than our
fundus dataset). Despite these advantages of the retina foundation model,
COMPRER shows superior performance. We also observed that on the prognosis
task (figure 4.b) almost all model runs except COMPRER’s are random. This is
interesting because COMPRER is the only model that had in its pretraining any
signal of future events - based on the temporal contrastive learning
objective.

Figure 5: Comparative Analysis of COMPRER and Retina Foundation Model
Performance in Predicting Diseases from Fundus Image Representations on an
External Dataset (UKBB). The prediction is done from frozen image embeddings
using logistic regression. We see that COMPRER outperforms the Retina
Foundation model in the prediction of ischaemic stroke, evidenced by a higher
AUC value. Conversely, the Retina Foundation model exhibits superior
performance in predicting chronic ischaemic heart disease, although COMPRER
also shows a test AUC above 0.6. For the conditions of stroke, heart failure,
and acute ischaemic heart disease, the COMPRER method slightly outperforms the
Retina Foundation model, though the margins are not significant. Its also
notable that the the retina foundation model was trained using $75\times$ more
fundus images, and with a $3.5\times$ larger model.

###  5.3 Performance on an OOD External Cohort

Validating the predictive power of a model on an out of distribution (OOD)
dataset is often considered the gold standard for demonstrating the real-world
applicability and robustness of a predictive framework. In this section, we
evaluate COMPRER’s performance on the external dataset - the UK Biobank
(UKBB), an extensive, well-characterized external cohort that has been at the
forefront of large-scale biomedical research (Sudlow et al., 2015). After data
filtering and cleaning, it comprises of 44K participants with fundus images.
The Retina Foundation model has previously showcased its traction on this
dataset, establishing a performance benchmark for the field. Figure 5
illustrates the comparison between COMPRER and the Retina Foundation model in
predicting various cardiovascular and related diseases from fundus image
representations. Our approach demonstrates competitive, if not superior,
predictive performance across numerous conditions. In the realm of ischaemic
stroke prediction, COMPRER conspicuously outperforms the Retina Foundation
model, denoting a higher AUC value. These results affirm the appropriateness
of COMPRER’s multi-modal, multi-objective pretraining paradigm, reinforcing
its utility in extracting salient features pertinent to disease states from
medical imagery. Moreover, they underscore our method’s efficiency; by
achieving these competitive performance metrics, COMPRER evidences that well-
conceived model architectures coupled with sophisticated pretraining
strategies can level the playing field against models with ostensibly more
advantageous training conditions (using $75\times$ the data, and $3.5\times$
the parameters).

##  6 Discussion

We presented COMPRER, a novel pretraining method targeted at extracting
medical features, specifically aimed at the enhancement of downstream tasks
such as disease diagnosis and prognosis. Our multi-modal, multi-objective
pretraining approach bridges the gap between advancing machine learning
architectures and their practical applications in clinical settings,
exhibiting superior predictive abilities compared to specialized models. In
the realm of image representation, COMPRER has demonstrated the ability to
distill meaningful insights from a large-scale, longitudinally derived
dataset, diminishing the reliance on arduously curated labels. The empirical
advancements evidenced by robust internal validation and insightful
applications in disease prediction advocate for COMPRER’s adoption. Notably,
the model provides competitive performance, transcending the need for vast
data volumes and extensive computational resources, highlighting a quality-
centric approach in medical data analytics. We have shown that the integration
of multiple objectives stretches the training duration but enriches the
model’s feature representations. This comprehensive approach, though
intensive, equips the model with a deepened understanding and enhanced
performance that may surpass models optimized for single objectives alone. The
superior performance in multimodal contrastive matching demonstrates this
phenomenon, suggesting that a more holistic, integrated training regimen can
indeed yield models with robust generalization capacities across various
tasks.

In addition, we have shown that COMPRER is capable of achieving higher test
AUC scores than models that have been trained with an order of magnitude more
data, and with larger models. We have shown this both in our HPP cohort, as
well as, in an external cohort - the UKBB, which shows our model can
generalize out of distribution, across diverse populations from different
continents. Interestingly, within the OOD UKBB dataset, a notable divergence
in the test AUC was observed for ischaemic stroke prediction between COMPRER,
and the Retina Foundation model. This finding is particularly intriguing given
that ischaemic stroke is a condition diagnosable through carotid ultrasound
(Zhang et al., 2014). It is worth noting that COMPRER was trained using both
fundus images and carotid ultrasound, suggesting that this multi-modality
approach may have played a role in the increase of COMPRER’s predictive
accuracy compared to the Retina Foundation model. It is also noteworthy that
in the internal cohort validation section (5.2), where we showed future
prognosis performance, the majority of model runs, with the exception of
COMPRER, exhibit random test AUC scores. This phenomenon might be interesting,
as it underscores the distinctive attribute of COMPRER’s pretraining
methodology. In contrast to the other models, COMPRER benefits from the
inclusion of a temporal information during its pretraining, notably - the
temporal, visit-based, contrastive learning objective. This objective provided
the model with insights into future events within the longitudinal data. We
hypothesize that this unique aspect of COMPRER’s training regimen is
contributing to its superior performance, even surpassing models like MMCL
that have encountered the same data but lack exposure to the temporal, visit-
based contrastive loss. This finding underscores the potential advantages of
incorporating temporal information in pretraining, shedding light on the
nuances of disease prognosis prediction and highlighting the efficacy of our
approach.

However, our work has several caveats, calling attention to the simultaneous
challenges and potential trajectories for improvement. Dataset scope and
representativeness remain pivotal for model generalization. While the HPP
dataset underpins our current findings, incorporating datasets with broader
demographic diversity is essential for enhancing model robustness and ensuring
its translational relevance across patient populations. A more diverse dataset
would help mitigate bias and uphold the model’s diagnostic integrity,
especially when encountered with dataset shifts in real-world scenarios. The
generalizability of COMPRER across different diseases and imaging modalities
is another frontier to be explored. Expanding the disease spectrum and
experimenting with a variety of modalities are key to consolidating the
framework’s applicability in diverse medical contexts. Moreover, the limited
computational resources constrained our model’s training to 40k steps, hinting
at the potential for further refinement. Investments in computational
infrastructure and collaborative efforts could uncover latent performance
enhancements and insights into the optimization dynamics of our model.
Interpretability and explainability are indispensable for clinician and
patient acceptance. Despite the strides made with our transformer-based model,
elucidating the AI’s decision-making processes remains crucial. Validation in
clinical environments could unravel the efficacy and adaptability of our model
and solidify its role within clinical workflows. It is important to note that
we conducted our research on only two modalities, however, we do see how this
method could easily extend beyond only two modalities. In the HPP dataset, we
have access to a rich variety of over 20 distinct data modalities, ranging
from visual information and time series data to textual records and tabular
measurements.

We recognize the untapped potential of leveraging multiple modalities within
our pretraining scheme. Future iterations of the methods we described here can
harness this diverse data landscape by incorporating additional losses for
multi-modal contrastive learning, introducing multi-visit losses that span
across these various modalities, and exploring other innovative techniques.
This multi-modal approach holds promise in further enriching the model’s
understanding of complex medical data, potentially leading to even more robust
generalization across a wide array of clinical tasks.

##  7 Availability of Code and Model Weights

In the interest of transparency and facilitating future research, we plan to
release the code and model weights associated with our study upon publication.

##  8 Bibliographical References

## References

  * Bajwa et al. [2021] J. Bajwa, U. Munir, A. Nori, and B. Williams.  Artificial intelligence in healthcare: transforming the practice of medicine.  _Future Healthcare Journal_ , 8:e188–e194, 2021.  doi:10.7861/fhj.2021-0095.
  * Esteva et al. [2019] Andre Esteva, Alexandre Robicquet, Bharath Ramsundar, Volodymyr Kuleshov, Mark DePristo, Katherine Chou, Claire Cui, Greg Corrado, Sebastian Thrun, and Jeff Dean.  A guide to deep learning in healthcare.  _Nature Medicine_ , 25(1):24–29, 2019.
  * Huang et al. [2023] S-C. Huang, A. Pareek, M. Jensen, M. P. Lungren, S. Yeung, and A. S. Chaudhari.  Self-supervised learning for medical image classification: a systematic review and implementation guidelines.  _npj Digital Medicine_ , 6:74, 2023.  doi:10.1038/s41746-023-00811-0.
  * Poplin et al. [2018] R. Poplin, A. V. Varadarajan, K. Blumer, Y. Liu, M. V. McConnell, G. S. Corrado, et al.  Prediction of cardiovascular risk factors from retinal fundus photographs via deep learning.  _Nature Biomedical Engineering_ , 2:158–164, 2018.  doi:10.1038/s41551-018-0195-0.
  * Spence [2006] J. D. Spence.  Technology insight: ultrasound measurement of carotid plaque–patient management, genetic research, and therapy evaluation.  _Nature Clinical Practice Neurology_ , 2:611–619, 2006.  doi:10.1038/ncpneuro0324.
  * Dai et al. [2021] L. Dai, L. Wu, H. Li, C. Cai, Q. Wu, H. Kong, et al.  A deep learning system for detecting diabetic retinopathy across the disease spectrum.  _Nature Communications_ , 12:3242, 2021.  doi:10.1038/s41467-021-23458-5.
  * Yan et al. [2019] Q. Yan, D. E. Weeks, H. Xin, H. Huang, A. Swaroop, E. Y. Chew, et al.  Deep-learning-based prediction of late age-related macular degeneration progression.  _medRxiv_ , 2019.  doi:10.1101/19006171.
  * Yu et al. [2021] J. Yu, Y. Zhou, Q. Yang, et al.  Machine learning models for screening carotid atherosclerosis in asymptomatic adults.  _Scientific Reports_ , 11:22236, 2021.  doi:10.1038/s41598-021-01456-3.  URL https://doi.org/10.1038/s41598-021-01456-3.
  * Siontis et al. [2021] K. C. Siontis, P. A. Noseworthy, Z. I. Attia, and P. A. Friedman.  Artificial intelligence-enhanced electrocardiography in cardiovascular disease management.  _Nature Reviews Cardiology_ , 18:465–478, 2021.  doi:10.1038/s41569-020-00503-2.
  * Zhou et al. [2023] Y. Zhou, M. A. Chia, S. K. Wagner, M. S. Ayhan, D. J. Williamson, R. R. Struyven, et al.  A foundation model for generalizable disease detection from retinal images.  _Nature_ , 622:156–163, 2023.  doi:10.1038/s41586-023-06555-x.
  * Shilo et al. [2021] S. Shilo, N. Bar, A. Keshet, Y. Talmor-Barkan, H. Rossman, A. Godneva, et al.  10 k: a large-scale prospective longitudinal study in israel.  _European Journal of Epidemiology_ , 36:1187–1194, 2021.  doi:10.1007/s10654-021-00753-5.
  * Moor et al. [2023] M. Moor, O. Banerjee, Z. S. H. Abad, H. M. Krumholz, J. Leskovec, E. J. Topol, et al.  Foundation models for generalist medical artificial intelligence.  _Nature_ , 616:259–265, 2023.  doi:10.1038/s41586-023-05881-4.
  * Grill et al. [2020] J-B. Grill, F. Strub, F. Altché, C. Tallec, P. H. Richemond, E. Buchatskaya, et al.  Bootstrap your own latent: A new approach to self-supervised learning, 2020.
  * Chen et al. [2020] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton.  A simple framework for contrastive learning of visual representations, 2020.
  * Radhakrishnan et al. [2023] A. Radhakrishnan, S. F. Friedman, S. Khurshid, K. Ng, P. Batra, S. A. Lubitz, et al.  Cross-modal autoencoder framework learns holistic representations of cardiovascular state.  _Nature Communications_ , 14:2436, 2023.  doi:10.1038/s41467-023-38125-0.
  * Dosovitskiy et al. [2020] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, et al.  An image is worth 16x16 words: Transformers for image recognition at scale, 2020.
  * Oquab et al. [2023] M. Oquab, T. Darcet, T. Moutakanni, H. Vo, M. Szafraniec, V. Khalidov, et al.  Dinov2: Learning robust visual features without supervision, 2023.
  * Radford et al. [2021] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, et al.  Learning transferable visual models from natural language supervision, 2021.
  * Kwasigroch et al. [2020] A. Kwasigroch, M. Grochowski, and A. Mikołajczyk.  Self-supervised learning to increase the performance of skin lesion classification.  _Electronics_ , 9:1930, 2020.  doi:10.3390/electronics9111930.
  * Tiu et al. [2022] E. Tiu, E. Talius, P. Patel, C. P. Langlotz, A. Y. Ng, and P. Rajpurkar.  Expert-level detection of pathologies from unannotated chest x-ray images via self-supervised learning.  _Nature Biomedical Engineering_ , 6:1399–1406, 2022.  doi:10.1038/s41551-022-00936-9.
  * Jaiswal et al. [2019] A. K. Jaiswal, P. Tiwari, S. Kumar, D. Gupta, A. Khanna, and J. J. P. C. Rodrigues.  Identifying pneumonia in chest x-rays: A deep learning approach.  _Measurement_ , 145:511–518, 2019.  doi:10.1016/j.measurement.2019.05.076.
  * Pereira et al. [2016] S. Pereira, A. Pinto, V. Alves, and C. A. Silva.  Brain tumor segmentation using convolutional neural networks in mri images.  _IEEE Transactions on Medical Imaging_ , 35:1240–1251, 2016.  doi:10.1109/TMI.2016.2538465.
  * Liu et al. [2021] H. Liu, J. Z. HaoChen, A. Gaidon, and T. Ma.  Self-supervised learning is more robust to dataset imbalance, 2021.
  * Bommasani et al. [2021] R. Bommasani, D. A. Hudson, E. Adeli, R. Altman, S. Arora, S. von Arx, et al.  On the opportunities and risks of foundation models, 2021.
  * Vaswani et al. [2017] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, et al.  Attention is all you need, 2017.
  * Ramesh et al. [2022] A. Ramesh, P. Dhariwal, A. Nichol, C. Chu, and M. Chen.  Hierarchical text-conditional image generation with clip latents, 2022.
  * Ramesh et al. [2021] A. Ramesh, M. Pavlov, G. Goh, S. Gray, C. Voss, A. Radford, et al.  Zero-shot text-to-image generation, 2021.
  * Brown et al. [2020] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, et al.  Language models are few-shot learners, 2020.
  * Holmberg et al. [2020] O. G. Holmberg, N. D. Köhler, T. Martins, J. Siedlecki, T. Herold, L. Keidel, et al.  Self-supervised retinal thickness prediction enables deep learning from unlabelled data to boost classification of diabetic retinopathy.  _Nature Machine Intelligence_ , 2:719–726, 2020.  doi:10.1038/s42256-020-00247-1.
  * Chefer et al. [2020] H. Chefer, S. Gur, and L. Wolf.  Transformer interpretability beyond attention visualization, 2020.
  * Ruder [2017] S. Ruder.  An overview of multi-task learning in deep neural networks, 2017.
  * Crawshaw [2020] M. Crawshaw.  Multi-task learning with deep neural networks: A survey, 2020.
  * Zhou et al. [2022] Yukun Zhou, Siegfried K Wagner, Mark A Chia, An Zhao, Moucheng Xu, Robbert Struyven, Daniel C Alexander, Pearse A Keane, et al.  Automorph: Automated retinal vascular morphology quantification via a deep learning pipeline.  _Translational vision science & technology_, 11(7):12–12, 2022.
  * Caron et al. [2021] M. Caron, H. Touvron, I. Misra, H. Jégou, J. Mairal, P. Bojanowski, et al.  Emerging properties in self-supervised vision transformers, 2021.
  * Balestriero et al. [2023] R. Balestriero, M. Ibrahim, V. Sobal, A. Morcos, S. Shekhar, T. Goldstein, et al.  A cookbook of self-supervised learning, 2023.
  * Ahadi et al. [2023] Sara Ahadi, Kenneth A Wilson, Boris Babenko, Cory Y McLean, Drew Bryant, Orion Pritchard, Ajay Kumar, Enrique M Carrera, Ricardo Lamy, Jay M Stewart, Avinash Varadarajan, Marc Berndl, Pankaj Kapahi, and Ali Bashir.  Longitudinal fundus imaging and its genome-wide association analysis provide evidence for a human retinal aging clock.  _eLife_ , 12:e82364, 2023.  doi:https://doi.org/.
  * Dinesen et al. [2021] Sebastian Dinesen, Pia S Jensen, Maria Bloksgaard, Søren Leer Blindbæk, Jo De Mey, Lars M Rasmussen, Jes S Lindholt, and Jakob Grauslund.  Retinal vascular fractal dimensions and their association with macrovascular cardiac disease.  _Ophthalmic Research_ , 64(4):561–566, 2021.  doi:10.1159/000514442.
  * Macgillivray et al. [2007] T J Macgillivray, N Patton, F N Doubal, C Graham, and J M Wardlaw.  Fractal analysis of the retinal vascular network in fundus images.  _Annu Int Conf IEEE Eng Med Biol Soc_ , 2007:6456–6459, 2007.  doi:10.1109/IEMBS.2007.4353837.
  * Sudlow et al. [2015] Cathie Sudlow, John Gallacher, Naomi Allen, Valerie Beral, Paul Burton, John Danesh, Paul Downey, Paul Elliott, Jane Green, Martin Landray, Bette Liu, Paul Matthews, Giok Ong, Jill Pell, Alan Silman, Alan Young, Tim Sprosen, Tim Peakman, and Rory Collins.  Uk biobank: An open access resource for identifying the causes of a wide range of complex diseases of middle and old age.  _PLOS Medicine_ , 12(3):e1001779, 2015.
  * Zhang et al. [2014] Y. Zhang, E. Guallar, Y. Qiao, and B.A. Wasserman.  Is carotid intima-media thickness as predictive as other noninvasive techniques for the detection of coronary artery disease?  _Arteriosclerosis, Thrombosis, and Vascular Biology_ , 34(7):1341–1345, Jul 2014.  doi:10.1161/ATVBAHA.113.302075.  URL https://doi.org/10.1161/ATVBAHA.113.302075.  Epub 2014 Apr 24.
  * Arcadu et al. [2019] F. Arcadu, F. Benmansour, A. Maunz, J. Willis, Z. Haskova, and M. Prunotto.  Deep learning algorithm predicts diabetic retinopathy progression in individual patients.  _npj Digital Medicine_ , 2:92, 2019.  doi:10.1038/s41746-019-0172-3.


# Dirac’s Equation in $R$-Minkowski Spacetime

T. Foughali 111E-mail<EMAIL_ADDRESS>and A. Bouda222E-mail:
<EMAIL_ADDRESS>
Laboratoire de Physique Théorique, Faculté des Sciences Exactes,
Université de Bejaia, 06000 Bejaia, Algeria

###### Abstract

We recently constructed the $R$-Poincaré algebra from an appropriate deformed
Poisson brackets which reproduce the Fock coordinate transformation. We showed
then that the spacetime of this transformation is the de Sitter one. In this
paper, we derive in the $R$-Minkowski spacetime the Dirac equation and show
that this is none other than the Dirac equation in the de Sitter spacetime
given by its conformally flat metric. Furthermore, we propose a new approach
for solving Dirac’s equation in the de Sitter spacetime using the Schrödinger
picture.

PACS numbers : 03.30.+p, 98.80.Jk, 03.65.Pm

Keywords : Fock’s transformation, de Sitter spacetime, Dirac’s equation.

##  1 Introduction

Deformed Special Relativity (DSR) [1, 2, 3, 4] and Fock’s transformation [5]
are two distinct approaches of nonlinear relativity which are developed for
completely different motivations. In addition to the speed of light, DSR keeps
invariant a minimal length on the order of the Planck length, while Fock’s
transformation keeps invariant a length which represents the universe radius.

By following the same way as in DSR [6], we recently proposed an appropriate
deformed Poisson brackets [7]

$\displaystyle\\{x^{\mu},x^{\nu}\\}$ $\displaystyle=0,$ (1)
$\displaystyle\\{x^{\mu},p^{\nu}\\}$
$\displaystyle=-\eta^{\mu\nu}+\frac{1}{R}\eta^{0\nu}x^{\mu},$ (2)
$\displaystyle\\{p^{\mu},p^{\nu}\\}$
$\displaystyle=-\frac{1}{R}[p^{\mu}\eta^{0\nu}-p^{\nu}\eta^{\mu 0}],$ (3)

from which we reproduced the Fock coordinate transformation

$t^{\prime}=\frac{\gamma(t-ux/c^{2})}{\alpha_{R}},\quad
x^{\prime}=\frac{\gamma(x-ut)}{\alpha_{R}},\quad
y^{\prime}=\frac{y}{\alpha_{R}},\quad z^{\prime}=\frac{p_{z}}{\alpha_{R}},$
(4)

where

${\alpha_{R}}=1+\frac{1}{R}\left[(\gamma-1)ct-\gamma\frac{ux}{c}\right],$ (5)

$R$ is the universe radius, $\gamma=(1-u^{2}/c^{2})^{-\frac{1}{2}}$,
$\eta^{\mu\nu}=(+1,-1,-1,-1)$ and $\mu,\nu=0,1,2,3$. Relations (4) define the
so-called $R$-Minkowski spacetime. From the above brackets, we established the
corresponding momentum transformation

$E^{\prime}={\alpha_{R}}{\gamma(E-up_{x})},\quad
p^{\prime}_{x}={\alpha_{R}}{\gamma(p_{x}-uE/c^{2})},\quad
p^{\prime}_{y}={\alpha_{R}}{y},\quad p^{\prime}_{z}={\alpha_{R}}{z}$ (6)

with which the four dimensional contraction $p_{\mu}x^{\mu}$ is an invariant,
allowing then a coherent description of plane waves. Here $c$ and $R$ are
invariant and in the limit $R\rightarrow\infty$, the above transformations
reduce to the Lorentz transformation for the coordinates as well as for the
energy-momentum vector.

The $R$-algebra, constituted by (1), (2) and (3), is completed in [8] by
involving pure rotation generators,

$M_{i}=\frac{1}{2}\epsilon_{ijk}J_{jk},$ (7)

and boost ones,

$\tilde{N}_{i}=J_{0i},$ (8)

where $J_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$ stands for the angular
momentum, $i,j,...=1,2,3$ and $\epsilon_{ijk}$ is the Levi-Civita
antisymmetric tensor ($\epsilon_{123}=1$). Following the Magpantay approach
[9], we proposed a modified expression of the boost generators

$N_{i}=J_{0i}-\frac{1}{2R}\eta_{\mu\nu}x^{\mu}x^{\nu}p_{i},$ (9)

that allows to construct the first Casimir invariant of the theory.

The quantization was done by the substitution of $x^{\mu}$ and $p^{\mu}$ by
the corresponding operators and the Poisson brackets by commutators. The
resulting phase space algebra of the $R$-Minkowski spacetime

$\displaystyle[x^{\mu},x^{\nu}]$ $\displaystyle=0,$ (10)
$\displaystyle[x^{\mu},p^{\nu}]$
$\displaystyle=-i\hbar(\eta^{\mu\nu}-\frac{1}{R}\eta^{0\nu}x^{\mu}),$ (11)
$\displaystyle[p^{\mu},p^{\nu}]$
$\displaystyle=-\frac{i\hbar}{R}(p^{\mu}\eta^{0\nu}-p^{\nu}\eta^{\mu 0}),$
(12)

and the underlying $R$-Poincaré algebra

$\displaystyle[N_{i},p_{0}]$ $\displaystyle=-i\hbar
p_{i}+i\hbar\frac{1}{R}N_{i},$ (13) $\displaystyle[N_{i},p_{j}]$
$\displaystyle=-i\hbar\delta_{ij}p_{0}-i\hbar\frac{1}{R}\varepsilon_{ijk}M_{k},$
(14) $\displaystyle[M_{i},p_{0}]$ $\displaystyle=0,$ (15)
$\displaystyle[M_{i},p_{j}]$ $\displaystyle=i\hbar\varepsilon_{ijk}p_{k},$
(16) $\displaystyle[M_{i},M_{j}]$
$\displaystyle=i\hbar\varepsilon_{ijk}M_{k},$ (17)
$\displaystyle[M_{i},N_{j}]$ $\displaystyle=i\hbar\varepsilon_{ijk}N_{k},$
(18) $\displaystyle[N_{i},N_{j}]$
$\displaystyle=-i\hbar\varepsilon_{ijk}M_{k},$ (19)

allowed us to obtain the following expression for the first Casimir

$C=p^{2}_{0}-{p^{i}}{p^{i}}+\frac{1}{R}({N}^{i}p^{i}+p^{i}{N}^{i})-\frac{1}{R^{2}}{M}^{i}{M}^{i},$
(20)

which obviously reduces, in the limit $R\rightarrow\infty$, to the first
Poincaré Casimir.

In the present work, we will focus in the $R$-Minkowski spacetime on the Dirac
equation which is already investigated in the context of the DSR [10, 11, 12].
Also, we aim to explore more the correspondence, already established in [8],
between de Sitter spacetime and the $R$-Minkowski spacetime. This may offer
more possibilities to handle the issue of constructing physical observables in
the de Sitter space [13, 14]. Furthermore, we propose a new approach for
solving the free Dirac equation in the de Sitter spacetime by using the
Schrodinger picture established in this context by Cotăescu [15].

The paper is organized as follows. In section 2, we establish the free Dirac
equation in the $R$-Minkowski spacetime and show that the obtained result is
identically the Dirac equation in a conformally flat de Sitter spacetime. In
section 3, we construct the Schrödinger picture version of this equation and
in section 4, we propose within this picture a new procedure to solve the free
Dirac equation in de Sitter spacetime. In section 5, we give some concluding
remarks.

##  2 Dirac equation in $R$-Minkowski spacetime

In [8], we proposed the following representation for the momentum

$\displaystyle p^{0}$
$\displaystyle=i\hbar(\partial^{0}-\frac{1}{R}x^{\mu}\partial_{\mu}),$ (21)
$\displaystyle p^{i}$ $\displaystyle=i\hbar\partial^{i},$ (22)

with which the complete algebra (10)-(19) is satisfied, and therefore showed
that expression (20) of the Casimir is exactly the Klein-Gordon operator of a
conformal flat metric of de Sitter spacetime. This result established a
correspondence between $R$-Minkowski spacetime and de Sitter spacetime.

For constructing the Dirac equation in the $R$-Minkowski spacetime, we impose
to the square of Dirac operator to reproduce partly Klein-Gordon operator
given by expression (20) of the Casimir. In view of this, let us express $C$
only in terms of the operators $x$ and $p$. With the use of expressions (7)
and (9) and taking into account commutators (10), (11) and (12), we can show
that

$\displaystyle{M}^{i}{M}^{i}$
$\displaystyle=\epsilon^{ijk}\epsilon^{ilm}x^{j}p^{k}x^{l}p^{m}=\vec{x}^{2}\vec{p}^{2}-(\vec{x}\cdot\vec{p})^{2}+i\hbar\vec{x}\cdot\vec{p},$
(23) $\displaystyle N^{i}p^{i}$
$\displaystyle=x^{0}\vec{p}^{2}-\vec{x}\cdot\vec{p}p^{0}-\frac{i\hbar}{R}\vec{x}\cdot\vec{p}-\frac{1}{2R}{x^{0}}^{2}\vec{p}^{2}+\frac{1}{2R}\vec{x}^{2}\vec{p}^{2},$
(24) $\displaystyle p^{i}N^{i}$
$\displaystyle=x^{0}\vec{p}^{2}-\vec{x}\cdot\vec{p}p^{0}-\frac{i\hbar}{R}\vec{x}\cdot\vec{p}-\frac{1}{2R}{x^{0}}^{2}\vec{p}^{2}+\frac{1}{2R}\vec{x}^{2}\vec{p}^{2}+3i\hbar
p_{0}.$ (25)

The above vectors are three-dimensional ones. Substituting these relations in
(20), we obtain

$C=p^{2}_{0}-\left(1-\frac{x^{0}}{R}\right)^{2}{\vec{p}}^{2}-\frac{2}{R}\vec{x}\cdot\vec{p}\,p^{0}+\frac{1}{R^{2}}(\vec{x}\cdot\vec{p})^{2}-\frac{3i\hbar}{R^{2}}\vec{x}\cdot\vec{p}+\frac{3i\hbar}{R}p_{0}.$
(26)

It is interesting to remark that by using this last expression and the fact
that

$\gamma^{0}\gamma^{i}p_{0}\left(1-\frac{x^{0}}{R}\right)p_{i}+\gamma^{i}\gamma^{0}\left(1-\frac{x^{0}}{R}\right)p_{i}p_{0}=0$
(27)

and

$\frac{\gamma^{i}\gamma^{0}}{R}p_{i}x^{j}p_{j}+\frac{\gamma^{0}\gamma^{i}}{R}x^{j}p_{j}p_{i}=-i\hbar\frac{\gamma^{0}\gamma^{i}}{R}p_{i},$
(28)

we can show

$\displaystyle\left[\gamma^{0}p_{0}+\left(1-\frac{x^{0}}{R}\right)\gamma^{i}p_{i}+\frac{\gamma^{0}}{R}x^{i}p_{i}+i\frac{3\hbar\gamma^{0}}{2R}+mc\right]\left[\gamma^{0}p_{0}+\left(1-\frac{x^{0}}{R}\right)\gamma^{i}p_{i}\right.\hskip
2.84526pt$
$\displaystyle\left.+\frac{\gamma^{0}}{R}x^{i}p_{i}+i\frac{3\hbar\gamma^{0}}{2R}-mc\right]=C-m^{2}c^{2}-i\hbar\left(1-\frac{x^{0}}{R}\right)\frac{\gamma^{0}\gamma^{i}}{R}p_{i}-\frac{9\hbar^{2}}{4R^{2}},$
(29)

$\gamma^{0}$ and $\gamma^{i}$ being Dirac matrices. Equation (29) suggests to
write the Dirac equation in $R$-Minkowski spacetime in the following form

$\left[\gamma^{0}p_{0}+\left(1-\frac{x^{0}}{R}\right)\gamma^{i}p_{i}+\frac{\gamma^{0}}{R}x^{i}p_{i}+i\frac{3\gamma^{0}}{2R}-mc\right]\Psi=0.$
(30)

Indeed, the last two terms in (29) that make the square of the Dirac operator
different from the Klein-Gordon operator are the manifestation of the spin
$1/2$ for the Dirac particle. In fact, in a curved spacetime, the solution to
the generally covariant Dirac equation is not a solution to the generally
covariant Klein-Gordon equation but to the generally covariant Pauli-
Schrödinger equation describing spin $1/2$ particles in a gravitational field
[16, 17]. In the square of the spinor covariant derivative, the spinorial
nature of the Dirac particle appears within the terms involving Fock-Ivanenko
coefficients. For more details, the presence of these two additional terms is
justified in Appendix A.

Using the representation given by (21) and (22), proposed in the $R$-Poincaré
algebra context [8], we obtain the differential form of (30)

$\left[i\left(1-\frac{x^{0}}{R}\right)\gamma^{0}\partial_{0}+i(1-\frac{x^{0}}{R})\gamma^{i}\partial_{i}+i\frac{3\gamma^{0}}{2R}-{mc/\hbar}\right]\Psi=0.$
(31)

This is exactly the Dirac equation in the conformally flat de Sitter spacetime
with a conformal factor $a(t)=(1-x^{0}/R)^{-1}$. Again, a correspondence
between $R$-Minkowski spacetime and de Sitter spacetime is established.

##  3 The Schrödinger picture

In order to solve equation (31), we will use the Schrödinger picture developed
in [15]. We will give an exact solution of the Dirac equation in the
$R$-Minkowski or de Sitter spacetime, that transforms to the form found by
Shishkin [18] and Cotăescu [19] in the natural picture.

Cotaĕscu showed that the transformation $\Psi(x)\to\Psi_{S}(x)=W(x)\Psi(x)$
leading to the Schrödinger picture is produced by the operator of time
dependent dilatations

$W(x)=\exp\left[-\ln(\alpha(t))(\vec{x}\cdot\vec{\partial})\right]\,,$ (32)

with the following properties:

$W(x)^{\dagger}=\alpha^{3}(t)\,W(x)^{-1}\,,$ (33)

and

$W(x)F(x)W(x)^{-1}=F\left(\frac{1}{\alpha(t)}x\right)\,,\quad
W(x)G(\vec{\partial})W(x)^{-1}=G\left(\alpha(t)\vec{\partial}\right),$ (34)

$F$ and $G$ being arbitrary functions. Setting $\alpha\equiv\xi^{-1}$, in the
$R$-Minkowski spacetime, we have $\xi=1-x^{0}/R$ and then

$W(x)=\exp\left[\frac{i}{\hbar}\ln(\xi)(\vec{x}\cdot\vec{p})\right]$ (35)

and

$W(x)G(\vec{p})W(x)^{-1}=G\left(\alpha(t)\vec{p}\right).$ (36)

Using the fact that $p_{0}$ commute with $\vec{x}\cdot\vec{p}$ and
$[p_{0},\xi]=-i\hbar\xi/R$, we have

$\displaystyle W(x)p_{0}W(x)^{-1}$ $\displaystyle=W(x)[p_{0},W(x)^{-1}]+p_{0}$
$\displaystyle=W(x)[p_{0},\xi]\frac{\partial W(x)^{-1}}{\partial\xi}+p_{0}$
$\displaystyle=p_{0}-\frac{\vec{x}\cdot\vec{p}}{R}.$ (37)

By substituting in (30) $\Psi(x)$ by $W^{-1}\Psi_{S}(x)$ and multiplying at
left the same equation by $W$, we obtain in the Schrödinger picture the free
Dirac equation in the $R$-Minkowski spacetime

$\left[\gamma^{0}p_{0}+\gamma^{i}p_{i}-2\frac{\gamma^{0}}{R}\vec{x}\cdot\vec{p}+i\frac{3\gamma^{0}}{2R}-mc\right]\Psi_{S}=0,$
(38)

where relations (36) and (37) have been used. With the representation given in
(21) and (22), the differential form of the last equation is

$\left[i\gamma^{0}\left(\left(1-\frac{x^{0}}{R}\right)\partial_{0}+\frac{x^{i}\partial_{i}}{R}+\frac{3}{2R}\right)+i\gamma^{i}\partial_{i}-{mc/\hbar}\right]\Psi_{S}=0.$
(39)

##  4 The solution

Making the substitution $\Psi_{S}=(1-x^{0}/R)^{3/2}\tilde{\Psi}$ in equation
(39), we obtain

$\left[i\gamma^{0}\left(\left(1-\frac{x^{0}}{R}\right)\partial_{0}+\frac{x^{i}\partial_{i}}{R}\right)+i\gamma^{i}\partial_{i}-{mc/\hbar}\right]\tilde{\Psi}=0.$
(40)

We put in what follows $c=\hbar=1$. Multiplying at left (40) by $\gamma^{0}$

$\left[i\left(1-\frac{x^{0}}{R}\right)\partial_{0}+i\left(\alpha^{i}+\frac{x^{i}}{R}\right)\partial_{i}-\gamma^{0}{m}\right]\tilde{\Psi}=0.$
(41)

and using the variable $\xi$, we obtain

$\left[{\xi}\partial_{\xi}-(R\alpha^{i}+{x^{i}})\partial_{i}-i\gamma^{0}{Rm}\right]\tilde{\Psi}=0.$
(42)

The function $\tilde{\Psi}$ must be a bispinor

$\tilde{\Psi}=\begin{pmatrix}\varphi\\\ \chi\end{pmatrix}$ (43)

where $\varphi$ and $\chi$ are two spinors to be determined. Using the
standard Dirac representation for $\gamma^{\mu}$ matrices, equation (42) leads
to the following system

$\displaystyle\big{(}{\xi}\partial_{\xi}-{x^{i}}\partial_{i}-i{Rm}\big{)}\varphi-R\sigma^{i}\partial_{i}\chi$
$\displaystyle=0,$ (44)
$\displaystyle\big{(}{\xi}\partial_{\xi}-{x^{i}}\partial_{i}+i{Rm}\big{)}\chi-R\sigma^{i}\partial_{i}\varphi$
$\displaystyle=0.$ (45)

Multiplying at left (45) by $R\sigma^{i}\partial_{i}$ and using (44), we get
to

$\left({\xi}^{2}\partial^{2}_{\xi}-2\xi{x^{i}}\partial_{i}\partial_{\xi}+2x^{i}\partial_{i}+x^{i}x^{j}\partial_{i}\partial_{j}-R^{2}\nabla^{2}+i{Rm}+{R^{2}m^{2}}\right)\varphi=0,$
(46)

which we can put in the form

$\left[({\xi}\partial_{\xi}-x^{i}\partial_{i})^{2}-({\xi}\partial_{\xi}-x^{i}\partial_{i})-R^{2}\nabla^{2}+i{Rm}+{R^{2}m^{2}}\right]\varphi=0.$
(47)

In the spherical coordinate system, $(r,\theta,\phi)$, $\nabla^{2}$ can be
separated as

$\nabla^{2}=\nabla^{2}_{r}+\frac{1}{r^{2}}\nabla^{2}_{(\theta,\phi)}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial r}\right)-\frac{{\vec{L}}^{2}}{r^{2}},$
(48)

where $\vec{L}=\vec{x}\times\vec{p}$ is the angular momentum generator

$\vec{L}^{2}=-\nabla^{2}_{(\theta,\phi)}=-\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta}-\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial^{2}\phi}.$
(49)

Knowing that $x^{i}\partial_{i}=r\partial_{r}$, equation (47) becomes

$\displaystyle\left[({\xi}\partial_{\xi}-r\partial_{r})^{2}-({\xi}\partial_{\xi}-r\partial_{r})-\frac{R^{2}}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial r}\right)\right.\hskip 85.35826pt$
$\displaystyle\left.+\frac{R^{2}}{r^{2}}\vec{L}^{2}+iRm+{R^{2}m^{2}}\right]\varphi=0.$
(50)

This equation is separable and gives a Sturm-Liouville system. So, we can
consider the following separation scheme

$\varphi=U(\xi,r)\Omega(\theta,\phi),$ (51)

which leads to the following system

$\vec{L}^{2}\Omega(\theta,\phi)-\lambda\Omega(\theta,\phi)=0,\\\ $ (52)
$\displaystyle\left[({\xi}\partial_{\xi}-r\partial_{r})^{2}-({\xi}\partial_{\xi}-r\partial_{r})-\frac{R^{2}}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial r}\right)\right.\hskip 56.9055pt$
$\displaystyle\left.+i{Rm}+{R^{2}m^{2}}+\frac{R^{2}}{r^{2}}\lambda\right]U(\xi,r)$
$\displaystyle=0,$ (53)

where $\lambda$ is a separation constant. The solution of (52) is given for
$\lambda=l(l+1)$ by the spherical harmonic spinors, constituted of the usual
spherical harmonics $Y_{lm}(\theta,\phi)$ and the base spinors $\chi(s_{3})$,

$\Omega(\theta,\phi)\equiv\Omega^{l}_{j,m}(\theta,\phi)=<j,m\mid
l,m^{\prime};\frac{1}{2},s_{3}>Y_{lm^{\prime}}(\theta,\phi)\chi(s_{3}),$ (54)

where $<j,m\mid l,m^{\prime};\frac{1}{2},s_{3}>$ are the Clebsch-Gordon
coefficients. Explicitly, these spherical harmonic spinors are given by [20]

$\displaystyle\Omega^{l}_{j,m}(\theta,\phi)$
$\displaystyle=\frac{1}{\sqrt{2j}}\begin{pmatrix}\sqrt{j+m}\;Y_{l,m-\frac{1}{2}}\\\
\sqrt{j-m}\;Y_{l,m+\frac{1}{2}}\end{pmatrix},\quad{\text{for}}\quad\quad
j=l+\frac{1}{2}$ (55) $\displaystyle\Omega^{l}_{j,m}(\theta,\phi)$
$\displaystyle=\frac{1}{\sqrt{2j+2}}\begin{pmatrix}-\sqrt{j-m+1}\;Y_{l,m-\frac{1}{2}}\\\
\sqrt{j+m+1}\;Y_{l,m+\frac{1}{2}}\end{pmatrix},\quad{\text{for}}j=l-\frac{1}{2}.$
(56)

Concerning equation (53), if we use the following variable change

$\eta={\xi}{r},\hskip 56.9055pt\quad\zeta={\xi},$ (57)

it will take the following separable form

$\displaystyle\frac{1}{R^{2}\zeta^{2}}\left(\zeta^{2}\partial^{2}_{\zeta}+i{Rm}+{R^{2}m^{2}}\right)U_{l}(\zeta,\eta)\hskip
113.81102pt$
$\displaystyle=\left(\frac{\partial^{2}}{\partial\eta^{2}}+\frac{2}{\eta}\frac{\partial}{\partial\eta}-\frac{l(l+1)}{{\eta^{2}}}\right)U_{l}(\zeta,\eta).$
(58)

Thus, if we set $U_{l}(\zeta,\eta)=V(\zeta)Z_{l}(\eta)$ , we can obtain the
following system

$\displaystyle\left(\zeta^{2}\partial^{2}_{\zeta}+{R^{2}\bar{\kappa}^{2}}{\zeta^{2}}+i{Rm}+{R^{2}m^{2}}\right)V(\zeta)$
$\displaystyle=0,$ (59)
$\displaystyle\left[{\eta^{2}}\frac{\partial^{2}}{\partial\eta^{2}}+2{\eta}\frac{\partial}{\partial\eta}+{\bar{\kappa}^{2}}{\eta^{2}}-l(l+1)\right]Z_{l}(\eta)$
$\displaystyle=0,$ (60)

where $\bar{\kappa}^{2}$ is a separation constant. Relation (59) is a Bessel’s
equation. Its solution can be expressed in term of Hankel function of first
kind [21, 22]

$V(\zeta)=\zeta^{\frac{1}{2}}H_{{}_{\nu_{-}}}(R\bar{\kappa}\zeta),$ (61)

where $\nu_{-}=1/2-iRm$. The general solution of Dirac’s equation with
$\nu_{-}$ is a linear combination that contains terms with each of the above
solutions. The other solution with $-\nu_{-}$ is obtained with the same
manner. Relation (60) is also a Bessel’s equation. Its solutions are the
spherical Bessel functions:
$j_{l}(\bar{\kappa}\eta)=j_{l}(\bar{\kappa}{r}{\xi})$,
$y_{l}(\bar{\kappa}\eta)=y_{l}(\bar{\kappa}{r}{\xi})$ and
$h^{1,2}_{l}(\bar{\kappa}\eta)=h^{1,2}_{l}(\bar{\kappa}{r}{\xi})$. For
$|\bar{\kappa}|<0$, there is no solution that is bounded at infinity and
regular at the origin. For $|\bar{\kappa}|>0$, the only solution that is
bounded everywhere is

$Z(\xi,r)=N_{l}j_{l}(\bar{\kappa}{r}{\xi}).$ (62)

where $N_{l}$ stands for a normalization constant. Thus, the physical solution
to (53) is

$U_{l}(\xi,r)=N_{l}\xi^{\frac{1}{2}}H_{{}_{\nu_{-}}}(R\bar{\kappa}\xi)j_{l}(\bar{\kappa}{r}{\xi}).$
(63)

Requiring that this solution be reduced in the limit $R\rightarrow\infty$ to
the usual one of Dirac’s equation in the Minkowski spacetime imposes
$\bar{\kappa}=p$, where $p$ is the four-momentum [23]. It follows that

$U_{l}(\xi,r)=N_{l}\xi^{\frac{1}{2}}H_{{}_{\nu_{-}}}(Rp\xi)\,j_{l}(p{r}{\xi}).$
(64)

It remains to determine the second spinor $\chi$. Introducing $X=p{r}{\xi}$
and using the property

$\vec{\sigma}.\vec{\nabla}f(r)\Omega^{l}_{j,m}=\left[\frac{df(r)}{dr}+\frac{1+\kappa}{r}f(r)\right](\vec{\sigma}.\hat{r})\Omega^{l}_{j,m},$
(65)

where

$\kappa=\mp(j+\frac{1}{2})=\left\\{\begin{array}[]{r@{~}r@{~}l}-(l+1)~{}&\text{for}~{}&j=l+\frac{1}{2}\\\
l~{}&\text{for}~{}&j=l-\frac{1}{2}\end{array}\right.\ \ \ \ \text{and}\ \ \ \
(\vec{\sigma}.\hat{r})\Omega^{\kappa}_{j,m}=-\Omega^{-\kappa}_{j,m},$ (66)

we have

$\displaystyle R\sigma^{i}\partial_{i}\varphi$
$\displaystyle={N_{l}}R\sigma^{i}\partial_{i}\left[j_{l}(X)\Omega^{l}_{j,m}(\theta,\phi)\right]\xi^{\frac{1}{2}}H_{{}_{\nu_{-}}}(Rp\xi)$
$\displaystyle={N_{l}}R\left\\{\left[\frac{dj_{l}(X)}{dr}+\frac{1+\kappa}{r}j_{l}(X)\right](\vec{\sigma}.\hat{r})\Omega^{l}_{j,m}\right\\}\xi^{\frac{1}{2}}H_{{}_{\nu_{-}}}(Rp\xi)$
$\displaystyle=-{N_{l}}R{p}\xi^{\frac{3}{2}}H_{{}_{\nu_{-}}}(Rp\xi)j_{\bar{l}}(X)\Omega^{\bar{l}}_{j,m},$
(67)

where we have used the properties of Bessel’s functions of integer order [21,
22]

$\frac{dj_{l}}{d\varrho}(\varrho)=\frac{l}{\varrho}j_{l}(\varrho)-j_{l+1}(\varrho),\quad\frac{dj_{l}}{d\varrho}(\varrho)=j_{l-1}(\varrho)-\frac{l+1}{\varrho}j_{l}(\varrho),\quad
j_{-l}=(-1)^{l}j_{l},$

and we have set $\bar{l}=2j-l$. With the use of (67) and of the fact that
$x^{i}\partial_{i}=r\partial_{r}$, equation (45) gives

$\big{(}{\xi}\partial_{\xi}-r\partial_{r}+i{Rm}\big{)}\chi=-{N_{l}}R{p}\xi^{\frac{3}{2}}H_{{}_{\nu_{-}}}(Rp\xi)j_{\bar{l}}(X)\Omega^{\bar{l}}_{j,m}.$
(68)

Introducing $\tilde{\chi}=\xi^{-1/2}\chi$, the last equation takes the form

$\left({\xi}\partial_{\xi}-r\partial_{r}+i{Rm}+\frac{1}{2}\right)\tilde{\chi}=-{N_{l}}R{p}\xi
H_{{}_{\nu_{-}}}(Rp\xi)j_{\bar{l}}(X)\Omega^{\bar{l}}_{j,m}.$ (69)

Setting $z=Rp\xi$, we get to the following equation:

$\left(z\partial_{z}-r\partial_{r}+i{Rm}+\frac{1}{2}\right)\tilde{\chi}=-{N_{l}}zH_{{}_{\frac{1}{2}-iRm}}(Rp\xi)j_{\bar{l}}(X)\Omega^{\bar{l}}_{j,m}.$
(70)

As $X=p{r}{\xi}$, it is easy to check that
$({\xi}\partial_{\xi}-{r}\partial_{r})j_{l}(X)=0$. It follows that by using
the recurrence relation for Bessel’s functions

$z\partial_{z}\mathscr{C}_{\nu}(z)+\nu\mathscr{C}_{\nu}(z)=z\mathscr{C}_{\nu-1}(z),$
(71)

where $\mathscr{C}\equiv J,Y,H$, and the properties of Hankel’s functions of
first kind

${H}^{(1)}_{-\nu}(z)=e^{i\pi\nu}H^{(1)}_{\nu}(z),\quad{H}^{(2)}_{-\nu}(z)=e^{-i\pi\nu}H^{(2)}_{\nu}(z),$
(72)

the solution for the spinor $\chi$ can be obtained as

$\chi=-{N_{l}}e^{-i\pi{\nu_{-}}}H^{(1)}_{{}_{\nu_{+}}}(Rp\xi)j_{\bar{l}}(pr\xi)\Omega^{\bar{l}}_{j,m},$
(73)

where $\nu_{+}=1/2+iRm$. Then, the solution for Dirac’s equation in its
reduced form (40) can be written as

$\tilde{\Psi}={N_{l}}\xi^{\frac{1}{2}}e^{-\frac{\pi}{2}Rm}\begin{pmatrix}e^{\frac{\pi}{2}Rm}H^{(1)}_{{}_{\nu_{-}}}(Rp\xi)j_{l}(p{r}{\xi})\Omega^{l}_{j,m}(\theta,\phi)\\\
ie^{-\frac{\pi}{2}Rm}H^{(1)}_{{}_{\nu_{+}}}(Rp\xi)j_{\bar{l}}(p{r}{\xi})\Omega^{\bar{l}}_{j,m}(\theta,\phi)\end{pmatrix}.$
(74)

Finally, one can write the solution of the Dirac equation, relation (39), as

${\Psi}_{S}=\xi^{\frac{3}{2}}\tilde{\Psi}={N_{l}}\xi^{2}e^{-\frac{\pi}{2}Rm}\begin{pmatrix}e^{\frac{\pi}{2}Rm}H^{(1)}_{{}_{\nu_{-}}}(Rp\xi)j_{l}(p{r}{\xi})\Omega^{l}_{j,m}(\theta,\phi)\\\
ie^{-\frac{\pi}{2}Rm}H^{(1)}_{{}_{\nu_{+}}}(Rp\xi)j_{\bar{l}}(p{r}{\xi})\Omega^{\bar{l}}_{j,m}(\theta,\phi)\end{pmatrix}.$
(75)

Let us now to determine the normalization constant ${N}_{l}$ by using the
condition

$\int_{\Sigma}d^{3}x\Psi^{\dagger}(x)\Psi(x)=1,$ (76)

where $d^{3}x=r^{2}drd\Omega$ and the integration must be done over the
spacelike hypersurface $\Sigma\equiv x^{0}=cst$. For more details, see
Appendix B. The standard normalization condition of the spherical Bessel’s
function being [24]

$\int^{\infty}_{0}r^{2}j_{l}(kr)j_{l^{\prime}}(k^{\prime}r)dr=\frac{\pi}{2k^{2}}\delta(k-k^{\prime})\delta_{ll^{\prime}},$
(77)

for $k={p}{\xi}$, we have

$\int^{\infty}_{0}r^{2}j_{l}({p}{\xi}r)j_{l^{\prime}}({p}{\xi}^{\prime}r)dr=\frac{\pi}{2{p^{2}}{\xi}^{3}}\delta(p-p^{\prime})\delta_{ll^{\prime}},$
(78)

and since the spherical harmonic spinors are normalized with respect to the
relation

$\int(\Omega^{*})^{l^{\prime}}_{j^{\prime},m^{\prime}}\Omega^{l}_{j,m}d\Omega=\delta_{jj^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}},$
(79)

it follows that

$\displaystyle|N_{l}|^{2}\frac{\pi}{2{p^{2}}{\xi}^{3}}\xi^{4}e^{-\pi
Rm}\left[e^{-\pi
Rm}(H^{(1)}_{\nu_{+}})^{*}(Rp\xi)H^{(1)}_{\nu_{+}}(Rp\xi)+\right.\hskip
56.9055pt$ $\displaystyle\left.e^{\pi
Rm}(H^{(1)}_{\nu_{-}})^{*}(Rp\xi)H^{(1)}_{\nu_{-}}(Rp\xi)\right]=1.$ (80)

By using the following properties of the Hankel functions of first kind

$(H^{(1,2)}_{\nu_{\pm}})^{*}=H^{(2,1)}_{\nu_{\mp}},\quad\quad\nu_{\pm}=\frac{1}{2}\pm
k,$ (81) $e^{\pm\pi k}H^{(1)}_{\nu_{\mp}}(z)H^{(2)}_{\nu_{\pm}}(z)+e^{\mp\pi
k}H^{(1)}_{\nu_{\pm}}(z)H^{(2)}_{\nu_{\mp}}(z)=\frac{4}{\pi z},$ (82)

equation (80) allows to deduce that

$|N_{l}|=\frac{{R^{\frac{1}{2}}{p}^{\frac{3}{2}}}}{2}e^{\frac{1}{2}\pi Rm}.$
(83)

Finally, the normalized wave function reads

${\Psi}_{S}=\frac{p}{2}\sqrt{\frac{\pi R}{2\xi
r}}\xi^{2}\begin{pmatrix}e^{\frac{\pi}{2}Rm}H^{(1)}_{{}_{\nu_{-}}}(Rp\xi)j_{l}(p{r}{\xi})\Omega^{l}_{j,m}(\theta,\phi)\\\
ie^{-\frac{\pi}{2}Rm}H^{(1)}_{{}_{\nu_{+}}}(Rp\xi)j_{\bar{l}}(p{r}{\xi})\Omega^{\bar{l}}_{j,m}(\theta,\phi).\end{pmatrix}$
(84)

Using the operator $W$ given by (35), one can easily obtain the free Dirac
equation solution ${\Psi_{NP}}$ in the natural picture

${\Psi_{NP}}=W^{-1}{\Psi}_{S}=\frac{p}{2}\sqrt{\frac{\pi
R}{2r}}\xi^{2}\begin{pmatrix}e^{\frac{\pi}{2}Rm}H^{(1)}_{{}_{\nu_{-}}}(Rp\xi)j_{l}(p{r})\Omega^{l}_{j,m}(\theta,\phi)\\\
ie^{-\frac{\pi}{2}Rm}H^{(1)}_{{}_{\nu_{+}}}(Rp\xi)j_{\bar{l}}(p{r})\Omega^{\bar{l}}_{j,m}(\theta,\phi).\end{pmatrix}$
(85)

If we perform a transition to the moving chart $\\{\tau,\vec{x}\\}$ with the
standard flat metric $ds^{2}=c^{2}d\tau^{2}-e^{2\omega\tau}d\vec{x}^{2}$ with
$\xi=e^{-\omega\tau}$ and $\omega=1/R$, one can easily check that
${\Psi_{NP}}$ is exactly the same solution found by Shishkin [18] and by
Cotăescu & al. [19].

##  5 Conclusion

In the present work, we constructed the free Dirac equation in the
$R$-Minkowski spacetime. After using a certain realization of the $R$-Poincaré
algebra, it turned out that the obtained equation is exactly the Dirac
equation in the conformally flat de Sitter spacetime. This is a further proof
of the correspondence between the $R$-Minkowski and the de Sitter space such
that the physics of $R$-Poincaré algebra is the same as in the de Sitter
relativity. So, this correspondence could be used to construct well-defined
physical observables in the de Sitter spacetime.

We also presented a new method for solving Dirac equation in the conformally
flat patch of de Sitter spacetime within the Schrödinger picture. The latter
was introduced by Cotăescu to investigate Dirac and Klein-Gordon equations in
the context of de Sitter spacetime.

### Appendix A: Squared Dirac equation in curved spacetime

In a curved spacetime, Dirac’s equation is given by [25]

$\left(i\,\gamma^{\mu}\,D_{\mu}-\frac{mc}{\hbar}\right)\psi(x)=0,$ (86)

where $D_{\mu}$ is the spinor covariant derivative,

$D_{\mu}\psi=(\partial_{\mu}+\Omega_{\mu})\psi,$ (87)

and

$\Omega_{\mu}(x)\equiv-\frac{i}{4}\,\omega_{ab\mu}(x)\,\sigma^{ab}=\frac{1}{8}\,\omega_{ab\mu}(x)\,[\,\gamma^{a},\,\gamma^{b}\,]$
(88) $\omega^{a}_{\hskip 3.5ptb\mu}=e_{\nu}^{\hskip
3.5pta}\,\left(\partial_{\mu}\,e^{\hskip 3.5pt\nu}_{b}+\,e_{b}^{\hskip
3.5pt\sigma}\,\Gamma^{\nu}_{\sigma\mu}\right),$ (89)

are respectively the Fock-Ivanenko coefficients[26] and the spin connection,
$\Gamma^{\nu}_{\sigma\mu}$ being the Christoffel symbols. It follows that the
square of Dirac’s equation [16]

$\bigl{(}i\gamma^{\mu}D_{\mu}+mc/\hbar\bigr{)}\,\bigl{(}i\gamma^{\nu}D_{\nu}-mc/\hbar\bigr{)}\psi=0\,$
(90)

can be rewritten in the form

$\left[g^{\mu\nu}D_{\mu}D_{\nu}-\frac{1}{2}\sigma^{\mu\nu}K_{\mu\nu}+(mc/\hbar)^{2}\right]\psi=0,$
(91)

where

$K_{\mu\nu}\equiv\frac{1}{2}(D_{\nu}\,D_{\mu}-D_{\mu}\,D_{\nu})=\partial_{\nu}\Omega_{\mu}-\partial_{\mu}\Omega_{\nu}+[\Omega_{\nu},\Omega_{\mu}]$
(92)

is the spin curvature. Equation (91) is the generally covariant extension of
the Pauli-Schrödinger equation that describes spin $1/2$ particles in a
gravitational field [16, 17]. The first term contains, in addition to the
Klein-Gordon operator, terms involving the Fock-Ivanenko coefficients
$\Omega_{\mu}$

$\displaystyle g^{\mu\nu}D_{\mu}D_{\nu}\psi$
$\displaystyle=(g^{\mu\nu}\partial_{\mu}\partial_{\nu}-g^{\mu\nu}\Gamma^{\lambda}_{\mu\nu}\partial_{\lambda})\psi+g^{\mu\nu}\left[(D_{\mu}\Omega_{\nu})+2\Omega_{\nu}\partial_{\mu}\right]\psi$
$\displaystyle=\Box_{KG}\psi+g^{\mu\nu}\left[(D_{\mu}\Omega_{\nu})+2\Omega_{\nu}\partial_{\mu}\right]\psi.$
(93)

In our case, the $R$-Minkowski space corresponds to the de Sitter spacetime
given by the metric

$ds^{2}=\displaystyle\frac{1}{(1-x^{0}/R)^{2}}\left[(dx^{0})^{2}-d\vec{x}^{2}\right]=\displaystyle\frac{1}{\xi^{2}}\left[(dx^{0})^{2}-d\vec{x}^{2}\right].$
(94)

In the chart with $x^{0}\in(-\infty,0],$ the tetrad field is given by $e_{\
a}^{\mu}=\xi\,\delta_{\ a}^{\mu}$ and the spin connection by $\omega_{\mu
ab}=\displaystyle{{(R\xi)^{-1}}}\left[\eta_{\mu b}\delta_{a}^{0}-\eta_{\mu
a}\delta_{b}^{0}\right]$. Then, the covariant derivative reads

$D_{\mu}=\partial_{\mu}+\displaystyle{\frac{1}{4R\xi}}\eta_{\mu
a}\left[\gamma^{0},\gamma^{a}\right],$ (95)

and a simple calculus gives for the sum of the terms involving the Fock-
Ivanenko coefficients $\Omega$

$\displaystyle
g^{\mu\nu}\left[(D_{\mu}\Omega_{\nu})+2\Omega_{\nu}\partial_{\mu}\right]$
$\displaystyle=g^{\mu\nu}\left[(\partial_{\mu}\Omega_{\nu})-\Gamma^{\sigma}_{\nu\mu}\Omega_{\sigma}+\Omega_{\mu}\Omega_{\nu}+2\Omega_{\nu}\partial_{\mu}\right]$
$\displaystyle=-\frac{1}{R}\xi\gamma^{0}\gamma^{i}\partial_{i}-\frac{3}{4R^{2}}.$
(96)

On the other hand, it is known that

$-\frac{1}{2}\sigma^{\mu\nu}K_{\mu\nu}=\frac{\mathcal{R}}{4},$ (97)

where $\mathcal{R}$ is the Ricci scalar and that in the de Sitter space, we
have $\mathcal{R}=12/R^{2}$. Taking into account equations (93), (96) and
(97), relation (91) can be rewritten as

$\left[\Box_{KG}-\frac{1}{R}\xi\gamma^{0}\gamma^{i}\partial_{i}+\frac{9}{4R^{2}}+(mc/\hbar)^{2}\right]\psi=0.$
(98)

Comparing to (29), it is clear that expressions of terms which make the square
of Dirac’s operator different from the one of Klein-Gordon are established.

### Appendix B: Remark on the integration measure

In the natural representation (NP), the normalization of the wave function,
expressed through the canonical variable $X^{\mu}=x^{\mu}/\xi$, is given by
the usual condition in special relativity

$\displaystyle<{\Psi}_{NP},{\Psi}_{NP}>$
$\displaystyle=\int_{\Sigma}d^{3}X\bar{\Psi}_{NP}(X)\gamma^{0}\Psi_{NP}(X)$
$\displaystyle=\int_{\Sigma}d^{3}X\Psi^{\dagger}_{NP}(X)\Psi_{NP}(X)=1,$ (99)

where the integration must be done over the spacelike hypersurface $\Sigma$,
determined by $X^{0}=cst$ meaning that $\xi=cst$. So, the spacelike
hypersurface at a constant time in the $R$-Minkowski spacetime is given by

$d^{3}X|_{T=cst}=dX\wedge dY\wedge dZ\,|_{t=cst}=d^{3}x\xi^{-3}.$ (100)

Thus, the normalization condition in the $R$-Minkowski spacetime is given by

$<{\Psi}_{NP},{\Psi}_{NP}>=\int_{\Sigma}d^{3}x\xi^{-3}\Psi^{\dagger}_{NP}(x)\Psi_{NP}(x)=1,$
(101)

where $d^{3}x=r^{2}drd\Omega$. Taking into account relation (33), one can
check that in the $R$-Minkowski spacetime the wave function,
$\Psi\equiv\Psi_{SP}$, in the Schrödinger representation, is normalized with
respect to the usual condition of special relativity

$\displaystyle<{\Psi}_{SP},{\Psi}_{SP}>$
$\displaystyle=<{\Psi}_{NP},{\Psi}_{NP}>$
$\displaystyle=\int_{\Sigma}d^{3}x\xi^{-3}\Psi^{\dagger}_{NP}(x)\Psi_{NP}(x)$
$\displaystyle=\int_{\Sigma}d^{3}x\Psi^{\dagger}_{SP}(x)\Psi_{SP}(x)=1.$ (102)

## References

  * [1] G. Amelino Camelia,  Nature 418, 34 (2002), arXiv e-print: gr-qc/0207049.
  * [2] G. Amelino Camelia,  Phys. Lett. B510, 255 (2001), arXiv e-print: hep-th/0012238.
  * [3] J. Magueijo and L. Smolin,  Phys. Rev. Lett. 88, 190403 (2002), arXiv e-print: hep-th/0112090.
  * [4] J. Magueijo and L. Smolin,  Phys. Rev.  D67, 044017 (2003), arXiv e-print: gr-qc/0207085.
  * [5] V. Fock,  The Theory of Space, Time and Gravitation, Pergamon Press, Oxford, London, New York, Paris (1964).
  * [6] S. Ghosh and P. Pal, Phys. Rev. D75, 105021 (2007), arXiv e-print: hep-th/0702159.
  * [7] A. Bouda and T. Foughali, Mod. Phys. Lett. A27, 1250036 (2012), arXiv e-print: 1204.6397.
  * [8] T. Foughali and A. Bouda, Can. J. Phys. 93, 734 (2015), arXiv e-print: 1605.01943.
  * [9] J. A. Magpantay, Phys. Rev. D84, 024016 (2011), arXiv e-print: 1011.3888.
  * [10] P. Gosselin, A. Bérard, H. Mohrbach and S. Ghosh,  Phys. Lett. B660, 267 (2008), arXiv e-print: 0709.0579.
  * [11] Z. Belhadi, F. Menas, A. Bérard, P. Gosselin and H. Mohrbach,  Int. J. Mod. Phys. A27, 1250031 (2012), arXiv e-print: 1106.1302.
  * [12] E. Harikumar, M. Sivakumar and N. Srinivas,  Mod. Phys. Lett. A26, 1103 (2011), arXiv e-print: 0910.5778.
  * [13] V. Balasubramanian, J. de Boer, and D. Minic, Phys.Rev. D65, 123508 (2002), arXiv e-print: hep-th/0110108.
  * [14] M. Spradlin, A. Strominger and A. Volovich, arXiv e-print: hep-th/0110007.
  * [15] I. I. Cotăescu, Mod. Phys. Lett. A22, 2965 (2007).
  * [16] T. Chapman and O. Cerceau, Am. J. Phys. 52, 994 (1984).
  * [17] P. M. Alsing, J. C. Evans and K. K. Nandi, Gen. Rel. Grav. 33, 1459 (2001), arXiv e-print: gr-qc/0010065.
  * [18] G. V. Shishkin, Class. Quant. Grav. 8, 175 (1991).
  * [19] I. I. Cotăescu, R. Racoceanu and C. Crucean, Mod. Phys. Lett. A21, 1313 (2006), arXiv e-print: hep-th/0602077.
  * [20] W. Greiner, Relativistic Quantum Mechanics - Wave Equations, 3rd edition (Springer 2000).
  * [21] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover (1964).
  * [22] I. S. Gradshteyn and I. M. Ryzhik,  Tables of Integrals, Series, and Products, Corrected and Enlarged Edition, Academic Press, Inc, New York (1980).
  * [23] A. Wachter, Relativistic Quantum Mechanics, Springer (2011).
  * [24] V. Berestetski, E. Lifchitz and L. Pitayevski, Théorie Quantique Relativiste, Mir, Moscow (1972).
  * [25] N. D. Birrel and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge University Press (1982).
  * [26] L. Parker and D.J. Toms,  Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity, Cambridge University Press (2009).