11institutetext: School of Electrical and Computer Engineering, College of
Engineering, University of Tehran, Tehran, Iran
<EMAIL_ADDRESS>
Department of Computer Science, University of Rochester, Rochester, USA
<EMAIL_ADDRESS>
Department of Computer science, Emory university, Atlanta, USA
<EMAIL_ADDRESS>

# Using BERT Encoding and Sentence-Level Language Model for Sentence Ordering

Melika Golestani 11 Seyedeh Zahra Razavi 22 Zeinab Borhanifard 11 Farnaz
Tahmasebian 33 Hesham Faili 11

###### Abstract

Discovering the logical sequence of events is one of the cornerstones in
Natural Language Understanding. One approach to learn the sequence of events
is to study the order of sentences in a coherent text. Sentence ordering can
be applied in various tasks such as retrieval-based Question Answering,
document summarization, storytelling, text generation, and dialogue systems.
Furthermore, we can learn to model text coherence by learning how to order a
set of shuffled sentences. Previous research has relied on RNN, LSTM, and
BiLSTM architecture for learning text language models. However, these networks
have performed poorly due to the lack of attention mechanisms. We propose an
algorithm for sentence ordering in a corpus of short stories. Our proposed
method uses a language model based on Universal Transformers (UT) that
captures sentences’ dependencies by employing an attention mechanism. Our
method improves the previous state-of-the-art in terms of Perfect Match Ratio
(PMR) score in the ROCStories dataset, a corpus of nearly 100K short human-
made stories. The proposed model includes three components: Sentence Encoder,
Language Model, and Sentence Arrangement with Brute Force Search. The first
component generates sentence embeddings using SBERT-WK pre-trained model fine-
tuned on the ROCStories data. Then a Universal Transformer network generates a
sentence-level language model. For decoding, the network generates a candidate
sentence as the following sentence of the current sentence. We use cosine
similarity as a scoring function to assign scores to the candidate embedding
and the embeddings of other sentences in the shuffled set. Then a Brute Force
Search is employed to maximize the sum of similarities between pairs of
consecutive sentences.

###### Keywords:

Sentence Ordering, Event Sequencing, Story Reordering, BERT Pretrained Model,
Sentences-Level Language Model

##  1 Introduction

Modeling text coherence and logical sequences of events is a fundamental
problem in natural language processing [25]. Coherence represents the logical
connections between the words, sentences and events in a text. An event is
generally considered to be the verb of a sentence along with its constellation
of arguments, such as subject and object [10]. In a coherent narrative text,
events follow a logical order, which makes it possible for a reader to make
sense of the text. In NLP tasks, “Sentence Ordering” models are suggested to
learn high-level structure that causes sentences to appear in a specific order
in human-authored texts and by learning to order sentences we can model text
coherence[25].

“Sentence ordering” refers to organizing a given shuffled set of sentences
into a coherent order [25]. Sentence ordering assists in modeling coherence in
text, which in turn can be applied in a variety of tasks such as multi-
document summarization [2, 21], story understanding [31], and retrieval based
QA systems [25]. A key factor to to solve the problem of sentence ordering is
to find a coherent sequence of events appeared in the input sentences. In
natural language processing, events are often meant to be the verbs of
sentences and their dependencies, such as the subject and the object [29]. A
number of studies have proposed methods to order events based on their
temporal and causal relations [31], or the semantic roles [9]. The researchers
developed a Skip-thought [18] encoder-decoder model utilizing an RNN. It takes
a sentence and encodes it into a constant-length vector, followed by previous
and subsequent sentences. Therefore, all input is first encrypted using a
hidden network vector; then its output is decrypted from that vector. Pichotta
& Mooney [34] replaced the RNN with an LSTM, and proposed a sentence-level
language model to predict the next sentence for the script inference task.
Nevertheless, skip-thought’s architecture’s simplicity results in a lower
quality of creating sentences embeddings compared to many newer methods and
networks like BERT [12]. The system uses an RNN-based encoder-decoder
architecture, whereas SBERT-WK [39] uses a Transformer-based architecture and
an attention mechanism to encode the sentences into vectors.

According to [9], a statistical script system model an event sequence as a
probabilistic model and infer additional events from a document events. This
system teaches the linguistic model of events; despite this, some critical
information in the sentence can be ignored because only verbs and their
dependents are used. Thus, the clue words (adverbs such as “then”, “before”,
and “after”) are omitted.

In this paper, we propose a sentence-level language model to solve the problem
of sentence ordering. Language models have been popular for years at the word
and letter levels [41, 19]. LSTMs [16] and RNNs [26] have been vastly used for
teaching language models at the word and letter levels is beneficial [28, 17,
17, 38, 5]. These models take a single word or character as input at time t,
update the hidden mode vector, and predict the next word or character at time
t+1. Kiros used a sentence level language model based on the RNN [18].
Pichotta in [34] used LSTM-based sentence-level models for script inference
due to the problem of RNNs, the vanishing gradients. RNNs and LSTMs encrypt
information about one sentence and decode information about the following
sentence. Despite this, these networks not use a self-attention mechanism[33]
for encoding sentences, and some information is lost.

We use a sentence level language model based on the universal transformers
(UT) [11]. A generalization of the Transformer model, the UT model is a
parallel-in-time recurrent sequence model. The UT model combines two aspects
of feed-forward models: parallelism and global receptive field. An additional
feature is the position-based halting mechanism. Using a self-attention
mechanism, the Universal Transformer refines its representations for all
sentences in the sequence in parallel.

Our model involves three components: the first component encodes sentences;
the second one uses a neural network to teach the language model at the
sentence level by universal transformers (UT) and to predict the next
sentence. The last component uses the cosine similarity and Brute Force search
to order sentences. We aim to design a model that can capture the sequence of
events by arranging a set of sentences.

The model we propose is designed to organize shuffled sentences of short
stories. By using this method, we encode sentences by the SBERT-WK model,
which causes us to pay attention to every word within a sentence equal to its
importance and transform the sentence into a vector. Thus the embedding
contains the clue words information. To capture the relationship between
Sentences, We use a UT-based language model, which predicts the next sentence
of the current sentence. Finally, we use a Brute Force search, using the
scoring function based on cosine similarity, which is scored between the
candidate sentence and other sentences, to maximize the sum of similarities
between two consecutive sentences. As dataset, we use ROCStories [30], which
includes around 100K short 5-sentence stories, written by human turkers. The
gold data follows the order similar to the original order in the stories.

The main applications of sentence ordering can be mentioned as: extractive and
multi-document text summarization [2], retrieval based QA systems [25],
storytelling [30], text coherence modeling [25], discourse coherence [13], and
text generation systems [7].

##  2 Related Work

Understanding relations between sentences has become increasingly important
for various NLP tasks, such as multi-document summarization [2], text
generation [7], and text coherence modeling [25]. Moreover, learning the order
of sentences can help in modeling text coherence [25]. In sentence ordering,
the goal is to arrange a set of unordered sentences in a cohesive order.
Ordering models aim to identify patterns resulting sentences to appear in a
specific order in a coherent text. Several previous studies have addressed the
task of sentence ordering for a set of data such as news articles [2, 21, 6].

Previous methods suggested on the task of sentence ordering fall into two
categories: traditional approaches, and deep learning-based approaches.
Traditional approaches to coherence modeling and sentence ordering often apply
probabilistic models [21]. Barzilay utilized content models to represent
topics as HMM states [4], then employed hand-crafted linguistic features,
including Entity Grid, to model the document structure [3]. [30] suggested
using n-gram overlapping to pick a final sentence from two human-made options
of the ROCStories stories. In [35], a similar method is used to arrange the
set of unordered sentences of the stories. As the next sentence, they pick the
sentence with the most n-gram overlaps with the current sentence. They
measured an overlap up to 4-gram using Smoothed-BLEU [24].

In recent years, researchers used neural approaches to solve sentence ordering
tasks. [1] proposed SkipThought + Pairwise model. The method involves
combining two points identified from the unary embedding of sentences without
considering context and a pairwise model of sentences based on their relative
composition in context. However, as SkipThought model [18] was applied to map
sentences into vector space, some information is missing. Another method
proposed in [23] is a productive model that is called Seq2Seq + Pairwise. A
Seq2Seq model based on an encoder-decoder architecture is used to predict the
following sentence, having one. This way, the model learns a probabilities
distributions over sentences.

Gong employed an end-to-end sentence ordering method, LSTM + PtrNet[14]. After
getting encrypted and decoded by LSTM models, pointer networks are being used
to arrange sentences. Another method proposed in [25], called LSTM + Set2Seq,
uses LSTM and attention mechanism to encode sentences and learn a
representation of context. Then a pointer network is applied to ranks
sentences. [40] proposed Hierarchical attention networks (HAN). The idea is to
capture the information of clue words to learn the dependency between
sentences. The system uses a word encoder with a BiLSTM architecture, an
attention layer that creates sentence embedding, and an attention mechanism to
encode and decode a group of sentences together. Although LSTM and BiLSTM-
based models show improvement, they are less efficient comparing to a BERT-
based encoder model. This is mostly because of the fact that BiLSTM models do
not look at both directions of a word simultaneously. In contrast, BERT is
trained to learn both left and right positions at the same time.

[27] proposed a CBoW model that generate sentence embedding using the average
of the word embedding vectors of the words that compose the sentence. This
embedding is used in [35] to suggest a baseline, where cosine similarity
between embeddings of pairs of sentences is used to order the pairs. This has
two essential defects: it does not include many sentence information such as
clue words and there is no way for modelling sentences’ dependencies.

[35] presents sentence correlation measure (SCM) for sentence ordering. This
measure has three main components:

  1. 1.

A sentence encoding component based on the pretrained SBERT-WK [39];

  2. 2.

A scoring component based on the cosine similarity;

  3. 3.

A ranking component based on a Brute Force search.

None of the above components need access to training data which provides an
advantage where limited data is available[35]. The method does not work as
well as HAN or LSTM + Set2Seq when enough training data is available, since
dependencies between sentences are not captured.

[20] proposed RankTxNet ListMLE, which is a pointer-based model. This method
relies on a pre-trained BERT model for encoding sentences and a self-attention
based transformer network for encoding paragraphs. To predict a relative score
for each sentence, they use a feed-forward network as a decoder which
determines each sentence’s position in a paragraph.

Our proposed model has some similarity with the methods presented in [14] and
[25] in using neural networks for for predicting sentence ordering in a
pairwise manner. However, we employ a BERT-based model for sentence encoding,
in contrast to the LSTM-based model they used. Moreover, by applying universal
transformers we encode a group of sentences and learn a language model.

##  3 Methodology

###  3.1 Task Formulation

The sentence ordering takes the story S as input, which its sentences are
probably unsorted:

$S:{s_{1},s_{2},...,s_{n}}$ (1)

Where n is the length of the sequence or the number of sentences in each
sequence. It outputs a permutation of the sentences like o so that o is equal
to $o^{*}$:

$s(o_{1}^{*})>s(o_{2}^{*})>\ldots>s(o_{5}^{*}),$ (2)

where $o^{*}$ is to the order of the sentences in the gold data.

###  3.2 Sentence-Level Language Model for Sentence Ordering (SLM)

We proposed Sentence-level Language Model (SLM) for Sentence Ordering compose
of Sentence Encoder, Story Encoder, and Sentence Organizer. The architecture
of the proposed model is shown in Figure1. Sentence encoder gets sentence as
input and encoded it into a vector using a fine-tuned pre-trained SBERT-
WK[39]. The embedding pays more attention to the sentence’s crucial parts,
such as the verbs and clue words. Furthermore, stop words are omitted. Then,
Story Encoder takes the sentence encoder as an input and learns the sentence
level language model using an encoder-decoder architecture based on universal
transformers (UT) [11]. UT couples parallelism with the global receptive field
in a feed-forward model. The UT model also includes position-based halting
[11]. The UT-based encoder-decoder component corrects vector representations
for each position (sentence) in parallel by combining information from
different positions using self-attention and applying a repetitive transfer
function [11]. Hence dependencies between the sentences are captured. The
vector is learned from the hidden state and then decodes the vector and
indicates the next sentence’s candidate.

What matters at runtime is that during training, information are transferred
to all nodes at time $t+1$ from all nodes at time $t$. Thus, parallelism is
created, which allows all tokens processed at the same time [11].

In the Sentence Organizer, we can rate each sentence and the candidate based
on their similarity. We use the cosine similarity as the scoring function. The
arrangement is made using a Brute Force search in all state spaces to maximize
the sum of the cosine similarity between two consecutive sentences like the
sentence ordering component in [35], as follow:

$\sum_{i=1,\ldots,4}S\left(s(o_{i}),s(o_{i+1}\right)=\max\sum_{\begin{subarray}{c}i,j=1,\ldots,5\\\
i\neq j\end{subarray}}S(s^{\prime}_{i},s_{j}),$ (3)

where $S(s^{\prime}_{i},s_{j})$ represents the cosine similarity between
$s^{\prime}_{i}$ and $s_{j}$, and $s^{\prime}_{i}$ is the candidate for the
next sentence of the $i$-th sentence, represented with $s_{i}$. $s(o_{i})$
represents the $i$-th sentence in the order of output, and $s_{i}$ is the
$i$-th sentence in the non-ordered permutation of sentences or the input’s
order.

Figure 1: Showing the SLM in the abstract. During model training, sentence
ordering is given to the network correctly to learn and model the
relationships and dependencies between the sentences. During testing, input
sentences are shuffled. Where $w_{j_{i}}$ is the jth word in the ith sentence
and $s(o_{i})$ is the ith sentence in output order. The goal is to output an
s(o) equal to $s(o^{*})$ (this represents the gold data ordering).

Our model is trained again using the universal sentence encoder (USE)
embeddings [8] to compare them to SBERT-WK embeddings, and additionally, we
employ the BiLSTM [36] network to learn the LM. Also, we apply the nearest
neighbor search, which is a greedy algorithm. The NN search calculations for
sentence ordering are presented in [35]. Consequently, we train, evaluate, and
test nine different models, including the following:

  * •

Fine-tuned pretrained SBERT-WK + UT + BFS,

  * •

Pretrained SBERT-WK + UT + BFS,

  * •

Pretrained SBERT-WK + BiLSTM + BFS,

  * •

Pretrained SBERT-WK + UT + NN,

  * •

Pretrained SBERT-WK + BiLSTM + NN,

  * •

USE + UT + BFS,

  * •

USE + BiLSTM + BFS,

  * •

USE + UT + NN,

  * •

USE + BiLSTM + NN,

So we can assess the effect of change on each component.

##  4 Experiment

###  4.1 Dataset

We used ROCStories dataset (a commonsense story dataset) [30]. The dataset
contains 98,162 five-sentence stories with an average word count of 50 words.
3,742 of stories have two choices as the fifth sentence or the final sentence.
The corpus has been presented for a shared task called LSDSem [32] where
models are supposed to choose the correct ending to a four-sentence story. All
the stories and options are generated by human. This dataset has some
essential characteristics that make it a fit for our task of learning
sequences of events: ROCStories contains causal and temporal relationships
among daily events. This makes it possible to learn the narrative structure of
a wide range of events. The dataset contains a collection of daily non-
fictional short stories suitable for the training of coherent text models.

###  4.2 Baselines and competitors

The proposed method is compared with five baselines as follows: Sentence
n-gram overlap [35], SkipThought + Pairwise [1], Seq2Seq + Pairwise [23],
Continues Bag of Words (CBoW) [35], Sentence Correlation Measure (SCM) [35].
The competitors methods follow as: Hierarchical Attention Networks (HAN) [40],
LSTM+PtrNet [14], LSTM+Set2Seq [25], and RankTxNet ListMLE [20].

###  4.3 Metrics

The metrics we use to evaluate story ordering outputs are “Kendal’s Tau” and
“PMR”, introduced below.

  1. 1.

Kendall’s Tau ($tau$)

Kendal’s Tau measures the quality of arrangements by Equation (4),

$\tau=1-((2\cdot\mathit{number\>of\>inversions}))/(N\cdot(N-1)/2)$ (4)

where $N$ represents the sequence length, and $number\>of\>inversions$ is
equal to how many binaries’ relative order is wrongly predicted. This measure
for sentence ordering is correlated with human judgment, according to [22].

The value of this criterion is in the range $[-1,1]$; the lower limit
indicates the worst case, and $\tau=1$ when the predicted order equals to the
order in the gold data are the same.

  2. 2.

Perfect Match Ratio (PMR)

PMR defined as following, calculates the ratio of exactly matching sentence
orders without penalizing incorrect ones.

$\mathrm{PMR}=((\mathit{\\#\>of\>Correct\>Pairs}))/(N\cdot(N-1)/2)$ (5)

where $N$ represents the sequence length. One can see that the PMR is always
in the range of 0 and 1, where PMR = 1 indicates the predicted order is
exactly the same as the gold order, and a PMR = 0 means that the predicted
order is precisely the contrary of the gold order.

###  4.4 Results and Analysis

As mentioned in section 3, SLM has three components. First, a sentence
encoder, which is designed based on SBERT-WK. Second, a story encoder, which
trains a sentence-level language model and learns the dependencies between the
sentences. In this component, the UT network is used. The third component is
sentences’ organizer which calculates the cosine similarity between sentence
embedding vectors and employs a Brute Force search to maximize the sum of
similarities among consecutive sentences.

In addition to SLM, we trained and tested nine other models by replacing each
component with other algorithms or architectures to select the best algorithm
and compare SLM with them. These nine models were constructed by replacing
SBERT-WK encoding with USE vectors, the UT network with a BiLSTM one, and the
BFS with a nearest neighbor search.

The parameters of the SLM are given in Table 1. The BERT vectors have 768
dimensions, while the USE ones have 512 dimensions. Following previous works
[25, 40], we randomly split the dataset into training (80%), test (10%), and
validation (10%) sets, where they contain 392645, 49080, and 49080 sentences
respectively.

Table 1: Parameters of the SLM. SLM encodes sentences with SBERT-WK embeddings and learns a language model with the UT network. Also, the USE embeddings with d$=$512 instead of the SBERT-WK ones are used to allow for a better choice. So hidden layer size of that is 4$\times$ 512. Method Components | Tau
---|---
Initial Learning Rate ($\alpha$) | 0.5
Regularization ($\lambda$) | $10^{-5}$
size of Embedding vectors (d) | 768
Hidden Layer size (h) | 4$\times$768

Based on table 2, SBERT-WK outperforms USE due to its better architecture for
sentence encoding. In all cases, UT is superior to BiLSTM, which was expected
as UT has an attention mechanism. Due to NN’s greedy nature, NN may not always
find the global optimal solution and may become stuck in the local optimal,
whereas BFS never will. So BFS is better than NN. That is why the best results
are happening where we use fine-tuned SBERT-WK embedding, UT to capture
dependencies between sentences, cosine similarity as a scoring function, and
searching the entire state space to maximize the total similarity of
consecutive sentences.

Table 2: SLM results by changing each component. Method Components | Tau | PMR
---|---|---
SLM (Fine tuned SBERT-WK + UT + Brute Force Search) | 0.7547 | 0.4064
SBERT-WK + UT + Brute Force Search | 0.7465 | 0.3893
USE + UT + Brute Force Search | 0.7206 | 0.373
SBERT-WK + BiLSTM + Brute Force Search | 0.7317 | 0.3762
USE + BiLSTM + Brute Force Search | 0.7044 | 0.3545
SBERT-WK + UT + Nearest Neighbor Search | 0.64 | 0.2755
USE + UT + Nearest Neighbor Search | 0.6162 | 0.267
SBERT-WK + BiLSTM + Nearest Neighbor Search | 0.6214 | 0.2576
USE + BiLSTM + Nearest Neighbor Search | 0.5980 | 0.2511

Figure 2 shows the results of the proposed model, SLM, along with the
baselines and competitors. SLM has improved by about 3.2% and 4.2% compared to
LSTM + PtrNet, and more than 4.3% and 4.8% compared to LSTM + Set2seq in
$\tau$ and PMR, respectively. This performance improvement, is due to the use
of a SBERT-WK model sentence encoders and employing an attention mechanism at
the story encoder utilizing universal transformers. Taking advantage of the
mentioned components, SLM can capture the “intradependecy” and
“interdependency” of sentences very well. Intradependency of a sentence refers
to the relations among each sentence’s words, while interdependency refers to
each sentence’s relations with other sentences of the story. Our method is
also superior to the HAN network, increasing $\tau$ criterion by more than 2.2
percent and PMR criterion by 1 percent. This could be because of the fact that
the sentences are encoded using the BERT-based model in the SLM. HAN, however,
uses a BiLSM encoder and multi-head attention layer.

Besides, SLM performed about 2.6 percent better than RankTxNet in PMR.
However, the Kendall’s $\tau$ score gives RankTxNet a slight superiority. The
difference is not significant, and can happen as a result of random parameters
of the network [37, 15].

Figure 2: Kendall’s tau ($\tau$) and perfect match ratio (PMR) on test set for
ROCStories datasets. The results of the model, compared to the baselines and
competitors.

##  5 Conclusion

We presented a language-model-based framework to solve sentence ordering task.
A sequence of shuffled sentences is inputted to our framework, where the
output should be a coherent order of the given set. We achieved state-of-the-
art performance in PMR scores on ROCStories dataset. We learned that SBERT-WK
is a suitable choice for sentence encoding. We analyzed how the method is
changed by using the USE to encode sentences. We also found that Universal
Transformers perform better for story encoding and learning the sentence-level
language model comparing to BiLSTM models. Moreover, we compared Brute Force
Search with Nearest Neighbor Search to order sentences. As future work, we
plan to develop more robust models to fulfil ordering of instances with longer
input sequences.

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[a]Anirban Das

# Self-interacting neutrinos as a solution to the Hubble tension?

Subhajit Ghosh

###### Abstract

Self-interaction among the neutrinos in the early Universe has been proposed
as a solution to the Hubble tension, a discrepancy between the measured values
of the Hubble constant from CMB and low-redshift data. However, flavor-
universal neutrino self-interaction is highly constrained by Big Bang
Nucleosynthesis and other laboratory experiments such as, tau and K-meson
decay, double-neutrino beta decay etc. We study the cosmology when only one or
two neutrino states are self-interacting. Such flavor-specific interactions
are less constrained by the laboratory experiments. Lastly, we address the
feasibility of resolving the Hubble tension within the framework of such
flavor-specific neutrino self-interaction.

##  1 Introduction

Since the early days of the Planck experiment, a discrepancy in the value of
the Hubble parameter from CMB observations and local (low redshift)
measurements have become a topic of active research [1]. This discrepancy has
been heightened further with the publication of the latest data from the
Planck experiment and the local $H_{0}$-measurement using cepheids and
supernovae by the SH0ES collaboration [2, 3, 4]. Planck measured value a
Hubble constant using the CMB data $H_{0}=67.36\pm
0.54\,\mathrm{km\,s^{-1}Mpc^{-1}}$, and the SH0ES collaboration measured
$H_{0}=74.03\pm 1.42\,\mathrm{km\,s^{-1}Mpc^{-1}}$. There is now a $4.4\sigma$
tension between these two values. An independent local measurement of $H_{0}$
performed in Ref.[5] found $H_{0}=69.8\pm 1.9\,\mathrm{km\,s^{-1}Mpc^{-1}}$.
Other than some unknown experimental systematics, this discrepancy could imply
a new physics beyond $\Lambda$CDM. Several such scenarios have been proposed
to address it. See Ref. [6] for a review.

The authors in Ref. [7] introduced a model of self-interacting neutrinos
(SINU) where the interaction is mediated by a heavy scalar $\phi$. At low
temperature, it leads to a flavor-universal four-Fermi interaction with a
coupling strength $G_{\rm eff}$. A Bayesian analysis of this model with the
CMB data yields two modes in the posterior distribution of $G_{\rm eff}$: a
strongly interacting (SI) mode with a large value ${\rm log}_{10}(G_{\rm
eff}/{\rm MeV^{-2}})\simeq-1.711^{+0.099}_{-0.11}$ ($68\%$ confidence limit),
and a moderately interacting (MI) mode with an upper bound ${\rm
log}_{10}(G_{\rm eff}/{\rm MeV^{-2}})<-3.57$ at $95\%$ confidence level when
the Planck 2015 and BAO data are used. However, SINU scenario faces strong
constraints from the laboratory experiments [8, 9]. They showed that a simple
model of flavor-universal SINU scenario is ruled out by laboratory constraints
from meson decay, $\tau$ decay, and double beta decay.

In Ref. [10], we performed a Bayesian analysis of flavor-specific neutrino
self-interaction scenario using latest cosmological data. Our goal was to
complement the flavor-specific SINU studies from laboratory experiments with
the cosmological data. We considered three massless neutrinos and fixed
$N_{\rm eff}=3.046$. The effect of neutrino mass can be speculated from our
results. We consider three scenarios depending on the number of interacting
species – 3-coupled ($\mathbf{3c+0f}$), 2-coupled + 1-free-streaming
($\mathbf{2c+1f}$), and 1-coupled + 2-free-streaming ($\mathbf{1c+2f}$)
respectively. We assumed same coupling strengths for the coupled species for
the first two cases.

##  2 The model

We consider scalar interactions between massless $\nu$ and $\phi$ as

$\mathcal{L}\supset g_{ij}\phi\bar{\nu}_{i}\nu_{j}\,,$ (1)

where $g_{ij}$ is the coupling between $\phi$ and the neutrino flavors $i$ and
$j$. At low temperature, a four-Fermi interaction among the neutrinos is
generated as follows,

$\mathcal{L}\supset G_{\rm
eff}^{(ijkl)}\bar{\nu}_{i}\nu_{j}\bar{\nu}_{k}\nu_{l},\qquad G_{\rm
eff}^{(ijkl)}\equiv\frac{g_{ij}g_{kl}}{M_{\phi}^{2}}\,.$ (2)

We consider only _diagonal_ interactions in the flavor space. The number of
neutrino flavors will be fixed to three with all of them having a same
temperature $T_{\nu}$. However, because of the complete equivalence of the
interacting states in the context of CMB, we need to consider only one common
coupling parameter $G_{\rm eff}$ for all the interacting states in a given
scenario. The comoving neutrino self-interaction opacity $\dot{\tau}_{\nu}$ is
defined as

$\dot{\tau}_{\nu}=-a(G_{\rm eff})^{2}T_{\nu}^{5}\,,$ (3)

where $a$ is the scale factor of the Universe.

Figure 1: CMB $TT$ and $EE$ angular power spectra for $\mathbf{3c+0f}$ (red,
solid), $\mathbf{2c+1f}$ (blue, dotted), and $\mathbf{1c+2f}$ (green, dashed)
scenarios are shown in the top panels. The $\Lambda$CDM power spectra are also
shown for comparison in solid black. The parameters for the $\Lambda$CDM
spectra correspond to the best-fit points for the TT,TE,EE+lowE dataset. The
bottom panels show the relative changes from the $\Lambda$CDM spectra. For
SINU plots, we have set ${\rm log}_{10}(G_{\rm eff}/{\rm MeV^{-2}})=-1$ and
the rest of the parameters are fixed to their $\Lambda$CDM best-fit values. We
also show the binned Planck 2018 data in both the plots as black circles with
errorbar.

##  3 Changes in the CMB power spectra

The self-interaction stops the neutrinos from free-streaming before
decoupling. The new interaction damps the perturbations for $\ell\geq 2$ and
impedes the growth of the anisotropic stress $\sigma$. The damping is maximum
for $\mathbf{3c+0f}$ where all three neutrinos are interacting, and gradually
decreases for $\mathbf{2c+1f}$ and $\mathbf{1c+2f}$ where the number of
interacting neutrino flavor is two and one respectively. The anisotropic
stress is related to the gravitational potentials $\phi$ and $\psi$ via

$k^{2}(\phi-\psi)=12\pi
Ga^{2}\sum_{i=\gamma,\nu}(\rho_{i}+P_{i})\sigma_{i}\,\simeq 16\pi
Ga^{2}\rho_{\rm tot}R_{\nu}\sigma_{\nu}\,,\quad
R_{\nu}=\dfrac{\rho_{\nu}}{\rho_{\nu}+\rho_{\gamma}}\;.$ (4)

In the last step in the equation, we ignored the small anisotropic stress of
photon $\sigma_{\gamma}$ before recombination. And, $R_{\nu}\simeq 0.41$ is
the fractional energy density of _free streaming_ neutrinos which, in
radiation domination in $\Lambda$CDM. The suppression of neutrino anisotropic
stress helps enhance the gravitational potentials $\phi$ and $\psi$. The
gravitational potentials in turn affect the evolution of the photon
perturbations.

In $\Lambda$CDM cosmology, the supersonic propagation of neutrino
perturbations creates a phase shift $\phi_{\nu}$ and an amplitude modification
$\Delta_{\nu}$ in the acoustic oscillations of the photon [11] , which in the
radiation domination, are given by

$\phi_{\nu}\simeq 0.19\pi R_{\nu}\,,\quad\Delta_{\nu}\simeq-0.27R_{\nu}\,.$
(5)

Together, they move the peaks in the TT and EE power spectra towards smaller
$\ell$, and suppress the amplitude of the power spectra. The self-interaction
stops the neutrinos from free-streaming and delays the neutrino decoupling. As
a result, the free-streaming neutrino fraction $R_{\nu}$ is decreased relative
to its $\Lambda$CDM value depending on the number of neutrino species which
are coupled.

##  4 Results

The posterior of $G_{\rm eff}$ shows a bimodal feature as expected. The MI
mode corresponds to smaller values of $G_{\rm eff}$ where the CMB spectra are
compatible with $\Lambda$CDM. The SI mode is characterized by a relatively
large value of $G_{\rm eff}$, and arises due to degeneracy with other
parameters, including $H_{0}$. The significance of the SI mode increases as
the number of interacting neutrinos goes down. This follows from the fact that
the changes in the CMB spectra in SINU are proportional to the number of
interacting neutrinos as can be seen from Eq.(5). This makes the SI mode
parameter values more compatible with data, and boosts the significance of the
mode.

Figure 2: The contours of 68%, 95%, and 99% confidence levels for ${\rm
log}_{10}(G_{\rm eff}/{\rm MeV^{-2}})$ and $H_{0}$ in $\mathbf{1c+2f}$
(green), $\mathbf{2c+1f}$ (blue), and $\mathbf{3c+0f}$ (red) for the Planck
TTTEEE+lowE+lensing dataset. The SI mode contour for $\mathbf{3c+0f}$ yields
the largest value of $H_{0}$ because of bigger phase shift. However, the
significance of the mode is much less compared to $\mathbf{2c+1f}$ and
$\mathbf{1c+2f}$.

The neutrino self-coupling strength $G_{\rm eff}$ has a positive correlation
with $H_{0}$ through the phase-shift. The mean value of $H_{0}$ shifts to
$H_{0}=69.46\pm 0.52\,\mathrm{km\,s^{-1}Mpc^{-1}}$ for the TTTEEE+lowE dataset
for the SI mode, reducing the tension with the SH0ES measurement to $\sim
3\sigma$. On the other hand, this value of $H_{0}$ is fully consistent with
the CCHP measurement $H_{0}=69.8\pm 1.9\,\mathrm{km\,s^{-1}Mpc^{-1}}$ [5].
However, the SI mode value of $H_{0}$ slightly decreases in the flavor-
specific $\mathbf{2c+1f}$ and $\mathbf{1c+2f}$ scenarios because of smaller
phase shift, but the significance of the mode is increased. This is evident
from figure 2. In table 1, we show $H_{0}$, $\Omega_{\Lambda}$,
$100\theta_{s}$, $r_{s}^{\ast}$, and $D_{A}^{\ast}$ for $\mathbf{3c+0f}$,
$\mathbf{2c+1f}$ and $\Lambda$CDM. The shift in $\theta_{s}$ due to the phase
shift in SINU decreases $D_{A}^{\ast}$ that helps increase the value of
$H_{0}$ and $\Omega_{\Lambda}$. Note that, the value of $r_{s}^{\ast}$ changes
by only about $1\sigma$ from $\Lambda$CDM, and plays a sub-dominant role in
changing the value of $H_{0}$.

##  5 Discussion

Neutrino self-interaction introduces a _phase shift_ and _enhance_ the CMB
anisotropy power spectra relative to $\Lambda$CDM. This scenario has been
proposed to ease the Hubble tension. However, the required strength of the
self-interaction to resolve the tension is strongly constrained in flavor-
universal scenario by other laboratory experiments. We studied the cases where
only two or one of the three neutrino species are interacting using the latest
CMB data from 2018 Planck experiment. We found a bimodal posterior as in the
previous studies with flavor-universal case. Moreover, we found that the
strongly interacting (SI) mode is boosted in the flavor-specific scenario.
Less number of interacting neutrino means less amount of changes in the CMB
power spectra which makes larger coupling strength more compatible with the
data boosting the SI mode.

|  SI: $\mathbf{3c+0f}$ |  SI: $\mathbf{2c+1f}$ |  $\Lambda$CDM
---|---|---|---
$H_{0}(\,\mathrm{km\,s^{-1}Mpc^{-1}})$ |  $69.47\pm 0.59$ |  $68.87\pm 0.58$ |  $67.90\pm 0.54$
$\Omega_{\Lambda}$ |  $0.7035\pm 0.0071$ |  $0.6989\pm 0.0072$ |  $0.6912\pm 0.0073$
$100\theta_{s}$ |  $1.0463\pm 0.00094$ |  $1.0447\pm 0.00079$ |  $1.04186\pm 0.00029$
$r_{s}^{\ast}({\rm Mpc})$ |  $144.58\pm 0.32$ |  $144.69\pm 0.31$ |  $144.87\pm 0.29$
$D_{A}^{\ast}({\rm Mpc})$ |  $12.69\pm 0.036$ |  $12.72\pm 0.034$ |  $12.773\pm 0.028$
Table 1: Parameter values and 68% confidence limits for SI mode in
$\mathbf{3c+0f}$ and $\mathbf{2c+1f}$, and $\Lambda$CDM in TTTEEE+lowE+lensing
data.

The neutrino coupling strength $G_{\rm eff}$ is degenerate with the Hubble
constant $H_{0}$, and the SI mode favors a slightly larger value of $H_{0}$
compared to $\Lambda$CDM, easing the Hubble tension. We, however, found that
the value of $H_{0}$ is decreased when only two or one neutrinos are
interacting. As a result, the SINU scenario cannot resolve the Hubble tension
when all laboratory and cosmological constraints are taken into consideration.
In our work, we did not include off-diagonal interactions as it requires one
to incorporate neutrino mass as well. Hence, it remains to be seen how the
addition of off-diagonal interaction affects the Hubble parameter.

## Acknowledgments

AD was supported by the U.S. Department of Energy under contract number DE-
AC02-76SF00515. SG was supported in part by the National Science Foundation
under Grant Number PHY-2014165 and PHY-1820860. SG also acknowledges the
support from Department of Atomic Energy, Government of India during the
initial stage of this project. This work used the computational facility of
Department of Theoretical Physics, Tata Institute of Fundamental Research.

## References

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# Logical inference approach to relativistic quantum mechanics:
derivation of the Klein-Gordon equation

H.C. Donker<EMAIL_ADDRESS>M.I. Katsnelson  H. De Raedt  K.
Michielsen  Radboud University, Institute for Molecules and Materials,
Heyendaalseweg 135, NL-6525AJ Nijmegen, The Netherlands  Zernike Institute for
Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747 AG
Groningen, The Netherlands  Institute for Advanced Simulation, Jülich
Supercomputing Centre, Forschungzentrum Jülich, D-52425 Jülich, Germany

###### Abstract

The logical inference approach to quantum theory, proposed earlier [Ann. Phys.
347 (2014) 45-73], is considered in a relativistic setting. It is shown that
the Klein-Gordon equation for a massive, charged, and spinless particle
derives from the combination of the requirements that the space-time data
collected by probing the particle is obtained from the most robust experiment
and that on average, the classical relativistic equation of motion of a
particle holds.

###### keywords:

logical inference, quantum theory, Klein-Gordon equation

††journal: Annals of Physics

##  1 Introduction

The inception of quantum theory was one of taking leaps. This is illustrated
by e.g. Schrödinger’s paper [1] in which he proposes his celebrated wave
equation. In this article [1] Hamilton’s principal function $S$ is postulated
to take the form $S=k\ln\psi$ with $k$ a constant and is then substituted in
the Hamilton-Jacobi equation (HJE). Upon variation of the resulting quadratic
functional with respect to $\psi$ (which Schrödinger later justifies using
Huygens’ principle [2]) an equation linear in $\psi$, now known as the
Schrödinger equation, was obtained. The derivation of the Klein-Gordon
equation [3, 4, 5, 6, 7] is essentially identical to that of the Schrödinger
equation namely, an action Ansatz is substituted in the relativistic Hamilton-
Jacobi equation, and after variation of the resulting quadratic functional
with respect to $\psi$, the relativistic analogue of the Schrödinger equation
is obtained [3, 4, 5, 6, 7].

Because of the ad hoc assumptions involved in obtaining these equations,
standard quantum mechanics textbooks usually present the formalism of quantum
theory as a set of postulates (see e.g. Refs. [8, 9, 10, 11]) and considerable
activity focuses on eliminating some of these postulates [12, 13, 14, 15, 16,
17, 18, 19]. Instead of starting from a set of postulates, the current work
presents an alternative derivation of the relativistic wave equation based on
the principles of logical inference (LI) [20, 21, 22, 23]. Specifically, we
demonstrate how the Klein-Gordon equation for a massive, charged and spinless
particle follows from LI based on the analysis of data recorded by a detector,
thereby extending earlier work [24, 25, 26, 27] to the relativistic domain.

The key concept in LI is the plausibility [23], a mental construct which
quantifies e.g. the chance that a detection event occurs. In general, the
degree of plausibility is expressed by a real number in the range of 0 and 1
[23]. The algebra of LI facilitates plausible reasoning in the presence of
uncertainty in a mathematically well-defined manner [20, 21, 22, 23]. In real
experiments there is not only uncertainty about the individual detection
events but there obviously is also uncertainty in the conditions under which
the experiments are carried out. Inevitably, the conditions of the experiment
will vary whenever the experiment is repeated. But if the experimental data is
to be reproducible, the experiment must be robust (to be quantified later)
with respect to small changes in the conditions under which the experiment is
being performed. Earlier work has shown that the equations of non-relativistic
quantum theory can be obtained by analyzing such robust experiments [24, 25,
26, 27]; most notable are the Schrödinger [25] and the Pauli equation [26].
Importantly, the requirement that the experiment is to be robust implies that
the plausibility must be viewed as an objective assignment (i.e. conditional
probability) rather that a subjective one [25]. The present work extends this
approach to the relativistic domain: it shows how the Klein-Gordon equation
[3, 7] for a massive, charged, and spinless relativistic particle emerges by
an analysis similar to the one employed in Refs. [25, 26].

##  2 Logical inference approach

###  2.1 Particle detection experiment

Consider an experiment in which a particle source and detectors are located at
fixed positions relative to the laboratory reference frame. The source emits a
particle that interacts with one of the detectors and triggers a detection
event that yields data in the form of three spatial coordinates
$\mathbf{r}=(x,y,z)$ of the detector and the clock time $t$ at which the event
occurred. The experiment is considered to be ideal in the sense that every
emitted particle triggers one and only one detector.

The experiment is repeated $N$ times, meaning that we let $N$ particles pass
through the detector. Each time a particle is created, the (laboratory) clock
time is reset. We label the particles and the corresponding data by the index
$n=1\dots N$ and denote the spatial and temporal resolution by $\Delta_{s}$
and $\Delta_{t}$, respectively. As particle $n$ passes through the detector,
the latter produces a time stamp $t_{n}$ and a vector of spatial coordinates
$\mathbf{r}_{n}=(x_{n},y_{n},z_{n})$, which because of the limited resolution,
correspond to the time-bin $j_{n}=\mathrm{ceiling}(t_{n}/\Delta_{t})$ and
space-bin $\mathbf{k}_{n}=\mathrm{ceiling}(\mathbf{r}_{n}/\Delta_{s})$ where,
element-wise, the function $\mathrm{ceiling}(x)$ returns the smallest integer
not smaller than $x$. In practice the number of time-bins and space-bins is
necessarily finite. Therefore we must have $0\leq j_{n}\leq J$ and
$(0,0,0)\leq\mathbf{k}_{n}\leq\mathbf{K}=(K_{x},K_{y},K_{z})$, where $J$,
$K_{x}$, $K_{y}$ and $K_{z}$ are (large) integer numbers.

The data collected after $N$ repetitions of the experiment is given by the set
of quadruples

$\Upsilon=\left\\{(j_{n},\mathbf{k}_{n})\;|\;0\leq j_{n}\leq
J;\;\mathbf{0}\leq\mathbf{k}_{n}\leq\mathbf{K};\;n=1\dots N\right\\}\,,$ (1)

or, denoting the total amount of clicks in bin
$\mathbf{j}=(j_{n},\mathbf{k}_{n})$ by $c_{\mathbf{j}}$, by the equivalent
data set

$\mathcal{D}=\\{c_{\mathbf{j}}\;|\;\sum_{\mathbf{j}}c_{\mathbf{j}}=N\\}\,.$
(2)

Note that at this stage, we have not yet assumed that there is a relation
between the space-time coordinates of the particle and the data set
$\mathcal{D}$.

###  2.2 Inference probability and Fisher information

Having specified the measurement scenario, the next step in the LI approach is
to encode the relation between the space-time coordinates of the particle and
the $n$-th detection event $\mathbf{j}_{n}=(j_{n},\mathbf{k}_{n})$ through the
inference-probability (i-prob) $P(\mathbf{j}_{n}|\theta,Z)$ where $\theta$ and
$Z$ specify the conditions under which the experiment is being performed [25].
The i-prob is, at this stage, a necessarily subjective number between zero and
one that expresses the uncertainty with which the $n$-th particle produces the
data $\mathbf{j}_{n}$. The particle is assumed to be characterized by its own
(unknown) clock time $\theta$ measured in a reference frame attached to the
particle The proposition $Z$ represents all other experimental conditions
(e.g. applied electromagnetic potentials) which are considered fixed for the
duration of the experiment but are deemed irrelevant for the problem at hand.

It is common practice to assume, as a first step, that events are independent,
meaning that knowing all earlier and future events, it is impossible to say
with certainty what the event will be. Following this practice, we assume that
the $N$ detection events are independent. Then, according to the algebra of LI
[20, 21, 22, 23], it follows immediately that the i-prob
$P(\Upsilon|\theta,N,Z)$ to observe data set $\Upsilon$ factorizes as

$P(\Upsilon|\theta,N,Z)=\prod_{n=1}^{N}P(\mathbf{j}_{n}|\theta,Z)\,,$ (3)

or, equivalently,

$P(\mathcal{D}|\theta,N,Z)=N!\prod_{\mathbf{j}}\frac{P(\mathbf{j}|\theta,Z)^{c_{\mathbf{j}}}}{c_{\mathbf{j}}!}\,.$
(4)

The salient feature of the experiment considered here is that there is
uncertainty about the individual detection events, that there is uncertainty
in the mapping from $\theta$ to the spatial coordinates and the time of the
detection events. However, if the experimental data is to increase our
capability to uncover relations among the observed events at different space-
time points, the experiment must be robust [25]. In the case at hand this
means that small changes in the unknown clock time $\theta$ do not lead to
erratic changes in the observed data $\mathcal{D}$, even though there is no
reproducibility on the level of individual events.

It is convenient to express the requirement of robustness as an hypothesis
test [25]. The evidence [22, 23] $\mathrm{Ev}$ for the hypothesis that
$\theta+\epsilon$ produces the data $\mathcal{D}$ relative to the hypothesis
that $\theta$ produces the same data is given by [22, 23, 25]

$\mathrm{Ev}=\ln\frac{P(\mathcal{D}|\theta+\epsilon,N,Z)}{P(\mathcal{D}|\theta,N,Z)}\,.$
(5)

The notion of a robust experiment then translates to the statement that for
all $\theta$ and arbitrary but small $\epsilon$, the evidence $|\mathrm{Ev}|$
should be as small as possible. In searching for the solution of the global
optimization problem, we exclude the trivial, non-informative experiment for
which $P(\mathcal{D}|\theta,N,Z)$ does not depend on $\theta$ [25]. Making use
of Eq. (4) and expanding Eq. (5) to second order in $\epsilon$ yields

$\displaystyle\mathrm{Ev}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{j}}c_{\mathbf{j}}\ln\frac{P(\mathbf{j}|\theta+\epsilon,Z)}{P(\mathbf{j}|\theta,Z)}=\sum_{\mathbf{j}}c_{\mathbf{j}}\left\\{\epsilon\frac{P^{\prime}(\mathbf{j}|\theta,Z)}{P(\mathbf{j}|\theta,Z)}-\right.\left.\frac{\epsilon^{2}}{2}\left[\left(\frac{P^{\prime}(\mathbf{j}|\theta,Z)}{P(\mathbf{j}|\theta,Z)}\right)^{2}-\frac{P^{\prime\prime}(\mathbf{j}|\theta,Z)}{P(\mathbf{j}|\theta,Z)}\right]\right\\}\,,$
(6)

where the primes indicate partial derivatives with respect to $\theta$.

Our goal is now to minimize $|\mathrm{Ev}|$ for all $\theta$ simultaneously.
First note that as $\sum_{j}P(\mathbf{j}|\theta,Z)=1$, all partial derivatives
of $\sum_{j}P(\mathbf{j}|\theta,Z)$ with respect to $\theta$ are zero.
Therefore the first and the third term in Eq. (6) vanish if we make the
assignment $c_{\mathbf{j}}=NP(\mathbf{j}|\theta,Z)$. This is an important
result: the criterion of robustness not only enforces the intuitively obvious
assignment $P(\mathbf{j}|\theta,Z)=c_{\mathbf{j}}/N$ but by doing so, it
changes the subjective nature of $P(\mathbf{j}|\theta,Z)$ into an objective,
physically measurable quantity (the relative frequency of outcomes). Thus, it
is at this point that the possibility to view the i-prob as a subjective
assignment is eliminated [25].

With this assignment, the expression for the evidence becomes

$\mathrm{Ev}=-\frac{\epsilon^{2}N}{2}\sum_{\mathbf{j}}\frac{1}{P(\mathbf{j}|\theta,Z)}\left(\frac{\partial
P(\mathbf{j}|\theta,Z)}{\partial\theta}\right)^{2}\,,$ (7)

and as $\epsilon$ is arbitrary, we can find the solution of the optimization
problem by minimizing the Fisher information

$I_{F}=\sum_{\mathbf{j}}\frac{1}{P(\mathbf{j}|\theta,Z)}\left(\frac{\partial
P(\mathbf{j}|\theta,Z)}{\partial\theta}\right)^{2}\,,$ (8)

for all $\theta$ simultaneously.

The basic equations of (relativistic) quantum theory are formulated in terms
of continuous space and time. Therefore, to derive such equations from a LI
approach, it is necessary to take the continuum limit of Eq. (8). This is
readily accomplished in the standard manner by letting the temporal resolution
$\Delta_{t}$ and spatial resolution $\Delta_{s}$ approach zero while keeping
the four dimensions of the four-dimensional volume fixed. Taking the continuum
limit and ignoring irrelevant prefactors, Eq. (8) becomes

$\displaystyle I_{F}$ $\displaystyle=$ $\displaystyle
c\int\mathrm{d}^{3}\mathbf{r}\int\mathrm{d}t\,\frac{1}{P(t,\mathbf{r}|\theta,Z)}\left(\frac{\partial
P(t,\mathbf{r}|\theta,Z)}{\partial\theta}\right)^{2}\equiv\int\mathrm{d}^{4}\mathbf{x}\frac{1}{P(\mathbf{x}|\theta,Z)}\left(\frac{\partial
P(\mathbf{x}|\theta,Z)}{\partial\theta}\right)^{2},$ (9)

where $\mathbf{x}=[ct,x,y,z]=[x^{0},x^{1},x^{2},x^{3}]$ denotes the four-
vector of a location in space-time and $c$ is the speed of light in vacuum.
Strictly speaking, Eq. (9) makes a slight abuse of notation: in the continuum
limit $P(\mathbf{x}|\theta,Z)$ is a probability density whilst
$P(\mathbf{x}|\theta,Z)\mathrm{d}\mathbf{x}$ is the corresponding
(dimensionless) i-prob. Henceforth it is assumed that this change of notation
is implicitly understood.

###  2.3 Special relativity

The above discussion focussed on the relation between a robust experiment and
the observed data but does not refer to any physical theory yet. The knowledge
or expectation about the physical behavior enters the LI approach by imposing
constraints on the minimization of Eq. (9). Generally speaking, in the absence
of uncertainty, we may expect to observe data that complies with the classical
mechanical description, Thus, in the case at hand, we require that in the
absence of uncertainty, the LI approach yields the results of the special
theory of relativity (STR).

Proper time, that is the time measured by a clock at rest, is a central notion
in the STR. In the measurement scenario described above, the i-prob
$P(\mathbf{x}|\theta,Z)\mathrm{d}\mathbf{x}$ was already assumed to depend on
the proper time $\theta$ of the particle. In the spirit of the STR, we assume
that

$P(\mathbf{x}|\theta,Z)=P(\tau|\theta,Z)\,,$ (10)

where $\tau$ ($c^{2}\tau^{2}\equiv c^{2}t^{2}-x^{2}-y^{2}-z^{2}$) is the
proper time in the reference frame of the detector. In words, we assume that
the i-prob to observe a space-time event depends on Lorentz scalars only. Note
that because the clock is being reset with each repetition of the experiment,
the proper times that enter our description are proper time intervals. In
addition, we assume that space-time is homogeneous, meaning that

$P(\tau+\delta|\theta+\delta,Z)=P(\tau|\theta,Z)\,,$ (11)

where $\delta$ is an arbitrary shift of proper time. A remark about the last
assumption may be in order. In the LI approach the measurement scenario and
the observed data are key to formulate the notion of a robust experiment.
Physics on the other hand is about connecting mental pictures, concepts about
the nature of the world around us, to the observed data. Viewed in this light,
Eq. (11) expresses our expectation that carrying out the experiment at another
point in space-time does not change our mental picture. From the definition of
the derivative, it follows directly from Eq. (11) that

$\frac{\partial P(\tau|\theta,Z)}{\partial\tau}=-\frac{\partial
P(\tau|\theta,Z)}{\partial\theta},$ (12)

and using this identity, Eq. (9) becomes

$I_{F}=\int\mathrm{d}^{4}\mathbf{x}\frac{1}{P(\mathbf{x}|\theta,Z)}\left(\frac{\partial
P(\mathbf{x}|\theta,Z)}{\partial\tau}\right)^{2}=\int\mathrm{d}^{4}\mathbf{x}\frac{1}{P(\tau|\theta,Z)}\left(\frac{\partial
P(\tau|\theta,Z)}{\partial\tau}\right)^{2}.$ (13)

Recall that the objective of the LI approach is to find
$P(\mathbf{x}|\theta,Z)$ that minimizes $I_{F}$ for all $\theta$
simultaneously, subject to constraints that we discuss next.

###  2.4 Motion of the particle

In the absence of uncertainty, successive observed detection events map one-
to-one on the relativistic motion of the particle, described by the laws of
the STR. In the LI approach, this limiting case enters through a
“correspondence principle” in terms of the HJE [25, 26]. This is not a
surprise: as mentioned in the introduction, the HJE is one of the key
ingredients in the derivation of the Schrödinger [1] and the Klein-Gordon
equation [3, 4, 5, 6, 7, 28] and it plays a similar role in the LI derivation
of these equations. In the present paper, we do not postulate a HJE but, in
analogy with the derivation of the non-relativistic HJE [29, 26], we follow an
alternative path and derive the relativistic HJE for a massive and charged
particle from a field description of the four-velocity
$\mathrm{d}\mathbf{x}/\mathrm{d}\tau$.

We start by assuming that there exists a (four-)vector field
$\mathbf{U}(\mathbf{x})$ such that

$\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\tau}=-U^{\mu}(\mathbf{x})\,,$ (14)

Here and in the following, we use the standard co/contra-variant notation and
the summation convention and denote the Minkowski metric by
$\eta=\mathrm{diag}(1,-1,-1,-1)$. Taking the derivative of Eq. (14) with
respect to $\tau$ yields

$\frac{\mathrm{d}^{2}x^{\mu}}{\mathrm{d}\tau^{2}}=-\frac{\partial
U^{\mu}}{\partial(ct)}\frac{\mathrm{d}(ct)}{\mathrm{d}\tau}-\sum_{i=1}^{3}\frac{\partial
U^{\mu}}{\partial
x^{i}}\frac{\mathrm{d}x^{i}}{\mathrm{d}\tau}=-\partial^{\nu}U^{\mu}\frac{\mathrm{d}x_{\nu}}{\mathrm{d}\tau}\,,$
(15)

where $\partial^{\mu}$ is the shorthand for $\partial/\partial x_{\mu}$. As
the norm of the four-velocity is $c$, we have $\mathbf{U}^{2}\equiv
U^{\alpha}U_{\alpha}=c^{2}$ is a constant and hence any derivative thereof is
zero. Therefore we have

$\partial^{\mu}\mathbf{U}^{2}=2\left(U^{0}\frac{\partial U^{0}}{\partial
x_{\mu}}-\sum_{i=1}^{3}U^{i}\frac{\partial U^{i}}{\partial
x_{\mu}}\right)=2U^{\nu}\partial^{\mu}U_{\nu}=0\,.$ (16)

Substitution of Eq. (14) into Eq. (16) yields

$(\partial^{\mu}U^{\nu})\frac{\mathrm{d}x_{\nu}}{\mathrm{d}\tau}=0\,.$ (17)

From Eqs. (15) and (17) it then follows that

$\frac{\mathrm{d}^{2}x^{\mu}}{\mathrm{d}\tau^{2}}=\left(\partial^{\mu}U^{\nu}-\partial^{\nu}U^{\mu}\right)\frac{\mathrm{d}x_{\nu}}{\mathrm{d}\tau}=F^{\mu\nu}\frac{\mathrm{d}x_{\nu}}{\mathrm{d}\tau}\,.$
(18)

Equation (18) has the same form as the Lorentz force equation of a particle
moving in an electrodynamic field [30] if we identify $F^{\mu\nu}$ with the
field-strength tensor of electrodynamics. In order that this identification
makes sense, it is necessary to assume that the particles in the particle
detection experiment are massive and charged.

If $S=S(\mathbf{x})$ represents a scalar field, the transformation
$A^{\mu}=U^{\mu}+\partial^{\mu}S$ yields

$(\bm{\partial}S-\mathbf{A})^{2}=c^{2}\,.$ (19)

where we introduced the shorthand notation
$(\bm{\partial}S)^{2}=(\partial_{\alpha}S)(\partial^{\alpha}S)$. As it is the
four-velocity d$\mathbf{x}$/d$\tau$ which corresponds to a physically relevant
quantity, imposing gauge invariance enforces introducing a non-vanishing
canonical momentum $p^{\mu}=\partial^{\mu}S$ in order keep the norm of the
four-velocity fixed to $c$. Equation Eq. (19) is the relativistic HJE in
disguise. Indeed, making use of $\bm{\partial}S=[{\partial S}/{\partial
x_{0}},{\partial S}/{\partial x_{1}},{\partial S}/{\partial x_{2}},{\partial
S}/{\partial x_{3}}]=[{\partial S}/{\partial x^{0}},-{\partial S}/{\partial
x^{1}},-{\partial S}/{\partial x^{2}},-{\partial S}/{\partial x^{3}}]$ and
$\mathbf{A}=[A^{0},A^{1},A^{2},A^{3}]=[A_{0},-A_{1},-A_{2},-A_{3}]$,
introducing the symbols $m$ and $q$ for the mass and charge of the particle,
respectively, and changing in Eq. (19) symbols according to
$\bm{\partial}S\rightarrow({\partial S}/{\partial ct},{\partial S}/{\partial
x},{\partial S}/{\partial y},{\partial S}/{\partial z})/m$ and
$\mathbf{A}\rightarrow q(\Phi,A_{x},A_{y},A_{z})/m$ (where $\Phi$ and
$(A_{x},A_{y},A_{z})$ are the usual scalar and vector potential, respectively
[30]), we find

$\left(\frac{\partial S(\mathbf{x})}{\partial
ct}-\frac{q}{c}\Phi(\mathbf{x})\right)^{2}=\left(\frac{\partial
S(\mathbf{x})}{\partial
x}+\frac{q}{c}A_{x}(\mathbf{x})\right)^{2}+\left(\frac{\partial
S(\mathbf{x})}{\partial
y}+\frac{q}{c}A_{y}(\mathbf{x})\right)^{2}+\left(\frac{\partial
S(\mathbf{x}|\theta,Z)}{\partial
z}+\frac{q}{c}A_{z}(\mathbf{x})\right)^{2}+m^{2}c^{2}\;,$ (20)

which is the relativistic HJE for a charged, massive particle in an
electromagnetic field [7, 3].

###  2.5 Derivation of the Klein-Gordon equation

As a first step, it is expedient to write Eq. (13) in an alternative form by
noting that $c\tau=\sqrt{\eta_{\mu\nu}x^{\mu}x^{\nu}}$ implies
$\partial^{\alpha}\tau=x^{\alpha}/(c^{2}\tau)$ such that

$\eta_{\mu\nu}\left(\partial^{\mu}P(\tau|\theta,Z)\right)\left(\partial^{\nu}P(\tau|\theta,Z)\right)=\left(\frac{\partial
P(\tau|\theta,Z)}{\partial\tau}\right)^{2}\eta_{\mu\nu}(\partial^{\mu}\tau)(\partial^{\nu}\tau)=\frac{1}{c^{2}}\left(\frac{\partial
P(\tau|\theta,Z)}{\partial\tau}\right)^{2}\,,$ (21)

and hence Eq. (13) can be written as

$I_{F}=c^{2}\int\mathrm{d}^{4}\mathbf{x}\frac{1}{P(\tau|\theta,Z)}\left(\bm{\partial}P(\tau|\theta,Z)\right)^{2}.$
(22)

The general guiding principle of the LI approach is that the experiment that
yields the most robust data is described by the probability density
$P(\mathbf{x}|\theta,Z)$ that minimizes $I_{F}$ for all $\theta$
simultaneously, subject to additional constraints that are deemed relevant to
the experiment at hand [25]. In the present case, we require that the
description is compatible with the special theory of relativity. For a
massive, charged particle and in the absence of uncertainty, the latter
requirement implies that the classical, relativistic HJE Eq. (19) should hold.
We can inject this requirement into the LI approach by considering the
functional

$F=c^{2}\int\mathrm{d}^{4}\mathbf{x}\left\\{\frac{\left(\bm{\partial}P(\tau|\theta,Z)\right)^{2}}{P(\tau|\theta,Z)}+\lambda\left[(\bm{\partial}S-\mathbf{A})^{2}-c^{2}\right]P(\tau|\mathbf{\theta},Z)\right\\}\,,$
(23)

where $\lambda$ is a weighting factor that reflects the importance of the
uncertainty and robustness relative to the contribution of the classical
dynamics. It is straightforward to show that the expression Eq. (23) is
invariant under Lorentz transformations.

Extremization of Eq. (23) can be carried out by the standard variational
calculus and yields a set of nonlinear partial differential equations for
$P(\mathbf{x}|\theta,Z)$ and $S(\mathbf{x})$. It is not difficult to show that
at an extremum, (i) the value of $F$ does not depend on the value of the
unknown proper time $\theta$ and that (ii) the value of $F$ is zero,
independent of $\lambda$. The latter result implies that the extrema describe
situations in which the uncertainty about the detection events is perfectly
balanced by the certainty that the classical HJE describes the motion of the
observed detection events.

It is now expedient to write Eq. (23) more explicitly as

$\displaystyle F$ $\displaystyle=$ $\displaystyle
c^{2}\int\mathrm{d}^{4}\mathbf{x}\bigg{\\{}\frac{1}{P(\mathbf{x}|\theta,Z)}\bigg{[}\left(\frac{\partial
P(\mathbf{x}|\theta,Z)}{\partial ct}\right)^{2}-\left(\frac{\partial
P(\mathbf{x}|\theta,Z)}{\partial x}\right)^{2}-\left(\frac{\partial
P(\mathbf{x}|\theta,Z)}{\partial y}\right)^{2}-\left(\frac{\partial
P(\mathbf{x}|\theta,Z)}{\partial z}\right)^{2}\bigg{]}$ (24)
$\displaystyle\hbox to42.67912pt{}+\lambda\bigg{[}\left(\frac{\partial
S(\mathbf{x})}{\partial ct}-A^{0}(\mathbf{x})\right)^{2}-\left(\frac{\partial
S(\mathbf{x})}{\partial x}+A^{1}(\mathbf{x})\right)^{2}-\left(\frac{\partial
S(\mathbf{x})}{\partial y}+A^{2}(\mathbf{x})\right)^{2}$ $\displaystyle\hbox
to71.13188pt{}-\left(\frac{\partial S(\mathbf{x})}{\partial
z}+A^{3}(\mathbf{x})\right)^{2}-c^{2}\bigg{]}P(\mathbf{x}|\theta,Z)\bigg{\\}}\,.$

We do not know of any direct method to solve the nonlinear set of equations
that results from searching for the extrema of Eq. (24) but, by analogy with
the non-relativistic case, we may consider a quadratic functional of a
complex-valued field $\varphi(\mathbf{x})$ and use the polar representation of
this field to construct the corresponding functional in terms of this
representation [25, 26, 31].

To this end, consider the quadratic functional

$\displaystyle Q$ $\displaystyle=$ $\displaystyle
4c^{2}\int\mathrm{d}^{4}\mathbf{x}\left\\{\left[\frac{\partial\varphi^{\ast}(\mathbf{x})}{\partial
ct}+\frac{i\sqrt{\lambda}}{2c}A^{0}(\mathbf{x})\varphi^{\ast}(\mathbf{x})\right]\left[\frac{\partial\varphi(\mathbf{x})}{\partial
ct}-\frac{i\sqrt{\lambda}}{2c}A^{0}(\mathbf{x})\varphi(\mathbf{x})\right]\right.$
$\displaystyle\hbox
to42.67912pt{}-\left[\frac{\partial\varphi^{\ast}(\mathbf{x})}{\partial
x}-\frac{i\sqrt{\lambda}}{2c}A^{1}(\mathbf{x})\varphi^{\ast}(\mathbf{x})\right]\left[\frac{\partial\varphi(\mathbf{x})}{\partial
x}+\frac{i\sqrt{\lambda}}{2c}A^{1}(\mathbf{x})\varphi(\mathbf{x})\right]$
$\displaystyle\hbox
to42.67912pt{}-\left[\frac{\partial\varphi^{\ast}(\mathbf{x})}{\partial
y}-\frac{i\sqrt{\lambda}}{2c}A^{2}(\mathbf{x})\varphi^{\ast}(\mathbf{x})\right]\left[\frac{\partial\varphi(\mathbf{x})}{\partial
y}+\frac{i\sqrt{\lambda}}{2c}A^{2}(\mathbf{x})\varphi(\mathbf{x})\right]$
$\displaystyle\hbox
to42.67912pt{}\left.-\left[\frac{\partial\varphi^{\ast}(\mathbf{x})}{\partial
z}-\frac{i\sqrt{\lambda}}{2c}A^{3}(\mathbf{x})\varphi^{\ast}(\mathbf{x})\right]\left[\frac{\partial\varphi(\mathbf{x})}{\partial
z}+\frac{i\sqrt{\lambda}}{2c}A^{3}(\mathbf{x})\varphi(\mathbf{x})\right]-\frac{\lambda
c^{2}}{4}\varphi^{\ast}(\mathbf{x})\varphi(\mathbf{x})\right\\}\,.$

Substituting the polar representation

$\varphi(\mathbf{x})=\sqrt{P(\mathbf{x}|\theta,Z)}e^{i\sqrt{\lambda}S(\mathbf{x})/2}\,,$
(26)

in Eq. (LABEL:Q) yields $Q=F$. Equations for the extrema of the functional $Q$
can be found by variation with respect to $\varphi^{\ast}(\mathbf{x})$,
yielding the linear partial differential equation

$\left[\frac{\partial}{\partial
ct}+i\frac{\sqrt{\lambda}}{2c}A^{0}(\mathbf{x})\right]^{2}\varphi(\mathbf{x})=\left\\{\left[\frac{\partial}{\partial
x}-i\frac{\sqrt{\lambda}}{2c}A^{1}(\mathbf{x})\right]^{2}+\left[\frac{\partial}{\partial
y}-i\frac{\sqrt{\lambda}}{2c}A^{2}(\mathbf{x})\right]^{2}+\left[\frac{\partial}{\partial
z}-i\frac{\sqrt{\lambda}}{2c}A^{3}(\mathbf{x})\right]^{2}-\frac{\lambda
c^{2}}{4}\right\\}\varphi(\mathbf{x})\;,$ (27)

which has the same mathematical structure as the KG equation [32]. This can be
made more explicit by changing symbols according to $\mathbf{A}\rightarrow
q(\Phi,A_{x},A_{y},A_{z})/m$ and $\lambda=4m^{2}/\hbar^{2}$, yielding

$\frac{1}{c^{2}}\left[i\hbar\frac{\partial}{\partial
t}-q\Phi(\mathbf{x})\right]^{2}\varphi(\mathbf{x})=\left\\{\left[\frac{\hbar}{i}\frac{\partial}{\partial
x}-\frac{q}{c}A_{x}(\mathbf{x})\right]^{2}+\left[\frac{\hbar}{i}\frac{\partial}{\partial
y}-\frac{q}{c}A_{y}(\mathbf{x})\right]^{2}+\left[\frac{\hbar}{i}\frac{\partial}{\partial
z}-\frac{q}{c}A_{z}(\mathbf{x})\right]^{2}+m^{2}c^{2}\right\\}\varphi(\mathbf{x})\;.$
(28)

Obviously, the weighting factor $\lambda=4m^{2}/\hbar^{2}$ cannot be
determined on the basis of logic only but has to follow from a comparison of
the outcome of calculations based on Eq. (27) with experimental data. It is of
interest to enquire to what extent Eq. (28) allows us to infer from the
observed data properties of the massive charged particles. The speed of light
in vacuum $c$ certainly does not depend on the properties of the massive
charged particle. Then, from Eq. (28), it is immediately clear that its
solutions are invariant under the transformation $\hbar\rightarrow\hbar\xi$,
$q\rightarrow q\xi$, and $m\rightarrow m\xi$. Hence, from the observed data we
may be able to determine two but not three of the constants that appear in Eq.
(28). For instance, by a suitable redefinition of the units of mass and
charge, $\hbar$ can be eliminated from Eq. (28) [33].

In practice, instead of solving the set of nonlinear equations in terms of
$P(\mathbf{x}|\mathbf{\theta},Z)$ and $S(\mathbf{x})$ that result from
minimizing Eq. (24), it is much easier to first solve Eq. (27) and then use
Eq. (26) to find
$P(\mathbf{x}|\theta,Z)=\varphi(\mathbf{x})^{\ast}\varphi(\mathbf{x})$. It is
important to recognize that the LI approach gives us the probability for
observing a space-time event $\mathbf{x}$ but does not yield an estimate of
the proper time of the particle $\theta$. The latter was and remains unknown.
The LI approach suggests that the wave function $\varphi(\mathbf{x})$ is only
a mathematical vehicle, be it an extraordinarily useful one, to transform a
set of nonlinear partial differential equations into a linear set of partial
differential equations. The interplay of the two real quantities
$S(\mathbf{x})$ and $P(\mathbf{x}|\theta,Z)$ which account for respectively,
the classical relativistic physics and the uncertainty on the collected data,
can be disentangled through the use of single complex wave function. But, as a
mathematical tool, the wave function does not need an interpretation: it is
$P(\mathbf{x}|\theta,Z)$ that is directly linked to the observed events.

##  3 Discussion

We have shown how the Klein-Gordon equation for massive, charged particles
derives from logical inference applied to experiments for which the observed
events are independent and for which the frequency distribution of these
events is robust with respect to small changes of the conditions under which
experiments are carried out. The present derivation is a logical
generalization of earlier work [24, 25, 26, 27] to the relativistic domain,
the fundamental difference being that the measured time is subject to
uncertainty.

Obviously, the transition from non-relativistic to relativistic quantum theory
is expected to bring in some radically new features. Landau and Peierls [34]
pointed out that in relativistic quantum theory the particle position cannot
be measured with an accuracy higher than its Compton wavelength. Measuring the
position of an electron with an accuracy higher than its Compton wavelength
requires an energy that exceeds the threshold for the creation of electron-
positron pairs [32], rendering meaningless the question which of the electrons
is the original one. Therefore, there is a common believe that relativistic
quantum theory cannot be a theory of individual particles but it must be a
field theory for a non-constant number of particles [35, 36]. The requirement
of a field theory description is also linked to the fact that the charge
density of the Klein-Gordon equation is not positive definite, as mentioned by
Dirac [37] and also stressed by Feshbach and Villars [32]. This is due to the
second order time derivative in the Klein-Gordon equation and indicates that
the wave function describes in fact two degrees of freedom instead of one
[32].

It is worth noting that there is no mention of the direction of time in the
logical inference approach. Indeed, when we derived Eq. (13) we allowed both
$\theta<\tau$ and $\theta>\tau$. This looks unusual from the point of view of
single-particle quantum mechanics (see, however, a discussion of time
(a)symmetry in Refs. [38, 39]). Within relativistic quantum mechanics it seems
more natural. As was suggested by Wheeler one might interpret anti-particles
as particles with the sign of the proper time reversed, i.e. as if the
particles are moving backward in time [40], and, at least, for non-interacting
particles this interpretation seems to be possible. In the measurement
scenario analyzed here, there is no way to discern the absorption of particles
from the emission of anti-particles. Both types of events contribute equally
well to the detection counts. This relates to the measurement scenario where
we make no distinction between detection events for which $\tau>\theta$ and
detection events for which $\tau<\theta$; causality is not a prerequisite in
the derivation presented here.

Naturally one might ask how to extent this approach to particles with non-zero
spin (e.g., Dirac equation). We leave this challenging program for future
research.

## Acknowledgments

MIK and HCD acknowledges financial support by the European Research Council,
project 338957
FEMTO/NANO.

## References

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